Static Hedging and Pricing American Exotic Options

San-Lin Chung*, Pai-Ta Shih, Wei-Che Tsai

Department of Finance, National Taiwan University, No. 1, Section 4, Roosevelt Road,

Taipei 10617, Taiwan, ROC

Abstract

This paper applies the static hedge portfolio (SHP) approach of Derman, Ergener, and

Kani (1995) and Carr, Ellis, and Gupta (1998) to price and/or hedge American exotic

options. We first show how to construct a static hedge portfolio to match the

complicated boundary conditions of American barrier options and lookback options.

Detailed analyses of the profit and loss distributions suggest that the static hedge

portfolio is far less risky than the dynamic delta-hedged portfolio. Moreover,

numerical results indicate that the efficiency of the proposed method is comparable to

Boyle and Tian (1999) for pricing American barrier options under the constant

elasticity of variance (CEV) model of Cox (1975) and comparable to Babbs (2000) for

pricing American floating strike lookback options under the Black-Scholes model. In

particular, the recalculation of the option prices and hedge ratios under the proposed

method is much easier and quicker than the tree methods.

JEL classification: G13

Keywords: American barrier options; American floating strike lookback option; Early

exercise boundary; Static hedge; Value-matching condition; Smooth-pasting condition

* Corresponding author: San-Lin Chung, Department of Finance, National Taiwan University, Room

707, No. 85, Section 4, Roosevelt Road, Taipei, 106, Taiwan, R.O.C. Tel: 886-2-33661084, Fax:

886-2-23660764, e-mail: chungs@management.ntu.edu.tw.

1

1. Introduction

Pricing and hedging American-style exotic options is an important yet difficult

problem in the finance literature. Many numerical methods and analytical

approximation formulae have been proposed to price these options as their market

size has grown substantially. For example, Boyle and Lau (1994), Ritchken (1995),

Cheuk and Vorst (1996), and Chung and Shih (2007) develop lattice methods and Gao,

Huang, and Subrahmanyam (2000), AitSahlia, Imhof, and Lai (2003), and Chang,

Kang, Kim, and Kim (2007) develop analytical approximation formulae for pricing

American barrier options under the Black-Scholes model.

On the other hand, since the seminal work of Derman, Ergener, and Kani (1995)

and Carr, Ellis, and Gupta (1998), static replication of exotic options, using standard

options, has been extended in several ways. First of all, it has been applied to hedge

barrier options statically beyond the Black-Scholes model, e.g. see Andersen,

Andreasen, and Eliezer (2002), Fink (2003), Nalholm and Poulsen (2006a), and

Takahashi and Yamazaki (2009). Secondly, several replication methods, such as the

optimal static-dynamic hedge method of Ilhan and Sircar (2006), the risk-minimizing

method of Siven and Poulsen (2009), and the line segments method of Liu (2010),

have been applied to form a static hedge portfolio for the exotic option. Thirdly, the

hedging performance or model risk of the static hedge portfolio are also widely

investigated, e.g. see Toft and Xuan (1998), Nalholm and Poulsen (2006b), and

Engelmann, Fengler, Nalholm, and Schwendner (2006). Finally, beyond the barrier

options, the static hedge approach has been applied to other types of options such as

American options (Chung and Shih (2009)), European-style Asian options (Albrecher,

Dhaene, Goovaerts, and Schoutens (2005)), and European-style installment options

(Davis, Schachermayer, and Tompkins (2001)).

2

In addition to the previous findings, this paper utilizes the static hedge portfolio

(SHP) approach to price and/or hedge American exotic options. In comparison to the

existing methods, the proposed method has at least three advantages. First of all, even

if the existing numerical or analytical approximation methods can price American

barrier options efficiently under the Black-Scholes model, it may be difficult to extend

them to other stochastic processes. In contrast, the proposed method is applicable for

more general stochastic processes, e.g. the constant elasticity of variance (CEV)

model of Cox (1975). Besides, the proposed method is applicable for other types of

exotic options beyond barrier options e.g. American floating strike lookback options.

Secondly, the hedge ratios, such as delta and theta, are automatically derived at the

same time when the static hedge portfolio is formed. Thirdly, when the stock price

and/or time to maturity instantaneously change, the recalculation of the prices and

hedge ratios for the American exotic options under the proposed method is quicker

than the initial computational time because there is no need to solve the static hedge

portfolio again. In contrast, the recalculation time of most existing American option

pricing methods is the same as the initial computational time.

We contribute to the static hedge literature in three ways. First of all, we show

how to construct static hedge portfolios for American barrier options and floating

strike lookback options.1 Due to the complexity of boundary conditions that must be

satisfied by an American exotic option, forming a static hedge portfolio is not a trivial

question. For instance, the static hedge portfolio of an American up-and-out put

option must satisfy the knock-out condition on the barrier and the value-matching and

smooth-pasting conditions on the early exercise boundary. To do so, one has to

carefully choose the types of standard European options (call or put) and their strike

1 We show how to form the SHP for an American floating strike lookback put option using the

hypothetic European options in Section 2.2. Although the obtained SHP is not a realistic hedge

portfolio, we show that its value provides a good price estimate for the lookback put in Section 4.2.

3

prices.

Secondly, we investigate the hedging performance of the proposed method and

compare with that of the dynamic hedge strategy for American up-and-out put options

under the Black-Scholes model. Specifically, we consider two static hedge portfolios

where one is composed of European options with standard strike prices (i.e. the strike

prices specified by the option exchange)2 and the other one is made of European

options with nonstandard strikes. The numerical results indicate that the hedging

performance of both static portfolios significantly outperform the dynamic hedge

strategy.

Thirdly, this article analyzes the efficiency (in terms of speed and accuracy) of

the proposed method for pricing American barrier options and floating strike lookback

options. Once the static hedge portfolio is solved, its value serves as a good estimate

of the American option price if the terminal and boundary conditions are matched well.

Our numerical results show that the proposed method is as efficient as the tree method

of Boyle and Tian (1999) for pricing American barrier options under the CEV model

and comparable to the tree method of Babbs (2000) for pricing American floating

strike lookback options under the Black-Scholes model. When the valuation needs to

be updated during the life of the option, the proposed method is especially attractive

because the recalculation is much quicker than the tree methods because there is no

need to solve the static hedge portfolio again. According to our numerical experiments

(not reported in the paper), solving the static hedge portfolio usually takes more than

95% of the total initial computational time. In other words, it takes less than 5% of the

initial computational time to update the option prices when the stock price and/or time

to maturity change. Thus the recalculation speed of the proposed method is at least

2 For example, according to CBOE’s equity option product specifications, the strike price interval is

$2.5 when the strike price is between $5 and $25, $5 when the strike price is between $25 and $200,

and $10 when the strike price is over $200.

4

twenty times quicker than Boyle and Tian (1999) and Babbs (2000).

The rest of the paper is organized as follows. Section 2 discusses how to

formulate the static hedge portfolio for American barrier options and American

floating strike lookback options. Section 3 compares the performance of static hedge

versus dynamic hedge for an American up-and-out barrier put option. Numerical

efficiency of the proposed method for pricing American up-and-out puts under the

CEV model and American floating strike lookback puts under the Black-Scholes

model is presented in Section 4. Section 5 concludes the paper.

2. Formulation of the static hedging portfolio

As emphasized before, forming the static hedge portfolio (SHP) for an American

exotic option is not a trivial question due to the complexity of boundary conditions of

the option. In this section, we demonstrate how to construct the SHP for an American

up-and-out put option and the SHP for an American floating strike lookback put

option under the Black-Scholes model. In the former case, the options used in the SHP

are standard European options traded in the option exchange. In contrast, the latter

case utilizes hypothetic European options, whose underlying asset is a non-tradable

asset, to form the SHP.

2.1. The static hedge portfolio for an American up-and-out put

Under the Black-Scholes model, it is well known that the price F of any

American (or European) barrier option written on a stock satisfies the following

partial differential equation (PDE):

5

2 21( )

2SS S tS F r q SF F rF , (1)

where S is the stock price. The volatility , the risk-free rate r , and the dividend

yield rate q are assumed constant. Particularly noteworthy is that the PDE only

holds in the continuation region and the holding value of the American barrier option

is determined by the terminal condition and boundary conditions. Additionally,

standard European options written on the same stock with different maturities or

different strike prices all satisfy the same PDE. Therefore, it is feasible to formulate a

static hedge portfolio of standard European options with different maturities and

strikes to match the terminal condition and boundary conditions of American barrier

options.

There are two boundary conditions for an American up-and-out put (AUOP)

option. One is the knock-out boundary and the other one is the early exercise

boundary. The first boundary implies that the SHP value must be zero when the stock

price equals the barrier. To match the second boundary, one particular difficulty arises

because it involves a free boundary problem, i.e. the early exercise boundary has to be

determined at the same time when the SHP is formulated. To overcome this problem,

we adopt the method of Chung and Shih (2009) by applying the value-matching and

smooth-pasting conditions on the early exercise boundary to solve the SHP.

Specifically, consider the construction of an n -point SHP which matches the

6

boundary conditions of the AUOP before maturity at n evenly-spaced time points,

i.e. 0 0t , 1t , …, 1nt T t , where t T n . Following Chung and Shih (2009),

we work backward to determine the number of the standard European options and

their strike prices for the n -point SHP. Our static hedge portfolio starts with one unit

of the corresponding European option to match the terminal condition of the AUOP.

At time 1nt , we add 1nw units of a European put, maturing at time nt T and

with a strike price equaling the early exercise boundary 1nB , and 1ˆ

nw units of a

European call, maturing at time nt T and with a strike price equaling the barrier

H , into the SHP (why both call and put options are used will be explained later). The

knock-out boundary condition on the barrier (i.e. when the stock price equals H ) and

the value-matching and smooth-pasting conditions on the early exercise boundary (i.e.

when the stock price equals 1nB ), respectively, imply that

1 1 1 1

1 1

0 , , , , , , , , , ,

ˆ , , , , , ,

E E

n n n n

E

n n

P H X r q T t w P H B r q T t

w C H H r q T t

(2)

1 1 1 1 1 1 1

1 1 1

, , , , , , , , , ,

ˆ , , , , , ,

E E

n n n n n n n

E

n n n

X B P B X r q T t w P B B r q T t

w C B H r q T t

(3)

1 1 1 1 1 1

1 1 1

1 , , , , , , , , , ,

ˆ , , , , ,

E E

p n n n p n n n

E

n c n n

B X r q T t w B B r q T t

w B H r q T t

(4)

where (.)EC , (.)EP , (.)E

c , and (.)E

p are the European call price, put price, call

delta, and put delta, respectively. Under the Black-Scholes model, their formulae are

1 2, , , , , ( ) ( ),E q rC S X r q Se N d Xe N d

(5)

7

2 1, , , , , ( ) ( ),E r qP S X r q Xe N d Se N d (6)

1, , , , , ( ),E q

c S X r q e N d (7)

1, , , , , ( ),E q

p S X r q e N d (8)

where is the time to maturity, )(N is the cumulative distribution function of the

standard normal distribution, 2

1

ln( ) ( 2)S X r qd

, and 2 1d d .

To solve three unknown variables ( 1nw , 1ˆ

nw , and 1nB ) from Equations (2) to

(4), we first solve Equations (3) and (4) and obtain the solutions of 1nw and 1ˆ

nw , as

functions of 1nB . Substituting 1nw and 1ˆ

nw into Equation (2) leads to a nonlinear

equation of 1nB , which can be solved numerically using the Newton-Raphson

method. Then we obtain 1nw and 1ˆ

nw from 1nB .

Similarly, at time it , we add iw units of a European put, maturing at time 1it

and with a strike price equaling iB , and ˆiw units of a European call, maturing at

time 1it and with a strike price equaling H , into the SHP. Note that the value of the

newly added options must be zero on the knock-out boundary and the early exercise

boundary at time 1it in order not to affect the existing solution of the SHP at time

1it .3 This is the reason why the European put with a strike price equaling iB is

utilized here because its payoff at maturity date (time 1it ) is zero both on the early

exercise boundary 1iB (due to the fact that 1i iB B ) and on the knock-out

boundary (since iB H ). Similarly, the payoff at maturity of the chosen European

call is also zero on the early exercise boundary and on the knock-out boundary. Again,

3 Otherwise we cannot work backward to solve the SHP. In this case, we have to solve all unknowns

simultaneously and thus the solution problem would become 3n-dimensional (there are three unknown

variables at each time point).

8

applying the knock-out boundary condition and the value-matching and

smooth-pasting conditions on the early exercise boundary at time it yields three

equations (similar to Equations (2)-(4)) to solve three unknowns ( iw , ˆiw , and iB ). In

summary, Figure 1 depicts the backward procedure of solving the SHP for the AUOP.

[Insert Figure 1 Here]

After solving all ˆiw s, iw s, and iB s ( 0, 1, , 1i n ), the value of the

n -point static hedge portfolio nSHP at time 0 is obtained as follows:4

0

1 0 1 1 0

2 0 2 1 2 0 1

0 0 0 1 0 0 1

, , , , ,

ˆ , , , , , , , , , ,

ˆ , , , , , , , , , ,

ˆ , , , , , , , , , , .

E

n

E E

n n n

E E

n n n n n

E E

SHP P S X r q T

w P S B r q T w C S H r q T

w P S B r q t w C S H r q t

w P S B r q t w C S H r q t

(9)

2.2. The static hedge portfolio for an American floating strike lookback put

The payoff of a lookback option depends on the realized maximum or minimum

of the underlying asset over a specified time. In this subsection, we consider a floating

strike lookback put option with the maturity payoff maxmax( ( ) )S T S , where

max 0( ) max( ; )uS t S t u t , 0t is the initial time, and T is the option’s maturity date.

According to Babbs (2000), the American floating strike lookback put price,

max( , , )A

FLP S S t , satisfies the following partial differential equation under the

Black-Scholes model:

22 2

2

1( )

2

A A AAFL FL FL

FL

P P PS r q S rP

S S t

, (10)

4 For the other diffusion models, the procedures for forming the SHP are similar except that the partial

differential equation satisfied by the option and the pricing formulae of the European options are

different.

9

when *

maxS S S and *S is the early exercise boundary.

Using the standard European options to form an SHP for the American floating

strike lookback put is difficult, if not impossible, because the early exercise boundary

depends on maxS and thus is path-dependent. To overcome the problem, as suggested

by Babbs (2000), we use the underlying asset as the numeraire and express the price

of an American floating strike lookback put option, ( ,1, )AV u t , as follows:5

max( , , )( ,1, )

AA FLP S S t

V u tS

, (11)

where max /u S S . From Equation (10), we know that ( ,1, )AV u t satisfies the

following partial differential equation:

22 2

2

1( )

2

A A AAV V V

u q r u qVu u t

, (12)

when *1 u u and * *

max /u S S . In addition, ( ,1, )AV u t satisfies the following

boundary conditions,

( ,1, ) max( 1,0)AV u T u , (13)

* *( ,1, ) 1AV u t u , (14)

*

( ,1, )1

A

u u

V u t

u

, (15)

1

( ,1, )0

A

u

V u t

u

.6 (16)

5 Because the pricing problem is homogeneous in the price of the underlying asset, the price of an

American floating strike lookback put option can be expressed in terms of a single state variable after

the change of numeraire from cash to the underlying asset. 6 Please refer to Theorem 3 of Goldman, Sosin, and Gatto (1979) for Equation (16). Their Theorem 3

states that when max( ) ( )S t S t , the probability distribution of max ( )S T is unaffected by the marginal

10

After the change of numeraire, ( ,1, )AV u t corresponds to the price of an

American call option with a strike price equaling 1 and the delta of the option is zero

when 1u . Since the strike price is no longer path-dependent, we can apply the

same procedure as described in Section 2.1 to formulate the SHP for ( ,1, )AV u t .

However, it should be emphasized that the component options used here are

hypothetic European options because the underlying asset ( max /u S S ) is not a

tradable asset. Moreover, Equation (12) implies that ( ,1, )AV u t is priced under

another risk-neutral world where the risk-free rate equals q and the continuous

dividend yield equals r. In such a world, the hypothetic European option has the same

closed-form solution as the Black-Scholes model except that r and q are exchanged,

e.g.

1 2,1, , , , ( ) ( ),E r qC u q r ue N d e N d

where 2

1

ln( 1) ( 2)u q rd

and 2 1d d .

3. The hedge performance of static hedge versus dynamic hedge for an

American up-and-out put option

The hedge effectiveness of the SHP is of great concern in the literature, e.g. see

Toft and Xuan (1998), Fink (2003), and Engelmann, Fengler, Nalholm, and

changes in the current maximum. Since the value of max( , , )A

FLP S S t depends upon the probability

distribution of max ( )S T , this theorem implies that

1

( ,1, )0

A

u

V u t

u

.

11

Schwendner (2006). As a demonstration, we investigate this issue for an American

up-and-out put option under the Black-Scholes model. We consider two types of SHPs.

The first SHP is formed by exactly following the procedure described in Section 2.1.

This portfolio is termed “SHP with nonstandard strikes” because it is consisted with

European put options with strike prices (i.e. iB s) not specified by the option

exchange. The second portfolio utilizes European put options with standard strikes

and thus is called “SHP with standard strikes”. As emphasized in Section 2.1, the

European put option added into the second SHP at time it must has a standard strike

price less than 1iB in order not to affect the existing solution of the SHP at time 1it .

Therefore, we choose a standard strike price closest to but less than 1iB . Moreover,

since there are only two unknowns ( iw and ˆiw ), we only apply the knock-out

boundary and the value-matching condition on the early exercise boundary ( iB is

given from the first portfolio) to solve the SHP with standard strikes.7

For ease of understanding, we show detailed components of two SHPs ( 6n )

for a one-year American up-and-out put (AUOP) option in Table 1. The parameters

are adopted from AitSahlia, Imhof, and Lai (2003) as follows: 0 100S , 100X ,

110H , 0.04r , 0q , and 0.2 . According to CBOE’s equity option product

specifications, the standard strikes suitable for our example include {80, 85, 90, 95,

7 Although the smooth-pasting condition is not applied to solve the SHP with standard strikes, the delta

values of the portfolio on the early exercise boundary are very close to -1.

12

100}. The benchmark price of the AUOP is computed using the trinomial tree method

of Ritchken (1995) with 52,000 time steps.

It is clear from Table 1 that the values of both SHPs are very close to the

benchmark value of the AUOP. The replication errors of both SHPs are only about

$0.01, which is around 0.20% of the AUOP price ($4.890921). Moreover, the

quantities of European call options and put options used in both SHPs are also of

similar magnitude.

[Insert Table 1 Here]

We further depict the replication mismatches of both SHPs ( 6n ) on the

knock-out boundary and the early exercise boundary.8 Generally speaking, Figure 2

suggests that the mismatches are small and that’s why the values of both SHPs are

close to the AUOP price. Moreover, Panel A of Figure 2 indicates that the mismatches

of both portfolios on the barrier are of the same magnitude and the maximum error is

only around $0.2, which is smaller than the typical bid-ask spread of the AUOP price.

In contrast, the mismatches of both portfolios on the early exercise boundary are

slightly different. It is evident that the mismatches of SHP with nonstandard strikes on

the early exercise boundary are slightly greater than the mismatches of SHP with

standard strikes.

8 The accurate early exercise boundary of the AUOP is calculated using the trinomial tree method of

Ritchken (1995) with 52,000 time steps per year.

13

[Insert Figure 2 Here]

Furthermore, we compare the profit and loss distributions of two SHPs ( 6n )

and the dynamic hedge strategy in Figure 3. We simulate 50,000 paths of the stock

prices, with 25,200 time steps per year. If the knock-out boundary or the early

exercise boundary is breached, we liquidate the static hedge portfolios and compute

the profit and loss value, i.e. the difference between the American up-and-out put

option value and the liquidation value of the static hedge portfolio. Formally, the

hedging error (profit and loss value) of the static hedge portfolio is defined as:

( ) ( ) ( )nHE S AUOP S SHP S , where is the simulated first hitting time, S

is the underlying asset price at time , and ( )AUOP S and ( )nSHP S are the

AUOP and the SHP values at time , respectively. Note that if the boundary is not

touched before maturity in a certain path, the hedging error would be zero because the

terminal condition is perfectly matched in the SHPs.

For the dynamic hedge strategy, we rebalance the delta-hedged portfolio (DHP)

daily until the boundary is breached or the option is expired.9 Then the hedging error

of the DHP is defined similarly as: ( ) ( ) ( )HE S AUOP S DHP S , where

( )DHP S is the DHP value at time .

From Panel A of Figure 3, we find that the maximum dollar loss of the SHP with

9 The hedge ratio (i.e. delta) of the AUOP is calculated from the extended tree method, proposed by

Pelsser and Vorst (1994), with 25,200 time steps per year.

14

nonstandard strikes is about $0.22, which is about 4.50% of the AUOP price. The

mean and the standard deviation of the profit and loss distribution are 0.024 and 0.039,

respectively, for the SHP with nonstandard strikes. Similarly, Panel B of Figure 3

shows that the maximum (0.22), mean (0.025), and standard deviation (0.036) of the

profit and loss distribution for the SHP with standard strikes are almost identical to

those of the SHP with nonstandard strikes. Comparing Panels A and B with Panel C of

Figure 3, it is evident that the hedging effectiveness of both SHPs outperforms that of

the DHP in terms of the maximum loss and the standard deviation of the hedging

errors. For instance, the maximum loss of the DHP can be greater than $2.5, which is

around 50% of the AUOP price. The standard deviation of the DHP (0.255) is also

more than six times the standard deviation of both SHPs.

[Insert Figure 3 Here]

Finally, we adopt four risk measures, suggested by Siven and Poulsen (2009), to

evaluate the profit and loss distributions of the SHPs and the DHPs. The first risk

measure, 2[ ( ) ]E HE S , represents the quadratic hedging error. The second risk

measure, [ ( ) max(0, ( ))]E HE S HE S , is the expected loss, which only concerns

the losses of the hedge portfolios, and thus is a one-sided risk measure. The third risk

measure, the 5% value-at-risk, is defined as 0.05 inf{ ;VAR z

Pr( ( ) ) 0.05}HE S z . This risk measure is one of the most widely used risk

15

measures in practice probably due to the Basel accords for banking regulations. The

fourth risk measure, the expected shortfall (also known as conditional value-at-risk),

is the mean loss beyond the 5% value-at-risk: 0.05 0.05[ ( ) | ( ) ]ES E HE S HE S VAR .

There are several points worth noting from Table 2. First of all, no matter which

risk measure is used, the hedge errors of the SHPs with nonstandard strikes have

almost the same risk as the hedge errors of the SHPs with standard strikes. The result

is consistent with Figures 2 and 3 which also suggest that the hedging effectiveness of

both SHPs is similar because their boundary mismatches and profit and loss

distributions are alike. Secondly, it is obvious that both SHPs outperform the DHP in

each risk measure. For instance, from Panels B and C of Table 2, we observe that a

quarterly hedged ( 4n ) SHP with standard strikes has at most 34% of the risk of a

daily hedged ( 260n ) DHP. Thirdly, the risks generally decrease in an order smaller

than (1 )O n when the hedging frequency increases for all hedge portfolios. For

example, for the SHP with standard strikes, when n increases six times (from 4 to

24), the 5% value-at-risk 0.05VAR only becomes halved (from 0.128 to 0.056).

[Insert Table 2 Here]

4. Efficiency of the static hedge portfolio for pricing American exotic options

As emphasized in the Introduction, the SHP value provides a good estimate of

the American exotic option price when the boundary conditions are matched well. In

16

this section, we will evaluate the numerical efficiency (in terms of speed and accuracy)

of the SHP approach for pricing American barrier options and American lookback

options, respectively. In the former case, we consider pricing American up-an-out put

(AUOP) options under the constant elasticity of variance (CEV) model of Cox (1975).

We choose the CEV model to show that the proposed method can be applied to more

general stochastic processes beyond the Black-Scholes model. In the latter case, we

consider valuing American floating strike lookback put options under the

Black-Scholes model. We choose the lookback options to demonstrate that the

proposed method is applicable for other types of exotic options beyond barrier options

(which are the most studied options in the static hedge literature.)

4.1. Pricing American up-and-out put options under the CEV model

Pricing barrier options under the CEV model is still a difficult task in the

literature, even for the European-style options. Recently, Davydov and Linetsky (2001)

successfully derive closed-form solutions for the European barrier option prices under

the CEV model. However, to the best of our knowledge, no attempt has been done for

American barrier options. In other words, the valuation of American barrier options

still relies on numerical methods such as the lattice approach of Boyle and Tian

(1999). Thus the proposed method (the SHP value) provides an alternative tool for

pricing these options.

17

In the CEV model, the stock price satisfies the following risk-neutral process:

2( )dS r q Sdt S dz , (17)

where r is the risk-free rate, q is the dividend yield, is the volatility parameter,

and is a positive constant.10

Solving the SHP for an AUOP under the CEV model

follows the same procedure as described in Section 2.1 except that the European

options are now priced under the CEV model. Thus the price formulae and the delta

formulae in Equations (2)-(4) must be changed. For example, when 2 , the

European call and put prices under the CEV model are given by

0

0

( , , , , , , )

2 2(2 ;2 ,2 ) [1 (2 ;2 ,2 )],

2 2

E

qT rT

C S X r q T

S e Q y x Xe Q x y

(18)

0

0

, , , , , ,

2 ;2 (2 ),2 1 2 ;2 2 (2 ),2 ,

E

rT qT

P S X r q T

Xe Q x y S e Q y x

(19)

where

22

2

2 1r q T

r qk

e

,

22

0

r q Tx kS e

,

2y kX ,

and Q is the complementary noncentral chi-square distribution function.11

10

The instantaneous variance of the stock price is given by 2S dt and thus its elasticity of return

variance with respect to price is a constant ( 2 ). When 2 (i.e. the elasticity is zero), the CEV

model degenerates to the Black-Scholes model. When 2 , the CEV model generates a negatively

skewed probability distribution that is widely observed in the equity option market. 11

The complementary noncentral chi-square distribution function is defined on the noncentral

chi-square density function, e.g.

18

We first investigate the convergence pattern of the SHP values as n increases.

We set 2

0 00.2S S and take beta parameter ( 4 3 ) from Schroder (1989).12

The other parameters are as follows: 0 40S , 45X , 50H , 1T , 0.0488r ,

and 0q . The benchmark price of the AUOP, 5.358829, is obtained from the tree

method of Boyle and Tian (1999) with 52,000 time steps per year. It is apparent from

Figure 4 that the SHP values accurately converge to the true price of the AUOP under

the CEV model and the convergence seems monotonic.

[Insert Figure 4 Here]

Next, we investigate the accuracy of the early exercise boundary obtained when

we solve the SHP backwardly. The benchmark values of the early exercise boundary

are derived from the tree method of Boyle and Tian (1999) with 52,000 time steps per

year. Table 3 reports the early exercise boundary calculated from the SHP with n = 52.

It is evident from Table 3 that the proposed method also provides accurate estimates

for the early exercise boundary, e.g. the maximum error is just 0.27% and the average

error is less than 0.05%.

[Insert Table 3 Here]

2 ;2 (2 ),2 [2 ;2 2 (2 ),2 ]x

Q x y p w y dw

,

where [2 ;2 2 (2 ),2 ]p w y is the noncentral chi-square density function with 2 2 (2 )

degrees of freedom and a noncentral parameter 2y . 12

In other words, the instantaneous volatility corresponds to 0.2 in the Black-Scholes model.

19

Finally, we conduct a detailed efficiency analysis by comparing the accuracy

and the computational time of our SHP method with those of the tree method of Boyle

and Tian (1999). We price 48 AUOPs with the following parameter set: 4 / 3,

45X , 50H , 0.0488r , 0q , 0 {40,42.5,45,47.5},S {0.5,0.75,1.5} ,

and {0.25,0.5,0.75,1}T . The accuracy is measured by the root of the mean squared

error (RMSE) or the root of the mean squared relative error (RMSRE), where

482

1

1

48i

i

RMSE e

, 48

2

1

1ˆ

48i

i

RMSRE e

, *

i i ie P P is the absolute error,

*ˆ ( ) /i i i ie P P P is the relative error, iP is the benchmark value of the i-th AUOP,

and *

iP is the estimated option price using the tree method or the SHP approach. The

computational time is the CPU time, based on an AMD Phenom II X2 550 PC,

required to calculate 48 AUOP prices.

Table 4 indicates that the numerical efficiency of our SHP method is comparable

to the tree method of Boyle and Tian (1999). When n is small (large), the SHP method

is inferior (superior) to the tree method. For example, the RMSE, the RMSRE, and the

computational time of the SHP method with 6n are greater than the tree method

with 50 steps. In contrast, the SHP method with 24n has the same accuracy as the

tree method with 200 steps but requires only one third the computational time of the

tree method (0.5732 seconds versus 1.6901 seconds). In addition, it is well known in

the literature (e.g. see Figlewski and Gao (1999)) that the total number of nodes of a

binomial (or trinomial) tree is of order 2( )O n . Therefore, when the number of time

steps doubles, the computational time becomes four times in the tree method. On the

other hand, the computational time of the SHP method is only proportional to n .13

This is the reason why the SHP method is superior to the tree method when n is

13

This is consistent with the fact that the number of options utilized in the SHP is 2 1n .

20

large. Finally, applying the SHP method to price American exotic options has one

specific advantage that unlike the tree method, the recalculation of the American

exotic option prices in the future is very easy because there is no need to solve the

SHP again.14

[Insert Table 4 Here]

4.2. Pricing American floating strike lookback put options under the Black-Scholes

model

In Section 2.2, we show how to form the SHP for an American floating strike

lookback put option using the hypothetic European options. Although the obtained

SHP is not a realistic hedge portfolio, its value still provides a good price estimate for

the lookback put when the boundary conditions are matched well. In this section, we

will analyze the numerical efficiency of the SHP method for pricing American

floating strike lookback puts. The numerical results are based on the parameter setting

of Chang, Kang, Kim, and Kim (2007) and are reported in Figure 5 and Table 5.

Our main findings can be summarized as follows. Figure 5 and Table 5

suggest that the SHP values monotonically converge to the true price of the American

floating strike lookback put, with a rate of convergence of order (1 )O n . For instance,

from the first row of Panel A in Table 5, the pricing errors of the SHP method with 6,

12, 24, and 48 steps are 0.0028, 0.0014, 0.0007, and 0.0003, respectively, which

14

Besides, it is well known in the literature (e.g. see Zvan, Vetzal, and Forsyth (2000)) that when the

stock price is close to the barrier, the tree method requires very large number of time steps to get

accurate price estimates for the barrier options.

21

suggest that the error is halved when n is doubled. In contrast, the price estimates

obtained from the tree method of Babbs (2000) converge to the benchmark price in an

oscillatory way. For example, from the first row of Panel A in Table 5, the pricing

errors of the tree method with 50, 100, 200, and 400 steps are 0.0008, -0.0006, 0.0009,

and 0.0002, respectively, which are obviously not monotonic. Moreover, Table 5

indicates that when n is small (large), the SHP method is inferior (superior) to the tree

method. This finding is similar to the case of pricing AUOPs under the CEV model.

Thus when accuracy is important (i.e. a large n is required), one should apply the

SHP method to price American exotic options. Finally, similar to the finding in

Section 4.1, we also observed that the computational time of the SHP method is only

proportional to n , but the computational time of the tree method increases in a

greater order (between 1.5( )O n and 2( )O n ).

[Insert Figure 5 Here]

[Insert Table 5 Here]

5. Conclusions

This paper extends the static hedge portfolio approach of Derman, Ergener, and

Kani (1995) and Carr, Ellis, and Gupta (1998) to price and/or hedge American exotic

options. The main contributions and results of this paper are as follows. First of all,

we successfully construct static hedge portfolios to match the terminal and boundary

22

conditions of American barrier options and lookback options. Secondly, we compare

the profit and loss distributions of the static hedge portfolios (SHPs) and the

delta-hedged portfolios (DHPs) for an American up-an-out put option. The results

indicate that the SHPs are far less risky than the DHPs no matter which risk measure

is used. Thirdly, we conduct detailed efficiency analyses and show that the proposed

method is as efficient as the numerical methods for pricing American barrier options

under the CEV model and American lookback options under the Black-Scholes

model.

The proposed method is attractive to price or hedge American exotic options in

several aspects. For instance, most option pricing methods are developed only for a

specific type of option and/or model. In contrast, the proposed method is applicable

for more general models or option types. Moreover, the hedge ratios, such as delta and

theta, can be easily computed at the same time when the static hedge portfolio is

found. Finally, the recalculation of the prices and hedge ratios for the American exotic

options is also easy and quick when the stock price and/or time to maturity are

changed.

23

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Andersen, L., Andreasen, J. and Eliezer, D., 2002. Static replication of barrier options:

Some general results, Journal of Computational Finance 5, 1-25.

Babbs, S., 2000. Binomial valuation of lookback options. Journal of Economic

Dynamics and Control 24, 1499-1525.

Boyle, P. P., and Lau, S. H., 1994. Bumping up against the barrier with the binomial

method. Journal of Derivatives 1, 6-14.

Boyle, P., and Tian, Y. S., 1999. Pricing lookback and barrier options under the CEV

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Carr, P., Ellis, K., and Gupta, V., 1998. Static hedging of exotic options. Journal of

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Chang G. H., Kang, J. K., Kim H. S., and Kim, I. J., 2007. An efficient approximation

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Davydov, D. and Linetsky, V., 2001. Pricing and hedging path-dependent options

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Derman, E., Ergener, D., and Kani, I., 1995. Static options replication. Journal of

Derivatives 2, 78-95.

24

Engelmann, B., Fengler, M. R., Nalholm, M., and Schwendner, P., 2006. Static versus

dynamic hedges: An empirical comparison for barrier options. Review of

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Figlewski, S. and Gao, B., 1999. The adaptive mesh model: A new approach to

efficient option pricing. Journal of Financial Economics 53, 313-351.

Fink, J., 2003. An examination of the effectiveness of static hedging in the presence

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Gao, B., Huang, J. Z., and Subrahmanyam, M., 2000. The valuation of American

barrier options using the decomposition technique. Journal of Economic

Dynamics and Control 24, 1783-1827.

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at the low, sell at the high". Journal of Finance 34, 1111-1127.

Ilhan, A., and Sircar, R., 2006. Optimal static-dynamic hedges for barrier options.

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Liu, Q., 2010. Optimal approximations of nonlinear payoffs in static replication.

Journal of Futures Markets, forthcoming.

Nalholm, M., and Poulsen, R., 2006a. Static hedging of barrier options under general

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Pelsser, A., and Vorst, T., 1994. The binomial model and the greeks. Journal of

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options. Journal of Economic Dynamics and Control 24, 1563-1590.

25

Figure 1. The Static Hedge Portfolio for an American Up-and-Out Put Option

This figure shows the backward procedure of formulating the static hedge portfolio (SHP) for an

American up-and-out put option. The SHP matches the knock-out boundary and the early exercise

boundary at n equally-spaced time points.

t0=0

X

Time tn-1 tn-2 T=tn

Early Exercise Boundary

0 1 0 1

0 0 0 1 1 0

0 0 1 0

( , , ) ( , , )

ˆ( , , ) ( , , )

ˆ ( , , )

E E

n n

E E

n

E

P B X T w P B B T

w P B B t w C B H T

w C B H t X B

▲ (tn-1, Bn-1)

▲ (tn-2, Bn-2) (t0, B0)▲

H ▲ (tn-1, H) ▲ (tn-2, H) ▲ (t0, H)

0 1 0 1

0 0 0 1 1 0

0 0 11

( , , ) ( , , )

ˆ( , , ) ( , , )

ˆ ( , , )

E E

P n P n

E E

P n C

E

C

B X T w B B T

w B B t w B H T

w B H t

1 1

1 1 1 1

1 1 1 1

( , , )

( , , )

ˆ ( , , )

E

n n

E

n n n n

E

n n n n

P B X T t

w P B B T t

w C B H T t X B

1 1

1 1 1 1

1 1 1

( , , )

( , , )

ˆ ( , , ) 1

E

P n n

E

n P n n n

E

n C n n

B X T t

w B B T t

w B H T t

1

1 1 1

1 1

( , , )

( , , )

ˆ ( , , ) 0

E

n

E

n n n

E

n n

P H X T t

w P H B T t

w C H H T t

1 1

0 0 1 1

0 1

( , , ) ( , , )

ˆ( , , ) ( , , )

ˆ ( , , ) 0

E E

n n

E E

n

E

P H X T w P H B T

w P H B t w C H H T

w C H H t

Barrier

Stock Price

26

Figure 2. The Mismatch Values on the Boundary for the Static Hedge Portfolios

Panel A. Mismatch Values on the Barrier

Panel B. Mismatch Values on the Early Exercise Boundary

This figure shows mismatch values of two SHPs (n = 6) on the knock-out boundary and the early

exercise boundary for an American up-and-out put (AUOP). The time to maturity of the AUOP is one

year and the other parameters are adopted from AitSahlia, Imhof, and Lai (2003) as follows: 0 100S ,

100X , 110H , 0.04r , 0q , and 0.2 . The accurate early exercise boundary of the

AUOP is calculated using the trinomial tree method of Ritchken (1995) with 52,000 time steps per

year.

27

Figure 3. The Profit and Loss Distributions

Panel A. SHP with Nonstandard Strikes

Panel B. SHP with Standard Strikes

28

Figure 3. continued

Panel C. Delta-Hedged Portfolio (DHP)

This figure shows the profit and loss distributions of two SHPs (n=6) and the DHP (n=260) for an

AUOP. The time to maturity of the AUOP is one year and the other parameters are adopted from

AitSahlia, Imhof, and Lai (2003) as follows: 0 100S , 100X , 110H , 0.04r , 0q , and

0.2 . We simulate 50,000 paths of the stock prices, with 25,200 time steps per year, to compute the

profit and loss distributions of the hedge portfolios. If the knock-out boundary or the early exercise

boundary is breached, we liquidate the hedge portfolio and compute the profit and loss value, i.e. the

difference between the AUOP value and the liquidation value of the hedge portfolio. Otherwise, we

compute the profit and loss value as the difference between the terminal payoff of the AUOP and the

terminal value of the hedge portfolio. The accurate early exercise boundary of the AUOP is calculated

using the trinomial tree method of Ritchken (1995) with 52,000 time steps per year.

29

Figure 4. The Convergence of the SHP Values to the AUOP Price under the CEV

model

This figure shows the convergence pattern of the SHP values to the accurate price of an AUOP under

the CEV model. We set /2

0 00.2S S and take beta parameter ( 4 3 ) from Schroder (1989). The

other parameters are as follows: 0 40S , 45X , 50H , 1T , 0.0488r , and 0q . The

benchmark price, 5.358829, is obtained from the tree method of Boyle and Tian (1999) with 52,000

time steps per year.

30

Figure 5. The Convergence of the SHP Values to the American Floating Strike

Lookback Put Price

This figure shows the convergence pattern of the SHP values to the accurate price of an American

floating strike lookback put under the Black-Scholes model. The parameters are adopted from Chang,

Kang, Kim, and Kim (2007) as follows: 0 50S , 0 max 0(0) / 1.02u S S , 0.05r , 0.05q ,

0.5T , and 0.2 . The benchmark price, 5.835480, is obtained from the tree method of Babbs

(2000) with 52,000 time steps per year.

31

Table 1. The Static Hedge Portfolio (SHP) for an American Up-and-Out Put

Panel A. SHP with Nonstandard Strikes

Quantity of European Call

Strike Quantity of

European Put Strike

Expiration (months)

Value for

0S = 100

-0.004415 110 0.046494 80.238013 2 -0.002381

-0.011848 110 0.047374 80.810245 4 -0.015407

-0.026374 110 0.048418 81.621783 6 -0.057712

-0.057201 110 0.052649 82.799137 8 -0.182052

-0.125989 110 0.007167 84.558963 10 -0.588633

-0.109803 110 0.166623 88.061471 12 -0.276521

1 100 12 6.003998

Net 4.881292

Panel B. SHP with Standard Strikes

Quantity of European Call Strike

Quantity of European Put Strike

Expiration (months)

Value for

0S = 100

-0.004334 110 0.041485 80 2 -0.002391

-0.011677 110 0.043703 80 4 -0.016377

-0.026126 110 0.042927 80 6 -0.062652

-0.057018 110 0.101540 80 8 -0.175055

-0.126408 110 0.044267 85 10 -0.545784

-0.109983 110 0.118888 90 12 -0.321408

1 100 12 6.003998

Net 4.880331

This table shows detailed components of two static hedge portfolios (n = 6) for an American up-and-out

put (AUOP). The time to maturity of the AUOP is one year and the other parameters are adopted from

AitSahlia, Imhof, and Lai (2003) as follows: 0 100S , 100X , 110H , 0.04r , 0q , and

0.2 . The benchmark value, 4.890921, is computed from the trinomial tree method of Ritchken

(1995) with 52,000 time steps per year. The standard strike prices chosen in Panel B are in accordance

to CBOE’s equity option product specifications.

32

Table 2. Hedge Performance of the SHPs and the DHPs

Risk Measures Panel A. SHP with Nonstandard Strikes

n = 4 n = 6 n = 12 n = 24

2[ ( ) ]E HE S 0.004745 0.002085 0.000841 0.000621

[ ( ) ]E HE S 0.055474 0.039733 0.027538 0.023364

0.05VAR 0.128371 0.091175 0.062225 0.054642

0.05ES 0.215289 0.133927 0.076160 0.070313

Risk Measures

Panel B. SHP with Standard Strikes

n = 4 n = 6 n = 12 n = 24

2[ ( ) ]E HE S 0.004467 0.001921 0.000775 0.000645

[ ( ) ]E HE S 0.052529 0.036423 0.024861 0.020837

0.05VAR 0.128446 0.090975 0.061704 0.055752

0.05ES 0.216490 0.134142 0.075698 0.071489

Risk Measures Panel C. DHP

n = 130 n = 260 n = 520 n = 1040

2[ ( ) ]E HE S 0.136446 0.066342 0.032267 0.015968

[ ( ) ]E HE S 0.251446 0.171417 0.119700 0.085317

0.05VAR 0.615136 0.418149 0.285075 0.200853

0.05ES 0.922256 0.635547 0.439287 0.307647

This table shows the risks of the profit and loss distributions for the static hedge portfolios (SHPs) and

the delta-hedged portfolios (DHPs) under the Black-Scholes model. The time to maturity of the

American up-and-out put (AUOP) is one year and the other parameters are adopted from AitSahlia,

Imhof, and Lai (2003) as follows: 0 100S , 100X , 110H , 0.04r , 0q , and 0.2 . We

simulate 50,000 stock price paths, with 52,000 time steps per year, to compute the profit and loss

distributions of the hedge portfolios. We adopt four risk measures used by Siven and Poulsen (2009) to

evaluate the profit and loss distributions. The first risk measure 2[ ( ) ]E HE S represents the quadratic

hedging errors, where ( )HE S is the hedge error, defined as the difference between the AUOP value

and the hedge portfolio value when the boundary condition or terminal condition is breached at time .

The second risk measure is the expected loss: [ ( ) max(0, ( ))]E HE S HE S . The third risk measure,

5% value-at-risk, is defined as 0.05 inf{ ; Pr( ( ) ) 0.05}VAR z HE S z . The fourth measure, the

expected shortfall, is the mean loss beyond 5% value-at-risk: 0.05 0.05[ ( ) | ( ) ]ES E HE S HE S VAR .

33

Table 3. The Early Exercise Boundary of the AUOP under the CEV Model

T-t Benchmark Value SHP Difference (%)

0 45.0000 45.0000 0.0000%

1/52 42.6295 42.7451 0.2712%

1/26 41.9174 41.8710 -0.1107%

3/52 41.4324 41.4148 -0.0424%

1/13 41.0384 41.0268 -0.0284%

5/52 40.7337 40.7064 -0.0670%

3/26 40.4305 40.4312 0.0018%

7/52 40.2149 40.1893 -0.0635%

2/13 40.0000 39.9730 -0.0676%

9/52 39.7859 39.7770 -0.0222%

5/26 39.6152 39.5980 -0.0434%

11/52 39.4449 39.4330 -0.0301%

3/13 39.3175 39.2802 -0.0949%

1/4 39.1481 39.1380 -0.0260%

7/26 39.0214 39.0049 -0.0422%

15/52 38.8950 38.8803 -0.0378%

4/13 38.7688 38.7630 -0.0149%

17/52 38.6848 38.6525 -0.0836%

. . . .

. . . .

. . . .

1/2 37.8932 37.8836 -0.0253%

3/4 37.2350 37.2154 -0.0527%

1 36.8275 36.8276 0.0002%

This table shows the accuracy of the early exercise boundary obtained when we solve the SHP

backwardly for an American up-and-out put (AUOP) under the CEV model. The number of nodes

matched on the early exercise boundary ( n ) in the SHP method is 52 per year. We set /2

0 00.2S S

and take beta parameter ( 4 3 ) from Schroder (1989). The other parameters are as follows:

0 40S , 45X , 50H , 1T , 0.0488r , and 0q . The benchmark value is obtained

from the tree method of Boyle and Tian (1999) with 52,000 time steps per year. The final column

shows the relative difference between the benchmark values and the estimated values from the SHP

method.

34

Table 4. The Valuation of American Up-and-Out Put Options under the CEV Model

0S T Tree SHP

Benchmark n = 50 n = 100 n = 200 n = 6 n = 12 n = 24

Panel A. 0.5

40 0.25 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

40 0.5 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

40 0.75 5.0000 5.0000 5.0000 5.0033 5.0026 5.0022 5.0018

40 1 5.0000 5.0089 5.0126 5.0170 5.0152 5.0141 5.0130

42.5 0.25 2.6265 2.6265 2.6266 2.6270 2.6267 2.6255 2.6265

42.5 0.5 2.8088 2.8074 2.8072 2.8078 2.8074 2.8051 2.8070

42.5 0.75 2.9345 2.9339 2.9339 2.9350 2.9342 2.9338 2.9335

42.5 1 3.0190 3.0186 3.0173 3.0203 3.0187 3.0179 3.0174

45 0.25 1.0475 1.0472 1.0476 1.0467 1.0473 1.0470 1.0476

45 0.5 1.3560 1.3569 1.3565 1.3548 1.3561 1.3555 1.3566

45 0.75 1.5276 1.5272 1.5273 1.5254 1.5267 1.5271 1.5272

45 1 1.6253 1.6274 1.6279 1.6267 1.6275 1.6277 1.6276

47.5 0.25 0.2923 0.2926 0.2926 0.2908 0.2917 0.2919 0.2922

47.5 0.5 0.5036 0.5031 0.5032 0.5000 0.5018 0.5021 0.5028

47.5 0.75 0.6107 0.6137 0.6129 0.6099 0.6117 0.6122 0.6125

47.5 1 0.6739 0.6745 0.6742 0.6724 0.6736 0.6739 0.6741

Panel B. 0.75

40 0.25 5.0776 5.0774 5.0779 5.0783 5.0781 5.0780 5.0779

40 0.5 5.2831 5.2820 5.2820 5.2812 5.2818 5.2819 5.2819

40 0.75 5.4273 5.4274 5.4270 5.4261 5.4269 5.4270 5.4269

40 1 5.5144 5.5182 5.5172 5.5181 5.5179 5.5176 5.5172

42.5 0.25 3.0869 3.0876 3.0864 3.0844 3.0854 3.0857 3.0858

42.5 0.5 3.4681 3.4689 3.4690 3.4638 3.4665 3.4674 3.4678

42.5 0.75 3.6714 3.6710 3.6701 3.6650 3.6680 3.6689 3.6693

42.5 1 3.7805 3.7804 3.7812 3.7783 3.7804 3.7808 3.7809

45 0.25 1.6325 1.6316 1.6319 1.6284 1.6305 1.6313 1.6317

45 0.5 2.0379 2.0383 2.0375 2.0316 2.0354 2.0367 2.0374

45 0.75 2.2213 2.2240 2.2234 2.2175 2.2212 2.2224 2.2229

45 1 2.3205 2.3195 2.3198 2.3160 2.3186 2.3193 2.3196

47.5 0.25 0.6665 0.6656 0.6655 0.6603 0.6632 0.6643 0.6648

47.5 0.5 0.9173 0.9172 0.9166 0.9107 0.9143 0.9156 0.9162

47.5 0.75 1.0258 1.0227 1.0232 1.0189 1.0218 1.0227 1.0231

47.5 1 1.0726 1.0773 1.0774 1.0747 1.0764 1.0769 1.0770

35

Table 4. continued

0S T Tree SHP

Benchmark n = 50 n = 100 n = 200 n = 6 n = 12 n = 24

Panel C. 1.5

40 0.25 5.3993 5.3980 5.3976 5.3948 5.3963 5.3968 5.3970

40 0.5 5.7775 5.7788 5.7778 5.7710 5.7750 5.7763 5.7769

40 0.75 5.9784 5.9771 5.9770 5.9701 5.9743 5.9755 5.9761

40 1 6.0873 6.0832 6.0835 6.0799 6.0824 6.0831 6.0833

42.5 0.25 3.5866 3.5837 3.5819 3.5760 3.5790 3.5800 3.5805

42.5 0.5 4.0458 4.0434 4.0418 4.0319 4.0374 4.0392 4.0401

42.5 0.75 4.2508 4.2473 4.2474 4.2388 4.2440 4.2456 4.2463

42.5 1 4.3484 4.3522 4.3517 4.3461 4.3495 4.3505 4.3509

45 0.25 2.1226 2.1237 2.1239 2.1168 2.1211 2.1226 2.1234

45 0.5 2.5311 2.5299 2.5303 2.5213 2.5272 2.5290 2.5300

45 0.75 2.6999 2.6975 2.6962 2.6888 2.6937 2.6952 2.6959

45 1 2.7755 2.7770 2.7768 2.7728 2.7758 2.7766 2.7769

47.5 0.25 0.9680 0.9682 0.9677 0.9602 0.9648 0.9664 0.9673

47.5 0.5 1.2059 1.2014 1.2014 1.1943 1.1987 1.2003 1.2010

47.5 0.75 1.2901 1.2926 1.2913 1.2869 1.2901 1.2911 1.2916

47.5 1 1.3345 1.3359 1.3349 1.3326 1.3343 1.3348 1.3349

RMSE 0.0029 0.0012 0.0006 0.0041 0.0014 0.0006 N/A

RMSRE 0.0013 0.0006 0.0004 0.0027 0.0009 0.0004 N/A

Time 0.1188 0.4524 1.6901 0.1619 0.2967 0.5732 N/A

This table shows the accuracy and the computational time of the SHP method and the tree method of

Boyle and Tian (1999) for pricing 48 AUOPs under the CEV model with the following parameter set:

4/ 3 , 45X , 50H , 0.0488r , 0q , 0 {40,42.5,45,47.5}S , {0.5,0.75,1.5} , and

{0.25,0.5,0.75,1}T . The accuracy is measured by the root of the mean squared error (RMSE) or the

root of the mean squared relative error (RMSRE), where 48

2

1

1

48i

i

RMSE e

, 48

2

1

1ˆ

48i

i

RMSRE e

,

*i i ie P P , *ˆ ( ) /i i i ie P P P , iP

is the benchmark value of the i-th AUOP, and *

iP

is the

estimated option price using the tree method or the SHP approach. The benchmark value is obtained

from the tree method of Boyle and Tian (1999) with 52,000 time steps per year. The computational

time is the CPU time (in seconds), based on an AMD Phenom II X2 550 PC, required to calculate 48

AUOP prices.

Table 5. The Valuation of American Floating Strike Lookback Put Options under the

Black-Scholes Model

σ T Tree SHP

Benchmark

n = 50 n = 100 n = 200 n = 400 n = 6 n = 12 n = 24 n = 48

Panel A. r = 0.025; q = 0.05

0.1 0.1 1.5963 1.5949 1.5964 1.5957 1.5983 1.5969 1.5962 1.5958 1.5955

0.1 0.3 2.5617 2.5580 2.5590 2.5580 2.5687 2.5631 2.5602 2.5587 2.5571

0.1 0.5 3.2757 3.2924 3.2852 3.2883 3.3091 3.2990 3.2937 3.2909 3.2880

0.2 0.1 2.7569 2.7658 2.7612 2.7601 2.7667 2.7637 2.7620 2.7612 2.7603

0.2 0.3 4.7600 4.7393 4.7457 4.7432 4.7647 4.7543 4.7487 4.7457 4.7426

0.2 0.5 6.1927 6.1760 6.1607 6.1700 6.2055 6.1875 6.1776 6.1725 6.1670

0.4 0.1 5.3873 5.3657 5.3595 5.3643 5.3791 5.3706 5.3660 5.3636 5.3611

0.4 0.3 9.5171 9.5070 9.4927 9.4810 9.5422 9.5145 9.4993 9.4882 9.4827

0.4 0.5 12.4666 12.4629 12.4545 12.4426 12.5415 12.4931 12.4665 12.4426 12.4373

Panel B. r = 0.05; q = 0.05

0.1 0.1 1.5152 1.5136 1.5152 1.5144 1.5145 1.5143 1.5142 1.5142 1.5141

0.1 0.3 2.3474 2.3425 2.3437 2.3423 2.3428 2.3421 2.3416 2.3414 2.3411

0.1 0.5 2.9283 2.9460 2.9368 2.9399 2.9427 2.9410 2.9402 2.9397 2.9393

0.2 0.1 2.6875 2.6967 2.6916 2.6904 2.6933 2.6920 2.6913 2.6910 2.6906

0.2 0.3 4.5622 4.5391 4.5453 4.5425 4.5514 4.5469 4.5444 4.5431 4.5417

0.2 0.5 5.8691 5.8465 5.8300 5.8392 5.8523 5.8447 5.8401 5.8380 5.8355

0.4 0.1 5.3218 5.2990 5.2926 5.2973 5.3075 5.3012 5.2977 5.2958 5.2939

0.4 0.3 9.3193 9.3078 9.2920 9.2796 9.3257 9.3050 9.2936 9.2853 9.2808

0.4 0.5 12.1358 12.1296 12.1202 12.1065 12.1801 12.1427 12.1220 12.1063 12.1004

Panel C. r = 0.05; q = 0.025

0.1 0.1 1.4629 1.4607 1.4619 1.4609 1.4579 1.4592 1.4595 1.4595 1.4604

0.1 0.3 2.2198 2.2136 2.2144 2.2124 2.2023 2.2054 2.2072 2.2079 2.2108

0.1 0.5 2.7405 2.7522 2.7427 2.7443 2.7260 2.7347 2.7379 2.7390 2.7432

0.2 0.1 2.6410 2.6492 2.6435 2.6421 2.6408 2.6416 2.6417 2.6418 2.6420

0.2 0.3 4.4431 4.4181 4.4230 4.4197 4.4128 4.4164 4.4177 4.4182 4.4183

0.2 0.5 5.6912 5.6626 5.6458 5.6532 5.6404 5.6456 5.6461 5.6469 5.6485

0.4 0.1 5.2801 5.2557 5.2487 5.2531 5.2577 5.2540 5.2519 5.2507 5.2493

0.4 0.3 9.2148 9.2014 9.1835 9.1703 9.2007 9.1832 9.1775 9.1749 9.1707

0.4 0.5 11.9848 11.9754 11.9647 11.9489 11.9947 11.9691 11.9551 11.9492 11.9417

37

Table 5. continued

σ T Tree SHP

Benchmark

n = 50 n = 100 n = 200 n = 400 n = 6 n = 12 n = 24 n = 48

Panel D. r = 0.05; q = 0

0.1 0.1 1.4185 1.4162 1.4173 1.4162 1.4102 1.4124 1.4145 1.4155 1.4157

0.1 0.3 2.1156 2.1093 2.1101 2.1079 2.0851 2.0956 2.1016 2.1051 2.1062

0.1 0.5 2.5924 2.5998 2.5908 2.5914 2.5557 2.5711 2.5813 2.5872 2.5902

0.2 0.1 2.5984 2.6061 2.6000 2.5985 2.5933 2.5959 2.5968 2.5976 2.5982

0.2 0.3 4.3375 4.3116 4.3156 4.3120 4.2913 4.3016 4.3069 4.3085 4.3103

0.2 0.5 5.5365 5.5036 5.4873 5.4932 5.4628 5.4732 5.4794 5.4844 5.4881

0.4 0.1 5.2404 5.2147 5.2075 5.2117 5.2104 5.2083 5.2079 5.2077 5.2076

0.4 0.3 9.1174 9.1029 9.0834 9.0698 9.0848 9.0710 9.0701 9.0698 9.0695

0.4 0.5 11.8458 11.8347 11.8232 11.8057 11.8263 11.8068 11.8028 11.8005 11.7977

RMSE 0.0264 0.0155 0.0086 0.0030 0.0308 0.0162 0.0083 0.0030 N/A

RMSRE 0.0042 0.0021 0.0010 0.0005 0.0047 0.0025 0.0011 0.0006 N/A

Time 0.0443 0.0979 0.2508 0.7391 0.0850 0.1558 0.3009 0.6014 N/A

This table shows the accuracy and the computational time of the SHP method and the tree method of

Babbs (2000) for pricing American floating strike lookback put options under the Black-Scholes model.

The parameters are adopted from Chang, Kang, Kim, and Kim (2007) as follows: 0 50S ,

0 max 0(0) / 1.02u S S . The accuracy is measured by the root of the mean squared error (RMSE) or

the root of the mean squared relative error (RMSRE), where 36

2

1

1

36i

i

RMSE e

,

362

1

1ˆ

36i

i

RMSRE e

, *i i ie P P , *ˆ ( ) /i i i ie P P P , iP

is the benchmark value of the i-th

American floating strike lookback put, and *iP

is the estimated option price using the tree method or

the SHP approach. The benchmark value is obtained from the tree method of Babbs (2000) with 52,000

time steps per year. The computational time is the CPU time (in seconds), based on an AMD Phenom II

X2 550 PC, required to calculate 36 lookback put prices.