levy models
TRANSCRIPT
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.
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Mohamed Raagi Student ID:1116027
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.
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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
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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
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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
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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
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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.
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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−).
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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)
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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,
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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
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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.
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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.
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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
.
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Mohamed Raagi Student ID:1116027
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
e−mTe−λ (mTe−λ)n
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
e−mTe−λ (mTe−λ)n
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
Mohamed Raagi Student ID:1116027
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)
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Mohamed Raagi Student ID:1116027
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,
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Mohamed Raagi Student ID:1116027
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-
Scholes formula. However, observe that unlike that of the Black-Scholes formula
25
Mohamed Raagi Student ID:1116027
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. 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.
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Mohamed Raagi Student ID:1116027
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,
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Mohamed Raagi Student ID:1116027
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)
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Mohamed Raagi Student ID:1116027
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-
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.
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Mohamed Raagi Student ID:1116027
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
Mohamed Raagi Student ID:1116027
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.
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Mohamed Raagi Student ID:1116027
Figure 2: Brownian Motion Implementation
Figure 3: Brownian Motion Implementation graph
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Mohamed Raagi Student ID:1116027
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
Mohamed Raagi Student ID:1116027
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.
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Mohamed Raagi Student ID:1116027
Figure 5: Poisson Implementation
Figure 6: Poisson Implementation graph
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Mohamed Raagi Student ID:1116027
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.
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Mohamed Raagi Student ID:1116027
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)
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Mohamed Raagi Student ID:1116027
•
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.
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Mohamed Raagi Student ID:1116027
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.
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Mohamed Raagi Student ID:1116027
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)
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Mohamed Raagi Student ID:1116027
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
Mohamed Raagi Student ID:1116027
33 end
42
Mohamed Raagi Student ID:1116027
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