levy models

Excess rate of return from jumprisks: geometric Levy models for

asset pricing

Mohamed Raagi

(Student Number 1116027)

Supervisor Dr. David Meier

MSc in Financial Mathematics Project

Brunel University London

October 15, 2015

Mohamed Raagi Student ID:1116027

Acknowledgements

First and foremost I would like to acknowledge that any success is due to Allah

the Creator of the worlds and I thank Him for showering me with His Mercy and

keeping me steadfast throughout my studies.

I would also like to express my greatest gratitude to my supervisor David Meir for

being there throughout my project, and assisting me whatever the time or day.

Last but not least I would like to thank my family and friends who supported me

throughout my university journey. For sticking with me through thick and thin

and always being there for me.

i


Abstract

When asset price can jump, the excess rate of return above the short

rate, which determines the investor-compensation for accommodating

jump risks, is no longer a linear function of the risk (volatility) or the

risk aversion. The form of the excess rate of return as a function of

these factors in the general context has been obtained recently.

The aim of the project is to review these recent developments, and to

simulate price processes entailing jumps so that the behaviour and

the impact of the excess rate of return can be analysed.

ii


Contents

Contents

1 Introduction 1

2 Literature Review 4

2.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Models 6

3.1 Brownian Motion Model . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Compound Poisson Model . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Geometric Gamma Model . . . . . . . . . . . . . . . . . . . . . . 15

4 Option Pricing 17

4.1 Option Pricing: Brownian Motion . . . . . . . . . . . . . . . . . . 18

4.2 Option Pricing: Poisson . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Option Pricing: Compound Poisson . . . . . . . . . . . . . . . . . 23

4.4 Option Pricing: Gamma . . . . . . . . . . . . . . . . . . . . . . . 27

5 Simulation 30

6 Conclusion 36

6.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Appendices 41

iii


List of Figures

1 Brownian motion Monte-carlo . . . . . . . . . . . . . . . . . . . . 30

2 Brownian Motion Implementation . . . . . . . . . . . . . . . . . . 32

3 Brownian Motion Implementation graph . . . . . . . . . . . . . . 32

4 Poisson Monte-carlo . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Poisson Implementation . . . . . . . . . . . . . . . . . . . . . . . 35

6 Poisson Implementation graph . . . . . . . . . . . . . . . . . . . . 35

iv


1 Introduction

In 1900 a PHD student discussed in his thesis how a Brownian motion can be

used within modern finance (Bachelier, Davis, & Etheridge, 2006). He specifically

looked at how this could be used to evaluate price stock options. He came with

the generalised formula for the price process of the asset to be,

St = S0 + σWt (1)

where σ is the volatility of the asset and Wtt≥0 represents the Brownian motion.

At a later point in time, using the foundation of the PHD student, Fisher Black

and Myron Scholes worked to improve the price process (1). The issue with the

process above was that it produced negative prices for the stock options. As a

result both Black and Scholes worked together in order to eradicate the problem

of negative values. They came up with the assumption that the underlying asset

price is a stochastic process Stt≥0. with;

dStSt

= µdt+ σdWt (2)

where µ represents the drift and Wt is the Brownian motion. Consequently, the

solution of the stochastic differential equation, (2), is given by

St = S0e(µ− 1

2σ2)t+σWt . (3)

Many financial theorists have come with plausible ways in pricing assets and

options; one of the more general methods was the geometric Levy model (GLM).

This model is one of many ways used to derive the Black-Scholes formula however,

we will be concentrating on the use of the pricing kernel approach, which simplifies

the whole derivation. The use of the pricing kernel ensures that we are working

in the physical measure unlike that of Black and Scholes who worked in the risk

neutral measure shown by the stochastic differential equation. Furthermore, the

use of the notion that prices of assets and stocks is viewed as a continuous function

1


of time has led onto the powerful belief of market completeness and unique pricing

of contingent claims by arbitrage. However, the validity of this was challenged as

an appropriate model of the stock returns, thus suggesting the approach of pure

jump models as a discontinuous model.

The work of (Brody, Hughston, & Mackie, 2012) looks at the general theory

of Levy models for dynamic asset pricing and it states that the GLM has four

parameters these are;

• Initial Price,

• Interest Rate,

• Volatility and

• Risk aversion

the parameters above are for one dimension Levy process once it has been speci-

fied.

For the following section we refer to the notes given by (Brody Hughston, 2014).

In recent times there has been an influx on the purchase of hybrid products. As a

consequence, new mathematical models have been requested to deal with pricing,

hedging and risk management of these products. As a result requirements are

needed for these models to meet.

1. Do not assume market is complete or derivatives are hedgeable

2. We require a modeling framework that is applicable to

(a) Pricing and hedging of hybrid derivatives

(b) Risk management and asset allocation

3. Modeling framework that allows for general calibration method

The most effective way in which we can meet these requirements for such models

is the use of a pricing kernel. In this method there is no opportunity for arbitrage

2


to occur, this is due to the existence of the pricing kernel. In order to define a

pricing kernel it must satisfy the following;

Let πt denote a pricing kernel such that πt > 0 for all t ≥ 0, exits if it satisfies,

Axiom 1. There exists an absolutely continuous, strictly increasing ”risk-free”

asset Bt (the money-market account).

Axiom 2. If St is the price process of any asset and Dt is the associated contin-

uous dividend rate, then the process Mt defined by

Mt = πtSt +

∫ t

0

πsDsds (4)

is a martingale. Thus E[|Mt|] <∞ for u > t > 0, and Et[|Mu|] = Mt.

Axiom 3. There exists an asset (a floating rate note) that offers a continuous

dividend rate sufficient to ensure that the value of the asset remains constant.

Axiom 4. A discount bond system PtT exists for 0 < t < T < ∞ satisfying

limT→∞

PtT = 0.

3


2 Literature Review

In this section we will take the works of previous learned scholars and critically

analyse and scrutinise their working. We will be looking at a range of authors who

have spoken about this topic who include (Applebaum, 2004; Schoutens, 2004),

we will also take the work of (Brody, Hughston, & Mackie, 2012).

2.1 Divisibility

In (Schoutens, 2004), he approached the issue of asset pricing from a more sta-

tistical standing point. He heavily criticised the method devised by Black and

Scholes by arguing that there is a unique flaw within their model as the log re-

turns are heavily skewed therefore returns are not normally distributed. He then

follows by saying that in order to price an option we require a distribution that

is more general and that has independent and stationary increments, as an ex-

ample a Brownian motion. To follow on the point of skewness and kurtosis, the

model/distribution chosen needs to be able to represent these properties by being

infinity divisible.

As a result Schoutens stops here and mathematically defines what infinitely di-

visible is. If the characteristic function φ(u) is the nth power of a characteristic

function then, we say that the distribution is infinitely divisible. So, we need

to figure out a distribution which satisfies such a definition and he came to the

conclusion of a Levy process. He then defines what a Levy process is just the as

we did in the above section. However, unlike our definition of a Levy process, he

then continues to define the process as a cadalag function.

Definition 1. For all t ∈ (a, b), the function f is right continuous and has a

left limit. If f is cadalag then we denote the left limit by f(t−) = lims→t

f(s), and

f(t−) = f(t) if and only if f is continuous at t denoting the jump at

∇f(t) = f(t)− f(t−).

4


In (Applebaum, 2004) he provides a more extensive definition of infinite divisibil-

ity, as follows;

Definition 2. Let X be a random variable taking values in Rd with law µX . We

say that X is infinitely divisible if, for all n ∈ N, there exist an independent and

identical distributed (i.i.d.) random variables Y(n)1 · · · Y (n)

n such that

X = Y(n)1 + · · ·+ Y (n)

n .

After the explanation of divisibility, Proposition 1.2.6 says that;

• X infinitely divisible;

• φX has an nth root that is itself the characteristic function of a random

variable, for each n ∈ N.

As we can see this point here is exactly the same as that of both Schoutens and

Applebaum emphasising the importance of infinite divisibility because this leads

onto the next important factor in Levy process.

In (Brody, Hughston, & Mackie, 2012), they do not touch on the infinite divis-

ibility within their book. This maybe due to them emphasising on the models

for Levy process than the actual definition. Interestingly, in order to fully under-

stand the concepts of the GLM we need to understand its definition and properties

held by a Levy process despite the fact that within their book they are heavily

interested with the outcome and risk of using a Levy process.

Schoutens countinue to talk about the characteristic function by defining what is

known as the cumulant characteristic function ψ(u) = log φ(u). This function is

sometimes called the characteristic exponent, which satisfies the following Levy-

Khintchine formula,

ψ(u) = iγu− 1

2σ2u2 +

∫ +∞

−∞(exp(iux)− 1− iux1|x|<1)ν(dx). (5)

5


3 Models

Before we explore the different types of Levy models let us first outline the defi-

nition of a Levy process.

At our first step we will construct a GLM in the general case forming a family

of GLM. Let us define the probability space for the GLM (we consider the one

dimensional case).

Definition 3. We remark Levy process on the probability space (Ω,F ,P) is a

process Xt such that,

• X0 = 0,

• Xt −Xs has independent increments F , and

• P (Xt −Xs ≤ y) = P (Xt+h −Xs+h ≤ y).

Hence, for Xt to provide a GLM we require that,

E[eαXt ] <∞. (6)

From the properties of independent increments and stationarity stated above, we

have a Levy exponent, ψ(α), such that

E[eαXt ] = etψ(α). (7)

Furthermore, we define a Levy Martingale as,

Definition 4. The process Mt is defined as

Mt = eαXt−tψ(α) (8)

is called the geometric Levy martingale associated with Xt and with volatility α

Then, using the properties of stationarity and independent increments, we can

further say that,

6


Es[Mt] = Ms. (9)

Let us continue onto obtaining the Levy models for each process. We will first

define the pricing kernel which, as we stated above, is the condition used to ensure

that there will be no arbitrage available. Hence the pricing kernel can be defined

as,

πt = e−rte−λXt−tψ(−λ). (10)

As we know that for option pricing theory we require that the product of the

pricing kernel and the price of the asset to be a martingale of the form,

πtSt = S0eβXt−tψ(β). (11)

From (11) we divide through by πt we have

St = S0erteσXt+tψ(λ)−tψ(σ−λ). (12)

where σ = β + λ.

If we rewrite the asset price as follows,

St = S0erteR(λ,σ)teσXt−tψ(σ), (13)

then we can write the risk premium as,

R(λ, σ) = ψ(σ) + ψ(−λ)− ψ(σ − λ). (14)

The risk premium is the excess rate of return above the interest rate, this is the

general formula for the risk premium. As we said earlier the GLM covers many

models and such models are

7


1. Brownian Motion,

2. Poisson,

3. Compound Poisson and

4. Gamma.

For us to use these models we must first obtain the Levy exponent by taking the

expectation from (7), afterwards we can take the Levy exponent and find the risk

premium of each of these models as well as the pricing kernel and asset price.

8


3.1 Brownian Motion Model

For the Brownian motion model we use the properties that a Brownian motion

is Xt ∼ N(0, t) hence, in order to solve for the Levy exponent we use the same

properties of a Brownian motion.

Below shows the definition and formula for a process Xt ∼ N(0, t) which we will

be using in order to solve for the Levy exponent.

N(x) =1√2π

∫ ∞−∞

exp[−1

2x2]. (15)

We will now solve for the Levy exponent,

E[αXt] =1√2π

∫ ∞∞

exp[αx− 1

2x2] (16)

=1√2π

∫ ∞∞

exp[−1

2(x2 − 2αx+ α2) +

1

2α2] (17)

=e

12α2

√2π

∫ ∞∞

exp[−1

2(x+ α)2]. (18)

By substituting y = x+ α, we then have a standard normal distribution function

with the integral being equal to 1 hence the final result of the integral will be

given by,

e12α2

, (19)

which means, using equation (7), that the Levy exponent for the Brownian motion

is

ψ(α) =1

2α2, (20)

where the excess rate of return is,

R(λ, σ) = ψ(σ) + ψ(−λ)− ψ(σ − λ) (21)

=1

2σ2 +

1

2λ2 − 1

2(σ − λ)2 (22)

= σλ. (23)

9


Finally, the general asset price formula in this model is given by,

ST = s0erT+σλT+σXT− 1

2Tσ2

, (24)

and the associated pricing kernel, using equation (10) is,

πT = e−rT−λXT−12Tλ2 . (25)

10


3.2 Poisson Model

To obtain the Levy exponent for a Poisson process we use the properties of a

Poisson distribution. This is done by taking the summation of the product of the

exponent and the Poisson process. We are able to do this due to the property that

a Poisson distribution is discrete, unlike a Brownian motion which is continuous.

Hence, the expectation of a discrete variable is to take the summation in product

with the process.

Definition 5. Let Nt be a standard Poisson process where the jump rate m > 0,

then Nt is given by

P(Nt = n) = e−mt(mt)n

n!. (26)

Thus, the expectation is calculated as follows,

E[eαNT ] =∞∑n=0

e−mT(mT )n

n!enα (27)

= e−mT∞∑n=0

(mTeα)n

n!(28)

= emT (eα−1). (29)

Taking this result we clearly see that the Levy exponent is,

ψ(α) = mT (eα − 1). (30)

Using this and (14) we are able to get the risk premium by applying the formula

which is positive and increasing,

R(λ, σ) = ψ(σ) + ψ(−λ)− ψ(σ − λ) (31)

= m(1− e−λ)(e−σ − 1). (32)

11


As a side note, as the jumps in this model are upwards then the risk the investor

encounters maybe greater as there are fewer jumps than wished for. This is

evident when we obtain the asset price of a non-dividend paying asset (done by

using (14)):

St = S0ert+σNT−mTe−λ(eσ−1). (33)

The associated pricing kernel is,

πt = exp[−rT − λNT −mT (eλ − 1)]. (34)

12


3.3 Compound Poisson Model

For the compound Poisson model let us first define the distribution that we will

use in order for us to solve the Levy exponent. From ’Asset Pricing’ by Dorje

Brody, it says that,

Definition 6. let NT be a standard Poisson process with the rate being m. Let

Ykk∈N be a collection of i.i.d. copies of a random variable Y with the property

φ(α) := E[eαY ] <∞. (35)

Hence the distribution will be of the form

Xt =

NT∑k=1

Yk. (36)

So, we take this distribution and find the Levy exponent by taking the expectation

outlined above,

E[eαXT

]= E

[eα

∑NTk=1 Yk

]. (37)

In the above expression we come across a problem in which we are taking the

expectation of Yk summed up to Nt. However, both these variables are indepen-

dent of each other. As a result, we use Lemma 2.3.4 in (Shreve, 2004). Applying

this Lemma we will taking another expectation within the current expectation as

shown below,

E[eαXT ] = E[E[eα

∑NTk=1 Yk | NT

]]. (38)

We will be solving this by using the property (35) to have the following,

g(Nt) = (φ(α))NT (39)

13


g(n) = E[eα∑nk=1 Yk ] = (φ(α))NT (40)

Taking this and the property and the knowledge that a compound Poisson is still

a Poisson process we then use the same method as that of a Poisson process to

have,

E[eα∑NTk=1 Yk = E

[(φ(α))NT

]e−m

∞∑n=1

λn

n!(φ(α))n (41)

= emeφ(α)−1. (42)

Hence the Levy exponent is

ψ(α) = m(φ(α)− 1). (43)

Finally, the price process for the asset is,

St = S0erT eσNT+tψ(λ)−Tψ(σ−λ) (44)

= S0erT eσNT+mT (φ(−λ)φ(σ−λ)). (45)

Then, the process for the pricing kernel for the compound Poisson process is

defined as,

πt = e−rT e−λNT−tψ(−λ) (46)

= e−rt−λNT −mt(φ(−λ)− 1). (47)

Using our Levy exponent (43) the excess rate of rate is,

R(λ, σ) = ψ(σ) + ψ(−λ)− ψ(σ − λ) (48)

= m(φ(σ) + φ(−λ)− φ(σ − λ)− 1). (49)

14


3.4 Geometric Gamma Model

We will now define a gamma process to be,

Definition 7.

P(γT ∈ dx) =

∫ ∞0

x(mt−1)e−x

Γ(mt)dx. (50)

As before, in order for us to find the Levy exponent we will undertake the expec-

tation of the exponential of the product between the process and α. In order to

do this we multiply our distribution with our gamma process so that we obtain,

E[eαγT ] =

∫ ∞0

αxx(mT−1)e−x

Γ(mt)dx. (51)

If we extract ΓT out of the the integral to have,

E[eαγT ] =1

ΓT

∫ ∞0

x(mT−1)e−x(1−α)dx, (52)

whilst noting1

θ= 1− α, (53)

we can substitute this into the integral so we obtain

E[eαγT ] =1

ΓT

∫ ∞0

x(mT−1)e−xθ dx. (54)

If we define the following integral over the domain of x and apply the definition

to have

E[eαγT ] =θmT

ΓT

∫ ∞0

x(mT−1)e−xdx = 1, (55)

then, using this property, we can apply at (51) to get,

E[eαγT ] = θmT . (56)

Substituting in, (53) we ascertain;

E[eαγT ] = (1− α)−mT . (57)

15


Applying the properties of exponentials and logarithms we get

E[eαγT ] = e−mT ln(1−α), (58)

thus, our Levy exponent becomes

ψ(α) = −m ln(1− α). (59)

Finally, the price process for the asset is,

ST = S0erT eσXT+tψ(λ)−tψ(σ−λ) (60)

= S0erT eσγT−m ln(1−λ)+m ln(1−σ+λ). (61)

Then, the process for the pricing kernel for the gamma process is defined as,

πt = e−rT e−λXT−Tψ(−λ) (62)

= S0erT e−λγT (1 + λ)mT . (63)

Hence, using (59), the risk premium can be defined as,

R(λ, σ) = ψ(σ) + ψ(−λ)− ψ(σ − λ) (64)

= m ln

[1− σ + λ

(1− sigma)(1 + lambda)

]. (65)

16


4 Option Pricing

The aim for us is to be able to take an option and price it where the underlying

asset is a Levy process. We will solve for the price of the option using the same

cases above when the random processes is either a Brownian motion, Poisson,

Compound Poisson or Gamma process.

We will consider the case in which the option, which we wish to price, is a call

option. We know that for a call option the payoff is

CT = (ST −K)+. (66)

As we know, in a call option we wish to buy the asset St for the strike price K.

However, if the St is less than K the price of the option will be zero. This is

denoted by,

CT = max(ST −K, 0). (67)

This ensures that there will be no negative value in which arbitrage may occur.

Also the ’max function’, which we will solve for in every case, signifies that if

St > K there is a positive payoff in which the investor receives St −K otherwise

the investor receives nothing. This is all done on the certainty that there is no

arbitrage because of the existence of the pricing kernel.

The function above is for the payoff for a call option, however we wish to ascertain

the price of the call option which is given by taking the expectation of the product

between the payoff of a call option and the pricing kernel. As we mentioned above,

the asset must be an exponential Levy process. The equation we will be evaluating

for each of the models is:

C0 = E[πTHT ]. (68)

17


4.1 Option Pricing: Brownian Motion

First let us consider the price of a European call option where the price process

of the underlying asset is a geometric Brownian motion. We recall that the value

of the derivative with payoff HT is given by,

C0 = E[πT (ST −K)+], (69)

and also the processes for both the asset price and the pricing kernel from (24)

and (25). Inputting both of these into equation (69) we have,

C0 = E[e−rT−12λ2T+λWT (S0e

rT− 12σ2T+λσT+σWT −K)+]. (70)

Since we know that a Brownian motion is normally distributed with N(0,T) then

we have W=X√T obtaining,

C0 = E[e−rT−12λ2T+λX

√T (S0e

rT− 12σ2T+λσT+σX

√T −K)+]. (71)

As a result we have our max function to be,

erT−12σ2T+λσT+σX

√T >

K

S0

(72)

X >ln K

S0erT− λσT + 1

2σ2T

σ√T

= x∗ (73)

From this we can see that this is positive as long as X > x∗ where x∗ is our

max function. Bearing this in mind, we will use the properties of Brownian

motion being Normally distributed to take the expectation (70). As we know

when computing the normal distribution we take the integral. In order to do this

we will split the integral into two parts, C0 = I1 + I2

I1 =S0√2π

∫ ∞x∗

e12(σ2−2σλ+λ2)

√T e(σ−λ)

√Txe−

12x2dx. (74)

As we can see the exponential can be factorized to have,

I1 =S0√2π

∫ ∞x∗

e−12(σ−λ)2

√T e(σ−λ)

√Txe−

12x2dx. (75)

18


This can be factorized even further to obtain,

I1 =S0√2π

∫ ∞x∗

e−12(x−(σ−λ)

√T )2dx. (76)

As we know the Integration for a normal distribution with mean 0 and variance

1 is,

N(x) =1√2π

∫ ∞−∞

e−12x2dx. (77)

As we use this property replacing x with y we have,

y = x− (σ − λ)√T ).

As a result, our integral I1 is,

I1 =S0√2π

∫ ∞x∗−(σ−λ)

√T

e12y2dy (78)

Keeping in mind the definition (77), we can rewrite our integral as follows,

I1 = S0N(−x∗ + (σ − λ)√T ). (79)

Let us now look at the integral of I2 applying the same method of our first integral.

I2 =Ke−rT√

2π

∫ ∞x∗

e−12λ2T+λ

√Tx− 1

2x2dx (80)

=Ke−rT√

2π

∫ ∞x∗

e−12(x+λ

√T )2dx. (81)

Substituting the following to obtain a normal distribution format we have

y = x+ λ√T

I2 =Ke−rT√

2π

∫ ∞x∗+λ

√T

e−12y2dy. (82)

As a result, the integral of I2 can be rewritten as,

I2 = Ke−rTN(−x∗ − λ√T ). (83)

Hence, the final price of the option is given by,

C0 = S0N(−x∗ + (σ + λ)√T )−Ke−rTN(−x∗ − λ

√T ). (84)

19


Now, if we consider our max function and apply it to our integrals I1 and I2, we

have, for I1,

−x∗ + (σ + λ)√T =

ln KS0ert

− λσT + 12σ2T

σ√T

+ σ√T − λ

√T (85)

=ln K

S0ert+ 1

2σ2T

σ√T

(86)

= d+. (87)

Applying the same logic to I2

−x∗ − λ√T =

ln KS0ert

− λσT + 12σ2T

σ√T

+ λ√T − λ

√T (88)

=ln K

S0ert− 1

2σ2T

σ√T

(89)

= d−. (90)

Hence, the price of a European call option driven by a Brownian motion is given

by,

C0 = S0N(d+)−Ke−rtN(d−). (91)

where

d± =ln K

S0ert± 1

2σ2T

σ√T

. (92)

Finally, we have shown that the price derived for a European call option under

the physical measure is the same as that derived under the risk neutral measure.

20


4.2 Option Pricing: Poisson

We will now consider the pricing of a European call option with the underlying

asset price being a Poisson process. Recalling the equation for the option pricing

formula and the process for the pricing kernel and asset price we have,

C0 = E[πT (ST −K)+] (93)

= E[(S0e(σ−λ)NT−mT (e(σ−λ)−1) −Ke−rT e−λNT−mT (−λ−1))+] (94)

In order for this to be positive the max function is as follows,

S0e(σ−λ)NT−mT (e(σ−λ)−1) −Ke−rT e−λNT−mT (−λ−1) > 0 (95)

eσNT+mTe−λ(1−eσ) >

Ke−rT

S0

. (96)

Hence from the this we get that NT > N∗,

NT >1

σ(K

S0erT+mTe−λ(eσ − 1)) = N∗, (97)

where N∗ is our max function.

Hence, using the ceiling function, dxe, where Z is the smallest integer larger than

N∗ we have,

C0 =∞∑n=Z

S0e(σ−λ)n−mT (e(σ−λ)−1)emT

(mT )n

n!−Ke−rT

∞∑n=z

e−mTe−λ (mTe−λ)n

n!. (98)

We will be splitting this into two sums that we solve for separately,

C0 = C1 + C2

.

21


Let us now consider C1, in order to solve for the expectation we use the definition

of a Poisson distribution as a sum which can be seen below.

C1 = S0

∞∑n=Z

emTe(σ−λ) (mTee

(σ−λ))n

n!. (99)

Then for C2, applying the same logic, we have

C2 = Ke−rT∞∑n=z


n!. (100)

Thus, we ascertain the final price of the option as

C0 = S0

∞∑n=Z

emTe(σ−λ) (mTee

(σ−λ))n

n!−Ke−rT

∞∑n=z


n!. (101)

If we define the cumulative Poisson distribution as,

PO(k, λ) =λke−λ

k!, (102)

then if take k = Z and λ = mTe(σ−λ) for the sum C1 and for the second sum C2

to take λ = mTe−λ and k = Z we ascertain the final price of a European call

option derived under the Poisson process to be of the form,

C0 = S0PO(Z,mTe(σ−λ))−Ke−rTPO(Z,mTe−λ) (103)

Finally, we noticed that the price of the call option is quite similar to the Black-

Scholes formula. However, observe that unlike that of the Black-Scholes formula

we noticed that the risk aversion parameter λ did not drop out suggesting that

the formula is dependent on the parameter λ along with the jump rate m.

22


4.3 Option Pricing: Compound Poisson

We will now consider the pricing of a European call option with the underlying

asset is a compound Poisson process. Taking the equation for the option pricing

formula and the process for the pricing kernel and asset price. Then we have

C0 = E[(S0e

(σ−λ)NT−mt(φ(σ−λ)−1) −Ke−rT e−λNT−mT (φ(−λ)−1))+]. (104)

As before, we need to obtain the max-function for each process, for the compound

Poisson process it goes as follows,

S0e(σ−λ)NT−mT (φ(σ−λ)−1) −Ke−rT e−λNT−mT (φ(−λ)−1) > 0, (105)

e(σ−λ)NT−mT (φ(σ−λ)−1)

e−λNT−mT (φ(−λ)−1)>KerT

S0

, (106)

e−σNT+mT (φ(−λ)−φ(σ−λ)) >KerT

S0

, (107)

σNT > ln(KerT

S0

)−mT (φ(−λ)− φ(σ − λ)), (108)

to obtain

NT >1

σ[ln(

KerT

S0

)−mT (φ(−λ)− φ(σ − λ))] = NT∗, (109)

where NT∗ is the max function.

Taking the max-function into consideration, we will evaluate the expectation

(104). In order to take this expectation we must use the definition of a com-

pound Poisson distribution that we defined in chapter 3.As before we will be

taking this expectation in two parts. Let us firstly consider the first part of the

expectation.

E[(S0e

(σ−λ)NT−mT (ψ(σ−λ)−1))]

= S0emT (ψ(σ−λ)−1)E

[e(σ−λ)NT

](110)

= S0emT (ψ(σ−λ)−1)E

[e(σ−λ)

∑NTk=1 Yk

]. (111)

23


If we only consider the expectation

E[e(σ−λ)

∑NTk=1 Yk

]= E

[e(σ−λ)

∑NTk=1 Yk | NT

]. (112)

We know that Yk is an i.i.d,as a result we can make the assumption that,

Yk ∼ N(0, 1). (113)

This then leads onto

n∑k=1

Yk ∼ N(0, n), (114)

then if we take the sum to be,

n∑k=1

Yk ∼ N(0, n) = x√n (115)

then our expectation becomes,

E[e(σ−λ)

∑NTk=1 Yk | NT ] = E[e(σ−λ)x

√n | NT > N∗T

]. (116)

If we use the compound Poisson distribution on our expectation we notice that

this is in the form of normal distribution as well, we gain

=∞∑n=0

emTmT n

n!

[1√2π

∫ ∞∞

e−12x2e(σ−λ)x

√ndx | NT >

N∗T√n

](117)

=∞∑n=0

emTmT n

n!e

(σ−λ)2n2

[1√2π

∫ ∞N∗T√n

e−12(x−(σ−λ)

√n)2

](118)

=∞∑n=0

emTmT n

n!e

(σ−λ)2n2

[N(−N

∗T√n

+ (σ − λ)√n)

]. (119)

Now if we consider the second part of the expectation

E[Ke−rT eλNT−mT (ψ(−λ)−1)] = Ke−rT−mT (ψ(−λ)−1)E[eλNT ]. (120)

As before if we just consider the expectation and start to apply the distribution

we have,

24


E[eλNT ] = E[e−λ∑NTk=1 Yk | NT ] (121)

= E[e−λx√n | NT ] (122)

=∞∑n=0

emTmT n

n!

[1√2π

∫ ∞∞

e−12x2e−λx

√ndx | NT >

N∗T√n

](123)

=∞∑n=0

emTmT n

n!e−λ2n

2

[1√2π

∫ ∞N∗T√n

e−12(x−λ

√n)2

]dx (124)

=∞∑n=0

emTmT n

n!e−λ2n

2

[N(−N

∗T√n

+ λ√n)

]. (125)

Hence the price of the call option is given by,

C0 = S0emT (ψ(σ−λ)−1)

∞∑n=0

emTmT n

n!e

(σ−λ)2n2

[N(−N

∗T√n

+ (σ − λ)√n)

]−Ke−rT e−mT (φ(−λ)−1)

∞∑n=0

emTmT n

n!e−λ2n

2

[N(−N

∗T√n

+ λ√n)

]. (126)

Finally writing the European call option derived under a Compound Poisson pro-

cess as,

C0 = S0emt(ψ(σ−λ)−1)

∞∑n=0

emTmT n

n!e

(σ−λ)2n2 N [d+]

−Ke−rT e−mT (φ(−λ)−1)∞∑n=0

emTmT n

n!e−λ2n

2 N [d−], (127)

where

d+ =1

σ[ln(

KerT

S0

)−mt(φ(−λ)− φ(σ − λ))]− λ√n, (128)

d− =1

σ[ln(

KerT

S0

)−mt(φ(−λ)− φ(σ − λ))]− (σ + λ)√n. (129)

Finally, we noticed that the price of the call Option is quite similar to the Black-


25



the formula is dependent on the parameter λ along with the jump rate m. Fur-

thermore, if we take n = 0 notice that the price of the option will be undefined

as d+ and d− are both undefined at that point.

26


4.4 Option Pricing: Gamma

We will now consider the pricing of a European call option where the price process

of the underlying asset is a Gamma process. If we recall the processes for the asset

price and the pricing kernel and use them for the following expectation we have,

C0 = E[(πTST −KπT )+] (130)

= E[(S0e(σ−λ)γT (1− σ + λ)mT − e−rTK − e−λγT (1 + λ)mT )+]. (131)

As before, we need to obtain the max-function for each process, for the Gamma

process it goes as follows,

S0e(σ−λ)γT (1− σ + λ)mT − e−rTK − e−λγT (1 + λ)mT > 0, (132)

e(σ−λ)γT (1− σ + λ)mT

e−λγT (1 + λ)mT>Ke−rT

S0

, (133)

eγT (1− σ

1 + λmT) >

Ke−rT

S0

, (134)

γT >1

σ[ln(

K

S0erT)−mt ln(1− σ

1 + λ)] = γ∗, (135)

where γ∗ is the max function.

Returning to the expectation, we know that we must take the the product of the

expectation and the distribution. If we split this into the sum of two integrals, as

we done in the section for Brownian motion,

I1 = S0

∫ ∞γ∗

(1− σ + λ)mT ex(σ−λ)x(mT−1)e−x

Γ(mT )dx. (136)

Then we collecting the like terms of powers of x, however for the power of mt− 1

we are required to multiply the integral by (1− σ+ λ) in order to ensure that we

do not change the integral as seen below,

27


I1 = S0(1− σ + λ)

∫ ∞γ∗

e−x(1−σ+λ)x(mT−1)(1− σ + λ)mT−1

ΓTdx (137)

= (1− σ + λ)

∫ ∞γ∗

S0e−x(1−σ+λ)(x(1− σ + λ))mT−1

ΓT. (138)

Now for the second part of the integration using the same method as above we

obtain,

I2 = Ke−rT∫ ∞γ∗

eλx(1 + λ)mTx(mT−1)e−x

Γ(mT )dx (139)

= Ke−rT (1 + λ)

∫ ∞γ∗

e−x(1+λ)(x(1 + λ))mT−1

ΓTdx. (140)

Hence, the price of the call option is as follows,

C0 = S0(1− σ + λ)

∫ ∞γ∗

e−x(1−σ+λ)(x(1− σ + λ))mT−1

ΓT

−Ke−rT (1 + λ)

∫ ∞γ∗

e−x(1+λ)(x(1 + λ))mT−1

ΓT. (141)

If we take y = x(1 − σ + λ) for the first part of the integral and the second we

have w = x(1 + λ if we substitute this in we have

C0 = S0

∫ ∞γ∗(1−σ+λ)

e−yymT−1

ΓT (mT )−Ke−rt

∫ ∞γ∗(1+λ)

e−wwmT−1

ΓT (mT ). (142)

If we take the cumulative gamma distribution which is defined as

G(x, k, θ) =

∫ ∞x

e−xθ xk−1

θkΓT (mT )(143)

28


Using this and taking x = γ∗(1 − σ + λ), k = mT and θ = 1 and for the second

integral x = γ∗((1 + λ), k = mT and θ = 1

C0 = S0G(γ∗(1− σ + λ),mT, 1)−Ke−rTG(γ∗(1 + λ),mT, 1) (144)

Finally, we noticed that the price of the call option is quite similar to the Black-



the formula is dependent on the parameter λ along with the jump rate m.

29


5 Simulation

For this section we will discuss the different types of simulations used to encompass

the theory of option pricing using Levy models. We applied the Monte-Carlo

method for all the different Levy process but also implemented the actual equation

used for the Brownian motion and the Poisson processes.

Let us first start with the Brownian motion and compare the two simulations that

we ran for this process. Below shows the script file for the Monte-Carlo method

that we used,

Figure 1: Brownian motion Monte-carlo

The term ’sfinal’ is the final asset price that was ascertained i.e. ST whilst the

30


term ’pt’ is our pricing kernel πT . We then took the expectation,

C0 = E[πt(ST −K)+]. (145)

and applied it to all the different models that we used. Since we know ST and

piT we input them in to solve for our call option. At the end notice that we have

the term C(i), this is purely so that we can run M simulations, which is 500 in

this case, to then take the average of them but to be discounted to today’s price,

which is what happens in the last line of the simulation.

For the next simulation of Brownian motion, we took the final option price calcu-

lated in chapter 4,

C0 = S0N(d+)−Ke−rtN(d−). (146)

where

d± =ln K

S0ert± 1

2σ2T

σ√T

, (147)

and implemented exactly as it seen in the equation above. Notice that in this

simulation that we have loop for different strike where, afterwards we plotted a

graph of the strike price against the call option prices. Below shows the simulation

that was implemented and the graph that was produced.

31


Figure 2: Brownian Motion Implementation

Figure 3: Brownian Motion Implementation graph

32


Table 1: Brownian motion

Monte-Carlo method Implementation

32.1779 31.9033

31.4545 30.9984

30.0095 30.0936

29.3310 29.1887

29.0216 28.2839

27.5953 27.3791

27.0816 26.4742

26.0563 25.5694

25.1838 24.6646

23.9418 23.7597

23.4861 22.8549

We can verify the outcome of these two implementation are similar to each other

by observing the outcome of the two methods which can be seen below,

Let us consider the Poisson process and compare the two simulations that we ran

for this process. Below shows the script file for the Monte-Carlo method that we

used,

33


Figure 4: Poisson Monte-carlo

Just as before for the Brownian motion, we know ST and piT and input them in

to solve for our call option. At the end recall that the term C(i) is purely used so

that we can run M simulations, which is 500 in this case, to then take the average

of them but to be discounted to today’s price, which is what happens in the last

line of the simulation.

For the implementation of the Poisson we used the final formula for the call option

price that we had which was,

C0 = S0

∞∑n=Z

emte(σ−λ) (mtee

(σ−λ))n

n!−Ke−rt

∞∑n=z

e−mte−λ (mte−λ)n

n!. (148)

Thus the simulation that had occurred from this can be seen below with the graph

that was produced with it.

34


Figure 5: Poisson Implementation

Figure 6: Poisson Implementation graph

35


6 Conclusion

Finally to summarise in this research project we have thoroughly covered the

concept of option pricing and specifically making mention to the pricing kernel

approach. In this conclusion we will summarise the outcomes of each chapter and

our final thoughts for the main points.

Firstly, in the first chapter we introduced the concept of option pricing mentioning

the most common method used, The Black and Scholes method. However, we

briefly mentioned that the Black and Scholes widely used formula, stemmed from

the work of a PHD student. Black and Scholes improved his method by eliminating

the possibility of negative values for the option prices by introducing the asset as

a stochastic process. However, we must recall that in order for Black and Scholes

to come up with this assumption they were working in the risk neutral measure

not in the physical measure. Furthermore, this highlighted that there were more

methods for option pricing that all had some root from the Black and Scholes

method, such a method was derived by using Levy models.

We noticed that whilst using these models the sense of arbitrage was eliminated

by the use of the pricing kernel. In using the pricing kernel it became extremely

clear that we were not working within the risk neutral measure but rather in the

physical one instead. From this we continued on to define four axioms that had

to be explicitly followed in order to use the pricing kernel method. The axioms

are,

Axiom 1. There exists an absolutely continuous, strictly increasing ”risk-free”

asset Bt (the money-market account).

Axiom 2. If ST is the price process of any asset, and DT is the associated con-

tinuous dividend rate, then the process Mt defined by

Mt = πtSt +

∫ t

0

πsDsds (149)

is a martingale. Thus E[|Mt|] <∞ for t > 0, and Et[|Mu|] = Mt.

36


Axiom 3. There exists an asset (a floating rate note) that offers a continuous

dividend rate sufficient to ensure that the value of the asset remains constant.

Axiom 4. A discount bond system PtT exists for 0 < t < T < ∞ satisfying

limT→∞

PtT = 0.

In chapter 3 we explored the different types of Levy models and their Levy ex-

ponents which allowed us to evaluate the formula for the risk premium. In each

of these cases we was able to determine the asset price and pricing kernel before

the evaluation of the risk premium. The models that we explored were; Brownian

motion, Poisson, Compound Poisson and gamma.

Once we had evaluated the the asset price and pricing kernel, alongside the Levy

exponent, we was able to price a European option. In this project we explored

and found how to evaluate a European call option when the underlying asset was

a Levy process. We done this by calculating the expectation,

C0 = E[πt(St −K)+]. (150)

As we knew the pricing kernel and the underlying asset we were able to find the

formula to obtain the option pricing for each of our models. This was done by

going back to the definition of each model and using its respected distribution. We

ensured that we did not violate the strictness of the expectation being positive,

i.e. St > K, by ensuring that we solved for our max function for each of the cases.

It is important to remember that we solved to obtain the prices of the call options

for each of the models, the result was,

•

C0 = S0N(d+)−Ke−rtN(d−), (151)

•

C0 = S0Po(Z,mte(σ−λ))−Ke−rtPo(Z,mte−λ), (152)

37


•

C0 = S0emt(ψ(σ−λ)−1)

∞∑n=0

emtmtn

n!e

(σ−λ)2n2 N [d+]

−Ke−rte−mt(φ(−λ)−1)∞∑n=0

emtmtn

n!e−λ2n

2 N [d−], (153)

•

C0 = S0G(γ∗(1− σ + λ),mt, 1)−Ke−rtG(γ∗(1 + λ),mt, 1). (154)

Finally, we came to the point of simulating our results using two different methods,

Monte-carlo and implementation of the formula. Both these methods produced

similar results and can be seen from the previous chapter. We then went on to

draw up a graph for the implementation method to see how the trend of the

prices looked. In this sense we gathered our simulation for different strike values

to obtain a graph of strike price against call option. The outcome can be seen in

the previous chapter, we done this for both the Brownian motion and the Poisson.

As a side note we also drew up the simulations for the gamma process and the

compound Poisson using the Monte-carlo method only.

In conclusion, the benefits of working with the pricing kernels was that it allowed

us to work in a framework where we assume that the market was incomplete whilst

working with a probability measure in the physical setting. Mathematically, the

method of pricing kernel allowed a very reliable and concise form for pricing

options under the axioms for the use of the pricing kernel.

38


6.1 Limitations

There a few aspects that we did not explore for many different purposes with time

being one of them, however, giving whether these factors favoured us we could have

looked at it at a much broader scale. The first of these would of been to analyse

the results had the interest rates not been fixed but constantly changing, this also

applies to the volatility and our risk aversion parameter. Another limitation that

we could of explored was the different types of Levy models that we did not talk

about within this dissertation and see if, like some of the models, the risk aversion

parameter does not drop out.

Another interesting concept to analyse was the conditions in which the risk aver-

sion drops out in and see if this is the same had we been working in a stochastic

level. From this we could analysed the outcomes of our current models and again

been working in the stochastic level also.

Finally, we could of explored the results of the implementation of each of our mod-

els when we was simulating. We were only able to simulate two of the four models

that we spoke about, however had we simulated the last two and compared it to

the Monte-Carlo method the outcomes would have made an interesting compari-

son and thus would allow for much further detail to observe. Once we had done

that we could of seen the outcome had we calibrated our results to real world data

and analyse those outcomes as well.

39


References

Applebaum, D. (2004). Levy processes and stochastic calculus. Cambridge, UK:

Cambridge University Press.

Bachelier L., Davis M. & Etheridge, A. (2006) Louis Bachelier’s theory of

speculation. Princeton University Press.

Brody, D. C. & Hughston, L. P. (2004) Chaos and coherence: a new framework

for interest-rate modelling. Proceeding of the Royal Society London A460,

85-110, (doi:10.1098/rspa.2003.1236).

Brody, D. C. & Hughston, L. P. (2014) Mathematical theory of dynamic asset

pricing. Lecture Notes, London: Brunel University.

Brody, D. C., Hughston, L.P. & Mackie, E. (2012) General theory of Geometric

Levy Models for Dynamic Asset Pricing. Proceedings of the Royal Society

of London. Series A, Mathematical and physical sciences, 468 (2142). pp.

1778 - 1798. doi: 10.1098/rspa.2011.0670

Brody, DC. , Hughston, LP. & Mackie, E. (2013) Lvy information and aggre-

gation of risk aversion. Proceedings of the Royal Society A469, 20130024.

Available at http://arxiv.org/pdf/1301.2964.pdf

F. Filo (2004) Option Pricing Under the Variance Gamma Process Unpublished

dissertation.

Shreve, S. (2004). Stochastic calculus for finance II. Springer.

Schoutens, W. (2004) Levy Processes in Finance: Pricing Financial Deriva-

tives (New York: Wiley)

40


Appendices

Compound Poisson

1 function [ OP ] = ECP( S0, K, r, T, t, M, s)

2

3 %Function used to calculate the price of a call option using

4 %the Monte -Carlo method.

5

6 S0 = 20; %Initial asset price.

7 K = 25; %Strike price.

8 s = 0.1; %Volatility value.

9 l = 0.1; %Risk aversion rate.

10 m = 5; %Jump rate.

11 r = 0.1; %Interest rate.

12 T =1; % Time at which call option will be excercised.

13 t = 1/365; %Time steps.

14

15 M = 500; %Number of simulations.

16

17 for i = 1:M

18

19 %Taking our actual stock price that was calculated after the expectation.

20 sfinal = S0.*exp(r*T)*exp(m*T*(exp (0.5*(l^2))-exp (0.5*(s-l)^2)))

21 *exp(s*sqrt(poissrnd(m*T))* randn );

22

23 %Taking our actual pricing that was calculated after the expectation.

24 pt = exp(r*T)*exp(-m*T*(exp (0.5*(l^2)) -1))

25 *exp(-l*sqrt(poissrnd(m*T))* randn );

26

27 C(i) = max(pt*(sfinal -K),0); %Payoff of call option at maturity.

28

29 end

30

31 OP = exp(-t*T)*mean(C); %payoff of option price dicounted.

32

41


33 end

42


Gamma

1 function [ OP ] = EG( S0, K, r, T, t, M, s)

2

3 %Function used to calculate the price of a call option using

4 %the Monte -Carlo method.

5

6 S0 = 20; %Initial asset price.

7 K = 25; %Strike price.

8 s = 0.1; %Volatility value.

9 l = 0.1; %Risk aversion rate.

10 m = 5; %Jump rate.

11 r = 0.1; %Interest rate.

12 T =1; % Time at which call option will be excercised.

13 t = 1/365; %Time steps.

14 M = 500; %Number of simulations.

15

16 for i = 1:M

17

18 %Taking our actual stock price that was calculated after the expectation.

19 sfinal = S0.*exp(r*T)*((1-s+l)/(1+l))^(m*T)*exp(s*gamrnd(m*T ,1));

20

21 %Taking our actual pricing that was calculated after the expectation.

22 pt = exp(-r*T)*exp(-l*gamrnd(m*T ,1))*(1+l)^(m*T);

23

24 C(i) = max(pt*(sfinal -K),0); %Payoff of call option at maturity.

25

26 end

27

28 OP = exp(-t*T)*mean(C); %payoff of option price dicounted.

29

30 end

43

levy models

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