a comparison of local volatility and implied volatility422086/fulltext02.pdf · a comparison of...

U.U.D.M. Project Report 2011:12

Examensarbete i matematik, 30 hpHandledare och examinator: Johan TyskJuni 2011

Department of MathematicsUppsala University

A Comparison of Local Volatility and Implied Volatility

Hui Ye

1

PrefacePublished in 1973, the Black-Scholes model has undoubtfully become one of the most frequently

used models in the financial derivative pricing field during the past few decades.As the option

pricing benchmark, the Black-Scholes model has shown an extensive application range, however,

some assumptions originally made in order to derive this model have been found unlikely to hold

in the reality. One of such controversial assumptions is that the volatility of underlying asset

remains constant throughout the entire option life. This has been criticized by more and more

researchers and practitioners as it is not in line with the research results on volatilities and the

observations accumulated from the real market data. Because of this shortcoming of Black-Scholes

model, people are eager to find such a model that incorporates the variability of the implied

volatility in the estabilishment of the model and still has similar volatility numerical values as

opposed to those implied by the Black-Scholes model. Our paper here takes a step in that direction

by investigating the replicating ability of a local volatility model. This study is featured by focusing

on the relationships between the implied volatility inferred by the Black-Scholes model, the local

volatility specified by the local volatility model and the volatility given by the Dupire's formula for

implied volatility.

2

Abstract

This paper mainly studies the relationships between the implied volatilities inferred by the Black-

Scholes model and the volatilities derived by the local volatility model. By studying the difference

between other volatilities and the implied volatilities, we can search for models that have similar

volatilities to those of Black-Scholes models, and yet still process more realistic and plausible price

processes that do not depend on a constant volatility, unlike the ones of the Black-Scholes models.

Our search for such models are illustrated by the local volatility model.

3

Acknowledgement

My gratitude and appreciation goes out to both of my supervisors. I would like thank Prof. Johan

Tysk for all the valuable advice during this study. Also I would like to thank Senior Lecturer Erik

Eström for the inspiration of the topic of this study. I would not be able to complete this study and

finish this paper without their constructive advice, kind help and sincere critiques. I personally

benefited a lot from this learning experience. I am really grateful for their help and this studying

opportunity. I am looking forward to working with them again someday.

4

ContentsChapter 1 Introduction-----------------------------------------------------------------------------------------5

1.1 Motivation---------------------------------------------------------------------------------------------------5

1.2 Objectives---------------------------------------------------------------------------------------------------5

1.3 Chapter Review---------------------------------------------------------------------------------------------6

Chapter 2. Background-----------------------------------------------------------------------------------------8

2.1 The Local Volatility Models-------------------------------------------------------------------------------8

2.2 The Dupire Model(Method)-----------------------------------------------------------------------------10

Chapter 3 Implied Volatility Models------------------------------------------------------------------------17

3.1 The Local Volatility Model------------------------------------------------------------------------------17

3.1.1 Option Pricing-------------------------------------------------------------------------------------------17

3.1.2 Implied Volatilities and The Local Volatilities------------------------------------------------------22

3.1.3 Implied Volatilities and The Dupire Volatilities-----------------------------------------------------41

3.1.4 Summary of Three Types of Volatilities-------------------------------------------------------------55

Chapter 4 Conclusions and Future Studies-----------------------------------------------------------------58

Notation---------------------------------------------------------------------------------------------------------60

AppendixA-----------------------------------------------------------------------------------------------------61

Appendix B-----------------------------------------------------------------------------------------------------89

Bibliography--------------------------------------------------------------------------------------------------130

5

Chapter 1. Introduction

1.1 Motivation

In general, volatility is a measure for variation of price of a certain financial instrument over time

in finance. There are many types of volatilities categorized by different standards. For example,

historical volatility is a type of volatility derived from time series based on the past market prices; a

constant volatility is an assumption of the nature of volatility that we usually make in deriving the

Black-Scholes formula for option prices. An implied volatility, however, is a type of volatility

derived from the market-quoted data of a market traded derivative, such as an option.

One of the most frequently used models, the Black-Scholes model which assumes a constant

volatility is used to derive the corresponding implied volatility for each quoted market price for

options. Indeed, the Black-Scholes model has been a great contribution to option pricing area

Nevertheless, there are still some facts that contradict the key assumptions in Black-Scholes model,

especially the constant volatility assumption. The evidence to this contradiction is a long-observed

pattern of implied volatilities, in which at-the money options tend to have lower implied volatilities

than in- or out-of-the-money options. This pattern is called "the volatility smile"(sometimes

referred to as "volatility skew") which was starting to show in American markets after the huge

stock market crash in 1987.

One explanation for this phenomenon is that in reality the volatility of an underlying asset is not

really a constant value throughout the lifespan of the derivative. That is why the volatility curve

plotted by the using of the values of implied volatility inferred by Black-Scholes model does not

appear to be horizontal, but displays a "volatility smile" in the plots. This, however, motivates us to

wonder whether such a model can be found, that gives a series of values of volatility close enough

to the volatility values in the volatility smile, i.e. , the implied volatility; or more specifically what

the difference between the volatility given by this alternative model and the corresponding implied

volatility inferred by Black-Scholes model is, if any.

Having this thought in mind, we can also apply this scheme of searching for suitable models to

testing among different types of models. Our demonstration in this paper uses the local volatility

model.

1.2 Objectives

In this paper, the alternative model type for the Black-Scholes model we use is the local volatility

model.

6

Our objective here is to set up the pricing model for options using the stock price processes and

other conditions specified by the local volatility model, solve the option values for this model,

calculate the corresponding implied volatilities for this model, thus to achieve our goal of

comparing these two volatilities, the implied volatilities and the local volatilities.

Besides the local volatility given by the local volatility model, we also want to compare the implied

volatilities to another local volatility, the dupire volatility. The Dupire volatility is a way of

calculating volatility under the Dupire model, which treats the strike price and the maturityKtime instead of the stock price and current time point as variables in the option valueT S tfunction . We will introduce this Dupire model and Dupire volatility in detail in),;,( tSTKVChapter 2. This additional analysis would give us some additional points of views to this local

volatility model here.

1.3 Chapter Review

Chapter 1, Introduction, mainly talks about the theoretical and practical reasons that motivate us to

write about this topic on implied volatility models in this paper, and sets straight the objectives of

our research as well.

Chapter 2, Background, introduces us to two types of models that will be focused on in the later

chapters of this paper. They are the the local volatility models and the Dupire local volatility model.

This chapter familiarize us to the basic knowledges of these models, and we will discuss these

models in detail including pricing option values and evaluating volatility values in Chapter 3.

Chapter 3, The Implied Volatility Models, concentrates on the difference between the implied

volatilities that are inferred by Black-Scholes model and the volatility factors that are specified by

the local volatility models, with all parameters of these two models staying the same. The former is

obtained by solving the volatility implied by the Black-Scholes formula for options reversely with

known option values. The latter in our paper here is the local volatility specified by the local

volatility model's price processes with known stock prices and time. And also, we are curiously that

what the difference between the implied volatility inferred by the Black-Scholes model evaluated

by a reverse calculation and the Dupire volatility computed by the Dupire method of calculating

volatility(which is called Dupire volatility) is, since both of the option values used in these two

models are essentially based on the Black-Scholes model. Hence, in short, between implied

volatility and local volatility(of the local volatility model), the implied volatility and the Dupire

volatility, we do two sets of cross-references by evaluating the distances between them to find their

inner connections. This is also the main idea of searching for a more realistic alternative asset price

processes for the Black-Scholes model(each different asset price processes correspond to a specific

7

option ricing model), which has the implied volatility close to that of an asset price process which

follows a Black-Scholes model. We finish this chapter by analysing our numerical results and plots.

Chapter 4, Conclusions, which sums up the conclusions for our research and the results in this paper.

8

Chapter 2. Background

In this chapter, we briefly introduce the models we use in this paper.

2.1 The Local Volatility Models

In the 1970s, when Black-Scholes formula was initially derived, most people were convinced that

the volatility of a certain asset given the current circumstance was a constant number. Then, later

on,after the economic crash in 1987, people were starting to doubt the constant volatility

assumption. Especially after more and more evidence of volatility smile was collected, people tend

to believe that the implied volatilities can not remain constant during the whole time. They

probably have some dependent relationships with some other factors in the option pricing model as

well. One of such guesses is that, the implied volatility could be depending on the stock price

and time . And if we study a model of price processes with a volatility that depends on the)(tS tstock price and time , we can try to explore the inner connection between the implied)(tS tvolatility , and the local volatility . The volatility in such models depends on theimpσ )),(( ttSσ

stock price and time . This is why we call these types of models the local volatility models,)(tS twhose volatilities are determined locally.

Hence, we take one example out of this category, and consider a case where the volatility is

decreasing with respect to the stock prices.

Given the local volatility model under an EMM(equivalent martingale measure, we use the same

acronym in the following) as following,Q

, (2.1)dtrdWStSdS t ⋅+⋅⋅= ),(σ

where we assume,

, (2.2)t

t SS 1)( =σ

. (2.3)0=rBy (2.2) and (2.3), the original model (2.1) is degenerated into the following form:

. (2.4)ttt dWSdS ⋅=

Denote the option value function as .),( tSV t

Hence, it follows from Ito formula and equation (2.4) that,

22

2

)(21 dSSVdS

SVdt

tVdV

∂∂

+∂∂

+∂∂

=

9

22

2

)(21 dWSSVdS

SVdt

tV

∂∂

+∂∂

+∂∂

=

(2.5)dSSVdtS

SV

tV

∂∂

+∂∂

+∂∂

= )21( 2

2

Thenwe consider delta-hedged portfolio,

. (2.6)SSVV∂∂

+−=π

Gven the martingale measure , thus under arbitrage-free condition, we will arrive at theQ

condition that,. (2.7)dtrd ⋅⋅= ππ

We re-write (2.6) in differential form that,

. (2.8)dSSVdVd∂∂

+−=π

Compare (2.8) with (2.7), then insert (2.5), the corresponding partial differential equation (PDE)

for model (2.4) takes the form,

. (2.9)021

2

2

=∂∂

+∂∂ S

SV

tV

If we let , and represent the option value, the first-order partial),( tSV t ),( tSV tt ),( tSV tSS

derivative with respect to variable , the second-order partial derivative with respect to variablet, respectively, (2.9) can be expressed in the following way,tS

. (2.10)0),(21),( =+ tSSVtSV tsstt

Model (2.4) is known as one of the local volatility models, whose form can be included into the

SDEMRDModel category inside the matlab database.

Creating the Local Volatility Model from Mean-Reverting Drift (SDEMRD) Models

The SDEMRD class derives directly from the SDEDDO class. It provides an interface in which the

drift-rate function is expressed in mean-reverting drift form:

, (2.11)tt

ttt dWtVXtDdtXtLtSdX )(),(])()[( )( ⋅+⋅−= α

where，

Xt is an NVARS-by-1 state vector of process variables;

S is an NVARS-by-NVARSmatrix of mean reversion speeds;

L is an NVARS-by-1 vector of mean reversion levels;

10

D is an NVARS-by-NVARS diagonal matrix, where each element along the main diagonal is the

corresponding element of the state vector raised to the corresponding power of α;

V is an NVARS-by-NBROWNS instantaneous volatility rate matrix;

dWt is an NBROWNS-by-1 Brownianmotion vector.

SDEMRD objects provide a parametric alternative to the linear drift form by reparameterizing the

general linear drift such that:

. (2.12))()(),()()( tStBtLtStA −==

Hence, we can create in matlab the model in

. (2.4)ttt dWSdS ⋅=

by inputing the following command in Matlab. SDEMRD objects display the familiar Speed and

Level parameters instead of A and B.

Table 2.1: The Local Volatility Model in Matlab

2.2 The Dupire Model(Method)

Frankly, this Dupire model is more of a method for calculating local volatilities than a pricing

model itself.

First of all, let us discuss some basic developments on the implied volatilities so far.

>> obj = sdemrd(0, 0, 0.5, 1) % (Speed, Level, Alpha, Sigma)

obj =

Class SDEMRD: SDE with Mean-Reverting Drift-------------------------------------------Dimensions: State = 1, Brownian = 1

-------------------------------------------StartTime: 0StartState: 1Correlation: 1

Drift: drift rate function F(t,X(t))Diffusion: diffusion rate function G(t,X(t))Simulation: simulation method/function simByEuler

Alpha: 0.5Sigma: 1Level: 0Speed: 0

11

One of the basic assumption in Black-Scholes is that the volatility of the underlying asset stays

constant during the entire time of option's lifespan. Hence, we can know from the Black-Scholes

formula for option prices, that, option prices has the following form

).,,;,( TKtSVV σ=

If we quote from the market date, the option price and underlying asset price of0VV = 0SS =an option with strike price and maturity at time point , we can obtain an0KK = 0TT = 0tt =equation from Black-Scholes formula for ,σ

. (2.13)),,;,( 00000 TKtSVV σ=

From Black-Scholes formula, we can calculate the Greeks, in particular, vega,

.0)()( 21 >==∂∂

= −− τφτφσ

ν ττ dKedSeV rq

Hence can be uniquely determined by equation (2.13).0σσ =

Since the volatility of the underlying asset is constant by assumption of the Black-Scholesσ

model. Then, theoretically the implied volatility derived from (2.13) should be a constant,0σσ =

i.e., independent of the strike price and maturity chosen here. However, in reality, this is0K 0T

contradicted by the existences of volatility smile and volatility skew. In fact, the implied volatility

inferred from option prices with different strike prices and expiration dates is a function ofσ

, [1].TK , ),( TKσσ =

The dependence on strike prices can be shown by the following figure 2.1 and figure. 2.2, given a

fixed maturity time and a fixed initial price at time point . The curve in0TT = 0SS = 0tt =figure 2.1 is called the volatility smile, the curve in figure 2.2 is called the volatility skew.

Similarly, the dependence on the maturity time can be illustrated by curve in figure 2.3, givenT

the stock prices and the strike prices stay unchanged. This shows the term structure ofS K

volatility.

12

Figure 2.1: Volatility Smile

Figure 2.2: Volatility Skew

Figure 2.3: The Term Structure of Volatility

13

To explore the characteristics of implied volatility in a more mathematical way, let us discuss the

model analytically.

Under risk-neutral measure, the underlying asset price process is

, (2.14)tdWtSdtqrSdS ),()( σ+−=

where is the risk-free interest rate, is the dividend yield, is the asset prices,r q S TttW ≤≤0}{

is a Brownian motion(Wiener process), is the asset's volatility that depends on asset pricesσ

and time .S tThus, by using the same approach as in section 2.1, we obtain the PDE for this option under Black-

Scholes model,

. (2.15)0)(),(21

2

222 =−

∂∂

−+∂∂

+∂∂ rV

SVSqr

SVStS

tV

σ

Adding the terminal and boundary conditions to equation (2.15), we can estabilish the following

value problem for option price, in particular, an European call option price.

Definition 2.1 is called the fundamental solution of the Black-Scholes equation, if),;,( TtSG ξ

it satisfies the following terminal value problem to the Black-Scholes equation:

⎪⎩

⎪⎨⎧

−=

=−∂∂

−+∂∂

+∂∂

=

)17.2(),(),(

)16.2(,0)(2 2

22

2

ξδ

σ

STSV

rVSVSqr

SVS

tvLv

where is the Dirac function.. □)(,0,0,0 xTtS δξ <<∞<<∞<<

Problem 2.2 Let be a call option price, satisfying the following terminal),,;,( TKtSVV σ=

value problem:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

∞<≤−=

≤≤∞≤≤

=−∂∂

−+∂∂

+∂∂

+ )19.2()0(.)(),()0,0(

)18.2(,0)(),(21

2

222

SKSTSVTtS

rVSVSqr

SVStS

tV

σ

Suppose at*SS = )0(, 1** Tttt <≤=

is given as the boundary condition,),0(),(),,;,( 21** TTTKTKFTKtSV ≤≤∞<<=σ

find . □),0(),,( 21 TtTStS ≤≤∞<≤=σσ

Before we are ready to discuss the Dupire method, let us familiarize ourselves with some

theoretical background [1] beforehand.

14

Theorem 2.3 If the fundamental solution is regarded as a function of , then),;,( ηξtSG ηξ ,

it is the fundamental solution of the adjoint equation of the Black-Scholes equation. That is, let

,),;,(),( ηξηξ tSGv =

then satisfies),( ηξv

⎪⎩

⎪⎨

⎧

−=

=−∂∂

−−∂∂

+∂∂

−=∗

)21.2(),(),(

)20.2(,0)()()(2

22

22

Stv

rvvqrvvvL

ξδξ

ξξ

ξξ

ση

where □.,0,0 1ηξ <∞<<∞<< tS

Corollary 2.4 Theorem 2.1 indicates, if the fundamental solution of equation (2.18) is

, then),;,(* tSG ηξ

□).,;,(),;,( * tSGtSG ηξηξ =

The proof of above theorem 2.1 and corollary 2.2 can be referred to Lishang Jiang(1994)[1].

Then, let us move on to discuss the Dupire method in detail.

We denote an European call option price as

, define the second derivative of the option prices with respect to strike prices),;,( TKtSVV =

. (2.22)),;,(2

2

TKtSGKV

=∂∂

By equation (2.22) and (2.23), satisfies the system thatG

⎪⎩

⎪⎨⎧

−=

=−∂∂

−+∂∂

+∂∂

)24.2(,)(),(

)23.2(,0)(),(21

2

222

KSTSG

rGSGSqr

SGStS

tG

δ

σ

where is the Dirac function. We know that , thus (2.24) can be written)( KS −δ )()( xx δδ =−

as,

. (2.25))()(),( SKKSTSG −=−= δδ

Then by Definition 2.1, we know that is the fundamental solution to equation),;,( TKtSG(2.18). By Theorem 2.3, is the fundamental solution, as a function of (),;,( TKtSG TK , tS,are paramters), similar to (2.23) and(2.24), thus satisfies the following system,

15

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

∞<≤−=<∞<≤

=−∂∂

−−∂∂

+∂∂

−

)27.2()0(.)(),;,()26.2(),0(

,0)()()),((21 22

2

2

KSKTKtSGTtK

rGKGK

qrGKTKKT

G

δ

σ

We substitute (2.22) into (2.26), (2.27), then integrate both sides twice with respect to K in interval

. Since we know that,],[ ∞K

i) given a certain , if , for a call option, the following items will all tend to 0, i.e.,S ∞→K

,0)(,,,, 2222 →∂∂

∂∂

∂∂ GK

KKGKGK

KVKV σσ

ii) ηηδηηηδξξ

dSKdSdKK

)()()( −−=− ∫∫ ∫∞∞ ∞

,)(

)()(0

+

+∞

−=

−−= ∫KS

dSK ηηδη

iii) ,),;,( 2

2

KVdVdTtSG

K K ∂∂

−=∂∂

=∫ ∫∞ ∞

ξξ

ξξ

iv) ),,;,(),;,( TKtSVdTtSVK

−=∂∂

∫∞

ξξξ

v) ,),;,( 2

2

VKVKdVdTtxG

KK+

∂∂

−=∂∂

= ∫∫∞∞

ξξ

ξξξξ

vi) .),()),(( 2

22222

2

2

KVKTKdGTd

K ∂∂

=∂∂

∫ ∫∞ ∞

σηηηση

ξξ

Thus, we can transform the system of (2.33), (2.34) based on into the following),;,( TKtSG

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

∞<≤−==

<∞<≤

=−∂∂

−−∂∂

+∂∂

−

+ )29.2()0(.)(),;,()28.2(),0(

,0)(),(21

2

222

KKStTKtSVTtK

qVKVKqr

KVTKK

TV

σ

From equation (2.28), we obtain the explicit expression for implied volatility

. (2.30)

2

22

21

)(),(

KVK

qVKVKqr

TV

TK

∂∂

+∂∂−+

∂∂

=σ

16

This idea of Dupire's of calculating volatility seems to be simple and nice in theory.

However, when it becomes to the reality, when traders want to apply this into real market, the first

obstacle we must overcome is calculating the derivatives of option price, i.e., .2

2

,,KV

KV

TV

∂∂

∂∂

∂∂

And in fact, there is no simple analytical way to do it but to resort to some numerical approach, for

example, finite difference method, etc. Nevertheless, as we are about to see in chapter 3 section 2,

the numerical approach is not good enough for calculating this Dupire volatility, as a slight amount

change in option value would lead to some significant change in the value of derivatives, thus the

volatility value. We can almost say that using (2.30) to calculate implied volatility is ill-posed

[1](we will go to details about this in Chapter 3).

17

Chapter 3. Implied Volatility Models

In this chapter, we compare two different types of volatilities, the local volatility and Dupire

volatility, with implied volatilities under the structure of local volatility model.

3.1 The Local Volatility Model

As we establish in Section 2.1 that, given an asset's price process under an EMM with the riskQfree interest rate that0=r

, (3.1)dWSdS ⋅=

we will have the option pricing problem for an European call option as

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

∞→→−→

−=

=

≤≤=+

+

)5.3(.1),(,),(

)4.3(,)(),()3.3(,0),0(

)2.3()0(0),(21),(

SastSVKStSV

KSTSVtV

TttSSVtSV

S

sst

3.1.1 Option Pricing

Since there is no simple analytical solution for the system (3.2)-(3.5), we then have to resort to the

numerical way to solve the option terminal value problem for this system.

We use software Matlab in this paper to solve numerical problems.

After a closer examination, we realize that we have a terminal boundary value problem here instead

of an initial one, hence in order to use the built-in initial boundary value solver function in Matlab,

we have to substitute some variables in the problem to shift the terminal boundary problem to an

initial boundary problem in order fit this problem into the solving range of the built-in function.

If we denote the time-to-maturity as , then it becomes obvious that if any one of thesetT -=τ

three variables( ) is fixed, the other two will either move in the same direction or in theTt,,τopposite ones. Thus, for every given , we have difference between and is fixed, writtenτ T tin the differential form, i.e.,

. (3.6)dtdT -=

Also, in our model here, we have the asset price not dependent on time as shown in (3.1), thus we

are ready to substitute the time point variable with the maturity time parameter , and regardt Tthe maturity time as a variable as well as treating time-to-maturity as a parameter fromT τ

now on .

18

Thus, system (3.2)-(3.5) can be transformed into the following system,

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

∞→→−→−==

=

≤≤==+−

+

)10.3(.1),(,),()9.3(,)(),()8.3(,0),0(

)7.3()0(0),(21),(

0

0

SasTSVKSTSVKStTSV

TV

tTtTSSVTSV

S

TssT

Then, we can apply the built-in function pdepe in Matlab to solve the above problem.

pdepe is a function that solves initial-boundary value problems for parabolic-elliptic Partial

Differential Equations (PDEs) in one-dimension.

pdepe solves PDEs of the form:

. (3.11)⎜⎜⎝

⎛⎟⎠⎞

∂∂

+⎜⎜⎝

⎛⎟⎠⎞

∂∂

∂∂

=∂∂

∂∂ −

xuutxs

xuutxfx

xx

tu

xuutxc mm ,,,),,,(),,,(

The PDE holds for and . The interval must be finite. can be 0,fttt ≤≤0 bxa ≤≤ ],[ ba m1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. If , then0>m amust be non-negative.

In (3.11), is a flux term and is a source term. The coupling of the),,,(xuutxf∂∂ ),,,(

xuutxs∂∂

partial derivatives with respect to time is restricted to multiplication by a diagonal matrix

. The diagonal elements of this matrix are either identically zero or),,,(xuutxc∂∂ ),,,(

xuutxc∂∂

positive. An element that is identically zero corresponds to an elliptic equation and otherwise to a

parabolic equation, and there must be at least one parabolic equation. An element of thatccorresponds to a parabolic equation can vanish at isolated values of if those values of arex xmesh points. Discontinuities in and/or due to material interfaces are permitted provided thatc s

a mesh point is placed at each interface[2].

For and all , the solution components satisfy initial conditions of the form0tt = x

. (3.12))(),( 00 xutxu =

For all and either or , the solution components satisfy boundary conditions oft ax = bx =the form

. (3.13)0),,,(),(),,( =∂∂

+xuutxftxqutxp

Particularly, in our PDE (3.2) here, if we denote in (3.2), as , as , asS x T t ),( TSV

, then (3.7)-(3.10) become),( txu

19

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

∞→→→−==

=

≤≤=⋅=

+

)17.3(.1),(,),()16.3(,)(),()15.3(,0),0(

)14.3()0(21

0

0

xastxuxtxuKxttxu

tu

tttuxu

x

Txxt

In fact, from a mere observation in the real market, we know that underlying stock price 100=Sis quite high for an option with strike price . Then we can replace the infinity requirement10=K

of limits in equation (3.17) by setting stock price to , given a strike price . Then,100=S 10=K

(3.10) and (3.17) become

, (3.18)100,1),(,-),( === SwhereTSVKSTSV S

. (3.19)100,1),(,-),( === xwheretxuKxtxu x

Then (3.14)-(3.17) take the new forms of (3.20)-(3.23),

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

===−==

=

≤≤=⋅=

+

)23.3(.100,1),(,-),()22.3(,)(),()21.3(,0),0(

)20.3()0(21

0

0

xwheretxuKxtxuKxttxu

tu

tttuxu

x

Txxt

Now, let us rearrange (3.20) in the following form

. (3.24)⎟⎠⎞

⎜⎝⎛ −⋅⋅

∂∂

=∂∂ − )(

2100 uuxx

xx

tu

x

Comparing (3.24) to (3.11), i.e.

, (3.11)⎜⎜⎝

⎛⎟⎠⎞

∂∂

+⎜⎜⎝

⎛⎟⎠⎞

∂∂

∂∂

=∂∂

∂∂ −

xuutxs

xuutxfx

xx

tu

xuutxc mm ,,,),,,(),,,(

we find out that., (3.25)0=m

, (3.26)1),,,( =∂∂xuutxc

, (3.27))(21),,,( uux

xuutxf x −⋅=∂∂

. (3.28)0),,,( =∂∂xuutxs

Next step, let us identify (3.21)-(3.23) to initial and boundary conditions of the form (3.12)-(3.13).

(3.22) itself already complies with the form of (3.12), i.e., the initial condition. Hence, identifying

(3.21) and (3.23) to the boundary conditions of the form (3.13), i.e.,

20

, (3.13)0),,,(),(),,( =∂∂

+xuutxftxqutxp

is equivalent to finding pairs of values of function and function , which satisfies),,( utxp ),( txq

the form in (3.13), given the flux function by (3.27).)(21),,,( uux

xuutxf x −⋅=∂∂

If we substitute (3.27) into (3.13), we have

. (3.29)0)(21),(),,( =−⋅⋅+ uuxtxqutxp x

And the boundary conditions (3.21) and (3.23) are

⎪⎩

⎪⎨⎧

===

=

)31.3(.100,1),(,-),(

)30.3(,0),0(

xwheretxuKxtxu

tu

x

We insert (3.30) into (3.29) at , then0=x

. (3.32)0)0(21),0(),,0( =−⋅⋅+ xuxtqutp

For (3.32) to hold, one option is to put and to 0, i.e.,),,0( utp ),0( tq

⎩⎨⎧

==

)34.3(.0),0()33.3(,0),,0(

tqutp

Similarly, we insert (3.31) into (3.29) at , then100=x

. (3.35)0|)(21),100(),,100( 100=−⋅⋅+ =xx uuxtqutp

We simplify (3.35), obtain

.0

)(21),100(),,100(

|))(1(21),100(),,100(

|)(21),100(),,100(

100

100

=

⋅+=

−−⋅⋅+=

−⋅⋅+

=

=

Ktqutp

Kxxtqutp

uuxtqutp

x

xx

This is to say,

. (3.36)021),100(),,100( =⋅+ Ktqutp

For (3.36) to hold, we can simply choose a pair of values of and ,),,100( utp ),100( tq

⎪⎩

⎪⎨⎧

=

=

)38.3(.1),100(

)37.3(,21-),,100(

tq

Kutp

In conclusion, our boundary conditions now take the form of

21

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

=

==

)42.3(.1),100(

)41.3(,21-),,100(

)40.3(,0),0()39.3(,0),,0(

tq

Kutp

tqutp

After specifying all the conditions and function forms, we are ready to gather together all the

thoughts stated above to write them into a program file pdex_u.m (which is included inAppendixA,

Table A.1)in Matlab. We set the values for each one of the variables and parameters, the initial

value of maturity , strike price , risk free interest rate , dividend yield00 == tT 10=K 0=r, using 201 mesh points in the option price range from 0 to 100 and 51 mesh points in the0=q

maturity range from 0 to 5, to simulate the numerical option value at each price level. The option

value curves, option values plotted against asset prices, under different time-to-maturity periods τ

are shown in Figure 3.1(the more complete series of curves of option value at different levels of

time-to-maturity is included in Appendix A). The option price surface with respect to the time-to-

maturity and asset price is shown in Figure 3.2.τ

Figure 3.1: The OptionValue Curves of EuropeanCalls and the Payoff Diagram at Maturity

22

Figure 3.2: The OptionValue Surface for EuropeanCall Options

As shown in figure 3.1 and figure 3.2, without any unexpected outcome , the option value curve

and surface under this local volatility model have no substantial difference to those of a vanilla

European call option under a generic Black-Scholes model. The longer the period of time-to-

maturity, the more valuable the call options; the higher the stock/asset price, the closer to payoff

the option values at maturity.

In the following subsection, we try to find out the internal connection between the implied

volatilities and the local volatilities.

3.1.2. Implied Volatilities and The Local Volatilities

With the preparations in Section 3.1.1, we can now move on to calculate the implied volatilities of

this local volatility model for each mesh point in the two-dimensional space consisted of asset

prices and time. As we mentioned in Section 2.2, the volatility is determined by the Black-Scholes

formula for option prices uniquely, given the other inputs, such as asset price , interest rate ,S r

dividend , maturity , time , strike , option price and so on. This is to say, we canq T t K V

derive a unique implied volatility from Black-Scholes model, in other words,impσ

exists and is unique.),,;,( VtKTSimp σσ =

23

The existence of the implied volatility can be observed from the corresponding relationship

between the option price and the implied volatility . This is true by the formulation of theV impσ

Black-Scholes formula for option pricing. The problem of the uniqueness of the implied volatility

can be solved by the monotonicity of the option price with respect to the maturity timeimpσ V

. For a call option value , where are variables. areT ),,;,( tKTSVVCall σ= TS , tK ,,σ

parameters. We know that the one of the Greeks in Black-Scholes formula for call options, vega,

[3]. Hence, given any value set of , we will find a unique0- <∂∂

=∂∂

=tV

TV

ν ),;,,( tKTSV

value for , which is called the implied volatility, denoted as . For example, , forσ impσ 0σσ =imp

an input set of .),;,,(),;,,( 0000 tKTSVtKTSV =

Therefore, we can regard as a function of , where are parameters, is alsoσ TS , tK , V

quoted from market price, i.e. . While at the same time, the local volatility),,;,( VtKTSσσ =

denoted as can be easily observed from the price processes of this local volatility model, thatlocσ

, for each mesh point in the price axis. Thus, the distance between two correspondingSloc1

=σ

volatilities can be easily calculated. The program for implied volatilities' calculation pdex_imp.m is

included inAppendixA TableA.2.

The plots of implied volatilities and local volatilities are shown in the following.

24

Figure 3.3: The Implied Volatility Curves(Plotted against the Stock Price )S

Figure 3.3 is the implied volatility curve plotted against the stock prices at three different time

points. As we can see in figure 3.3, the implied volatility of the option is quite large (In fact, when

the stock price is close to 0, the implied volatility tends to infinity. We will discuss this in detail at

this end of Section 3.1.2) at those points where the stock prices are close to 0, and as the stockS

price goes up, the implied volatilities gradually fall back to a relatively low and stable level.S

The implied volatility decreases at a decreasing speed as the stock price increase. From an

economic point view, if the stock prices drop to a level close to 0, then the options based on the

same stock will be extremely risky, thus the indicator of riskiness will be extremely large, i.e.

as . On the contrary, the higher the stock price , the less risky the call,∞→impσ 0→S S

option value . However, the decreasing of the riskiness of the underlying asset that the option isV

based on, is not enough to reduce all of the risks that the option is facing, some of which are some

systematic risks, such as the macroeconomic status and so on. Hence, as the stock price increases,

eventually, the implied volatilities will tend to a stable non-negative level in general. However, in

our stock price process here, we assume the risk-free interest rate is 0. This means that investors are

25

not rewarded at an interest for taking the systematic risk, this means that the systematic risk0=r

is 0. Hence, in our special case here(the risk-free interest rate is 0), when the stock price tends to

infinity, the implied volatility tends to 0. The term structure of the implied volatility is shown in

figure 3.4.

Figure 3.4: The Implied Volatility Curves(Plotted against the Time-to-maturity )τ

Figure 3.4 shows the term structure of the implied volatilities at different stock price levels. As the

maturity time comes closer, the option value will become more volatile, hence the implied

volatility will become higher.And as the stock price increases, the curve of implied volatilities

plotted against the time-tom-maturity will shift downward as a whole.

26

Figure 3.5: The Local Volatility CurveSloc1

=σ

Figure 3.5 shows the local volatility curve that given by which only depends on theSloc1

=σ

stock price . And, we know that as , as well asS ∞→=Sloc1

σ 0→S 01→=

Slocσ

as .∞→S

From the illustration of above figure 3.3-3.5, we find out that the implied volatility and theimpσ

local volatility almost have the same tendency of change. Then, we are more curious to findlocσ

out exactly how far away they are from each other.

The Distance between Implied Volatilities and Local Volatilities

Since we have calculated the value of implied volatilities and are aware of that the localimpσ

27

volatility has the form , then we can find out the distance between and bySloc1

=σ impσ locσ

distance function .S

d implocimp1

−=−= σσσ

We put our theory here into practice by program file pdex_dis_imp_loc.m written in matlab(this

program is include in Appendix A). All the parameters and indicators that need to be specified are

gathered in the following table 3.1.

Table 3.1: The Initial Variable Set-up for Program pdex_dis_imp_loc.m

The plots of this section is shown in the following figure 3.6 and figure 3.7.

Price(stock/asset price) 201 mesh points, from 0 to 100.Strike(option strike price) 10Rate(risk-free interest rate) 0Time(time-to-maturity) 51 mesh points, from 0 to 5Value(option value) 51×201 values, calculated in Section 3.1.1

Limit(the upper bound for volatility searchinginterval)

10 times

Yield(dividend yield) 0Tolerance(calculation accuracy) 10-16

Class(option type) call option

28

Figure 3.6: The Comparison of Implied Volatility and Local Volatility ( )10=K

Figure 3.6 is a demonstration of how the distance between the implied volatility and the local

volatility changes as the stock price increases.At first, the the local volatility curve is above the

implied volatility curve, then as the stock price increases, the local volatility decreases more rapidly,

then at a certain stock price level, they intersect, and after that the implied volatility curve lies

above the the local volatility curve. The change of distance between them is shown by the distance

curve marked in black in figure 3.6. Before the stock price reaches the strike price, as the stock

prices increases, the distance decreases rapidly; then at the point when the stock price is equal to the

strike price, the distance reaches 0; after the stock price exceeds the strike price, the distance

gradually increases to a certain relatively low level and stays that way as the stock price continues

to increase.

29

Figure 3.7: TheAbsolute Difference between Implied Volatility and the Local Volatility Curve

(PlottedAgainst , )imploc σσ − S 10=K

When plotting distance curves between the implied volatilities and the local volatilities solely, we

obtain the three curves shown in figure 3.7, each of which represents a different time-to-maturity

level. We can hardly tell them apart without magnifying them since they are lying very close to

each other in figure 3.7. Nevertheless, we can almost say affirmatively that the distance between

these two volatilities are essentially 0 at the point where the stock price is equal the optionSstrike price ; while at those points where the price doesn't reach the strike price level from theKbelow, the distance between them are relatively far away. On the other hand, as the stock price goes

up from above the strike price, the distance between them then will be maintained at a quite stable

level. The more accurate and actual computation results can be read from numerical results

included inAppendix B.

30

Figure 3.8: TheAbsolute Difference between Implied Volatility and the Local Volatility Curve

(PlottedAgainst )imploc σσ − T

In order to show how the distance between the implied volatility and the local volatility changes as

time goes by, we plot figure 3.8. Notice in the distance function that

,S

d implocimp1

−=−= σσσ

the only time-sensitive factor in it is the implied volatility . Hence, at the parts of curvesimpσ

where the maturity time is far away, the distance curves in figure 3.8 reveal some similarT

nature to implied volatility curves in figure 3.4, as they both tend to stay at a relative stable level,

almost parallel to the time-to-maturity axis. Another interesting fact can be observed in figure 3.8 as

well is that: when the stock price is below the strike price , the distance curve shiftsS K

downward as the stock price goes up; when the stock price is above the strike price ,S S K

the distance curve shifts upward as the stock price continues to increase. The distance curveS

hits the bottom when the stock price is equal to the strike price . This observation again, isS K

consistent with our conclusions in figure 3.6 and figure 3.7.

31

The Limits of Implied Volatilities and Local Volatilities at S=0.

One vague statement that we have not really explained in this section is that we say the implied

volatility is very large for those points at which the stock prices are close to 0.SAlthough the plots of the implied volatility curves have indicated that the implied volatilities would

more than likely to go to infinity when the stock price tends to 0. However, we still need more

concrete evidence to prove our speculation here.

We know from initial condition of option pricing that when the stock price falls back to 0, the

option value is also 0, meaning that the ownership of this asset is worthless. Then it makes no sense

to talk about the implied volatility of the option value. Thus, we choose a small neighbourhood of 0

on the stock price axis with its left side end open. For example, we choose . ByS ]10,0( 10−=Susage of the option pricing scheme described in Section 3.1.1, we calculate the option values within

this small interval. We modify our previous program for option pricing by equally choosing 201

mesh points on interval and adding a single point to the collection of]10,0[ 10−=S 100=Smesh points. In this way, we can both achieve the pricing for option prices at small stock price

points and keep our boundary conditions unchanged. This altered program for option pricing is

named as pdex_u_small_s.m, which is included inAppendixA. Then, we use the same scheme for

implied volatility calculations as before. The program file of implied volatility computation for

small , pdex_imp_small_s.m is included inAppendixA as well. The more detailed initialSparameter set-up for program pdex_imp_small_s.m is displayed in the following table 3.2.

Table 3.2: The Initial Variable Set-up for Program pdex_imp__small_s.m

The plots of this program are shown in figure 3.9-3.12. Figure 3.9 depicts the implied volatility

curves plotted against the stock prices at different time-to-maturities. However, according to figure

Price(stock/asset price) 202 mesh points, 100 points from 0 to 10-10 ,and 1.

Strike(option strike price) 10Rate(risk-free interest rate) 0Time(time-to-maturity) 51 mesh points, from 0 to 5Value(option value) 51×202 values


109 times



32

3.9, the implied volatility at is not a very large number, although the slopeimpσ 1010.10 −⋅=S

is quite steep in the neighbourhood of . Notice that the scale of the volatility axis and the0=S

scale of the stock price axis are obviously different, the latter is enormously larger comparing to the

former. So, if we put both axis to the same measure scale, then the implied volatility curve will be

extremely steep in this interval [0,10-10]. From the above argument, we realize that the steepness of

the implied volatility curve is not necessary an accurate way to determine whether the implied

volatility tends to infinity at .0=S

As an alternative of graphical analysis, let us consider the derivative of implied volatility with

respect to the stock price . Notice that our partition of the interval [0,10-10] is enough small,Simp

∂∂σ

hence we can use the implied volatility value at each mesh point and the step size of the partition to

approximate the derivatives at each point. For example, we select some results of impliedSimp

∂∂σ

volatility from program's data(the numerical results of the implied volatilities calculated by

pdex_imp_small_s.m, which are included inAppendix B).

Table 3. 3: The Slopes of Implied Volatility Curves

As we can see in table 3.3, that the derivative is a really large negative number whenSimp

∂∂σ

τ0impσ

1impσ 100 - SSS =∆0

10

SSimpimpimp

∆

−=

∂

∂ σσσ

2.8 3.71288296174587 3.6618307127546212105.0- −⋅

1110.021-82.4861021044979-

⋅≈

3.8 3.23962470398632 3.1956475324642212105.0- −⋅

1110-0.874.21168795434304-

⋅≈

4.8 2.92041541613200 2.8811834011648012105.0- −⋅

1110.780-4.40027846402993-

⋅≈

33

the stock price is . And we know that the implied volatility is a positiveS 1010−impσ

number at . Hence, if the value of the slope of the implied volatility curve stays1010−=SSimp

∂∂σ

at the current amount, the implied volatility will eventually go to plus infinity as the stock price

continues to decrease from below . This is to say, if holds, then1010−=S 02

2

≥∂∂Simpσ

. Now, all we have to do make sure that is true. We plot the slopes of+∞=→ impSσ

0lim 02

2

≥∂∂Simpσ

implied volatilities into curves in figure 3.10, with respect to the stock prices . And indeed, as itS

is shown in figure 3.10, the slopes of the "slope curves" are truly non-positive, i.e., . In02

2

≥∂∂Simpσ

fact, we can also read from the numerical results of slopes of implied volatility curves to arrive at

the same conclusion(see table 3.4). The slopes of implied volatility curves are decreasing as the

stock price decreases, i.e., . Then, we can say surely that the implied volatility02

2

≥∂∂Simpσ

impσ

tends to infinity as the stock price goes to 0(The more complete numerical results of slopesS

are included inAppendix B).

In short, this is to say, because

i) for ;0|0>=SSimpσ )10( 10

0−=οS

ii) , for ;)10(1| 100

οσ

⋅−=∂

∂=SS

imp

S)10( 10

0−=οS

iii) for , ;02

2

≥∂∂Simpσ

00 SS ≤≤ )10( 100

−=οS

that we have

for , and thus .∞=→ impSσ

0lim 00 SS ≤< ∞=

→ impSσ

0lim

34

Table 3. 4: The Slopes of Implied Volatility Curves

Figure 3.9: The Implied Volatility Curves(Against Stock/Asset Price )S

\Sτ 12105.0 −⋅ 12100.1 −⋅ 12105.1 −⋅ 1210.02 −⋅

2.8 -102104497982.486 -60234023483.1758 -42968495088.3899 -33463371536.9570

3.8 -87954343044.2116 -51889446183.0994 -37017182609.4161 -28829357814.5602

4.8 -78464029934.4002 -46292532890.8126 -33025342546.6038 -25721008566.0331

35

Figure 3.10: The Slope Curves of Implied Volatility Curves(Against Stock/Asset Price )S

Now that we know for a fact that the implied volatility tends to infinity as stock price goes to 0, and

also that the local volatility tends to infinity as the stock price goes to 0, since . Then,Sloc1

=σ

we are even more curious about the their speeds of converging to infinity, in order to make better

judgements when approximating the implied volatility by local volatility at small stock prices.

First of all, we draw figures for the local volatility curve and the slope curve, which are shown in

figure 3.11 and figure 3.12.

36

Figure 3.11: The Local Volatility Curve (Against Stock/Asset Price ).50-1 SS==σ S

Figure 3.12: The Slope Curve of Local Volatility Curve .51-

3 21-1

21- S

SS ⋅==σ

37

We know from , that And the shape of curves in figureSloc1

=σ .0limlim0

=∞=∞→→ locSlocSσσ ，

3.11 and 3.122 confirms that. Notice that the order of magnitude of the vertical axis in figure 3.11

and 3.12 are completely different from that of figure 3.9 and 3.10. So, we can say affirmatively that

the speed of the local volatility's converge to infinity is much more faster than that of the implied

volatility's. But the question is that how much faster the former is. We know in general, that the

implied volatility curve is below the local volatility curve on interval [0,10-10], i.e.,

. We also know that, the corresponding slope curves of the implied5.00 −=<<< Slocimp σσ

volatilities are above those of the local volatilities due to the fact that their signs are negative, i.e.,

. These motivate us to find out the exact magnitude of implied05.0 5.1 <∂

∂=

∂∂

=⋅− −

SSS imploc σσ

volatility and its 1st order derivative , or at least the range of it. One way to do it isimpσSimp

∂∂σ

by fitting and into some certain potential functions' forms. More specifically,impσSimp

∂∂σ

we find the value ranges of and , andα β upperlower ααα −<−<−<0

that satisfy and0<<< upperlower βββ upperlower SS impαα σ <<<0

.05.05.0 <∂

∂⋅−<

∂∂

<∂

∂⋅−

SS

SSS upperlower

impββ σ

We establish our algorithm for searching such ranges by the following way. First of all, from a

simple observation of previous plots and some tests on the matlab program, we choose a rough

lower bound of searching region of , -0.5(motivated by the degree of S in local volatility), aα

upper bound -0.01(motivated by some simple testing on matlab). Also, we choose a rough lower

bound of searching region of , -1.5(motivated by the degree of S in slope of local volatility), aβ

upper bound -0.01(motivated by some simple testing matlab). Set the searching step size to 0.01,

then we can begin our search for such suitable values of and . The search programα β

pdex_imp_slope_small_s_fit.m is included inAppendixA.And the results of our searching is

38

.90.0,00.1,12.0,03.0

−=−=

−=−=

upper

lower

upper

lower

ββ

αα

Thus, we can estimate the implied volatility and its 1st order derivatives as

and , for .21.03.000 −− <<< SS impσ 05.05.0 09.00.01 <⋅−<∂

∂<⋅− −− S

SS impσ

]10,0( 10−∈S

The plots for demonstrating such estimations are shown in figure 3.13 and figure 3.14.

The result of this estimation is to say that if we were to express the implied volatility in aimpσ

potential form, that , where . Since the local volatilityασ Simp = 03.02.10 −<<− α

is given, hence , i.e., .5.0−= Slocσ 0limlimlim0 5.0

12.0

005.0

03.0

0=≤≤= −

−

→→−

−

→ SS

SS

Sloc

imp

SS σσ

0lim0

=→

loc

imp

S σσ

Similarly, if we present the 1st order derivative of the implied volatility with respect to stock/asset

price in a potential form, that , where . Also, knowingSimp

∂∂σ ασ Simp = 90.00.01 −<<− β

the 1st order derivative of a local volatility , we then can compute the limit5.15.0 −⋅−= Slocσ

, i.e., .0.50-.50-limlim

.50-

.50-lim0 5.1

9.0

005.1

0.1

0=

⋅⋅

≤

∂∂∂

∂

≤⋅⋅

= −

−

→→−

−

→ SS

S

SSS

Sloc

imp

SS σ

σ

0lim0

=

∂∂∂

∂

→

S

Sloc

imp

S σ

σ

In addition, we can write that . Again, let us look back on the distance between the∞=→

imp

locS σ

σ0

lim

implied volatility and the local volatility. Facts are , ,5.0−= Slocσ 5.15.0 −⋅−=∂∂ SSlocσ

, and for .21.03.000 −− <<< SS impσ 05.05.0 09.00.01 <⋅−<∂

∂<⋅− −− S

SS impσ

]10,0( 10−∈S

Hence the distance function satisfieslocimpd σσ −=

⎪⎩

⎪⎨

⎧

−⋅−=⋅−−⋅−>∂

∂−∂=

∂∂

>−>=−=

−−−−

−−

),(5.0)5.0(5.0

,0-

9.05.19.05.1

12.05.0

SSSSSS

d

SSd

imploc

imploclocimp

σσ

σσσσ

where, .]10,0( 10−∈S

39

And we know that and−∞→−⋅=−⋅− −−− )1(.50-)(5.0 6.05.19.05.1 SSSS

as , which means as∞→−⋅=− −−− )1( 38.05.012.05.0 SSSS 0→S ∞→−= locimpd σσ

.0→S

This is to say that the distance between these two volatilities is infinitely far away. So, when the

stock price is really close to 0, it would be not suitable to approximate the implied volatility by the

local volatility due to the fact that their distance function tends to infinity at 0.

40

Figure 3.13: The Fitting to Potential Function of The Implied Volatility Curves

Figure 3.14: The Fitting to Potential Function of The Implied Volatility Slope Curves

41

3.1.3. Implied Volatilities and The Dupire Volatilities

In this section, we concentrate on finding the inner connections between the implied volatilities and

the Dupire volatilities. The approach and ideas to solve the implied volatilities for the local

volatility model are the same as shown in Section 3.1.2 stated above. So, our focus here is mainly

concentrated on finding the Dupire volatilities.

The Dupire method is an approach of calculating volatilities by using market-quoted inputs, such

as , , , and so on. This local volatility is called the Dupire volatility.TV∂∂

KV∂∂ V 2

2

KV

∂∂

From formula (2.30) in Chapter 2, we know that

. (3.43)

2

22

21

)(),(

KVK

qVKVKqr

TV

TK

∂∂

+∂∂−+

∂∂

=σ

And also , thus0== qr

. (3.44)

2

22

21

),(

KVK

TV

TK

∂∂

∂∂

=σ

Up till now, the option value we have discussed is a function of asset price and maturity timeS, i.e., , where strike price is regarded as a parameter, relatively fixed, comparing toT ),( TSV Kand .S T

According to formula (3.44), in order to calculate the Dupire volatility, we have to obtain the values

for the derivative and first.2

2

KV

∂∂

TV∂∂

To calculate the derivatives, our plan here is to choose a spectrum of strike prices , and for eachK, we use the same scheme of pricing options as before while the strike price is treated as if itK K

is a constant. Thenwe collect all the option values calculated in this way and rearrange them not

only by the time-to-maturity indexes and the stock price indexes, but also by the strike price

indexes.

Hence, the option prices calculated in this way will be a 3-dimensional space composed of time,

price and strike. As abstract as it is, this would enable us to implement the finite difference method

to calculate the two deorivatives mentioned in (3.44).

Then, let us discuss the practical way of implementing the finite difference method.

42

Finite difference method is a numerical method that approximates the solutions to differential

equations by replacing derivative expressions with approximately equivalent difference quotients.

We use the forward difference algorithm here. Suppose function 's 1st and 2nd order)(xfderivatives at exist, then we can use the following method to approximate andax = )(af ′

.)(af ′′

; (3.45)0,)()()( →−+

=′ hash

xfhfaf

. (3.46)0,)()(2)2()( 2 →++−+

=′′ hash

xfhxfhfaf

If we apply this algorithm to our call option here, using the same notation, that,

for a given asset price and a maturity time ,S T

, (3.47)2

);,();,(2)2;,(),(h

KtxuhKtxuhKtxuTKCKK++−+

=

, (3.48)h

KtxuKhtxuTKCT);,();,(),( −+

=

where the variable represents asset price , variable stands for the maturity time , andx S t Tis a parameter in function .K u

First of all, let us compute the option values. The parameters' set-up in this subsection is organized

in table 3.5.

Table 3.5: The Initial Parameter Set-up for Program pdex_dupire.m

The numerical results of option values are calculated by program pdex_dupire_option.m(the m file

is included inAppendixA). The plot of option value surface is shown in figure 3.15 and the option

value curve plotted against the strike price is shown in figure 3.16.KFigure 3.15 and figure 3.16 are in accordance with our intuitions, that the option value isV

Price(stock/asset price) 101 grid points, from 0 to 100.Strike(option strike price) 21 mesh points, from 9.18① to 11.Rate(risk-free interest rate) 0Time(time-to-maturity) 21 mesh points, from 0 to 2Value(option value) 21×101×21 values


100 times per annum



① The starting point of the mesh point setseems like an unconventional choice. However, this enables us to placethe middle point of 21 mesh points on a point with value K=10, i.e., K21=11, K11=10 K1=9.18. The value of→middle point of strike price spectrum is in line with the strike price we use in calculating the implied volatility.

43

positively correlated with the stock price and negatively correlated with the strike price .S KThe latter is not quite obvious shown in figure 3.15, but can be observed from figure 3.16.Although

the strike price's impact on the option value is not as much as the stock price , still a higher strikeSprice level will lead to a decrease in option value to a certain extent.

Figure 3.15: The OptionValue Surface

44

Figure 3.16: The OptionValue Curves(Against the Strike Price )K

Then, we can use the numerical results of the option prices and the algorithms for derivatives

calculations to compute the Dupire volatilities. The program code is pdex_dupire.m (included in

AppendixA). The plots from this program are shown in figure 3.17 and figure 3.18.

The dependence of the Dupire volatility on the time-to-maturity is reflected in figure 3.17.Asτ

the maturity time approaches, the Dupire volatility increases at an increasing speed. This pattern,

however, is in consistence with the implied volatility curves in figure 3.4. They all display a

property of the volatility curves that (if the derivatives exist).0,0 2

2

<∂∂

<∂∂

TTσσ

Besides the above, curves in figure 3.17 also show that the Dupire volatility is sensitive to strike

price K as well. A small increment of 0.1 in strike price K leads to a huge upward shift of about 1 in

the volatility curve.

45

Figure 3.17 The Dupire Volatility Curve(PlottedAgainst Time-to-maturity )τ

All other things equal, panel a of figure 3.18 is plotted with a strike , while panel b is.99=Kplotted with a strike . However, this small difference in choosing strike prices leads to a10=Khuge difference in the shapes of Dupire volatility curves, as shown in panel a and panel b of figure

3.18.

Figure 3.18, panel a, describes the Dupire volatility curve at , . Figure 3.18,9.9=K 9.1=τ

panel b, depicts the Dupire volatility curve at , . Their shapes are quite different10=K 9.1=τ

from the the implied volatility's and the local volatility's. The Dupire volatility curves change the

convexity at least once throughout the the range within which the stock price changes.As for the

implied volatility and local volatility curves, their convexity don't change on the interval for stock

prices, this can be observed from curves in figure 3.6. Besides the differenceswith other volatility

curves, the Dupire volatility curves themselves are different from one another. Graphically, we can

see that the convexity changes at least twice in panel a while it only changes at least once in panel b.

Moreover, it's quite obvious that the curve is more peaked in panel b since the difference between

the extreme point and the lowest point in the Dupire volatility curve is much larger.Without ruling

out the impact resulted from choosing the finite difference method in calculation, we can still tell

that the Dupire volatility is extremely sensitive to strike prices K.

46

Figure 3.18 a : The Dupire Volatility Curve(K=9.9)

Figure 3.18 b : The Dupire Volatility Curve(K=10)

47

Although, it appears that the Dupire volatility will drop to 0 rather rapidly as the stock price

increases.But in fact, the Dupire is not as nice as it seems in the diagram. Recall the algorithms of

derivatives and the formula for calculating the Dupire volatility,

, (3.47)2

);,();,(2)2;,(),(h

KtxuhKtxuhKtxuTKCKK++−+

=

. (3.48)hKtxuKhtxuTKCT);,();,(),( −+

=

. (3.44)

2

22

21

),(

KCK

TC

TK

∂∂

∂∂

=σ

computed by (3.48) is a positive real number. This is guaranteed by the monotonicity),( TKCTof option value with respect to the time-to-maturity . However, we can not say the same forV Tthe 2nd order derivative because (3.47) is not necessarily a positive number since),( TKCKK

function is not necessarily a convex function with respect to in the interval we),,( Ktxu Kchoose. Hence, the numerical solution of (3.44) could be a complex number with a non-zero

imaginary part. In such scenarios, the Dupire volatility can not be computed by the formulation

(3.44) proposed by Dupire.

However, the numerical existence problem of the Dupire volatility while using the finite difference

method is only part of the difficulties we might encounter in Dupire volatility calculation here.

We notice that, in figure 3.18, after the stock price reaches a certain high level, the Dupire volatility

begins to display some irregularity and inconsistency in the movement of the volatility curve as the

stock price continues to increase.And eventually, the Dupire volatility vanishes when the stockSprice reaches 80(take figure 3.18 for an example). In retrospect, during our previous discussion in

section 3.1.2, the implied volatility curve in figure 3.3 shows a relatively stable property when the

stock price is at a relatively high level, and this property continues to remain so even when theSstock price continues to increase.And naturally, we would wonder why is that these twoSdifferent volatilities have so extraordinarily different behavior at high stock prices. To solve this

mystery, we need to go back to the Dupire formula for volatility calculations, equation (3.44).

Dupire formula states that, the volatility of an option can be calculated by the following,

. (3.44)

2

22

21

),(

KCK

TC

TK

∂∂

∂∂

=σ

48

The derivatives (for computing the Dupire volatility) required here, are the essential2

2

KC

TC

∂∂

∂∂

，

inputs for determining the Dupire volatility's value.

Let us first examine the 1st order derivative of option price with respect to the maturity-time, .TC∂∂

We know from experience and intuition that is a positive number, our interpretation is thatTC∂∂

longer time-to-maturity produces a high option value. In fact, from an observation of the numerical

lincluded inAppendix B), we find out that, when the price of underlying asset/stock is high enough,

the option value curves lie parallel to the payoff diagram at maturity. This is shown by figure 3.19.

And, another fun fact is that, at large stock prices, the increment between two option values whose

maturity-times are adjacent to each other is a constant. This is to say, , wherekKTSTC

=∂∂ ),,(

is a constant and is quite large comparing to the strike price . Besides our numerical0>k S K

results included inAppendix B, we can conclude the same results by the curve in figure 3.20.

49

Figure 3.19 : The OptionValue Curve( )1.0:0.2:0=τ

Figure 3.20 : The DerivativeTCCT ∂∂

=

50

Figure 3.21 : The DerivativeKCCK ∂∂

=

As for the derivative , we consider its corresponding 1st order derivative .2

2

KCCKK ∂

∂=

KCCK ∂∂

=

We can read from the numerical results of program pdex_dupire_option.m that, the 1st oder

derivative tends to be piecewise constant when the stock price is relatively quite highKCCK ∂∂

=

comparing to the strike price (say from 81 to 100). The graph to illustrate this is shown in figureK

3.20. Hence, the 2nd order derivative is 0 in the region that . This2

2

KCCKK ∂

∂= ]100,81[∈S

indicates that, the Dupire volatility

can not be evaluated by the finite difference numerical method for

2

22

21

),(

KCK

TC

TK

∂∂

∂∂

=σ

[81,100].∈S

This explains why the Dupire volatility can not be computed for large stock prices when using the

51

finite difference numerical method.

And notice that in figure 3.18 that, there is a small jump around , after examining the]80,70[∈S

numerical results around those points, we find out that this is caused by the incompatibility between

and . To be more specific, incompatibility is referred to that at thoseTCCT ∂∂

= 2

2

KCCKK ∂

∂=

jump points, the value of still exists and is positive under the evaluation by finiteTCCT ∂∂

=

difference method, while the value of has already become 0. In conclusion, this2

2

KCCKK ∂

∂=

explains why the Dupire volatility curve behaves in this way in figure 3.18.

Distance Between Implied Volatilities and Dupire Volatilities

By distance function , we are able to compute the distance between the implieddupireimpd σσ −=

volatility and the Dupire volatility.

Then, we canmove on to calculate the distance between implied volatilities and Dupire volatilities.

We include the program file pdex_dis_dupire.m in AppendixA. The plots for illustrating the

distance between the implied volatilities and the Dupire volatilities are shown in the following

figure 3.22 and 3.22. Panel a and panel b in figure 3.22, are the distance curves plotted against the

stock price at , and at , . Figure 3.23 is a plot of theS 9.1=τ 9.9=K 9.1=τ 10=Kdistance value of mesh points within the interval , plotted against the strike price]49,40[∈S

(The plots for the other groups are included inAppendixA).KFor the curve in figure 3.22, panel a, on one hand, the distance curve has some similar patterns as

opposed to the Dupire volatility curve in figure 3.18 a, i.e., the distance curve is truncated for large

stock price. The distance between these two volatilities can not be measured by distance function

using finite difference method due to the fact that the Dupire volatility can not be calculated from

the finite difference method for large stock prices. The reason of this is stated in the illustration of

figure 3.18 a. On the other hand, the distance curve in figure 3.22 has also inherited some properties

from the implied volatility curve demonstrated in figure 3.3, that as the stock price goes up the

distance between these two volatilities decreases as long as the stock price does not exceed the

certain upper boundary for which the Dupire volatility is truncated by using the finite difference

numerical method.

As for the curve in figure 3.22, panel b, resembles some similar features of the curve in figure 3.18

b. The distance between the Dupire volatility and the implied volatility reaches its maximum point

52

around , and hits its minimum around at which the difference between the47=S 70=SDupire volatility and the implied volatility is almost 0.And the curve in panel b is more peaked than

the curve in panel a. Because of the sensitivity of the Dupire volatility curve to strike price K, hence

the absolute difference curve between the Dupire volatility and the implied volatility is highly

sensitive to the strike price K as well.

Because of these properties, we have to impose more constraints on the usage of approximating the

implied volatility by Dupire volatility, since we not only have to make sure the distance between

these two volatilities is controllable but also in numerical existence under the finite difference

method we use here. Besides these, in fact, there are still more obstacles in applying this

approximation scheme. This is because the Dupire volatility calculated by the finite difference

numerical method does not always show a nice consistency between adjacent mesh points. We can

observe this problem from the numerical results included inAppendix B.

Figure 3.22 a: TheAbsolute Difference between Implied Volatility and The Dupire Volatility

(PlottedAgainst Stock Price )dupireimp σσ − S

53

Figure 3.22 b: TheAbsolute Difference between Implied Volatility and The Dupire

Volatility(PlottedAgainst Stock Price )S

54

Figure 3.23 : TheAbsolute Difference between Implied Volatility and The Dupire Volatility at

MeshPoints(PlottedAgainst )K

Figure 3.23 shows the distance between the volatilities of each mesh point. The circled points are

sparce in figure 3.23, due to the same reason that we stated in the illustration of figure 3.18 that the

Dupire volatilities' numerical existence is not guaranteed by our algorithms of finite difference

method. We can not evaluate numerically for some points with our current algorithm.

To sum up our investigation into the inner connections between the implied volatility and the

Dupire volatility, we draw the following conclusions. If we can make the option value to be convex

with respect to the strike price on a certain finite interval, then our scheme of volatilityK

calculation by finite difference method is practical. Hence, we can compare the volatility calculated

by the Dupire method to the one computed by using Black-Scholes formula rebersely. Even if we

can obtain a convex function of option value with respect to the strike price by imposingK

conditions on the local volatility model or choosing a suitable interval, the distance between the

implied volatility still will not be of some specific regularity. As we can see in figure 3.22, the

lower bound of the distance seems to be unstable, since every time we change the upper bound of

stock prices, the lower bound will be very likely to change accordingly.And as for the upper bound

55

of the distance, we can determine it by comparing the value of the left endpoint of the curve and the

value of the peak point of the curve. But still, this upper bound of distance is not fixed either. And

according our conclusions at the end of Section 3.1.2 that the implied volatility tends to infinity as

, unless we can prove the Dupire volatility has the same speed of convergence to infinity, it0→S

would be really hard to approximate the implied volatility by the Dupire volatility. More

importantly, the distance curve does not converge to anything or have a certain pattern of changing.

It is described in LiShangJiang (1994) [1] that the Dupire volatility is ill-posed. Based on all of the

research in section 3.1.3, the Dupire volatility does seem to a method not very appropriate to

simulate the implied volatility under our local volatility model here.

3.1.4. Summary of Three Types of Volatilities

After previous subsections of discussions with three types of volatilities, the local volatility, the

implied volatility, the Dupire volatility, we now summarize their features comprehensively.

Apparently, the local volatility we discuss here, is determined only by the stock priceSloc1

=σ

. Given a certain time-to-maturity and a strike price , the implied volatility, however, isS τ K

determined by the stock price process (provided that all other parameters, such as risk-freeS

interest rate and dividend yield is known). Or, in other words, with all other parameters inr q

the option pricing model given, the stock price process and the time-to-maturity co-S τ

determine the implied volatility . As for the Dupire volatility, it is determined),( τσσ Simpimp =

by the maturity time and the strike price , while the stock price and the time pointT K S t

and all other parameters influence the Dupire volatility merely as parameters.And it's worth

mentioning that the Dupire volatility is very sensitive to the strike price .K

We draw these three types of volatilities in the same diagram. Figure 3.24, panel a, is plotted with a

strike price , and panel b is drawn with a strike price ..99=K 10=K

Throughout the entire interval for stock price , we conclude the following.S

For the local volatility,

i) when the price of stock/underlying asset is below the option's strike price ,(the option isS K

out-of-the-money) the local volatility is larger than the implied volatility; conversely, when the

price of stock/underlying asset is above the option's strike price ,(the option is in-the-S K

56

money) the local volatility is smaller than the implied volatility.

ii) as the stock price tends to 0, the local volatility tends to infinity; as the stock price tends to

infinity, the local volatility eventually tends to 0.

iii) as the stock price increases, the local volatility decreases at a decreasing speed.

For the implied volatility,

i) as the stock price tends to 0, the implied volatility tends to infinity at a much slower speed than

the local volatility; as the stock price tends to infinity, the implied volatility tends to a stable non-

negative level, particularly, when the risk-free interest rate is set to be 0, then the impliedr

volatility tends to 0 as the stock price tends to infinity.

ii) as the stock price increases, the implied volatility decreases at a decreasing speed.

For the Dupire volatility,

i) the Dupire volatility curve changes its convexity at least once on the stock price interval.

ii) the Dupire volatility is highly sensitive to the value of strike price .K

iii) judging by the numerical approach, the finite difference method we use, as the stock price goes

up to a relatively high level, the Dupire volatility can not be numerically evaluated after it reaches 0

at certain point in the stock price axis.

iv) based on our investigation in this paper, the Dupire volatility is bounded within the range of

variation of the stock price .S

57

Figure 3.24. a The Volatility Curves( )dupireimploc σσσ ,,

Figure 3.24. b The Volatility Curves( )dupireimploc σσσ ,,

58

Chapter 4. Conclusions and Future Studies

In this paper, we mainly consider a local volatility model with the local volatility . We5.0−= Slocσ

evaluate the model numerically after deriving this option pricing model by using its specified

underlying asset's price processes.

We focus the study on three types of volatilities, the local volatility specified by the model itself

, the implied volatility inferred by the Black-Scholes model , and the so-called Dupirelocσ impσ

volatility given by the Dupire formula for implied volatilities . By showing the absolutedupireσ

difference between the implied volatility and the local volatility, and between the implied volatility

and the Dupire volatility, we illustrate the inner connections between these volatilities. From our

research here, we conclude that the difference between the local volatility and the implied volatility

is bounded for large values of stock price ; however, the difference between them tends toSinfinity since the implied volatility tends to infinity at a much slower speed than the local volatility

as the stock price tends to 0. At the same time, we also investigate the5.0−= Slocσ Srelationships between the implied volatility and the volatility given by the Dupire formula. For

large values of stock price , the Dupire volatility tends to 0 as well as the implied volatility does;Sfor small small values of stock price , the results shown by this numerical study is inconclusive.SBesides, the Dupire volatility is also time-dependent and highly sensitive to the change of the

values of strike price .KWe know that, our research here is based on an option pricing model(featured by its stock price

processes) that has a implied volatility similar to the one inferred by the Black-Scholes model, yet

still processes a price process that does not depend on the assumption of constant volatility, which

is more plausible and realistic. Hence, intuitively, we would expect these three volatilities possess

some similarities. And our results of this paper verify this speculation, all three types of volatilities

tend to 0 for large values of stock price ; the local volatility and the implied volatility tend toSinfinity for small values of stock price although they have difference speeds of convergenceStowards infinity.

However, procedures stated and demonstrated above may have some limitations as well. For

example, if we choose another local volatility model other than this one we use in this paper, one

question may arise naturally is that whether it is possible to solve the option values for such a model,

if the mechanics of this model does not allow the option value to be solved by any built-in function

of programming software. An obvious alternative is to resort to some other numerical methods to

treat the PDE we might encounter discretely, such as the finite difference method and so on.

Nevertheless, the accuracy of such approximations might be questionable. To avoid such "vains",

59

one useful shortcut is that we can observe the situation in the real market, make guesses such as the

one in Heston model[4] that the stock's volatility might be negatively correlated to the stock price,

to find suitable models guided by experiential knowledge and intuitions.

For future studies, we can also studies other local volatility models other than this simple one we

use here in this paper. Moreover, we can always search for another numerical option pricing

approach other than the one we use here. Besides the alternative approaches and models for implied

volatility study. Another future study direction can be looked into is the investigation of the

tendency of change of the Dupire volatility as the stock price tends to 0, whether it is really

bounded on the stock price axis, whether it tends to infinity as the implied volatility and the local

volatility do, still needs to be verified. Furtherore, other numerical approaches that can fix the

truncation problem of the Dupire volatility at large values of stock price can be studied as well.SThis concludes our conclusions of this paper and directions for future study references to this topic.

60

NotationsSDE Stochastic Differential EquationPDE Partial Differential EquationEMM Equivalent Martingale MeasureK Strike PriceS Stock/Asset Pricet Time pointT Maturity Timeτ Time-to-maturity

TttW ≤≤0}{ Brownian Motion, Wiener Process

V Option ValueG The 2nd Order Derivative of Option Price with

Respect to the StrikeC Call Option Value

impσ The Implied Volatility

locσ The Local Volatility

locσ The Dupire Volatility

r The Risk-free Interest Rate

tνThe Stochastic Volatility

ρ The Correlation between Brownian Motions

61

AppendixA

Program codes and plots

1. pdex_u.m

Program code

TableA.1pdex_u.mfunction value = pdex_u(m, T, x, tau)

%Function calculates the difference between two types of volatilities.

i = 51; %The number of meshpoints in the range of the maturity time T.

t0 = 0; %The initial value of maturity time T.

tT = 5; %The endpoint value of maturity time T.

j = 201; %The number of meshpoints in the range of the asset prices S.

K = 10; %The strike price K.

x = linspace(0,100,j); %The meshpoints in the range of asset prices S.

T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.

t = linspace(t0,tT,i); %The actual variable notation we use in pdepe

tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0

m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

%Using the built-in function pdepe to solve the PDE for option value V.

u = sol(:,:,1);

%Solve the PDE, denote the first solution as the option value.

value =u;

% -------------------------------------------------------------------------

% The curve of distance plotting against the asset price

figure;

plot(x,value);

colormap hsv;

xlabel('Stock Prices');

ylabel('Option Value');

title('European Call Options');

figure;

plot(x,value(51,:),'Color','red'); hold on;

plot(x,value(11,:),'Color','blue'); hold on;

plot(x,value(1,:),'Color','black'); hold on;

62

Plots




% -------------------------------------------------------------------------

% The surface consists of the distance between two kinds of volatilities

figure;

surf(x,tau,value);

colormap hsv;

title('Distance between two kinds of volatilities');


ylabel('Time-to-maturity \tau');

zlabel('Implied Volatilities');

title('The European Call Option Value Surface');

% -------------------------------------------------------------------------

function [c,f,s] = pdex1pde(x,t,u,DuDx)

c = 1;

f = 1/2*(x*DuDx-u);

s = 0;

end

% -------------------------------------------------------------------------

function u0 = pdex1ic(x)

u0 = max((x-K),0);

end

% -------------------------------------------------------------------------

function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)

K = 10;

pl = ul;

ql = 0;

pr = -1/2*K;

qr = 1;

end

end

63

Figure A.1

2. pdex_imp.m

Program code

TableA.2

pdex_imp.mfunction value = pdex_imp(m, T, x, tau)









t = linspace(t0,tT,i);


imp_v = ones(i,j);%Implied volatilities


64



u = sol(:,:,1);


Limit = 100;

Yield = 0;

Tolerance = 1e-18;

Class={'Call'};

for i=1:51

for j=1:201

imp_v(i,j) = blsimpv(x(j), K, 0, tau(i), u(i,j), Limit, Yield,

Tolerance, Class);

end

end

value =imp_v;

% -------------------------------------------------------------------------

% The curve of implied volatilities

figure;

plot(x,value(29,:),'.','Color','black','LineWidth',2);hold on;

plot(x,value(39,:),'Color','blue','LineWidth',2);hold on;

plot(x,value(49,:),'Color','magenta','LineWidth',2);


ylabel('Implied Volatilities');


% -------------------------------------------------------------------------

% The Term Structure of Implied Volatilities for European Call Options

figure;

plot(tau,value(:,2),'Color','black','LineWidth',2);hold on;

plot(tau,value(:,12),'Color','cyan','LineWidth',2);hold on;

plot(tau,value(:,22),'Color','red','LineWidth',2);hold on;

plot(tau,value(:,32),'Color','blue','LineWidth',2);hold on;


plot(tau,value(:,52),'.','Color','magenta','MarkerSize',6);

colormap hsv;

xlabel('Time-to-maturity \tau');


title('The Term Structure of Implied Volatilities for European Call

Options');

% -------------------------------------------------------------------------

% The implied volatility surface

figure;

surf(x,tau,value);

65

colormap hsv;




title('The Implied Volatility Surface of European Call Options');

% -------------------------------------------------------------------------


c = 1;

f = 1/2*(x*DuDx-u);

s = 0;

end

% -------------------------------------------------------------------------


u0 = max((x-K),0);

end

% -------------------------------------------------------------------------


K = 10;

pl = ul;

ql = 0;

pr = -1/2*K;

qr = 1;

end

end

66

Plots

Figure A.2

3. pdex_dis_imp_loc.m

Program code

TableA.3

pdex_dis_imp_loc.mfunction value = pdex_dis_imp_loc(m, T, x, tau)











67

sig = ones(i,j);% The local volatilities


dis = ones(i,j);% The distance between local & implied volatilities




u = sol(:,:,1);


Limit = 10;

Yield = 0;

Tolerance = 1e-16;

Class={'Call'};

for i=1:51

for j=1:201

sig(i,j)=sqrt(1/x(j));


Tolerance, Class);

dis(i,j)=abs(imp_v(i,j)-sig(i,j));

end

end

value =dis;

% -------------------------------------------------------------------------


figure;

plot(x,sig);


ylabel('Local Volatility');

title('European Call Options')

% -------------------------------------------------------------------------


figure;

plot(x,value(31,:),'.','Color','red','LineWidth',2);hold on;

plot(x,value(41,:),'Color','blue','LineWidth',2);hold on;

plot(x,value(51,:),'Color','black','LineWidth',2);hold on;


ylabel('Distance between Two Types of Volatilities');


% -------------------------------------------------------------------------

% The curve of distance plotting against the time-to-maturity

figure;

plot(tau,value(:,2),'Color','red','LineWidth',2);hold on;

68

plot(tau,value(:,3),'Color','blue','LineWidth',2);hold on;

plot(tau,value(:,12),'Color','cyan','LineWidth',2);hold on;

plot(tau,value(:,21),'Color','green','LineWidth',2);hold on;


xlabel('Time-to-maturity');

ylabel('Distance between Two Types of Volatilities');


% -------------------------------------------------------------------------


figure;

surf(x,tau,value);

title('Distance between Two Types of Volatilities');


ylabel('Time-to-maturity');

zlabel('Distance between Two Types of Volatilities');

title('The Distance Surface of European Call Options');

% -------------------------------------------------------------------------


c = 1;

f = 1/2*(x*DuDx-u);

s = 0;

end

% -------------------------------------------------------------------------


u0 = max((x-K),0);

end

% -------------------------------------------------------------------------


K = 10;

pl = ul;

ql = 0;

pr = -1/2*K;

qr = 1;

end

end

69

Plots

Figure A.3: The Distance Surface of EuropeanCall Options

4. pdex_u_small_s.m

Program code

TableA.4

pdex_u_small_s.m

function value = pdex_u_small_s(m, T, x, tau)







x = ones(1,j);

xx = linspace(0,10^-100,j-1);

for j=1:201

x(1,j)=xx(j);

end

x(1,202)=100;

70

%The meshpoints in the range of asset prices S.


t = linspace(t0,tT,i); %The actual variable notation we use in pdepe





u = sol(:,:,1);


u_small_s = ones(i,j-1);

for i=1:51

for j=1:201

u_small_s(i,j)= u(i,j);

end

end

value =u_small_s;

% -------------------------------------------------------------------------


figure;

plot(xx,value);

colormap hsv;




figure;

plot(xx,value(51,:),'Color','red'); hold on;

plot(xx,value(11,:),'Color','blue'); hold on;

plot(xx,value(1,:),'Color','black'); hold on;




% -------------------------------------------------------------------------


figure;

surf(x,tau,u);

colormap hsv;

title('Distance between two kinds of volatilities');




71

5. pdex_imp_small_s.m

Program code

TableA.5

title('The European Call Option Value Surface');

% -------------------------------------------------------------------------


c = 1;

f = 1/2*(x*DuDx-u);

s = 0;

end

% -------------------------------------------------------------------------


u0 = max((x-K),0);

end

% -------------------------------------------------------------------------


K = 10;

pl = ul;

ql = 0;

pr = -1/2*K;

qr = 1;

end

end

pdex_imp_small_s.m

function value = pdex_imp_small_s(m, T, x, tau)






K = 10; %The strike price K. K=1,S=20 imp_1 K=10,S=100,imp_2

x = ones(1,j);

xx = linspace(0,10^-10,j-1);

for j=1:201

x(1,j)=xx(j);

end

72

x(1,202)=100;

%The meshpoints in the range of asset prices S.








u = sol(:,:,1);


Limit = 1000000000;

Yield = 0;

Tolerance = 1e-18;

Class={'Call'};

for i=1:51

for j=1:202


Tolerance, Class);

end

end

imp_v_small_s = ones(i,j-1);

for i=1:51

for j=1:201

imp_v_small_s(i,j)= imp_v(i,j);

end

end

value =imp_v_small_s;

% -------------------------------------------------------------------------

% The curve of implied volatilities

figure;

plot(xx,value(29,:),'Color','black','LineWidth',2);hold on;

plot(xx,value(39,:),'Color','blue','LineWidth',2);hold on;

plot(xx,value(49,:),'Color','magenta','LineWidth',2);



title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2');

% -------------------------------------------------------------------------

slope = ones(i-1,j-2);

73

xxx = ones(1,j-2);

for i=1:50

for j=1:199

slope(i,j)=(imp_v_small_s(i+1,j+1)-imp_v_small_s(i+1,j+2))/(xx(j)-

xx(j+1));

xxx(1,j)=xx(j+1);

end

slope(i,200)=slope(i,199);

end

xxx(1,200)=xx(201);

% -------------------------------------------------------------------------

% The curve of implied volatility curves' slope

figure;

plot(xxx,slope(29,:),'Color','black','LineWidth',2);hold on;

plot(xxx,slope(39,:),'Color','blue','LineWidth',2);hold on;

plot(xxx,slope(49,:),'Color','magenta','LineWidth',2);


ylabel('The Slope of Implied Volatility Curves');


% -------------------------------------------------------------------------

loc = ones(i,j-1);

for i=1:51

for j=1:201

loc(i,j)=sqrt(1/(xx(j)));

end

end

% -------------------------------------------------------------------------

% The curve of local volatilities

figure;

plot(xx,loc,'Color','blue','LineWidth',2);


ylabel('Local Volatilities');


% -------------------------------------------------------------------------

slope_loc = ones(i-1,j-2);

for i=1:50

for j=1:200

slope_loc(i,j) =-(1/2)*(xxx(1,j)^(-3/2));

end

end

% -------------------------------------------------------------------------

% The curve of local volatility curves' slope

figure;

plot(xxx,slope_loc,'.','Color','black','LineWidth',2);

74

6. pdex_imp_slope_small_s_fit.m

Program code

TableA.6


ylabel('The Slope of Local Volatility Curves');


% -------------------------------------------------------------------------


c = 1;

f = 1/2*(x*DuDx-u);

s = 0;

end

% -------------------------------------------------------------------------


u0 = max((x-K),0);

end

% -------------------------------------------------------------------------


K = 10;

pl = ul;

ql = 0;

pr = -1/2*K;

qr = 1;

end

end

pdex_imp_slope_small_s_fit.mload local_volatility_imp_small_s.mat

imp= ans;

xx = linspace(0,10^-10,201);

load local_volatility_imp_small_s_slope.mat

xxx = linspace(xx(2),10^-10,200);

x=ones(1,8);

% -------------------------------------------------------------------------

%lower bound for implied volatility curve,tau=0.1

x(1)=-0.5;

for j=1:201

while imp(2,j)<xx(j)^x(1)

x(1)=x(1)+0.01;

end

75

end

% -------------------------------------------------------------------------

%upper bound for implied volatility curve,tau=0.1

x(2)=-0.01;

for j=1:201

while imp(2,j)>xx(j)^x(2)

x(2)=x(2)-0.01;

end

end

% -------------------------------------------------------------------------

%lower bound for implied volatility curve,tau=5

x(3)=-0.5;

for j=1:201

while imp(51,j)<xx(j)^x(3)

x(3)=x(3)+0.01;

end

end

% -------------------------------------------------------------------------

%upper bound for implied volatility curve,tau=5

x(4)=-0.01;

for j=1:201

while imp(51,j)>xx(j)^x(4)

x(4)=x(4)-0.01;

end

end

% -------------------------------------------------------------------------

%lower bound for implied volatility curve,tau=0.1

x(5)=-1.5;

for j=1:200

while slope(1,j)>-0.5*(xxx(j))^(x(5));

x(5)=x(5)+0.01;

end

end

% -------------------------------------------------------------------------

%upper bound for implied volatility curve,tau=0.1

x(6)=-0.01;

for j=1:200

while slope(1,j)<-0.5*(xxx(j))^(x(6));

x(6)=x(6)-0.01;

end

end

% -------------------------------------------------------------------------

%lower bound for implied volatility curve,tau=5

x(7)=-1.5;

76

for j=1:200

while slope(50,j)>-0.5*(xxx(j))^(x(7));

x(7)=x(7)+0.01;

end

end

% -------------------------------------------------------------------------

%upper bound for implied volatility curve,tau=5

x(8)=-0.01;

for j=1:200

while slope(50,j)<-0.5*(xxx(j))^(x(8));

x(8)=x(8)-0.01;

end

end

% -------------------------------------------------------------------------

value=x;

% -------------------------------------------------------------------------

% The curves of implied volatilities

figure;

plot(xx,imp(2,:),'.','Color','cyan','LineWidth',2);hold on;

plot(xx,imp(51,:),'.','Color','black','LineWidth',2);hold on;




% -------------------------------------------------------------------------

loc=ones(1,201);

% -------------------------------------------------------------------------

for j=1:201

loc(j)=(xx(j))^(x(1));

end

plot(xx,loc,'Color','green','LineWidth',4);hold on;

for j=1:201

loc(j)=(xx(j))^((x(1)+x(2))/2);

end

plot(xx,loc,'Color','magenta','LineWidth',2);hold on;

for j=1:201

loc(j)=(xx(j))^(x(2));

end

plot(xx,loc,'Color','blue','LineWidth',4);hold on;

for j=1:201

loc(j)=(xx(j))^(x(3));

end

plot(xx,loc,'Color','blue','LineWidth',2);hold on;

for j=1:201

loc(j)=(xx(j))^((x(3)+x(4))/2);

77

7. pdex_dupire_option.m

Program code

TableA.7

end

plot(xx,loc,'Color','red','LineWidth',2);hold on;

for j=1:201

loc(j)=(xx(j))^(x(4));

end

plot(xx,loc,'Color','green','LineWidth',2);hold on;

% -------------------------------------------------------------------------

% The curve of slopes

figure;

plot(xxx,slope(1,:),'Color','black','LineWidth',4);hold on;

plot(xxx,slope(50,:),'Color','blue','LineWidth',4);hold on;


ylabel('The Slope of Implied Volatility Curves');


% -------------------------------------------------------------------------

locc=ones(1,200);

for j=1:200

locc(j)=-0.5*(xxx(j))^(x(5));

end

plot(xxx,locc,'Color','red','LineWidth',2);hold on;

for j=1:200

locc(j)=-0.5*(xxx(j))^(x(6));

end

plot(xxx,locc,'Color','magenta','LineWidth',2);hold on;

% -------------------------------------------------------------------------

for j=1:200

locc(j)=-0.5*(xxx(j))^(x(7));

end

plot(xxx,locc,'Color','green','LineWidth',2);hold on;

for j=1:200

locc(j)=-0.5*(xxx(j))^(x(8));

end

plot(xxx,locc,'Color','cyan','LineWidth',2);hold on;

% -------------------------------------------------------------------------

pdex_dupire_option.mfunction value = pdex_dupire_option(m, K, x, tau)


78





h = 21; %The number of meshpoints in the range of the strike prices K.





K = linspace(9.18,11,h); %The meshpoints in the range of the strike prices K.

sig_2 = ones(i,j,h);%The square of implied volatility

imp_v = ones(i,j,h);%The implied volatilities.

sig_dupire = ones(i,j,h);%The Dupire volatility

dis = ones(i,j,h);%The distance between Dupire&implied volatilities.

u = ones(i,j,h);


for h=1:21



u(:,:,h) = sol(:,:,1);


end

value = u;

% -------------------------------------------------------------------------

%The Option Value Surface

figure;

uu = ones(21,101);

for h=1:21

for j=1:101

uu(h,j)=u(21,j,h);

end

end

surf(x,K,uu);

xlabel('Stock/Asset Prices');

ylabel('Strike Prices');

zlabel('Option Value');

title('The Option Value Surface,\tau=2');

% -------------------------------------------------------------------------

%The Option Value Curve(Agaist the Strike K)

79

8.pdex_dupire.m

Program code

TableA.8

figure;

uu = ones(21,101);

for h=1:21

for j=1:101

uu(h,j)=u(21,j,h);

end

end

plot(K,uu(:,1),'Color','red','LineWidth',2);hold on;

plot(K,uu(:,51),'Color','blue','LineWidth',2);hold on;

plot(K,uu(:,101),'Color','black','LineWidth',2);hold on;

xlabel('Strike Prices K');

ylabel('Option Value V');

title('The Option Value Curve,\tau=2');

% -------------------------------------------------------------------------


c = 1;

f = 1/2*(x*DuDx-u);

s = 0;

end

% -------------------------------------------------------------------------


u0 = max((x-K(h)),0);

end


pl = ul;

ql = 0;

pr = -1/2*K(h);

qr = 1;

end

end

pdex_dupire.mfunction value = pdex_dupire(m, K, x, tau)






80







u = ones(i,j,h); %The option values




dis = ones(i,j,h);%The distance between Dupire&implied volatilities.


Limit = 100;

Yield = 0;

Tolerance = 1e-18;

Class={'Call'};

C_T = ones(i,j,h);

C_KK = ones(i,j,h);

for h=1:21



u(:,:,h) = sol(:,:,1);


end

for j=1:101

for i=1:20

for h=1:19

C_T(i,j,h) = (u(i+1,j,h)-u(i,j,h))/(t(i+1)-t(i));

C_KK(i,j,h)=(u(i,j,h+2)+u(i,j,h)-2*u(i,j,h+1))/((K(h+1)-K(h))^2);

sig_2(i,j,h)=2*C_T(i,j,h)/(K(h)^2*C_KK(i,j,h));

if sig_2(i,j,h)>=0

sig_dupire(i,j,h)=sqrt(sig_2(i,j,h));

else

sig_dupire(i,j,h)=NaN;

end

end

end

81

end

dupire = ones(i-1,j-1,h-2);

for i=1:20

for j=1:100

for h=1:19

dupire(i,j,h)=sig_dupire(i,j+1,h);

end

end

end

value = dupire;

vol= ones(j-1,h-2);

xx = linspace(x(2),100,100);

KK = linspace(9.18,K(h-2),h-2);

TT= linspace(T(1),T(i-1),i-1);

% -------------------------------------------------------------------------

for j=1:100

for h=1:19

vol(j,h)=dupire(20,j,h);

end

end

figure;

plot(xx,vol(:,9),'.','Color','red'); hold on;

xlabel('Stock Prices S');

ylabel('Dupire Volatility');

title('European Calls,Dupire Volatility Curve, \tau=1.9, K=9.9');%K=9.908

figure;

plot(xx,vol(:,10),'.','Color','red'); hold on;

xlabel('Stock Prices S');


title('European Calls,Dupire Volatility Curve, \tau=1.9, K=10');%K=9.999

% -------------------------------------------------------------------------

vol_t=ones(20,19);

for h=1:19

for i=1:20

vol_t(i,h)=dupire(i,20,h);

end

end

tt=linspace(T(1),T(20),20);

figure;

plot(tt,vol_t(:,9),'.','Color','red'); hold on;

plot(tt,vol_t(:,10),'.','Color','blue'); hold on;

82

9.pdex_dis_imp_dupire.mProgram code

TableA.9

xlabel('Time-to-maturity \tau');


title('European Calls,Dupire Volatility Curve, S=20');

% -------------------------------------------------------------------------


c = 1;

f = 1/2*(x*DuDx-u);

s = 0;

end

% -------------------------------------------------------------------------


u0 = max((x-K(h)),0);

end


pl = ul;

ql = 0;

pr = -1/2*K(h);

qr = 1;

end

end

pdex_dis_imp_dupire.mfunction value = pdex_dis_imp_dupire(m, T, x, tau)












u = ones(i,j,h); %The option values V.


83



dis = ones(i-1,j,h-2);%The distance between Dupire&implied volatilities.


Limit = 100;

Yield = 0;

Tolerance = 1e-18;

Class={'Call'};

C_T = ones(i,j,h);

C_KK = ones(i,j,h);

for h=1:21



u(:,:,h) = sol(:,:,1);


for i=1:21

for j=1:101

imp_v(i,j,h) = blsimpv(x(j), K(h), 0, tau(i), u(i,j,h), Limit,

Yield, Tolerance, Class);

end

end

end

for h=1:19

for j=1:101

for i=1:20

C_T(i,j,h) = (u(i+1,j,h)-u(i,j,h))/(t(i+1)-t(i));

C_KK(i,j,h)=(u(i,j,h+2)+u(i,j,h)-2*u(i,j,h+1))/((K(h+1)-K(h))^2);

sig_2(i,j,h)=2*C_T(i,j,h)/(K(h)^2*C_KK(i,j,h));

if sig_2(i,j,h)>=0

sig_dupire(i,j,h)=sqrt(sig_2(i,j,h));

else

sig_dupire(i,j,h)=NaN;

end

dis(i,j,h)=abs(imp_v(i,j,h)-sig_dupire(i,j,h));

end

end

end

84

10.Dupire_plots_dis.mProgram code

TableA.10

distance= ones(i-1,j-1,h-2);


xx = linspace(x(2),100,100);

KK = linspace(9.18,K(h-2),h-2);

for i=1:20

for j=1:100

for h=1:19

distance(i,j,h)=dis(i,j+1,h);

end

end

end

value = distance;

% -------------------------------------------------------------------------


c = 1;

f = 1/2*(x*DuDx-u);

s = 0;

end

% -------------------------------------------------------------------------


u0 = max((x-K(h)),0);

end


pl = ul;

ql = 0;

pr = -1/2*K(h);

qr = 1;

end

end

dupire_plots_dis.m

for type=1:2

if type==1

load local_volatility_dis_dupire.mat;

distance;

%The distance between the implied volatility and the dupire

85

%volatility

% -------------------------------------------------------------------------








% -------------------------------------------------------------------------


xx = linspace(x(2),100,100);

% -------------------------------------------------------------------------

distance_1 = ones(20,19);

%For plotting distance surface,S=29.











ttau = linspace(0,1.9,20);

%Time-to-maturity,time-to-maturity=linspace(0.1,2,20).

KK = linspace(9.1,10.9,19);

%Strike prices, from K(2) to K(20),strike=linspace(9.1,10.9,19).

% -------------------------------------------------------------------------

for i=1:20

for h=1:19

distance_1(i,h)=distance(i,55,h);%S=55






end

end

% In order to demonstrate the relationships between

%strike prices K and distancetance between the volatilities, we select some

%effective data from the computing results by picking column 1-20 in time

%meshpoints, column 11-70 in asset price meshpoints(eliminating those

86

%out-of-money points), column 2-20 in strike prices meshpoints, i.e.,

%time-to-maturity=linspace(0.1,2,20),

%price=linspace(10,69,60),strike=linspace(9.1,10.9,19).

% -------------------------------------------------------------------------

figure;

plot(KK,distance_1,'o','Color','red'); hold on;



xlabel('Strike Prices K');ylabel('Distance');title('European Calls,

S=55:65:1, \tau=0:1.9:0.1');

figure;

plot(KK,distance_4,'o','Color','blue'); hold on;




S=70:80:1, \tau=0:1.9:0.1');

% -------------------------------------------------------------------------

figure;

plot(xx,distance(20,:,10),'.','Color','blue'); hold on;

xlabel('The Stock Prices S');

ylabel('The Distance');

title('The Distance Curve \tau=1.9, K=9.9');

% -------------------------------------------------------------------------

end

if type==2

load local_volatility_dis_dupire1.mat;

distance=ans;

%The distance between the implied volatility and the dupire

%volatility

% -------------------------------------------------------------------------








% -------------------------------------------------------------------------


xx = linspace(x(2),100,100);

% -------------------------------------------------------------------------



87











ttau = linspace(0,1.9,20);

%Time-to-maturity,time-to-maturity=linspace(0.1,2,20).

KK = linspace(9.1,10.9,19);

%Strike prices, from K(2) to K(20),strike=linspace(9.1,10.9,19).

% -------------------------------------------------------------------------

for i=1:20

for h=1:19







end

end

% In order to demonstrate the relationships between

%strike prices K and distancetance between the volatilities, we select some

%effective data from the computing results by picking column 1-20 in time

%meshpoints, column 11-70 in asset price meshpoints(eliminating those

%out-of-money points), column 2-20 in strike prices meshpoints, i.e.,

%time-to-maturity=linspace(0.1,2,20),

%price=linspace(10,69,60),strike=linspace(9.1,10.9,19).

% -------------------------------------------------------------------------

figure;





S=55:65:1, \tau=0:1.9:0.1');

figure;


88

Plots

Figure A.4




S=70:80:1, \tau=0:1.9:0.1');

% -------------------------------------------------------------------------

figure;

plot(xx,distance(20,:,10),'.','Color','blue'); hold on;

xlabel('The Stock Prices S');

ylabel('The Distance');

title('The Distance Curve \tau=1.9, K=10');

% -------------------------------------------------------------------------

end

end

89

Appendix B

Numerical results

1. pdex_dis_european.m

Table B.1 Distance between Implied Volatilities and Local Volatilities

tau=0.1:1:10; K=10; S=0.5:100:200; r=0，q=0.

S\tau 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.5 0.38121710463

0.56202628141

0.63764349171

0.68138362441

0.70930400844

0.72911719886

0.74291556128

0.75364093769

0.76192500543

0.76835603124

1 0.1568472335

0.29727630276

0.35479060181

0.38721567973 4.07E-01 0.421240

722490.43071234359

0.43781149542

0.44313212615

0.44716781166

1.5 0.079851112732

0.19668822929

0.2435547783

0.26928166874 2.85E-01 0.295276

198120.30227281675

0.30734066609

0.3110442074

0.31380231805

2 0.043809557089

0.14377621278

0.18298753735

0.20390049192 2.16E-01 0.224242

335570.22953945373

0.23325742891

0.23591591075

0.23786393361

2.5 0.02474594886

0.11143558702

0.1446000573

0.16173964357 1.72E-01 0.177780

360960.18181887993

0.18457521156

0.18650862371

0.18790345719

3 0.014203857135

0.0898260792

0.11794522421

0.13198601408 1.40E-01 0.144623

02340.14769458832

0.14974048042

0.15115022207

0.15215057439

3.5 0.0084491081817

0.074479836628

0.098240347451

0.10966933365 1.16E-01 0.119555

947560.12187045214

0.12337902905

0.1243997755

0.12511008085

4 0.0055858472887

0.063066055416

0.082975577613

0.092172906451 9.70E-02 0.099811

4427840.10152720649

0.10262216219

0.1033476191

0.10383974336

4.5 0.0045533637491

0.054246253429

0.070705784348

0.077984865619 8.17E-02 0.083773

8806830.085013609228

0.085785923341

0.086283607658

0.086609017504

5 0.0047126469886

0.047194877022

0.060540937843

0.066169042355 6.89E-02 0.070431

24010.071291874146

0.071810535036

0.072130824443

0.072327792519

5.5 0.0056538076325

0.041372734601

0.05190326799

0.05611261 5.81E-02 0.059114

3919010.059673824716

0.059992805149

0.060174579524

0.060272535422

6 0.0070967849401

0.036409200015

0.044400245524

0.047395403057 4.87E-02 0.049361

9884590.049683789077

0.049846465072

0.049920628511

0.049942812564

6.5 0.0088342403076

0.032035861076

0.03775363071

0.039718598468 4.05E-02 0.040845

369120.040982073006

0.041023930092

0.041015623378

0.040980723531

7 0.01069303823

0.028046981074

0.031758792998

0.032864376975

0.033225244523

0.033324246804

0.033319580777

0.03326987766

0.03319990549

0.033123053761

7.5 0.012501325247

0.024276244109

0.026261945981

0.026672000066

0.026701507477

0.026619142471

0.02650945314

0.026392418869

0.026278157234

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0.02058576494

0.021148046921

0.02102232082

0.020805855795

0.020593371306

0.020408605068

0.020244541551

0.02010037908

0.019973545563

8.5 0.015063536815

0.016865705996

0.016334160232

0.015826438994

0.015439353051

0.015140736457

0.014905557566

0.014711667948

0.014549148614

0.014409806611

9 0.015129590141

0.013041900052

0.011764958072

0.011016796266

0.010525756632

0.010177103033

0.0099124032809

0.0097028458776

0.0095310413388

0.0093855427985

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0.0090831345279

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0.0060045460935

0.0056349789657

0.0053587646869

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10 0.010406040359

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0.0018257235286

0.001459545036

0.0011868861975

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0.00080694483241

0.00066398776611

10.5 0.005160389942

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0.0023946005087

0.0026511013317

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11 0.0013059134515

0.003422241281

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0.0056617628258

0.0059653676034

0.006195482706

0.0063771527103

0.0065259062484

0.0066518516811

90

11.5 0.0083133524762

0.0076521224059

0.0082398686139

0.0087000494039

0.0090337945192

0.0092848025767

0.0094802628419

0.0096398184484

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12 0.01540850842

0.011833934932

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0.011975683396

0.012192629057

0.012380042065

0.012534511146

0.012667475864

0.012781791537

0.012881106418

12.5 0.022340759103

0.015928928666

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0.01507486044

0.015159820894

0.015274164483

0.015382940499

0.015485458329

0.015578761692

0.015662291003

13 0.028990507838

0.019906798096

0.018363816975

0.018011322582

0.01795345945

0.01798704686

0.018046501523

0.018115557493

0.018185686826

0.018252226043

13.5 0.035313118715

0.023746903873

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0.020588740211

0.020535916997

0.020543130669

0.020576512578

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14 0.041303919531

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0.024410298008

0.023442098458

0.02307863793

0.022935706712

0.022888355026

0.022884468891

0.022903157154

0.022932735512

14.5 0.046977771882

0.030975367047

0.027231831216

0.025955655362

0.025434416284

0.025199369031

0.025095698235

0.02505341487

0.025044670193

0.025054551154

15 0.052357625122

0.034361798768

0.029925332871

0.028345716354

0.027666001814

0.027338190131

0.027176979634

0.027095552705

0.027058752796

0.027048355536

15.5 0.057468438526

0.037603089227

0.032497202861

0.030619778034

0.02978227942

0.029362075098

0.029142570478

0.029021593306

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0.031791333255

0.031279794296

0.031001624261

0.030840991838

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16.5 0.066976740184

0.043685927164

0.037306699812

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0.033700640722

0.033099189668

0.032762280519

0.032562141424

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17 0.071415265637

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0.035517223359

0.034827345203

0.034431840259

0.034192536166

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0.033948988042

17.5 0.075666552972

0.049302265676

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0.037247757471

0.03647072641

0.036016913411

0.035738909582

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0.040499124418

0.038898647128

0.038035293137

0.037523540525

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0.036867975407

18.5 0.083663649072

0.054526706727

0.045826433953

0.042225937761

0.040476062846

0.039526589252

0.038957291722

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0.041985950732

0.040949811999

0.040323345954

0.039932165472

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19.5 0.091062448777

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0.043434018692

0.042309863306

0.041626553283

0.041198311677

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0.047028882183

0.044825707651

0.043611385185

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0.042406201744

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20.5 0.097779148496

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0.046166156076

0.044858781467

0.044062454724

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21 0.10093642391

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0.047460165238

0.046056228459

0.045203568095

0.044663209776

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21.5 0.10305275848

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0.048712170879

0.047207677394

0.046298709579

0.045719718077

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22 0.21320071636

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0.049926224817

0.048316851668

0.047351561625

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22.5 0.21081851068

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0.054100156339

0.051105987844

0.049387241651

0.048365602408

0.047705648547

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23 0.20851441406

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0.050422099457

0.049344103011

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23.5 0.20628424925

0.076442776523

0.063195151912

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0.053375362192

0.051424435469

0.050290123418

0.049542510406

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24 0.20412414523

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0.054470420906

0.052397017712

0.05120650898

0.050412038581

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24.5 0.20203050891

0.080175842065

0.066235881994

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0.055542130522

0.053342374657

0.052095888596

0.051252364462

0.05068938413

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25 0.2 0.081665618899

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0.060404233472

0.056592414445

0.054262801387

0.052960675352

0.052065833123

0.051468507385

0.051054648916

25.5 0.1980295086

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0.069164544621

0.061597419351

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0.055160368978

0.05380306995

0.052854627699

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26 0.19611613514

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0.058635097848

0.056036936075

0.054625066841

0.053620771397

0.052949264065

0.052488860464

91

26.5 0.19425717247

0.19425717247

0.071986435886

0.063926484615

0.059630129607

0.05689416345

0.055428462707

0.054366130433

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27 0.19245008973

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0.073353039093

0.065063994666

0.060609062924

0.05773352652

0.056214867309

0.055092418078

0.054339066633

0.053828814658

27.5 0.19069251785

0.19069251785

0.074723470441

0.066184813187

0.061572782694

0.058556335534

0.05698571505

0.055801200064

0.055004041225

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28 0.1889822365

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0.067289024716

0.062522035096

0.059363744843

0.057742278546

0.0564939005

0.055651059925

0.055087770135

28.5 0.18731716232

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0.077387514192

0.068377709068

0.063457453351

0.060156774282

0.058485681704

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29 0.18569533818

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0.077291740107

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0.064379512251

0.06093632214

0.059216913379

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0.056896545357

0.056277441081

29.5 0.18411492358

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0.079305588343

0.070505697352

0.065288693754

0.061703181863

0.059936840246

0.058487793162

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30 0.18257418584

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0.066185665853

0.062458016563

0.060646220899

0.059127850789

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0.057408057357

30.5 0.18107149209

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0.087004839025

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0.067070697091

0.063201508114

0.061345718665

0.059757102508

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31 0.17960530203

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0.073576639701

0.067942853421

0.063934100385

0.062035904292

0.060376292176

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31.5 0.17817416127

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0.074532949771

0.068805512361

0.064656251906

0.062717278902

0.060986083227

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32 0.1767766953

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0.075309773773

0.069654694696

0.065368491046

0.06339025895

0.061587061534

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32.5 0.17541160386

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0.076204601211

0.070495334886

0.066070952805

0.064055281853

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33 0.17407765596

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0.076149242699

0.071310695742

0.066764938185

0.064712643061

0.062764594774

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0.060527345506

33.5 0.17277368512

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0.074931194264

0.072135775894

0.06744919491

0.0653625928

0.063342028945

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34 0.17149858514

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0.083644537939

0.072939759719

0.068123077224

0.066005329381

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34.5 0.17025130615

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0.17025130615

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0.073725928047

0.0687899149

0.066641224116

0.064475963429

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35 0.16903085095

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0.074367925746

0.069441304727

0.067269863433

0.065033106861

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35.5 0.16783627166

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0.075779983861

0.070106580939

0.067892760186

0.065583981173

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36 0.16666666667

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0.075869508203

0.070734183419

0.068509905904

0.066128817679

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36.5 0.16552117772

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0.068854173639

0.071381934794

0.069117767653

0.066667793878

0.064861540358

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37 0.16439898731

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0.16439898731

0.0720379876

0.069721961366

0.067201005372

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37.5 0.16329931619

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0.072599266602

0.070320211563

0.067728658784

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38 0.16222142113

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0.072923478096

0.070901149696

0.068251626786

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38.5 0.16116459281

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0.073503225535

0.071491226542

0.068766172864

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39 0.16012815381

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0.16012815381

0.16012815381

0.074246994014

0.072066964969

0.069279044305

0.067169293713

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39.5 0.15911145684

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0.15911145684

0.15911145684

0.075161968969

0.072650702883

0.069786597029

0.067616018201

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40 0.15811388301

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0.15811388301

0.074124956738

0.073200110884

0.070297676537

0.068057464037

0.066858053506

40.5 0.15713484026

0.15713484026

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0.15713484026

0.15713484026

0.15713484026

0.073846194044

0.070786011646

0.068495172483

0.067278371312

41 0.15617376189

0.15617376189

0.15617376189

0.15617376189

0.15617376189

0.15617376189

0.074188108361

0.071288388058

0.06892806489

0.067695193941

92

41.5 0.15523010514

0.15523010514

0.15523010514

0.15523010514

0.15523010514

0.15523010514

0.07434639182

0.071758409278

0.069358568427

0.068109253444

42 0.15430334996

0.15430334996

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0.15430334996

0.15430334996

0.15430334996

0.075731467452

0.072233767643

0.069786044746

0.06852143279

42.5 0.15339299777

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0.15339299777

0.15339299777

0.15339299777

0.15339299777

0.072844024387

0.072674047892

0.070202323234

0.068926471925

43 0.15249857033

0.15249857033

0.15249857033

0.15249857033

0.15249857033

0.15249857033

0.074552266009

0.073004255216

0.070606217121

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43.5 0.15161960872

0.15161960872

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0.15161960872

0.15161960872

0.15161960872

0.15161960872

0.073455491996

0.071047490899

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44 0.15075567229

0.15075567229

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0.15075567229

0.15075567229

0.15075567229

0.15075567229

0.073631819994

0.071438422195

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44.5 0.1499063378

0.1499063378

0.1499063378

0.1499063378

0.1499063378

0.1499063378

0.1499063378

0.072619744278

0.071856634599

0.070529594396

45 0.1490711985

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0.1490711985

0.1490711985

0.1490711985

0.1490711985

0.074670154139

0.072348638598

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45.5 0.14824986333

0.14824986333

0.14824986333

0.14824986333

0.14824986333

0.14824986333

0.14824986333

0.075871271662

0.072861995302

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46 0.14744195615

0.14744195615

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0.14744195615

0.14744195615

0.14744195615

0.14744195615

0.07588134685

0.072990044664

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46.5 0.14664711502

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0.14664711502

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0.14664711502

0.14664711502

0.078938853965

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47 0.1458649915

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0.1458649915

0.1458649915

0.1458649915

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0.07311011719

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47.5 0.14509525002

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0.14509525002

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48 0.1443375673

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0.1443375673

0.072354695642

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48.5 0.14359163172

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0.14359163172

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0.14359163172

0.074692942297

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49 0.14285714286

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0.14285714286

0.14285714286

0.14285714286

0.14285714286

0.14285714286

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0.073546797388

49.5 0.1421338109

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50 0.14142135624

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50.5 0.14071950895

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51 0.1400280084

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0.075535325825

51.5 0.13934660286

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52 0.13867504906

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52.5 0.13801311187

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0.13801311187

0.13801311187

0.13801311187

0.13801311187

0.13801311187

0.13801311187

0.13801311187

53 0.13736056395

0.13736056395

0.13736056395

0.13736056395

0.13736056395

0.13736056395

0.13736056395

0.13736056395

0.13736056395

0.13736056395

53.5 0.1367171854

0.1367171854

0.1367171854

0.1367171854

0.1367171854

0.1367171854

0.1367171854

0.1367171854

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54 0.13608276349

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0.13608276349

0.13608276349

0.13608276349

0.13608276349

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54.5 0.1354570923

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0.1354570923

0.1354570923

0.1354570923

0.1354570923

0.1354570923

0.1354570923

0.1354570923

0.1354570923

55 0.13483997249

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0.13483997249

0.13483997249

0.13483997249

0.13483997249

0.13483997249

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55.5 0.13423121104

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0.13423121104

0.13423121104

0.13423121104

0.13423121104

0.13423121104

0.13423121104

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56 0.13363062096

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0.13363062096

0.13363062096

0.13363062096

0.13363062096

0.13363062096

0.13363062096

0.13363062096

0.13363062096

93

56.5 0.13303802105

0.13303802105

0.13303802105

0.13303802105

0.13303802105

0.13303802105

0.13303802105

0.13303802105

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57 0.13245323571

0.13245323571

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0.13245323571

0.13245323571

0.13245323571

0.13245323571

0.13245323571

0.13245323571

0.13245323571

57.5 0.13187609468

0.13187609468

0.13187609468

0.13187609468

0.13187609468

0.13187609468

0.13187609468

0.13187609468

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58 0.13130643286

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0.13130643286

0.13130643286

0.13130643286

0.13130643286

0.13130643286

0.13130643286

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58.5 0.13074409009

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0.13074409009

0.13074409009

0.13074409009

0.13074409009

0.13074409009

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59 0.13018891098

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0.13018891098

0.13018891098

0.13018891098

0.13018891098

0.13018891098

0.13018891098

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59.5 0.12964074471

0.12964074471

0.12964074471

0.12964074471

0.12964074471

0.12964074471

0.12964074471

0.12964074471

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0.12964074471

60 0.12909944487

0.12909944487

0.12909944487

0.12909944487

0.12909944487

0.12909944487

0.12909944487

0.12909944487

0.12909944487

0.12909944487

60.5 0.12856486931

0.12856486931

0.12856486931

0.12856486931

0.12856486931

0.12856486931

0.12856486931

0.12856486931

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0.12856486931

61 0.12803687993

0.12803687993

0.12803687993

0.12803687993

0.12803687993

0.12803687993

0.12803687993

0.12803687993

0.12803687993

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61.5 0.12751534261

0.12751534261

0.12751534261

0.12751534261

0.12751534261

0.12751534261

0.12751534261

0.12751534261

0.12751534261

0.12751534261

62 0.127000127

0.127000127

0.127000127

0.127000127

0.127000127

0.127000127

0.127000127

0.127000127

0.127000127

0.127000127

62.5 0.12649110641

0.12649110641

0.12649110641

0.12649110641

0.12649110641

0.12649110641

0.12649110641

0.12649110641

0.12649110641

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63 0.12598815767

0.12598815767

0.12598815767

0.12598815767

0.12598815767

0.12598815767

0.12598815767

0.12598815767

0.12598815767

0.12598815767

63.5 0.12549116103

0.12549116103

0.12549116103

0.12549116103

0.12549116103

0.12549116103

0.12549116103

0.12549116103

0.12549116103

0.12549116103

64 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125

64.5 0.12451456127

0.12451456127

0.12451456127

0.12451456127

0.12451456127

0.12451456127

0.12451456127

0.12451456127

0.12451456127

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65 0.12403473459

0.12403473459

0.12403473459

0.12403473459

0.12403473459

0.12403473459

0.12403473459

0.12403473459

0.12403473459

0.12403473459

65.5 0.12356041264

0.12356041264

0.12356041264

0.12356041264

0.12356041264

0.12356041264

0.12356041264

0.12356041264

0.12356041264

0.12356041264

66 0.12309149098

0.12309149098

0.12309149098

0.12309149098

0.12309149098

0.12309149098

0.12309149098

0.12309149098

0.12309149098

0.12309149098

66.5 0.1226278679

0.1226278679

0.1226278679

0.1226278679

0.1226278679

0.1226278679

0.1226278679

0.1226278679

0.1226278679

0.1226278679

67 0.12216944436

0.12216944436

0.12216944436

0.12216944436

0.12216944436

0.12216944436

0.12216944436

0.12216944436

0.12216944436

0.12216944436

67.5 0.12171612389

0.12171612389

0.12171612389

0.12171612389

0.12171612389

0.12171612389

0.12171612389

0.12171612389

0.12171612389

0.12171612389

68 0.12126781252

0.12126781252

0.12126781252

0.12126781252

0.12126781252

0.12126781252

0.12126781252

0.12126781252

0.12126781252

0.12126781252

68.5 0.12082441867

0.12082441867

0.12082441867

0.12082441867

0.12082441867

0.12082441867

0.12082441867

0.12082441867

0.12082441867

0.12082441867

69 0.12038585309

0.12038585309

0.12038585309

0.12038585309

0.12038585309

0.12038585309

0.12038585309

0.12038585309

0.12038585309

0.12038585309

69.5 0.11995202878

0.11995202878

0.11995202878

0.11995202878

0.11995202878

0.11995202878

0.11995202878

0.11995202878

0.11995202878

0.11995202878

70 0.11952286093

0.11952286093

0.11952286093

0.11952286093

0.11952286093

0.11952286093

0.11952286093

0.11952286093

0.11952286093

0.11952286093

70.5 0.11909826684

0.11909826684

0.11909826684

0.11909826684

0.11909826684

0.11909826684

0.11909826684

0.11909826684

0.11909826684

0.11909826684

71 0.11867816582

0.11867816582

0.11867816582

0.11867816582

0.11867816582

0.11867816582

0.11867816582

0.11867816582

0.11867816582

0.11867816582

94

71.5 0.1182624792

0.1182624792

0.1182624792

0.1182624792

0.1182624792

0.1182624792

0.1182624792

0.1182624792

0.1182624792

0.1182624792

72 0.1178511302

0.1178511302

0.1178511302

0.1178511302

0.1178511302

0.1178511302

0.1178511302

0.1178511302

0.1178511302

0.1178511302

72.5 0.1174440439

0.1174440439

0.1174440439

0.1174440439

0.1174440439

0.1174440439

0.1174440439

0.1174440439

0.1174440439

0.1174440439

73 0.1170411472

0.1170411472

0.1170411472

0.1170411472

0.1170411472

0.1170411472

0.1170411472

0.1170411472

0.1170411472

0.1170411472

73.5 0.1166423687

0.1166423687

0.1166423687

0.1166423687

0.1166423687

0.1166423687

0.1166423687

0.1166423687

0.1166423687

0.1166423687

74 0.11624763874

0.11624763874

0.11624763874

0.11624763874

0.11624763874

0.11624763874

0.11624763874

0.11624763874

0.11624763874

0.11624763874

74.5 0.11585688927

0.11585688927

0.11585688927

0.11585688927

0.11585688927

0.11585688927

0.11585688927

0.11585688927

0.11585688927

0.11585688927

75 0.11547005384

0.11547005384

0.11547005384

0.11547005384

0.11547005384

0.11547005384

0.11547005384

0.11547005384

0.11547005384

0.11547005384

75.5 0.11508706753

0.11508706753

0.11508706753

0.11508706753

0.11508706753

0.11508706753

0.11508706753

0.11508706753

0.11508706753

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76 0.11470786694

0.11470786694

0.11470786694

0.11470786694

0.11470786694

0.11470786694

0.11470786694

0.11470786694

0.11470786694

0.11470786694

76.5 0.1143323901

0.1143323901

0.1143323901

0.1143323901

0.1143323901

0.1143323901

0.1143323901

0.1143323901

0.1143323901

0.1143323901

77 0.11396057646

0.11396057646

0.11396057646

0.11396057646

0.11396057646

0.11396057646

0.11396057646

0.11396057646

0.11396057646

0.11396057646

77.5 0.11359236685

0.11359236685

0.11359236685

0.11359236685

0.11359236685

0.11359236685

0.11359236685

0.11359236685

0.11359236685

0.11359236685

78 0.11322770341

0.11322770341

0.11322770341

0.11322770341

0.11322770341

0.11322770341

0.11322770341

0.11322770341

0.11322770341

0.11322770341

78.5 0.1128665296

0.1128665296

0.1128665296

0.1128665296

0.1128665296

0.1128665296

0.1128665296

0.1128665296

0.1128665296

0.1128665296

79 0.11250879009

0.11250879009

0.11250879009

0.11250879009

0.11250879009

0.11250879009

0.11250879009

0.11250879009

0.11250879009

0.11250879009

79.5 0.11215443082

0.11215443082

0.11215443082

0.11215443082

0.11215443082

0.11215443082

0.11215443082

0.11215443082

0.11215443082

0.11215443082

80 0.11180339887

0.11180339887

0.11180339887

0.11180339887

0.11180339887

0.11180339887

0.11180339887

0.11180339887

0.11180339887

0.11180339887

80.5 0.11145564252

0.11145564252

0.11145564252

0.11145564252

0.11145564252

0.11145564252

0.11145564252

0.11145564252

0.11145564252

0.11145564252

81 0.11111111111

0.11111111111

0.11111111111

0.11111111111

0.11111111111

0.11111111111

0.11111111111

0.11111111111

0.11111111111

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81.5 0.11076975512

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0.11076975512

0.11076975512

0.11076975512

0.11076975512

0.11076975512

0.11076975512

0.11076975512

82 0.11043152607

0.11043152607

0.11043152607

0.11043152607

0.11043152607

0.11043152607

0.11043152607

0.11043152607

0.11043152607

0.11043152607

82.5 0.11009637651

0.11009637651

0.11009637651

0.11009637651

0.11009637651

0.11009637651

0.11009637651

0.11009637651

0.11009637651

0.11009637651

83 0.10976425999

0.10976425999

0.10976425999

0.10976425999

0.10976425999

0.10976425999

0.10976425999

0.10976425999

0.10976425999

0.10976425999

83.5 0.10943513103

0.10943513103

0.10943513103

0.10943513103

0.10943513103

0.10943513103

0.10943513103

0.10943513103

0.10943513103

0.10943513103

84 0.10910894512

0.10910894512

0.10910894512

0.10910894512

0.10910894512

0.10910894512

0.10910894512

0.10910894512

0.10910894512

0.10910894512

84.5 0.10878565864

0.10878565864

0.10878565864

0.10878565864

0.10878565864

0.10878565864

0.10878565864

0.10878565864

0.10878565864

0.10878565864

85 0.10846522891

0.10846522891

0.10846522891

0.10846522891

0.10846522891

0.10846522891

0.10846522891

0.10846522891

0.10846522891

0.10846522891

85.5 0.10814761409

0.10814761409

0.10814761409

0.10814761409

0.10814761409

0.10814761409

0.10814761409

0.10814761409

0.10814761409

0.10814761409

86 0.1078327732

0.1078327732

0.1078327732

0.1078327732

0.1078327732

0.1078327732

0.1078327732

0.1078327732

0.1078327732

0.1078327732

95

86.5 0.10752066611

0.10752066611

0.10752066611

0.10752066611

0.10752066611

0.10752066611

0.10752066611

0.10752066611

0.10752066611

0.10752066611

87 0.10721125348

0.10721125348

0.10721125348

0.10721125348

0.10721125348

0.10721125348

0.10721125348

0.10721125348

0.10721125348

0.10721125348

87.5 0.10690449676

0.10690449676

0.10690449676

0.10690449676

0.10690449676

0.10690449676

0.10690449676

0.10690449676

0.10690449676

0.10690449676

88 0.10660035818

0.10660035818

0.10660035818

0.10660035818

0.10660035818

0.10660035818

0.10660035818

0.10660035818

0.10660035818

0.10660035818

88.5 0.10629880069

0.10629880069

0.10629880069

0.10629880069

0.10629880069

0.10629880069

0.10629880069

0.10629880069

0.10629880069

0.10629880069

89 0.105999788

0.105999788

0.105999788

0.105999788

0.105999788

0.105999788

0.105999788

0.105999788

0.105999788

0.105999788

89.5 0.10570328452

0.10570328452

0.10570328452

0.10570328452

0.10570328452

0.10570328452

0.10570328452

0.10570328452

0.10570328452

0.10570328452

90 0.10540925534

0.10540925534

0.10540925534

0.10540925534

0.10540925534

0.10540925534

0.10540925534

0.10540925534

0.10540925534

0.10540925534

90.5 0.10511766625

0.10511766625

0.10511766625

0.10511766625

0.10511766625

0.10511766625

0.10511766625

0.10511766625

0.10511766625

0.10511766625

91 0.10482848367

0.10482848367

0.10482848367

0.10482848367

0.10482848367

0.10482848367

0.10482848367

0.10482848367

0.10482848367

0.10482848367

91.5 0.1045416747

0.1045416747

0.1045416747

0.1045416747

0.1045416747

0.1045416747

0.1045416747

0.1045416747

0.1045416747

0.1045416747

92 0.10425720703

0.10425720703

0.10425720703

0.10425720703

0.10425720703

0.10425720703

0.10425720703

0.10425720703

0.10425720703

0.10425720703

92.5 0.10397504898

0.10397504898

0.10397504898

0.10397504898

0.10397504898

0.10397504898

0.10397504898

0.10397504898

0.10397504898

0.10397504898

93 0.10369516947

0.10369516947

0.10369516947

0.10369516947

0.10369516947

0.10369516947

0.10369516947

0.10369516947

0.10369516947

0.10369516947

93.5 0.103417538

0.103417538

0.103417538

0.103417538

0.103417538

0.103417538

0.103417538

0.103417538

0.103417538

0.103417538

94 0.10314212463

0.10314212463

0.10314212463

0.10314212463

0.10314212463

0.10314212463

0.10314212463

0.10314212463

0.10314212463

0.10314212463

94.5 0.10286889997

0.10286889997

0.10286889997

0.10286889997

0.10286889997

0.10286889997

0.10286889997

0.10286889997

0.10286889997

0.10286889997

95 0.10259783521

0.10259783521

0.10259783521

0.10259783521

0.10259783521

0.10259783521

0.10259783521

0.10259783521

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95.5 0.10232890202

0.10232890202

0.10232890202

0.10232890202

0.10232890202

0.10232890202

0.10232890202

0.10232890202

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0.10232890202

96 0.10206207262

0.10206207262

0.10206207262

0.10206207262

0.10206207262

0.10206207262

0.10206207262

0.10206207262

0.10206207262

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96.5 0.10179731971

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0.10179731971

0.10179731971

0.10179731971

0.10179731971

0.10179731971

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97 0.10153461651

0.10153461651

0.10153461651

0.10153461651

0.10153461651

0.10153461651

0.10153461651

0.10153461651

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0.10153461651

97.5 0.10127393671

0.10127393671

0.10127393671

0.10127393671

0.10127393671

0.10127393671

0.10127393671

0.10127393671

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0.10127393671

98 0.10101525446

0.10101525446

0.10101525446

0.10101525446

0.10101525446

0.10101525446

0.10101525446

0.10101525446

0.10101525446

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98.5 0.10075854437

0.10075854437

0.10075854437

0.10075854437

0.10075854437

0.10075854437

0.10075854437

0.10075854437

0.10075854437

0.10075854437

99 0.10050378153

0.10050378153

0.10050378153

0.10050378153

0.10050378153

0.10050378153

0.10050378153

0.10050378153

0.10050378153

0.10050378153

99.5 0.10025094142

0.10025094142

0.10025094142

0.10025094142

0.10025094142

0.10025094142

0.10025094142

0.10025094142

0.10025094142

0.10025094142

100 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

S\tau 2.8 3.8 4.8 2.8 3.8 4.8 2.8 3.8 4.8

0.5 0.7970325885

0.79872839101

0.80006203177

0.058421198807

0.058380448734

0.058430848972

0.069773410164

0.068321552429

0.068097528298

96

1 0.46282569744

0.46313236767

0.46318236462

0.058753630368

0.058707446687

0.05875419211

0.06988048598

0.068390357442

0.068162873428

1.5 0.32345231995

0.32333041299

0.32303825858

0.059075216352

0.059023522235

0.059066604492

0.069987022884

0.068458419318

0.068227549034

2 0.24405483574

0.2437453126

0.2433328894

0.059386443722

0.059329134921

0.05936854327

0.07009317364

0.068525766881

0.068291586583

2.5 0.19188449292

0.19147978716

0.19102240988

0.059687776563

0.059624720747

0.059660442242

0.070198722454

0.068592428209

0.068355016057

3 0.15462971908

0.1541739141

0.15370235096

0.059979657554

0.059910693765

0.059942713431

0.07030395884

0.068658427841

0.068417866157

3.5 0.12652695587

0.12604404722

0.12557224877

0.060262509297

0.06018744753

0.060215748537

0.070408761787

0.068723793608

0.068480164515

4 0.10448596707

0.10399020347

0.10352482309

0.060536735531

0.060455356443

0.060479920269

0.070513201694

0.068788549773

0.068541937398

4.5 0.086688056293

0.086188421911

0.085732642971

0.060802722234

0.060714776987

0.060735583578

0.070616965989

0.068852718816

0.068603210011

5 0.071987223162

0.071489628078

0.071044897361

0.06106083862

0.060966048861

0.060983076785

0.070720372395

0.068916322703

0.068664006419

5.5 0.059622553137

0.059131000071

0.058697887289

0.061311438055

0.06120949603

0.061222722627

0.070823534596

0.068979381334

0.068724349686

6 0.049067446384

0.048584688748

0.048163334936

0.061554858881

0.061445427695

0.06145482922

0.070926340254

0.069041919614

0.068784261573

6.5 0.039944996607

0.03947292392

0.03906327575

0.061791425171

0.061674139182

0.061679690944

0.071028913007

0.069103951854

0.068843763214

7 0.031977810076

0.031517686311

0.031119614033

0.062021447422

0.061895912771

0.06189758927

0.0711307718

0.069165502281

0.068902874942

7.5 0.024956866985

0.024509493605

0.024122846768

0.062245223183

0.062111018457

0.062108793515

0.071232548174

0.069226583523

0.068961615223

8 0.018721457908

0.018287295929

0.017911926472

0.062463037628

0.062319714659

0.062313561547

0.0713342782

0.06928720334

0.069020002788

8.5 0.013145840819

0.012725104192

0.012360872561

0.062675164099

0.06252224887

0.062512140431

0.071434993754

0.069347399666

0.069078055119

9 0.0081301196913

0.007722847002

0.0073696201315

0.062881864575

0.062718858265

0.062704767039

0.071534820421

0.069407176576

0.069135788791

9.5 0.0035938566773

0.0031999664173

0.0028576128752

0.063083390149

0.062909770261

0.062891668599

0.071635504547

0.069466541119

0.069193219623

10 0.00052849552442

0.00090916394868

0.0012407785741

0.063279981431

0.063095203036

0.063073063221

0.071735817388

0.069525521056

0.069250363191

10.5 0.0042909201189

0.0046585667695

0.0049795951881

0.063471868948

0.063275366016

0.063249160372

0.071833899015

0.069584123948

0.069307233824

11 0.0077383170743

0.0080931712271

0.0084037724928

0.063659273532

0.063450460315

0.063420161326

0.07193452838

0.069642358668

0.069363845521

11.5 0.010908342901

0.011250636425

0.011550991424

0.063842406621

0.06362067916

0.063586259577

0.072031408726

0.069700245088

0.069420211811

12 0.013832831985

0.014162790673

0.014453100694

0.064021470646

0.063786208272

0.063747641223

0.072131840759

0.069757787997

0.069476345954

12.5 0.016538889462

0.016856725864

0.017137213164

0.064196659315

0.063947226225

0.063904485332

0.072227498006

0.069815006398

0.069532259917

13 0.01904974439

0.019355654211

0.019626561604

0.064368157908

0.064103904784

0.064056964268

0.072318038847

0.069871894132

0.069587967844

13.5 0.021385422911

0.021679584354

0.021941173743

0.064536143609

0.064256409217

0.064205244007

0.072411504823

0.06992847107

0.069643480616

14 0.02356328469

0.023845859918

0.024098409891

0.064700785703

0.064404898582

0.064349484427

0.07250356851

0.069984740788

0.069698811531

14.5 0.025598454097

0.025869592259

0.02611339497

0.06486224599

0.064549526002

0.064489839582

0.072616610327

0.070040745594

0.069753971711

15 0.027504169631

0.027764011011

0.027999368541

0.065020678853

0.064690438919

0.064626457948

0.072710784096

0.070096472577

0.069808977763

15.5 0.029292069539

0.029540750196

0.02976797058

0.065176231734

0.064827779321

0.064759482668

0.072826263573

0.070151878099

0.069863841942

97

16 0.030972427271

0.031210083358

0.031429476433

0.065329045144

0.06496168398

0.064889051771

0.072904124626

0.070207035343

0.069918577857

16.5 0.032554346957

0.032781118079

0.032992991328

0.065479253123

0.065092284651

0.065015298376

0.072987033894

0.070261956313

0.069973204824

17 0.034045926488

0.034261957841

0.03446661244

0.065626983419

0.065219708267

0.065138350894

0.073105938967

0.070316720054

0.070027744743

17.5 0.035454394061

0.035659837469

0.035857564779

0.065772357638

0.065344077128

0.065258333207

0.073220864796

0.070371140671

0.070082219399

18 0.036786222859

0.036981237033

0.037172315753

0.065915491524

0.065465509068

0.065375364839

0.073290866093

0.070425233643

0.070136657834

18.5 0.038047227759

0.038231978124

0.038416672232

0.066056495118

0.065584117627

0.065489561121

0.073389373454

0.070479216953

0.070191097093

19 0.039242647262

0.039417305615

0.039595863127

0.06619547323

0.065700012199

0.065601033346

0.073444310266

0.070533195945

0.070245575068

19.5 0.040377213294

0.040541957433

0.040714609899

0.066332525277

0.065813298193

0.065709888904

0.073656475566

0.070586474256

0.070300150905

20 0.041455210996

0.04161022438

0.041777186983

0.066467745864

0.065924077146

0.06581623143

0.073680148242

0.070640208746

0.070354895192

20.5 0.042480530204

0.042626001646

0.042787473732

0.066601224801

0.066032446875

0.065920160927

0.073660220464

0.070693449213

0.070409892393

21 0.043456710001

0.043592833362

0.043748999225

0.066733047232

0.066138501622

0.066021773891

0.073642095562

0.070746687704

0.070465262829

21.5 0.044386977409

0.044513951261

0.044664981072

0.066863293505

0.066242332122

0.066121163429

0.073836654874

0.070799731695

0.070521144415

22 0.045274281133

0.045392308372

0.045538359121

0.06699204058

0.066344025782

0.066218419364

0.073994014592

0.070852875144

0.070577718799

22.5 0.046121321077

0.046230608445

0.046371824852

0.067119360691

0.066443666749

0.066313628352

0.073833779299

0.070906273465

0.070635244718

23 0.04693057424

0.047031331758

0.04716784711

0.067245322895

0.066541336012

0.066406873973

0.074222847592

0.070959917134

0.070694027358

23.5 0.047704317496

0.047796757781

0.047928694682

0.067369991702

0.066637111577

0.066498236836

0.074520462211

0.071013381636

0.070754452294

24 0.048444647691

0.048528985162

0.048656456202

0.067493429115

0.066731068441

0.066587794669

0.07433241794

0.071068364456

0.070817050865

24.5 0.049153499411

0.049229949373

0.049353057718

0.067615692091

0.066823278856

0.066675622411

0.07489299361

0.071123396518

0.070882464253

25 0.049832660728

0.049901438345

0.050020278269

0.06773683877

0.06691381221

0.066761792291

0.073325784302

0.071181633726

0.070951565154

25.5 0.050483787184

0.050545106343

0.050659763709

0.067856916685

0.067002735305

0.066846373919

0.073051447299

0.071239071682

0.071025406178

26 0.051108414239

0.051162486315

0.051273039018

0.067975979579

0.067090112396

0.066929434374

0.072176217389

0.071301149908

0.071105382548

26.5 0.051707968368

0.051755000905

0.05186151929

0.068094068491

0.067176005187

0.067011038262

0.075347758329

0.071366117092

0.071193188956

27 0.052283776974

0.052323972294

0.052426519547

0.068211231302

0.067260473037

0.067091247804

0.071478477472

0.071436441766

0.071291064253

27.5 0.052837077263

0.052870631014

0.052969263542

0.068327505479

0.067343572903

0.067170122905

0.072975846373

0.071513843508

0.071401683167

28 0.053369024192

0.053396123857

0.053490891655

0.068442933921

0.067425359571

0.067247721241

0.0732542702

0.071605788105

0.071528358541

28.5 0.053880697614

0.053901520994

0.053992467994

0.068557555509

0.067505885712

0.067324098291

0.10232890202

0.071702504331

0.071675192652

29 0.054373108696

0.054387822376

0.054474986785

0.0686713922

0.067585201718

0.067399307441

0.073808061064

0.071827538701

0.071846953945

29.5 0.054847205699

0.054855963517

0.054939378153

0.068784488574

0.067663356032

0.067473400038

0.077940810257

0.071974327726

0.072049298247

30 0.055303879197

0.055306820723

0.055386513326

0.06889688007

0.067740395316

0.067546425457

0.10153461651

0.072151358172

0.072288530015

30.5 0.055743966782

0.055741215824

0.055817209359

0.069008583601

0.067816363929

0.067618431117

0.075601441211

0.07237247855

0.072571458586

98

2. pdex_dupire_option.m

Table B.2 The Option Prices(Dupire)

31 0.056168257321

0.056159920459

0.056232233414

0.069119631144

0.067891304966

0.067689462627

0.10101525446

0.072647791442

0.072905307897

31.5 0.05657749481

0.056563659982

0.056632306639

0.069230027803

0.067965259395

0.067759563788

0.10075854437

0.072977284618

0.073296905316

32 0.05697238186

0.056953116995

0.0570181077

0.069339840422

0.06803826654

0.067828776641

0.078907327269

0.073390069657

0.07375240366

32.5 0.057353582869

0.05732893458

0.057390275996

0.069449047301

0.068110363663

0.067897141567

0.075723867936

0.073888660394

0.074276446336

33 0.057721726885

0.057691719237

0.057749414584

0.069557711285

0.068181587697

0.06796469731

0.075998458814

0.074476375812

0.074871910211

33.5 0.058077410219

0.058042043569

0.058096092854

0.069665831699

0.068251972963

0.06803148099

K=9.9

S\tau 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0 0 0 0 0 0 0 0 0 0

1 0 1.91E-14 1.16E-11 4.51E-10 5.69E-09 3.76E-08 1.66E-07 5.62E-07 1.56E-06 3.72E-06 7.91E-06

2 0 3.34E-12 1.05E-09 2.73E-08 2.53E-07 1.31E-06 4.76E-06 1.36E-05 3.23E-05 6.75E-050.00012713157451

3 0 2.73E-10 4.44E-08 7.67E-07 5.24E-06 2.15E-05 6.43E-050.00015506859276

0.00032059671056

0.00059068170724

0.00099584894721

4 0 1.37E-08 1.14E-06 1.31E-05 6.68E-050.00021812284945

0.00054199907023

0.0011195159533

0.0020276522572

0.0033319099577

0.0050823861226

5 0 4.67E-07 1.99E-05 0.0001529940418

0.00058462302286

0.0015362392532

0.0032066480711

0.0057394902939

0.0092186656366

0.013675218611

0.019099755736

6 0 1.13E-050.00024602192531

0.00127843345

0.003724382041

0.0079833820041

0.01420233193

0.022346168737

0.032273776654

0.043794600344

0.056707864116

7 00.00019519023949

0.0022104430681

0.0079067555402

0.017892985955

0.031870633401

0.049188119641

0.069161427185

0.091179873385

0.11474412381

0.13946397533

8 0 0.0024228679091

0.01459628447

0.036921319922

0.066552088053

0.10079280743

0.1377580672

0.1762236819

0.21541159448

0.25483849606

0.29419521557

9 0 0.021201244573

0.071080129065

0.13214181253

0.19599456598

0.2593537999

0.32102819311

0.38061737414

0.438087612

0.49352514727

0.54704515082

10 0 0.12718251664

0.25539614581

0.36803720922

0.46751818768

0.55710878826

0.63909929461

0.71508657869

0.7861952856

0.85322620406

0.91679131498

11 0.2 0.50254658508

0.68054072381

0.8148529031

0.92722378072

1.0258019866

1.114702802

1.1963371659

1.2721940973

1.3433349919

1.4105439168

12 1.2 1.270918019

1.3790047998

1.4822973701

1.5769386601

1.6640002525

1.7447679443

1.8203265546

1.8915331414

1.9590200488

2.0232886323

13 2.2 2.2131523359

2.2564469337

2.3146117901

2.3777518806

2.4417128517

2.504837372

2.5664322316

2.6262854401

2.684366814

2.7407090605

14 3.2 3.2020504235

3.2155427474

3.2418413202

3.2769971539

3.3174670412

3.3609453015

3.4060050077

3.4517705118

3.4977150331

3.5435047105

15 4.2 4.2002800583

4.2038304393

4.2139479811

4.230940928

4.2536140062

4.2805643801

4.310620708

4.3428741778

4.3766577964

4.4114885486

99

16 5.2 5.2000345827

5.2008614544

5.2043014327

5.2116385076

5.2231300041

5.2384204904

5.2569487148

5.2781301403

5.3014400373

5.3264431079

17 6.2 6.2000039579

6.2001796243

6.2012407488

6.2041304536

6.2094875721

6.2175348948

6.2282078995

6.2412896199

6.2565004065

6.2735560509

18 7.2 7.2000004282

7.2000351994

7.2003379194

7.2013927363

7.2037197328

7.2076904905

7.2134874775

7.2211441007

7.2305936167

7.2417139098

19 8.2 8.2000000445

8.2000065599

8.2000876153

8.2004490089

8.2014006567

8.2032534083

8.2062441326

8.2105182613

8.2161381489

8.2231023717

20 9.2 9.2000000045

9.2000011748

9.2000217855

9.2001392154

9.2005087933

9.2013321126

9.2028066765

9.2050944859

9.2083091978

9.2125159653

21 10.2 10.2 10.200000204

10.200005229

10.200041737

10.200179037

10.200529587

10.201228011

10.202407585

10.204183346

10.206643087

22 11.2 11.2 11.200000035

11.200001219

11.200012161

11.200061268

11.200205032

11.200524264

11.201112385

11.202062876

11.203459329

23 12.2 12.2 12.200000006

12.200000277

12.20000346

12.200020465

12.200077521

12.200218896

12.200503439

12.200997913

12.201769738

24 13.2 13.2 13.200000001

13.200000062

13.200000965

13.200006695

13.2000287

13.200089584

13.200223589

13.200474283

13.200890572

25 14.2 14.2 14.2 14.200000014

14.200000265

14.200002152

14.200010431

14.200036013

14.20009762

14.20022179

14.200441364

26 15.2 15.2 15.2 15.200000003

15.200000072

15.200000682

15.200003731

15.20001425

15.200041971

15.200102193

15.200215674

27 16.2 16.2 16.2 16.200000001

16.200000019

16.200000213

16.200001316

16.200005561

16.200017799

16.20004646

16.200104032

28 17.2 17.2 17.2 17.2 17.200000005

17.200000066

17.200000459

17.200002144

17.200007458

17.200020868

17.200049588

29 18.2 18.2 18.2 18.2 18.200000001

18.20000002

18.200000158

18.200000818

18.200003092

18.200009273

18.200023383

30 19.2 19.2 19.2 19.2 19.2 19.200000006

19.200000054

19.20000031

19.20000127

19.200004081

19.200010919

31 20.2 20.2 20.2 20.2 20.2 20.200000002

20.200000018

20.200000116

20.200000518

20.200001781

20.200005054

32 21.2 21.2 21.2 21.2 21.2 21.200000001

21.200000006

21.200000043

21.20000021

21.200000772

21.200002321

33 22.2 22.2 22.2 22.2 22.2 22.2 22.200000002

22.200000016

22.200000084

22.200000332

22.200001059

34 23.2 23.2 23.2 23.2 23.2 23.2 23.200000001

23.200000006

23.200000034

23.200000142

23.20000048

35 24.2 24.2 24.2 24.2 24.2 24.2 24.2 24.200000002

24.200000014

24.200000061

24.200000216

36 25.2 25.2 25.2 25.2 25.2 25.2 25.2 25.200000001

25.200000005

25.200000026

25.200000097

37 26.2 26.2 26.2 26.2 26.2 26.2 26.2 26.2 26.200000002

26.200000011

26.200000043

38 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.200000001

27.200000005

27.200000019

39 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.200000002

28.200000009

40 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.200000001

29.200000004

41 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.200000002

42 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.200000001

100

43 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2

44 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2

45 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2

46 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2

47 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2

48 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2

49 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2

50 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2

51 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2

52 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2

53 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2

54 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2

55 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2

56 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2

57 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2

58 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2

59 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2

60 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2

61 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2

62 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2

63 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2

64 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2

65 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2

66 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2

67 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2

68 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2

69 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2

101

70 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2

71 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2

72 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2

73 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2

74 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2

75 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2

76 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2

77 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2

78 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2

79 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2

80 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2

81 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2

82 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2

83 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2

84 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2

85 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2

86 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2

87 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2

88 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2

89 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

90 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2

91 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2

92 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2

93 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2

94 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2

95 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2

96 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2

102

97 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2

98 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2

99 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2

100 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2

S\tau 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0 0 0 0 0 0 0 0 0 0 0

1 1.53E-05 2.75E-05 4.64E-05 7.41E-050.00011341175203

0.00016697550108

0.00023782522063

0.00032910653414

0.00044401435868

0.00058576931504

20.00022067999625

0.00035840307213

0.00055120139488

0.00081008693919

0.0011457390057

0.001568229717

0.0020868021025

0.0027097940127

0.003444434106

0.0042969095229

3 0.0015652088983

0.00232488863

0.0032977621876

0.0045025225042

0.0059534442069

0.0076607407391

0.0096310373156

0.011867604641

0.014371152736

0.017139996515

4 0.0073133693226

0.010045431675

0.013286442016

0.01703414809

0.021278814468

0.026005274828

0.031194706009

0.03682532307

0.042874400968

0.049318134385

5 0.025456259835

0.032693663426

0.040748602748

0.049553884084

0.059043489635

0.069153917451

0.079825036475

0.091000508178

0.10262822289

0.11466010092

6 0.070821922558

0.08596101798

0.10196671941

0.1187013653

0.13604750413

0.15390438334

0.17218507854

0.19081564588

0.20973150689

0.22887783337

7 0.1650343899

0.19121380706

0.21781945044

0.244711038

0.27177798073

0.29893261882

0.32610538537

0.35324261563

0.3803011638

0.40724838531

8 0.33327949703

0.37196102974

0.41016857603

0.44786096342

0.48501096665

0.52160413185

0.55763640856

0.59311159408

0.6280385518

0.66242930213

9 0.59878015632

0.6488643985

0.69742761247

0.74458271417

0.79042856811

0.835056123

0.8785485402

0.92098086942

0.96242126459

1.0029302018

10 0.97737133948

1.0353450975

1.0910074945

1.1446013934

1.1963357375

1.246387318

1.2949061453

1.3420193722

1.3878375375

1.432456734

11 1.4743978845

1.5353516444

1.5937491291

1.6498755595

1.7039728958

1.7562427195

1.8068548214

1.8559517387

1.9036571594

1.9500779387

12 2.0847498321

2.1437333738

2.2005035423

2.2552793736

2.308250494

2.3595793557

2.4094053611

2.4578479574

2.5050117684

2.5509881865

13 2.7953881903

2.8484986869

2.9001440864

2.9504172772

2.999401267

3.047175449

3.0938144107

3.1393870433

3.1839567533

3.2275805384

14 3.5889229355

3.6338343919

3.678168249

3.7218850034

3.7649586179

3.8073768611

3.8491388833

3.8902521136

3.9307292634

3.9705858084

15 4.4470076526

4.4829445508

4.5191075823

4.5553567004

4.5915836472

4.6277049374

4.6636572088

4.6993942632

4.7348815406

4.7700950423

16 5.3527833394

5.3801667899

5.4083592574

5.4371739808

5.4664602629

5.4960951561

5.5259776824

5.5560266707

5.5861743215

5.6163666786

17 6.2921923489

6.3121705847

6.3332792344

6.3553365626

6.3781888092

6.4017048966

6.4257721849

6.4502952711

6.4751909441

6.5003889063

18 7.254359612

7.2683800178

7.2836227218

7.2999449248

7.3172181824

7.335327155

7.3541682318

7.3736493369

7.3936876814

7.4142100109

19 8.2313682922

8.2408703412

8.2515239824

8.2632385601

8.2759247159

8.2894962838

8.3038714703

8.3189734649

8.3347310098

8.3510781727

20 9.2177404412

9.2239806698

9.2312102531

9.2393880442

9.2484651028

9.2583881461

9.2691021847

9.2805515674

9.2926824661

9.3054424593

21 10.209846408

10.213829142

10.218605226

10.224171708

10.230513553

10.237607175

10.245423382

10.253928492

10.26308757

10.27286409

103

22 11.205369943

11.207846592

11.210925443

11.214627948

11.21896314

11.223930318

11.229521484

11.23572229

11.242515045

11.249878617

23 12.202880984

12.204384885

12.206325569

12.208736578

12.211641147

12.215053697

12.218981403

12.223424839

12.228380201

12.23383944

24 13.201522162

13.202415682

13.203613966

13.205153583

13.207063892

13.209367458

13.21208073

13.215214376

13.218774537

13.222763121

25 14.200792817

14.201313162

14.202039117

14.20300468

14.204239682

14.20576946

14.207614843

14.209792224

14.212313954

14.215188679

26 15.200407476

15.200704971

15.20113713

15.20173263

15.202518419

15.203519013

15.204756054

15.206248202

15.20801091

15.210056714

27 16.200206849

16.20037408

16.200627212

16.200988825

16.201481432

16.202126695

16.202944793

16.203954223

16.205171219

16.206609951

28 17.200103807

17.200196363

17.200342433

17.20055888

17.200863456

17.201274106

17.20180834

17.202482997

17.203313542

17.204314154

29 18.200051547

18.200102049

18.200185186

18.200313024

18.200498933

18.200757064

18.201101821

18.201547638

18.202108308

18.202796983

30 19.200025349

19.200052548

19.200099271

19.200173848

19.200285972

19.200446367

19.200666393

19.200957864

19.201332495

19.201801837

31 20.200012355

20.200026832

20.200052787

20.200095798

20.200162673

20.200261268

20.200400237

20.200588895

20.200836814

20.201153725

32 21.200005974

21.200013597

21.200027863

21.200052409

21.200091885

21.200151884

21.200238806

21.200359771

21.200522351

21.200734472

33 22.200002867

22.200006843

22.200014609

22.200028482

22.200051563

22.200087733

22.200141609

22.200218484

22.200324191

22.200465003

34 23.200001367

23.200003423

23.200007613

23.200015386

23.200028761

23.200050377

23.200083487

23.200131938

23.200200114

23.200292861

35 24.200000648

24.200001703

24.200003946

24.200008266

24.200015955

24.200028769

24.200048956

24.200079255

24.200122892

24.200183532

36 25.200000306

25.200000843

25.200002036

25.200004419

25.200008806

25.200016345

25.200028563

25.200047373

25.200075105

25.200114477

37 26.200000144

26.200000416

26.200001046

26.200002352

26.200004838

26.200009244

26.200016588

26.200028186

26.200045692

26.200071088

38 27.200000067

27.200000204

27.200000535

27.200001247

27.200002647

27.200005206

27.200009592

27.200016698

27.20002768

27.20004396

39 28.200000031

28.2000001

28.200000273

28.200000659

28.200001443

28.20000292

28.200005525

28.200009853

28.200016702

28.200027078

40 29.200000015

29.200000049

29.200000139

29.200000347

29.200000784

29.200001632

29.200003171

29.200005793

29.200010041

29.200016618

41 30.200000007

30.200000024

30.200000071

30.200000182

30.200000425

30.20000091

30.200001814

30.200003394

30.200006016

30.200010163

42 31.200000003

31.200000012

31.200000036

31.200000096

31.20000023

31.200000506

31.200001035

31.200001983

31.200003593

31.200006196

43 32.200000001

32.200000006

32.200000018

32.20000005

32.200000124

32.20000028

32.200000589

32.200001155

32.20000214

32.200003766

44 33.200000001

33.200000003

33.200000009

33.200000026

33.200000067

33.200000155

33.200000334

33.200000671

33.200001271

33.200002283

45 34.2 34.200000001

34.200000005

34.200000014

34.200000036

34.200000086

34.200000189

34.200000389

34.200000753

34.200001381

46 35.2 35.200000001

35.200000002

35.200000007

35.200000019

35.200000047

35.200000107

35.200000225

35.200000445

35.200000833

47 36.2 36.2 36.200000001

36.200000004

36.20000001

36.200000026

36.20000006

36.20000013

36.200000263

36.200000501

48 37.2 37.2 37.200000001

37.200000002

37.200000006

37.200000014

37.200000034

37.200000075

37.200000155

37.200000301

104

49 38.2 38.2 38.2 38.200000001

38.200000003

38.200000008

38.200000019

38.200000043

38.200000091

38.200000181

50 39.2 39.2 39.2 39.200000001

39.200000002

39.200000004

39.200000011

39.200000025

39.200000054

39.200000108

51 40.2 40.2 40.2 40.2 40.200000001

40.200000002

40.200000006

40.200000014

40.200000031

40.200000065

52 41.2 41.2 41.2 41.2 41.2 41.200000001

41.200000003

41.200000008

41.200000018

41.200000039

53 42.2 42.2 42.2 42.2 42.2 42.200000001

42.200000002

42.200000005

42.200000011

42.200000023

54 43.2 43.2 43.2 43.2 43.2 43.2 43.200000001

43.200000003

43.200000006

43.200000014

55 44.2 44.2 44.2 44.2 44.2 44.2 44.200000001

44.200000002

44.200000004

44.200000008

56 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.200000001

45.200000002

45.200000005

57 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.200000001

46.200000001

46.200000003

58 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.200000001

47.200000002

59 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.200000001

60 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.200000001

61 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2

62 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2

63 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2

64 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2

65 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2

66 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2

67 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2

68 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2

69 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2

70 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2

71 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2

72 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2

73 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2

74 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2

75 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2

105

76 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2

77 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2

78 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2

79 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2

80 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2

81 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2

82 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2

83 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2

84 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2

85 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2

86 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2

87 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2

88 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2

89 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

90 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2

91 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2

92 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2

93 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2

94 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2

95 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2

96 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2

97 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2

98 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2

99 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2

100 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2

106

3. pdex_dupire.m

Table B.3 The Dupire Volatilities

4. pdex_dis_dupire.m

Table B.4 Distance between Implied Volatilities and the Dupire Volatilities

S=1:100 .9.1.99 == τ，K

S=1:10 S=11:20 S=21:30 S=31:40 S=41:50 S=51:60 S=61:70 S=71:80 S=81:90 S=91:100

dupireσ0.1103875

1351

0.0997995

08443

0.1008571

9287

0.1054613

1299

0.1063929

1169

0.1005420

3662

0.0927723

37099

0.0851126

23971NaN NaN

dupireσ0.1076708

6022

0.0997230

90679

0.1011916

7192

0.1058786

9067

0.1060496

9887

0.0997735

64282

0.0920217

31358

0.0731887

71445NaN NaN

dupireσ0.1056060

4521

0.0997211

77438

0.1015750

5219

0.1062409

3047

0.1056361

0557

0.0989937

26871

0.0913724

05838

0.0814369

46953NaN NaN

dupireσ0.1041218

578

0.0997553

88519

0.1020028

7325

0.1065384

4185

0.1051580

4199

0.0982061

27539

0.0904617

37785

0.0677596

35682NaN NaN

dupireσ0.1030239

5962

0.0998121

17284

0.1024683

1984

0.1067633

7199

0.1046217

8906

0.0974167

90179

0.0899351

55032

0.0714249

27393NaN NaN

dupireσ0.1021527

2677

0.0998936

68878

0.1029625

3212

0.1069097

8776

0.1040338

4167

0.0966223

86341

0.0892097

0572Inf NaN NaN

dupireσ0.1014207

1648

0.1000051

7119

0.1034749

6498

0.1069737

5558

0.1034008

074

0.0958330

65625

0.0882457

96623

0.0451730

90455NaN NaN

dupireσ0.1007983

6181

0.1001518

7943

0.1039938

0133

0.1069533

181

0.1027290

1956

0.0950573

49174

0.0863168

13945Inf NaN NaN

dupireσ0.1003138

6197

0.1003395

3131

0.1045064

1173

0.1068483

8764

0.1020247

7141

0.0942440

41036

0.0868018

02914Inf NaN NaN

dupireσ0.0999892

99656

0.1005734

5679

0.1049998

4282

0.1066605

6525

0.1012934

5784

0.0934875

21274

0.0834390

478030 NaN NaN

tau=1.9;K=10.

x Distance x Distance x Distance x Distance x Distance

1 0 8 0 15 0.038940004132 22 0.068873

053982 29 0.10268879475

2 0 9 0 16 0.0096295917963 23 0.075617

101167 30 0.10557478798

3 0 10 0.0055731734841 17 0.011647

847367 24 0.081526241573 31 0.107758

81617

4 0 11 0.033925944179 18 0.028040

893949 25 0.086754277933 32 0.108238

5101

5 0 12 0.64329959398 19 0.041173

752426 26 0.091414513072 33 0.138504

78093

107

5. pdex_imp_small_s.m

Table B.5 Implied Volatilities Volatilities for Small Stock Prices

6 0 13 0.16942858353 20 0.051991

380509 27 0.095590702894 34 NaN

7 0 14 0.08371187034 21 0.061090

850366 28 0.099342677508

impσ 1.0:1:.10=τ

S(10-12) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5 17.067374456

12.374808848

10.261732387

8.9885985531

8.1130805179

7.4628010286

6.9547831538

6.5434512861

6.2012658136

5.9107854141

1 16.807915572

12.18978099

10.109905584

8.8566500881

7.9947402955

7.3545311423

6.8543562506

6.4493577404

6.1124270276

5.82639813

1.5 16.654951304

12.08068344

10.020377406

8.7788393657

7.9249514453

7.2906788408

6.7951276005

6.3938628779

6.060030152

5.7766257294

2 16.545877707

12.002883154

9.9565293611

8.7233457874

7.8751775815

7.2451379284

6.7528835684

6.3542812586

6.0226576181

5.7411246589

2.5 16.460958653

11.942308056

9.9068155815

8.6801358652

7.8364204939

7.209676281

6.7199886499

6.3234591426

5.9935553914

5.7134794798

3 16.391368347

11.892664854

9.8660723467

8.6447221475

7.8046556713

7.1806119612

6.6930277045

6.2981968286

5.9697025155

5.6908206547

3.5 16.332384569

11.850586173

9.8315366027

8.6147033947

7.777729559

7.1559747029

6.6701732129

6.2767820665

5.949482378

5.671612589

4 16.281181594

11.814056928

9.8015548616

8.588642614

7.7543533712

7.1345853953

6.6503314813

6.2581901385

5.9319274897

5.6549362737

4.5 16.235932943

11.781774575

9.7750582899

8.5656109033

7.7336940002

7.1156818209

6.6327955033

6.2417586221

5.9164124196

5.6401976148

5 16.195389228

11.75284809

9.7513157264

8.5449728167

7.7151815258

7.0987425504

6.6170816156

6.2270343513

5.9025093054

5.6269901858

5.5 16.158657868

11.726640897

9.7298047959

8.5262743574

7.6984087626

7.0833950405

6.6028442555

6.2136935511

5.8899124495

5.6150236033

6 16.125078854

11.702682275

9.7101392194

8.5091798063

7.6830746052

7.0693638011

6.5898278831

6.2014967921

5.8783957896

5.6040831291

6.5 16.094150304

11.680614273

9.6920252539

8.4934338658

7.6689500762

7.056439333

6.5778381697

6.1902619962

5.8677874123

5.5940054608

7 16.065481626

11.660158307

9.6752342837

8.4788378405

7.6558569635

7.0444585829

6.5667238684

6.1798474509

5.8579535156

5.584663498

7.5 16.038762836

11.641093281

9.6595848691

8.4652340351

7.6436538279

7.0332921417

6.5563649398

6.1701406833

5.848787905

5.5759563647

8 16.01374378

11.623240771

9.6449305857

8.4524951865

7.6322265243

7.0228355728

6.5466645087

6.1610509242

5.8402048775

5.56780265

8.5 15.990219649

11.606454692

9.6311515484

8.4405170969

7.6214815954

7.013003367

6.5375432565

6.1525038543

5.8321342619

5.5601356968

9 15.968020631

11.590613922

9.6181483551

8.4292133712

7.6113415506

7.0037246238

6.5289354154

6.144437851

5.8245178747

5.5529002353

9.5 15.947004359

11.57561691

9.6058376668

8.4185115747

7.6017414208

6.9949398993

6.5207858447

6.1368012483

5.8173069336

5.5460499256

10 15.927050293

11.561377673

9.5941489195

8.4083503759

7.5926261953

6.9865988605

6.513047855

6.1295502995

5.8104601329

5.5395455306

10.5 15.908055465

11.547822766

9.5830218366

8.3986773845

7.5839488832

6.9786585124

6.5056815634

6.1226476348

5.8039421884

5.5333535325

11 15.889931225

11.534888951

9.5724045197

8.3894474919

7.5756690245

6.9710818353

6.4986526298

6.116061078

5.7977227194

5.5274450719

11.5 15.872600695

11.522521389

9.562251963

8.3806215802

7.5677515323

6.9638367257

6.4919312744

6.109762726

5.7917753797

5.5217951215

108

12 15.855996774

11.510672215

9.5525248828

8.3721655049

7.5601657807

6.9568951616

6.4854915033

6.1037282226

5.7860771722

5.5163818357

12.5 15.840060545

11.499299399

9.5431887858

8.364049285

7.552884878

6.9502325377

6.4793104922

6.0979361817

5.780607904

5.5111860324

13 15.824739993

11.488365839

9.5342132201

8.356246452

7.5458850831

6.9438271315

6.4733680904

6.0923677221

5.7753497476

5.5061907765

13.5 15.809988968

11.477838611

9.5255711662

8.3487335209

7.5391453309

6.9376596685

6.4676464186

6.0870060912

5.7702868848

5.5013810419

14 15.795766334

11.467688371

9.5172385399

8.3414895578

7.5326468443

6.9317129682

6.4621295395

6.0818363556

5.7654052155

5.4967434341

14.5 15.782035267

11.457888849

9.5091937808

8.3344958225

7.5263728141

6.9259716503

6.4568031857

6.076845147

5.7606921171

5.4922659625

15 15.76876267

11.448416435

9.5014175118

8.3277354725

7.5203081328

6.9204218916

6.4516545346

6.0720204503

5.7561362449

5.4879378502

15.5 15.755918691

11.439249833

9.4938922532

8.3211933142

7.5144391725

6.9150512225

6.4466720189

6.0673514271

5.7517273652

5.4837493756

16 15.743476305

11.430369763

9.4866021826

8.3148555948

7.5087535969

6.9098483558

6.4418451678

6.0628282662

5.747456214

5.4796917388

16.5 15.731410973

11.421758719

9.4795329316

8.3087098251

7.5032402038

6.904803041

6.437164473

6.0584420582

5.743314379

5.4757569481

17 15.719700342

11.413400755

9.4726714126

8.3027446291

7.497888789

6.8999059416

6.4326212733

6.0541846876

5.7392941974

5.4719377245

17.5 15.708323998

11.405281307

9.4660056711

8.2969496158

7.492690032

6.8951485293

6.4282076574

6.0500487415

5.7353886696

5.4682274187

18 15.697263239

11.397387033

9.4595247579

8.2913152681

7.487635396

6.8905229928

6.4239163794

6.0460274302

5.731591385

5.4646199408

18.5 15.686500898

11.389705684

9.4532186198

8.2858328474

7.4827170428

6.8860221601

6.4197407862

6.0421145198

5.7278964572

5.461109699

19 15.676021172

11.382225989

9.4470780041

8.280494311

7.477927758

6.8816394303

6.4156747543

6.0383042725

5.7242984684

5.4576915469

19.5 15.665809486

11.374937546

9.4410943757

8.2752922401

7.473260887

6.877368714

6.4117126351

6.0345913955

5.720792421

5.4543607368

20 15.655852365

11.367830745

9.4352598448

8.2702197764

7.4687102781

6.8732043826

6.407849207

6.030970996

5.7173736954

5.4511128803

20.5 15.64613733

11.360896681

9.429567103

8.2652705676

7.4642702332

6.8691412222

6.4040796333

6.0274385421

5.7140380122

5.4479439121

21 15.636652797

11.35412709

9.4240093681

8.2604387181

7.459935464

6.8651743938

6.4003994254

6.0239898278

5.7107813999

5.4448500597

21.5 15.627387999

11.347514289

9.4185803342

8.2557187466

7.4557010538

6.8612993988

6.3968044098

6.0206209424

5.7076001661

5.4418278152

22 15.618332905

11.341051124

9.4132741281

8.2511055481

7.4515624233

6.8575120469

6.3932906996

6.0173282438

5.7044908718

5.4388739118

22.5 15.60947816

11.334730919

9.4080852709

8.2465943599

7.4475153005

6.8538084292

6.3898546685

6.0141083342

5.7014503087

5.4359853017

23 15.600815022

11.328547436

9.4030086432

8.2421807319

7.4435556937

6.8501848932

6.3864929283

6.0109580387

5.6984754789

5.433159137

23.5 15.592335309

11.322494839

9.398039454

8.2378604999

7.4396798674

6.8466380209

6.3832023081

6.0078743862

5.695563577

5.4303927528

24 15.584031354

11.316567658

9.3931732139

8.2336297612

7.4358843208

6.843164609

6.3799798366

6.0048545923

5.692711974

5.4276836515

24.5 15.575895964

11.31076076

9.3884057094

8.2294848534

7.4321657689

6.8397616515

6.3768227251

6.0018960441

5.6899182025

5.4250294893

25 15.567922378

11.30506932

9.3837329814

8.225422335

7.4285211246

6.8364263237

6.3737283533

5.9989962862

5.6871799442

5.4224280638

25.5 15.560104234

11.299488802

9.3791513048

8.2214389679

7.4249474832

6.8331559679

6.3706942555

5.9961530084

5.6844950176

5.4198773026

26 15.552435541

11.294014929

9.3746571705

8.2175317018

7.4214421085

6.8299480802

6.3677181093

5.9933640342

5.6818613678

5.4173752536

26.5 15.544910647

11.288643669

9.3702472687

8.2136976597

7.41800242

6.8268002995

6.364797724

5.9906273111

5.6792770565

5.4149200757

109

27 15.537524216

11.283371216

9.3659184747

8.2099341254

7.4146259809

6.823710396

6.3619310313

5.9879409009

5.6767402538

5.4125100304

27.5 15.530271205

11.27819397

9.3616678345

8.2062385311

7.4113104881

6.8206762622

6.3591160762

5.9853029719

5.6742492298

5.4101434744

28 15.523146842

11.273108527

9.3574925534

8.2026084474

7.4080537624

6.8176959042

6.3563510089

5.9827117905

5.6718023479

5.4078188529

28.5 15.516146605

11.268111661

9.3533899841

8.1990415732

7.4048537394

6.8147674331

6.353634077

5.9801657149

5.6693980578

5.4055346932

29 15.509266211

11.263200316

9.3493576169

8.1955357269

7.4017084624

6.8118890581

6.3509636195

5.9776631885

5.6670348896

5.4032895989

29.5 15.502501592

11.258371588

9.34539307

8.192088838

7.398616074

6.80905908

6.3483380599

5.9752027342

5.6647114485

5.4010822451

30 15.495848886

11.253622723

9.3414940813

8.1886989401

7.3955748104

6.8062758846

6.3457559009

5.9727829489

5.6624264097

5.3989113732

30.5 15.48930442

11.248951099

9.3376585001

8.1853641639

7.3925829946

6.8035379375

6.3432157189

5.9704024986

5.6601785138

5.3967757868

31 15.482864701

11.244354225

9.3338842803

8.1820827305

7.3896390311

6.8008437787

6.3407161595

5.9680601144

5.6579665623

5.3946743473

31.5 15.476526403

11.239829727

9.3301694734

8.1788529462

7.3867414004

6.7981920179

6.3382559331

5.9657545877

5.6557894141

5.3926059709

32 15.470286354

11.235375342

9.3265122225

8.175673197

7.3838886548

6.7955813303

6.3358338106

5.9634847667

5.6536459818

5.3905696243

32.5 15.464141531

11.230988913

9.3229107567

8.1725419433

7.3810794132

6.7930104521

6.3334486198

5.9612495531

5.6515352281

5.3885643223

33 15.458089048

11.226668381

9.3193633857

8.1694577159

7.3783123577

6.7904781775

6.3310992418

5.9590478986

5.6494561632

5.3865891246

33.5 15.452126149

11.222411779

9.3158684953

8.1664191114

7.3755862294

6.7879833545

6.3287846081

5.9568788017

5.6474078416

5.3846431329

34 15.446250199

11.218217227

9.3124245424

8.1634247887

7.3728998253

6.7855248822

6.3265036973

5.9547413056

5.6453893596

5.3827254887

34.5 15.440458679

11.214082927

9.3090300513

8.1604734648

7.3702519944

6.7831017076

6.3242555326

5.9526344947

5.6433998529

5.380835371

35 15.434749178

11.21000716

9.3056836097

8.157563912

7.3676416356

6.7807128229

6.3220391791

5.9505574933

5.6414384943

5.3789719941

35.5 15.429119385

11.205988276

9.302383865

8.1546949547

7.3650676941

6.7783572631

6.3198537416

5.9485094622

5.6395044917

5.3771346052

36 15.42356709

11.202024698

9.2991295212

8.1518654661

7.3625291593

6.7760341032

6.3176983621

5.9464895975

5.6375970859

5.3753224834

36.5 15.418090168

11.19811491

9.2959193355

8.1490743659

7.3600250621

6.7737424567

6.315572218

5.9444971286

5.6357155491

5.3735349372

37 15.412686586

11.194257462

9.2927521157

8.1463206177

7.3575544728

6.7714814728

6.31347452

5.9425313157

5.6338591831

5.3717713032

37.5 15.40735439

11.190450958

9.2896267173

8.1436032267

7.355116499

6.7692503348

6.3114045107

5.9405914491

5.6320273174

5.3700309446

38 15.402091701

11.18669406

9.286542041

8.1409212373

7.3527102834

6.7670482584

6.3093614623

5.9386768469

5.6302193083

5.3683132498

38.5 15.396896718

11.182985479

9.2834970306

8.1382737312

7.3503350023

6.7648744898

6.3073446756

5.9367868538

5.628434537

5.366617631

39 15.391767706

11.179323981

9.2804906703

8.1356598255

7.3479898638

6.7627283042

6.3053534782

5.9349208399

5.6266724088

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7.9696654714

7.1990572491

6.6264259757

6.1788906819

5.8164056069

5.5147527601

5.2586123059

88.5 15.063852819

10.945202734

9.0882455484

7.9685017638

7.19801311

6.6254703527

6.1780040188

5.8155746454

5.5139680258

5.2578667407

89 15.061583188

10.943582082

9.0869146749

7.967344501

7.1969747529

6.6245200211

6.1771222649

5.8147482844

5.5131876359

5.2571253027

89.5 15.059326064

10.941970358

9.0855911317

7.9661936114

7.1959421135

6.6235749221

6.1762453657

5.8139264728

5.512411542

5.2563879462

90 15.057081309

10.940367464

9.0842748377

7.9650490245

7.1949151287

6.6226349978

6.1753732674

5.8131091603

5.5116396966

5.2556546259

90.5 15.054848787

10.938773301

9.0829657132

7.9639106711

7.1938937363

6.6217001914

6.1745059174

5.8122962974

5.5108720532

5.2549252977

91 15.052628364

10.937187776

9.08166368

7.962778483

7.1928778753

6.620770447

6.1736432637

5.8114878356

5.5101085657

5.2541999178

91.5 15.050419908

10.935610793

9.0803686609

7.9616523933

7.1918674854

6.6198457095

6.1727852553

5.810683727

5.5093491891

5.2534784433

92 15.048223291

10.934042261

9.07908058

7.960532336

7.1908625075

6.6189259249

6.171931842

5.8098839246

5.5085938788

5.252760832

92.5 15.046038384

10.932482089

9.0777993629

7.9594182462

7.1898628835

6.6180110398

6.1710829744

5.809088382

5.5078425912

5.2520470425

93 15.043865063

10.930930187

9.076524936

7.9583100602

7.1888685561

6.6171010019

6.1702386039

5.8082970537

5.5070952833

5.2513370338

93.5 15.041703204

10.929386467

9.0752572272

7.9572077151

7.187879469

6.6161957597

6.1693986827

5.807509895

5.5063519127

5.2506307657

94 15.039552687

10.927850843

9.0739961654

7.9561111491

7.1868955668

6.6152952624

6.1685631636

5.8067268617

5.5056124378

5.2499281987

94.5 15.037413391

10.926323229

9.0727416806

7.9550203015

7.1859167948

6.61439946

6.1677320005

5.8059479103

5.5048768176

5.2492292939

95 15.035285199

10.924803542

9.071493704

7.9539351124

7.1849430994

6.6135083036

6.1669051476

5.8051729982

5.5041450118

5.2485340128

95.5 15.033167995

10.923291699

9.0702521677

7.9528555228

7.1839744277

6.6126217445

6.1660825602

5.8044020834

5.5034169806

5.2478423178

96 15.031061666

10.921787619

9.0690170052

7.9517814748

7.1830107277

6.6117397354

6.165264194

5.8036351243

5.502692685

5.2471541718

96.5 15.0289661

10.920291222

9.0677881507

7.9507129113

7.1820519481

6.6108622292

6.1644500056

5.8028720804

5.5019720865

5.2464695381

97 15.026881186

10.918802429

9.0665655395

7.9496497759

7.1810980386

6.6099891798

6.163639952

5.8021129113

5.5012551472

5.2457883808

97.5 15.024806815

10.917321162

9.065349108

7.9485920135

7.1801489495

6.6091205418

6.1628339913

5.8013575777

5.5005418297

5.2451106645

98 15.022742881

10.915847346

9.0641387934

7.9475395694

7.1792046317

6.6082562704

6.1620320817

5.8006060406

5.4998320973

5.2444363542

98.5 15.020689278

10.914380904

9.0629345341

7.9464923899

7.1782650373

6.6073963215

6.1612341825

5.7998582617

5.4991259139

5.2437654155

99 15.018645903

10.912921763

9.0617362691

7.9454504223

7.1773301187

6.6065406518

6.1604402534

5.7991142033

5.4984232438

5.2430978146

99.5 15.016612652

10.911469851

9.0605439386

7.9444136143

7.1763998293

6.6056892184

6.1596502547

5.7983738281

5.4977240519

5.2424335182

100 15.014589426

10.910025094

9.0593574835

7.9433819148

7.1754741229

6.6048419793

6.1588641473

5.7976370996

5.4970283037

5.2417724933

S\tau 2.8 3.8 4.8 2.8 3.8 4.8 2.8 3.8 4.8

0.5 3.7128829617

3.239624704

2.9204154161

3.3932013313

2.9641916224

2.674665125

3.3390876908

2.917554361

2.6330446465

1 3.6618307128

3.1956475325

2.8811834012

3.3920560889

2.9632046514

2.6737843522

3.3385017658

2.9170493644

2.6325939567

114

1.5 3.631713701

3.1697028094

2.8580371347

3.3909270415

2.9622316355

2.6729160318

3.3379200574

2.9165480014

2.6321465095

2 3.6102294535

3.1511942181

2.8415244634

3.3898137336

2.9612721822

2.6720598133

3.3373425048

2.9160502197

2.6317022581

2.5 3.5934977677

3.1367795392

2.8286639592

3.3887157285

2.9603259151

2.6712153612

3.3367690483

2.9155559679

2.6312611567

3 3.5797829147

3.1249636574

2.8181218757

3.3876326076

2.9593924735

2.6703823534

3.3361996299

2.915065196

2.6308231606

3.5 3.5681559727

3.1149464047

2.809184396

3.386563969

2.9584715114

2.6695604813

3.3356341925

2.9145778547

2.630388226

4 3.558060975

3.1062488657

2.8014242819

3.3855094273

2.9575626964

2.6687494482

3.3350726803

2.914093896

2.6299563098

4.5 3.5491384974

3.0985614243

2.7945653231

3.3844686121

2.9566657095

2.6679489695

3.3345150387

2.9136132729

2.6295273703

5 3.5411426427

3.0916722561

2.7884185479

3.3834411674

2.9557802439

2.6671587716

3.3339612142

2.9131359393

2.6291013661

5.5 3.5338977152

3.085430009

2.7828489317

3.3824267511

2.9549060049

2.6663785914

3.3334111544

2.9126618499

2.6286782571

6 3.5272737832

3.0797227528

2.7777566183

3.381425034

2.9540427088

2.6656081757

3.3328648081

2.9121909607

2.6282580038

6.5 3.521172034

3.0744653643

2.7730656661

3.3804356993

2.9531900826

2.664847281

3.3323221249

2.9117232283

2.6278405675

7 3.5155155582

3.0695915881

2.7687169661

3.3794584418

2.9523478633

2.6640956727

3.3317830556

2.9112586102

2.6274259104

7.5 3.5102433131

3.0650488376

2.764663603

3.3784929676

2.9515157978

2.6633531249

3.331247552

2.9107970649

2.6270139953

8 3.5053060336

3.0607946715

2.7608677128

3.3775389934

2.9506936419

2.6626194196

3.3307155667

2.9103385515

2.626604786

8.5 3.5006633816

3.0567943391

2.7572982924

3.3765962461

2.9498811603

2.661894347

3.3301870535

2.9098830303

2.6261982466

9 3.4962819094

3.0530190253

2.7539296341

3.3756644621

2.9490781259

2.6611777043

3.3296619667

2.909430462

2.6257943424

9.5 3.492133573

3.0494445701

2.7507401831

3.3747433871

2.9482843195

2.6604692961

3.329140262

2.9089808081

2.6253930391

10 3.488194626

3.046050516

2.7477116878

3.3738327756

2.9474995296

2.6597689336

3.3286218956

2.9085340312

2.624994303

10.5 3.4844447832

3.042819387

2.7448285567

3.3729323903

2.9467235518

2.6590764343

3.3281068247

2.9080900942

2.6245981012

11 3.4808665784

3.0397361357

2.7420773646

3.3720420021

2.9459561886

2.6583916222

3.3275950072

2.9076489609

2.6242044013

11.5 3.4774448643

3.0367877122

2.7394464688

3.3711613895

2.9451972491

2.6577143268

3.3270864019

2.9072105958

2.6238131718

12 3.4741664188

3.0339627257

2.7369257066

3.3702903382

2.9444465488

2.6570443834

3.3265809684

2.9067749641

2.6234243815

12.5 3.4710196322

3.0312511737

2.7345061539

3.3694286409

2.943703909

2.6563816328

3.326078667

2.9063420315

2.6230379998

13 3.4679942542

3.0286442256

2.7321799319

3.3685760971

2.9429691569

2.6557259207

3.3255794587

2.9059117646

2.6226539969

13.5 3.4650811899

3.0261340456

2.7299400498

3.3677325127

2.9422421252

2.6550770977

3.3250833054

2.9054841303

2.6222723434

14 3.462272332

3.0237136489

2.7277802758

3.3668976995

2.941522652

2.6544350194

3.3245901695

2.9050590964

2.6218930104

14.5 3.4595604226

3.0213767827

2.7256950309

3.3660714755

2.9408105801

2.6537995457

3.3241000142

2.9046366312

2.6215159695

15 3.4569389386

3.0191178268

2.723679301

3.365253664

2.9401057576

2.6531705408

3.3236128034

2.9042167034

2.6211411931

15.5 3.4544019954

3.0169317116

2.7217285628

3.3644440941

2.939408037

2.6525478732

3.3231285016

2.9037992825

2.6207686538

16 3.4519442663

3.0148138476

2.7198387216

3.3636425996

2.9387172752

2.6519314152

3.3226470738

2.9033843384

2.6203983248

115

16.5 3.4495609138

3.0127600672

2.718006059

3.3628490197

2.9380333336

2.6513210433

3.322168486

2.9029718418

2.6200301798

17 3.4472475317

3.0107665741

2.7162271879

3.3620631982

2.9373560777

2.6507166371

3.3216927045

2.9025617635

2.6196641929

17.5 3.4450000951

3.0088299006

2.7144990143

3.3612849835

2.9366853766

2.6501180802

3.3212196961

2.9021540752

2.6193003388

18 3.4428149173

3.0069468704

2.7128187044

3.3605142284

2.9360211038

2.6495252595

3.3207494286

2.9017487488

2.6189385925

18.5 3.4406886135

3.0051145672

2.7111836562

3.3597507899

2.9353631359

2.648938065

3.3202818701

2.901345757

2.6185789295

19 3.4386180682

3.0033303068

2.7095914747

3.3589945293

2.9347113534

2.6483563899

3.3198169891

2.9009450726

2.6182213256

19.5 3.4366004073

3.0015916131

2.7080399507

3.3582453114

2.9340656399

2.6477801305

3.3193547549

2.9005466693

2.6178657573

20 3.4346329743

2.9998961971

2.7065270419

3.3575030054

2.9334258824

2.647209186

3.3188951374

2.9001505209

2.6175122013

20.5 3.4327133082

2.9982419386

2.7050508566

3.3567674835

2.9327919712

2.6466434583

3.3184381067

2.8997566019

2.6171606347

21 3.4308391251

2.9966268697

2.703609639

3.3560386218

2.9321637992

2.6460828521

3.3179836336

2.8993648869

2.6168110351

21.5 3.4290083014

2.9950491609

2.7022017568

3.3553162997

2.9315412627

2.6455272746

3.3175316894

2.8989753513

2.6164633803

22 3.4272188595

2.9935071082

2.7008256895

3.3546003997

2.9309242604

2.6449766356

3.3170822458

2.8985879708

2.6161176486

22.5 3.4254689541

2.9919991218

2.6994800188

3.3538908078

2.9303126939

2.6444308472

3.3166352751

2.8982027213

2.6157738187

23 3.4237568613

2.9905237163

2.6981634192

3.3531874126

2.9297064674

2.643889824

3.31619075

2.8978195794

2.6154318695

23.5 3.4220809675

2.9890795016

2.6968746506

3.3524901059

2.9291054876

2.6433534828

3.3157486435

2.8974385219

2.6150917804

24 3.4204397608

2.9876651749

2.6956125504

3.3517987823

2.9285096637

2.6428217423

3.3153089293

2.897059526

2.6147535311

24.5 3.4188318222

2.9862795137

2.6943760279

3.351113339

2.927918907

2.6422945237

3.3148715813

2.8966825693

2.6144171016

25 3.4172558183

2.984921369

2.693164058

3.3504336759

2.9273331314

2.6417717499

3.3144365741

2.8963076298

2.6140824723

25.5 3.4157104944

2.9835896599

2.6919756762

3.3497596954

2.9267522528

2.641253346

3.3140038823

2.8959346859

2.6137496237

26 3.4141946687

2.9822833681

2.6908099738

3.3490913025

2.9261761893

2.6407392388

3.3135734814

2.8955637161

2.6134185369

26.5 3.4127072264

2.9810015331

2.689666094

3.3484284045

2.9256048609

2.6402293571

3.3131453469

2.8951946995

2.6130891931

27 3.411247115

2.979743248

2.6885432276

3.3477709108

2.9250381897

2.6397236313

3.3127194549

2.8948276155

2.6127615739

27.5 3.4098133397

2.9785076557

2.6874406097

3.3471187333

2.9244760997

2.6392219935

3.3122957817

2.8944624437

2.6124356611

28 3.4084049589

2.977293945

2.6863575166

3.3464717859

2.9239185167

2.6387243776

3.3118743043

2.8940991641

2.612111437

28.5 3.4070210812

2.9761013476

2.6852932629

3.3458299847

2.9233653685

2.6382307191

3.3114549996

2.8937377571

2.6117888839

29 3.405660861

2.9749291352

2.6842471986

3.3451932476

2.9228165842

2.6377409549

3.3110378452

2.8933782031

2.6114679845

29.5 3.4043234961

2.9737766165

2.6832187069

3.3445614947

2.9222720951

2.6372550234

3.310622819

2.8930204832

2.6111487219

30 3.4030082245

2.972643135

2.6822072021

3.3439346478

2.9217318337

2.6367728646

3.3102098992

2.8926645786

2.6108310791

30.5 3.4017143218

2.9715280665

2.6812121272

3.3433126306

2.9211957344

2.6362944199

3.3097990642

2.8923104707

2.6105150398

31 3.4004410986

2.970430817

2.6802329522

3.3426953685

2.9206637328

2.6358196319

3.309390293

2.8919581414

2.6102005876

116

Table B.6 Slopes of Implied Volatility Curves for Small Stock Prices

31.5 3.3991878987

2.9693508211

2.6792691726

3.3420827888

2.9201357663

2.6353484445

3.3089835647

2.8916075726

2.6098877066

32 3.3979540965

2.9682875397

2.6783203075

3.3414748203

2.9196117735

2.6348808032

3.3085788588

2.8912587468

2.609576381

32.5 3.3967390955

2.9672404589

2.6773858983

3.3408713935

2.9190916944

2.6344166544

3.3081761551

2.8909116464

2.6092665952

33 3.3955423261

2.966209088

2.6764655071

3.3402724404

2.9185754707

2.6339559459

3.3077754337

2.8905662545

2.6089583339

33.5 3.3943632447

2.9651929584

2.6755587159

3.3396778944

2.9180630448

2.6334986266

Simp

∂∂σ 1.0:1:.10=τ

S(10-12) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5-5.1891776734E+11

-3.7005571602E+11

-3.0365360519E+11

-2.6389692989E+11

-2.3668044473E+11

-2.1653977265E+11

-2.008538064E+11

-1.8818709149E+11

-1.7767757202E+11

-1.6877456831E+11

1-3.0592853667E+11

-2.1819510081E+11

-1.7905635691E+11

-1.5562144493E+11

-1.3957770057E+11

-1.2770460292E+11

-1.1845730022E+11

-1.1098972492E+11

-1.0479375114E+11

-99544801152

1.5-2.1814719272E+11

-1.5560057232E+11

-1.2769608959E+11

-1.1098715654E+11

-99547727451

-91081824878

-84488064142

-79163238721

-74745067703

-71002140968

2-1.6983810811E+11

-1.2115019628E+11

-99427559128

-86419844277

-77514175180

-70923294871

-65789836985

-61644231863

-58204453405

-55290358241

2.5-1.3918061196E+11

-99286402507

-81486469644

-70827435477

-63529645284

-58128639510

-53921890733

-50524628104

-47705751931

-45317650112

3-1.1796755617E+11

-84157362651

-69071487939

-60037505613

-53852224503

-49274516618

-45708983254

-42829524107

-40440274981

-38416131555

3.5-1.0240594972E+11

-73058489273

-59963482306

-52121561486

-46752375764

-42778615259

-39683463169

-37183856060

-35109776531

-33352630546

4-90497302476

-64564707543

-52993143305

-46063421283

-41318741958

-37807148729

-35071956076

-32863032862

-31030140182

-29477317880

4.5-81087430244

-57852970040

-47485127035

-41276173147

-37024948785

-33878541010

-31427775409

-29448541587

-27806228353

-26414857962

5-73462720576

-52414386066

-43021860936

-37396918673

-33545526285

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126

97-4148741241.4

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S\tau 2.8 3.8 4.8 2.8 3.8 4.8 2.8 3.8 4.8

0.5-1.0210449798E+11

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127

7-10544490198

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16.5-4626764155

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128

17-4494873349.8

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17.5-4370355515.4

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18.5-4141090693.9

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19-4035321627

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19.5-3934866023.4

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21.5-3578883887.6

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22-3499810712.8

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24.5-3152007858.6

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25-3090647749.1

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25.5-3031651447.1

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26-2974884573

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26.5-2920222756.8

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129

27-2867550722.7

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27.5-2816761471.6

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28-2767755551.4

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28.5-2720440402.3

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29-2674729769.5

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29.5-2630543174.9

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30-2587805441.1

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30.5-2546446261.6

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31-2506399812.6

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31.5-2467604400.7

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32-2430002143.9

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32.5-2393538681.5

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33-2358162909.8

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3.3077754337

2.8905662545

2.6089583339

33.5-2323826742.2

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-1017367684.7

-907960224.36

130

Bibliography

[1]Lishang Jiang(1994)Mathematical Modeling and Methods of Option Pricing, 312, Tongji University.[2] Matlab Indexesfunction pdepe.[3]Wikipedia,Greeks(Finance), http://en.wikipedia.org/wiki/Greeks_(finance) .[4] Steven L. Heston (1993)A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond andCurrency Options, Yale University.

http://en.wikipedia.org/wiki/Greeks_(finance)

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