Greg

Estimating Parameters of a Jump-Diﬀusion Model with an
Application to Option Pricing: The Case of the Dar es Salaam Stock
Exchange.
Gregory BOADU-SEBBE(gboadusebbe@aims.ac.tz)
AIMS141500143
African Institute for Mathematical Sciences (AIMS)
Supervised by: Dr Olivier MENOUKEU PAMEN
University of Liverpool, U.K
Co-supervised by: Dr Wilson MAHERA CHARLES
African Institute for Mathematical Sciences (AIMS), Tanzania
12th July 2015
Submitted in partial fulﬁllment of a structured masters degree at AIMS Tanzania

Abstract
Based on the assumption that, the stock prices satisfy a jump-diffusion process, the Merton jump-
diffusion process for pricing contingent claims (options) is discussed. Empirical studies have suggested
the need to move away from the classical log normal dynamics of the Black-Scholes model framework. In
addition, we show that, the direct maximum likelihood procedure in estimating jump-diffusion processes
is not appropriate. The reason is that, log-return is equivalent to a discrete mixture of M normally
distributed variables in jump-diffusion models with a sufficiently large M. Thus, from modern litera-
ture of ”mixture of distributions”, it is known that the likelihood function is unbounded and leads to
inconsistency in parameter estimation. Derivation of an efficient and robust method, which provides
consistent and asymptotically normally distributed estimators will be made. This method is then applied
to some actively traded stocks on the Dar es Salaam Stock Exchange to price contingent claims in a
jump-diffusion setting. Comparison results will be made with the case of the Black-Scholes model.
Keywords: Merton Jump-Diffusion Model, Log-likelihood Function, Expectation Maximisation (EM)
Procedure, Option Pricing.
Declaration
I, the undersigned, hereby declare that the work contained in this research project is my original work, and
that any work done by others or by myself previously has been acknowledged and referenced accordingly.
Gregory Boadu-Sebbe, 12th July 2015
i

Contents
Abstract i
1 Introduction 1
1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Preliminaries 3
2.1 Basic Definitions and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Jump-Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Merton Jump-Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Option Pricing using Martingale Approach . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Model Calibration and Results 22
3.1 Discrete -Time Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Expectation Maximisation (EM) Procedure . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Conclusion 38
5 APPENDIX. 39
5.1 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Appendix C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4 Appendix D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
References 47
ii

1. Introduction
Option pricing in the standard Black-Scholes model has received a lot of attention since the pioneering
work of Black and Scholes (1973). In their seminal work on “The pricing options and corporate liabil-
ities”, Black and Scholes (1973) developed a classical model of the financial market process, that is,
log-normal diffusion process, whereby the log-return process has normal distribution. They based on the
assumption that, log-returns are normally distributed with constant volatility, resulting in a closed form
formula for the plain-vanilla options. Despite the success of the Black-scholes model in normal distri-
bution as well as geometric Brownian motion, in reality, behaviour of returns in the financial markets
diverge or depart from this ideal. For instance, in pricing stocks, the market distribution should possess
some realistic properties not found in the ideal log-normal classical model of the Black-Scholes. Such
properties are;
i) the model should allow random fluctuations such as sudden upsurges and crashes(jumps).
ii) there should be some skewness in the log-return distribution since downward outliers are larger
than upward outliers. In addition, the model distribution should incorporates high peak and heavy
tails (kurtosis). These are asymmetric and leptokurtic features respectively. In the Black-Scholes
model, the marginal distribution of the assets is assumed to be normal. According to Kou (2008),
many standard empirical studies have suggested that the distribution is skewed to the left, and
has two heavier tails with a higher peak than those of the classical models.
iii) volatility smile. Suppose the assumption of the Black-Scholes model on volatility is correct, then
this would mean that, the implied volatility should be constant. However, it has been recognized
in empirical studies that the implied volatility curve is convex of the strike price, thus, it resembles
a “smile”.
In view of these, modern standard literature such as (Cont and Tankov, 2004) and (Elliot and Kopp,
2005) have suggested the need for modern financial modelling to move away from the classical log-
normal dynamics of the Black-Scholes model framework. Alternative approaches to the geometric
Brownian motion and normal distribution within continuous-time dynamic financial time series models
of Black-Scholes model are the jump-diffusion processes. These processes have been used to model
stock prices (Merton, 1976) among other applications, whereby the reference cited by no means is
exhaustive. In this case, the classical models have been extended to incorporate changing volatility
and leptokurtic features. Several models have been proposed to accommodate the above mentioned
properties, especially the leptokurtic and asymmetric features. These include;
(a) the t-model, the log variance gamma model and the hyperbolic model, which are referred to as
the generalised hyperbolic models; see, for example, Barndorff-Nielsen and Shephard (2001) and
Samoradnitsky and Taqqu (1994).
(b) the time changed Brownian motion; see, for example, Clark (1973) and Madan and Seneta (1990).
(c) the factal Brownian motion, stable process and the Chaos theory; see, for example, (Mandelbrot,
1997).
(d) jump-diffusion models proposed by Merton (1976) and Kou (2008).
(e) models based on Lévy processes; see, for example Cont and Tankov (2004).
Although useful, there are difficulties in obtaining analytical solutions with the aim of pricing option using
these models. The merits of these jump-diffusion models are that they can account for discrete jumps in
1

Section 1.1. The Problem Page 2
the sample paths of the processes, and also provide a simplified way of replacing Gaussian distributions in
the geometric Brownian motion by Gaussian mixture distributions. The Gaussian mixture distributions
lead to meaningful and appropriate models for heavy-tailed distributions and leptokurtic features in
finance.
The classical models of Black and Scholes (1973) are straightforward to estimate the model param-
eters. On the other hand, jump-diffusion models, such as the Merton jump-diffusion model, are not
easy and straightforward to implement (that is, to estimate the model parameters) from discrete data
sample. The use of good fitting procedures is thus not only very important but also necessary to
avoid model misspecification. Although, the Maximum Likelihood Estimation is efficient in estimating
diffusions whose dynamics are described by homogeneous stochastic differential equation (for example
Black-scholes model) especially, with large sample size but, several researches have shown that, the
direct Maximum Likelihood Estimation procedure is invalid in estimating jump-diffusion models. This
is because, the log-return is equivalent to a discrete mixture of M normally distributed variables in
jump-diffusion models, where M becomes very large. Especially, direct maximum likelihood estimation
procedures can be sensitive to the choice of initial starting values. In addition, it is known (Honore, 1998)
that, the likelihood function for some parametric specifications is unbounded and leads to inconsistency
of the direct Maximum Likelihood Estimation.
1.1 The Problem
The problem under consideration is option pricing in the Merton jump-diffusion model. Our task is to
investigate an alternative method for determining maximum likelihood estimates of the model parameters
based on the Expectation Maximisation (EM) algorithm developed by Duncan et al. (2009).
1.2 The Purpose
The main purpose of this report are
(1) to derive and solve the Merton jump-diffusion model. Also, we show that the log-return distribution
of the Merton jump-diffusion model incorporates the leptokurtic and asymmetric features.
(2) to use the daily closing prices data from the Dar es Salaam Stock Exchange to estimate parameters
of the Merton jump-diffusion model and to price contingent claims in a jump diffusion setting.
(3) to make comparison with the case of Black-Scholes model.
1.3 Organisation
The rest of the report is organised as follows: In chapter 2, basic definitions and examples of the jump-
diffusion processes are studied in the first section. The main model under consideration is proposed
in the subsequent section. Finally, studies in option pricing (European option) are carried out. Model
calibration (estimation of model parameters) and results are presented in chapter 3. Also, chapter 3
discusses the comparison of the results from the proposed model and that of Black-Scholes. The last
chapter is the conclusion.

2. Preliminaries
We shall consider a complete probability space (Ω, Ft, P), where Ω is a set of events in a financial
market, Ft is market information filtration over a time horizon [0, T], and P is a data-generating
measure. The complete probability space describes financial market uncertainty. Throughout this
report, we will denote S(t) as the underlying asset price (in our case stock prices) at time t under P
and it’s continuously compounded return as d(ln(S(t)) for all t ∈ [0, T].
We will model our underlying asset prices to evolve continuously as a jump-diffusion process that is
adapted to information filtration Ft. Due to this, it will be most useful to start with some basic
definitions and concepts in jump-diffusion processes which have some application in finance, especially
option pricing. The definitions, theorems (propositions) and examples presented in the section below
can be found in modern literature such as Cont and Tankov (2004) and Elliot and Kopp (2005).
2.1 Basic Definitions and Theorems
2.1.1 Definition (Stochastic Process).
A Stochastic process is a collection of random variables
X = {X(t) : t ∈ τ} , X = {X(t, ω) : t ∈ τ; ω ∈ Ω}
defined on the complete probability space (Ω, Ft, P).
2.1.2 Definition (Lévy process).
A stochastic (cadlág) process X(t) = {X(t) : t ≥ 0} is said to be a Lévy process if it satisfies the
following properties;
i. X(0) = 0 almost surely.
ii. X(t) has stationary increments, that is, for any s < t, X(t) − X(s) is equal in distribution to
X(t − s).
iii. X(t) has independent increments; that is, for any 0 ≤ t1 < t2 < . . . < tn < ∞,
X(t2) − X(t1), X(t3) − X(t2), . . . , X(tn) − X(tn−1) are independent.
iv. X(t) is “stochastic continuous”, that is, the probability of a jump at some fixed time t is zero.
This means that, for any ε > 0 and t ≥ 0 it holds that
lim
h→0
P (|X(t + h) − X(t)| ≥ ε) = 0, where h = t, change in time.
However, since X(t) is a cadlag process then it is continuous from the right and it’s limit from the left
at every point in time t exist. In other words, for every t ∈ T, t → X(t) almost surely.
From the definition 2.1.2, a continuous process with independent increments is a Gaussian process and
stationary increments imply that X(t) is a Brownian motion with drift. X(t) is called a continuous
-time process if τ is an interval, such that τ = [0, T], and we call it a discrete-time process if τ is a
finite or countably infinite set, such as τ = {0, 1, 2, . . .} .
3

Section 2.1. Basic Definitions and Theorems Page 4
2.1.3 Remark.
(i) for a fixed t, X(t, ω) is a random variable such that, X(t) = X(t, ω); ω ∈ Ω, ω is a number .
(ii) for a fixed ω, X(t, ω) is s function of time; X(t, ω) = ω(t), t ∈ [0, T] . This is called the
sample path of the process X (or realisation ). ω is a function in this case.
Therefore, a stochastic process is a function of two variables t and ω representing time and uncertainty
respectively.
2.1.4 Definition (Martingale).
A stochastic process X(t) is said to be a martingale with respect to a filtration Ft if it satisfies the
following conditions;
1. E [|X(t)|] < ∞ for each t. Thus, X(t) is integrable.
2. X(t) is adapted to the filtration Ft
3. E [X(t)|Fs] = X(s) for s < t.
Note: If E [X(t)|Fs] ≥ X(s) and E [X(t)|Fs] ≤ X(s) then, X(t) is a submartingale and supermartin-
gale respectively.
Martingale is very crucial in contingent claims, which is the main purpose for this report.
2.1.5 Definition (Characteristic Function).
Let X be any random variable. Then, the characteristic function of X is its probability distribution. On
the real line it is given by
φX(u) = E eiuX
=
∞
−∞
eiux
f(x)dx
where u is a real number, i is the imaginary unit, E is the expected value and f is the probability density
function of X.
2.1.6 Definition (Poisson Process).
Let {τi}i≥1 be a sequence of independent exponential random variables with parameter λ and stopping
time
Tn =
n
i=1
τi.
The process {N(t), t ≥ 0} defined by
N(t) =
n≥1
1(t) ≥ Tn (2.1.1)
is called a Poisson process with parameter λ called intensity (is the average number of jumps per unit of
time t ). The Poisson process counts the number of jumps until time t. It has the following properties
which we state without proof:
2.1.7 Proposition.
Let {N(t)}t≥0 be a Poisson process.
(a) for any t > 0, N(t) is almost surely finite.

Section 2.1. Basic Definitions and Theorems Page 5
(b) for any ω, the sample path t → N(t, ω) is right continuous with left limit (Cadlag).
(c) N(t) is continuous in probability; ∀ t > s, N(s) −→ N(t).
(d) for any ω, the sample path t −→ N(t, ω) is piecewise constant and increases by jumps of size 1.
(e) for any t > 0, N(t) follows a Poisson distribution with parameter λt:
∀ n ∈ N, P(N(t) = n) = e−λt (λt)n
n!
.
(f) the characteristic function of N(t) is given by
E eiuN(t)
= eλt(eiu−1)
, ∀ u ∈ R.
(g) N(t) is a Markov process with stationary and independent increments, it has the Markov property:
∀ t > s, E [f(N(t))|N(u), u ≤ s] = E [f(N(t))|N(s)] .
For proof of the above proposition 2.1.7, see (Cont and Tankov, 2004).
Note: The measure of the Poisson Process is it intensity, λ.
2.1.8 Definition (Compound Poisson Process).
A stochastic process is called a compound Poisson process if it is defined by
X(t) =
N(t)
m=1
Ym (2.1.2)
where N(t) is a Poisson process and Ym are independent and normally distributed random variables
with distribution function F.
The measure of the compound Poisson process X(t) is given by
ν(B) = λ
R
dF(x) = λ
∞
−∞
f(x)dx (2.1.3)
where λ is the intensity of N(t).
The characteristic function of the compound Poisson process is also given by
E eizX(t)
= exp λt E eizYm
− 1 (2.1.4)
where E eizYm denotes the characteristic function of F.
2.1.9 Proposition. (Itô formula for jump - diffusion processes ).
Let X be a diffusion process with jumps, defined as the sum of a drift term (a Brownian Stochastic
integral) and a compound Poisson process:
X(t) = X(0) +
t
0
a(s)ds +
t
0
σ(s)dB(s) +
N(t)
m=1
Xm.

Section 2.2. Jump-Diffusion Models Page 6
where a(t) and σ(t) are continuous non-anticipating processes with
E
T
0
σ2
(t)dt < ∞.
Then, for any function f ∈ C1,2([0, T]×R), the process Yt = f(t, X(t)) can be represented in differential
form as:
df(X(t), t) =
∂f(X(t), t)
∂t
dt + a(t)
∂f(X(t), t)
∂x
dt +
σ(t)2
2
∂2f(X(t), t)
∂x2
dt +
σ(t)
∂f(X(t), t)
∂x
dB(t) + [f(X(t) + X(t)) − f(X(t))] .
2.2 Jump-Diffusion Models
Jump-diffusion models are special cases of Lévy models where the frequency of the jump is finite. In
jump-diffusion processes, the underlying stock price, S(t), is mostly described by a continuous diffusion
part and a discontinuous jump part. The usual fluctuation in the stock prices is accounted for by the
continuous diffusion part (determined by a Brownian motion). The discontinuous jump part is responsible
for the extreme events and is determined by an impulse function, η. This impulse function causes changes
in the underlying asset price, and is also determined by a distribution function. According to Merton
(1976), “the jump part enables us to model sudden and unexpected price jumps of the underlying asset”.
The general formula for the jump-diffusion models is given by
dS(t) = αS(t)dt + σS(t)B(t) + ηS(t)dN(t) (2.2.1)
where S(t) is the stock price at time t, B(t) is a standard Brownian motion, N(t) is a Poisson process
with an intensity λ and η is an impulse function which causes a jump of S(n) to S(n + 1). Examples
of the jump-diffusion processes are:
1. the Merton Jump-diffusion model where
η(x) = N(α, ω).
2. the Kou model with double exponential distributions, where
η(x) = pη1 exp(−η1x)1(x≥0) + qη2 exp(η2x)1(x<0)
where p is the probability that a jump occurs, q is the probability that a jump does not occurs
and 1 is an indicator function.
3. the variance Gamma model, where
η(x) =
C eGx
−x , x < 0
C e−Mx
x , x > 0
In our case, we will consider the first example (Merton Jump-Diffusion model) against the classical
model of Black and Scholes.

Section 2.3. Merton Jump-Diffusion Model Page 7
2.3 Merton Jump-Diffusion Model
The Merton Jump-Diffusion model is an exponential Lévy process of the form;
St = S0 eL(t)
,
where the stock price process {S(t)}t≥0 is modelled as an exponential Lévy process {L(t)}t≥0 of finite
jumps. Merton’s choice of the Lévy process was based on the assumption that, an underlying asset
price, S(t), follows the continuous-dynamic model with instantaneous return given by the Stochastic
Differential Equation (SDE);
dS(t)
S(t)
= αdt + σdB(t) + dJ(t) (2.3.1)
where α is the instantaneous expected return on the stock, σ is the volatility of the stock price, B(t) is
the Brownian motion and J(t) is the discontinuous jump part. The term αdt + σdB(t) is a Brownian
motion with a drift process (that is, the continuous diffusion part) and
J(t) =
N(t)
m=1
Ym
is a compound Poisson process, where N(t) is a Poisson process. The jump sizes Ym are independent
and identically distributed with distribution F. We shall assume that Ym > −1, which ensures non-
negative stock prices. The difference between the Black-Scholes and Merton Jump-Diffusion models is
the addition of the jump part in equation (2.3.1).
2.3.1 Derivation of Merton Jump-Diffusion Model.
We derive Merton jump-diffusion model (2.3.1) based on the assumption that, jumps in the stock
prices occur identically and independently. In Merton’s model, the continuous diffusion part is normally
distributed which is mostly modelled by a Brownian motion with a drift process. The discontinuous
jump part is modelled by a compound Poisson process.
Using a Poisson process dN(t), the probabilities that an asset price jumps within a small time interval
dt are as follows;
• the probability that an asset price jumps once in the time interval dt is
P {dN(t) = 1} ∼= λdt,
• the probability that an asset price does not jump in the time interval dt is
P {dN(t) = 0} ∼= 1 − λdt,
• the probability that an asset price jumps more than once in the time interval dt is
P {dN(t) ≥ 2} ∼= 0,
where the parameter λ is the average number of jumps per unit of time (intensity) and is independent
of time t.

Now, we denote yt as the absolute price jump size. Assuming that in the small time interval dt, the
underlying asset price jumps from S(t) to ytS(t), then the percentage change in the asset price caused
by the jump (relative price jump size) is given by
dS(t)
S(t)
=
ytS(t) − S(t)
S(t)
= yt − 1 (2.3.2)
where yt is a non-negative random variable (as assumed by (Merton, 1976)) drawn from log-normal
distribution. Thus, ln(S(t)) is log-normal distributed with mean µ and variance δ2, then,
E [yt] = eµ+1
2
δ2
E (yt − E [yt])2
= E y2
t − 2eµ+1
2
δ2
E [E [yt]] + E [yt]2
= e2µ+2δ2
− 2e2µ+δ2
+ e2µ+δ2
= eδ2
− 1 e2µ+δ2
.
Incorporating the above properties into the Merton jump-diﬀusion dynamic of asset price, we obtain a
SDE of the form;
dS(t)
S(t)
= (α − λk)dt + σdB(t) + (yt − 1)dN(t). (2.3.3)
Here, B(t), N(t) and yt are assumed to be independent processes. Hence, the relative price jump size
in equation (2.3.2) is log normally distributed with mean;
E [yt − 1] = E [yt] − 1 = eµ+1
2
δ2
− 1 ≡ k
and variance;
E (yt − 1 − E [yt − 1])2
= E (yt − 1)2
− 2E [yt − 1] E [E [yt − 1]] + E [yt − 1]2
= E y2
t − 2yt + 1 − 2k2
+ k2
= e2µ+2δ2
− 2k + 1 − k2
= e2µ+2δ2
− 2 e2µ+1
2
δ2
− 1 + 1 − e2µ+δ2
+ 2e2µ+1
2
δ2
− 1
= e2µ+2δ2
− e2µ+δ2
+ 2
= eδ2
− 1 e2µ+δ2
+ 2.
Equivalently, we can say that, the log-return jump size ln ytS(t)
S(t) is a normal random variable such
that
ln
ytS(t)
S(t)
= ln(yt) ≡ Yt ∼ Normal(µ, δ2
).
Since
E [(yt − 1)dN(t)] = E [(yt − 1)] E [dN(t)] = kλdt

then, using equation (2.3.3), the expected relative change becomes
E
dS(t)
S(t)
= E [(α − λk)dt] + E [σB(t)] + E [(yt − 1)dN(t)]
= (α − λk)dt + 0 + kλdt, since B(t) is independent E(B(t)) = 0
= αdt.
This is by far the predictable part of the jump. This provides the reason why the expected return of
the asset αdt is adjusted by −λkdt in the drift term of equation (2.3.3) to make the jump part an
unpredictable event.
In a small time interval dt, if the asset price does not jump ( dN(t) = 0), then the jump-diffusion
process is a simple Brownian motion with a drift (that is, Black-Scholes model) and is given by
dS(t)
S(t)
= (α − λk)dt + σdB(t).
If the asset price jumps in small time interval dt (dN(t) = i), then we have the Merton jump-diffusion
for the relative price jump of (yt − 1) given by
dS(t)
S(t)
= (α − λk)dt + σdB(t) + (yt − 1)dN(t). (2.3.4)
2.3.2 Solution to the Merton Jump-Diffusion.
We now look for the solution to the SDE in equation (2.3.4), that is
dS(t)
S(t)
= (α − λk)dt + σdB(t) + (yt − 1)dN(t)
Applying the Itô formula for jump-diffusion model provided by Cont and Tankov (2004) in proposition
2.1.9 we obtain;
d(ln S(t)) =
∂(ln S(t))
∂t
dt + (α − λk)S(t)
∂(ln S(t))
∂s
dt +
σ2
2
S(t)2 ∂2(ln S(t))
∂s2
dt +
σS(t)
∂(ln S(t))
∂s
dB(t) + [ln(ytS(t)) − ln(S(t))]
= 0 + (α − λk)S(t)
1
S(t)
dt +
σ2
2
S(t)2 −1
S(t)2
dt + σS(t)
1
S(t)
dB(t) +
[ln yt + ln S(t) − ln S(t)]
d(ln S(t)) = (α − λk)dt −
σ2
2
dt + σdB(t) + ln yt
d(ln S(t)) = (α −
σ2
2
− λk)dt + σdB(t) + ln yt (2.3.5)

Integrating equation (2.3.5) over the time interval 0 ≤ s ≤ t, we obtain
t
0
d(ln S(s)) =
t
0
(α −
σ2
2
− λk)ds +
t
0
σdB(s) +
N(t)
i=1
ln yi
ln S(t) − ln S(0) = (α −
σ2
2
− λk)(t − 0) + σ(B(t) − B(0)) +
N(t)
m=1
ln y.
But B(0) = 0 from the definition of the Brownian motion and ln ym ≡ Ym from the definition of the
log-return jump size. So
ln S(t) = ln S(0) + (α −
σ2
2
− λk)t + σB(t) +
N(t)
m=1
Ym (2.3.6)
S(t) = S(0) exp



(α −
σ2
2
− λk)t + σB(t) +
N(t)
m=1
Ym



.
Or
S(t) = S(0) exp (α −
σ2
2
− λk)t + σB(t) exp



N(t)
m=1
ln ym



= S(0) exp (α −
σ2
2
− λk)t + σB(t)
N(t)
m=i
ym. (2.3.7)
This shows that the price process {S(t) : 0 ≤ t ≤ T} is modelled as an exponential Lévy process of the
form
S(t) = S(0)eL(t)
,
in which S(t) is a Lévy process of finite jumps such that;
ln
S(t)
S(0)
= L(t) = (α −
σ2
2
− λk)t + σB(t) +
N(t)
m=1
Ym.
2.3.3 Remark.
For no jumps within the time interval [0, t], the compound Poisson process in log-price scale becomes
N(t)
m=1
ln ym =
N(t)
m=1
Ym = 0
or in absolute price scale becomes
N(t)
m=1
ym = 1

if N(t) = 0. This results in the Black-Scholes case of log-return ln S(t)
S(0) which is normally distributed
thus, ln S(t)
S(0) ∼ Normal (α − σ2
2 )t, σ2t . In Merton Jump-diffusion case, due to the presence
of compound Poison jump process the log-return is non-normal. In this respect, Merton made a simple
assumption that, the log-return jump size (Ym) is normally distributed with mean µ and variance δ2,
that is, (Ym) ∼ N(µ, δ2) or ln yt ∼ N(µ, δ2). Hence, the distribution of the log-return is Gaussian.
This assumption helps us to obtain the probability density of the log-return st = ln S(t)
S(0) as a
converging series of the form;
F(st) =
∞
i=1
P(N(t) = i)f(st | N(t) = i)
F(st) =
∞
i=1
e−λt(λt)i
i!
φ st ; (α −
σ2
2
− λk)t + iµ, σ2
t + iδ2
(2.3.8)
where
∞
i=1
e−λt(λt)i
i!
= P(N(t) = i)
is the probability that the asset price jump i times during the time interval of length t, and
φ st ; (α −
σ2
2
t + iδ2
= f(st | N(t) = i)
f(st | N(t) = i) =
1
2π(σ2t + iδ2)
exp



−
st − (α − σ2
2 − λk)t + iµ
2
2(σ2t + iδ2)



. (2.3.9)
For the Black-Scholes, the log-return assuming that the asset price jumps i times, has the normal density
φ st ; (α − σ2
2 − λk)t + iµ, σ2t + iδ2 in the time interval of t. Matsuda (2004) interprets the log-
return density in the Merton jump-diffusion model as the weighted average or mean of the Black-Scheles
normal density.
Applying Fourier Transform (FT) with parameters (a, b) = (1, 1) in definition 2.1.5, the Merton

log-return density function has the following characteristic function
φS(t)(ω) = E eiωS(t)
=
∞
∞
exp(iωst)F(st)dst
=
∞
∞
exp(iωst)
∞
i=0
e−λt(λt)i
i!



1
2π(σ2t + iδ2)
exp



−
st − (α − σ2
2 − λk)t + iµ
2
2(σ2t + iδ2)





 dst
= exp λ t exp
1
2
ω(2iµ − δ2
ω) − λ t(1 + iωk) −
1
2
tω −2iα + σ2
(i + ω)
= exp λ t exp iωµ −
δ2ω2
2
− λt(1 + iωk) + itωα −
σ2
2
tωi −
σ2
2
tω2
= exp t λ exp iωµ −
δ2ω2
2
− λ(1 + iωk) + iωα −
σ2
2
ωi −
σ2
2
ω2
= exp t λ exp iωµ −
δ2ω2
2
− 1 + iω α −
σ2
2
− λk −
σ2ω2
2
= exp [t Ψ(ω)]
where
Ψ(ω) = λ exp iωµ −
δ2ω2
2
− 1 + iω α −
σ2
2
− λk −
σ2ω2
2
, (2.3.10)
k = eµ+1
2
δ2
− 1, µ is the expected log-return jump size. The sign of µ determines the type of skewness
of the characteristic function of the Merton jump-diffusion density. ω is the frequency of the character-
istic function (thus, ω = 2π).
2.3.4 Definition (Cumulant).
Let φ(u) be the characteristic function defined in terms of the Fourier Transform (FT) of the probability
density function F(x). Using FT parameters (a, b) = (1, 1) we have
φ(u) =
∞
−∞
exp(iux)F(x)dx.
The Cumulants kn are then defined by
ln φ(t) =
∞
n=1
kn
(it)n
n!
.
Definition 2.3.4 can be found in Kou (2008).
Using Maclaurin series we have
ln φ(t) = (it)µ1+
1
2!
(it)2
(µ2−µ
2
1 )+
1
3!
(it)3
(2µ
3
1 −3µ1µ2+µ3)+
1
4!
(it)4
(−6µ
4
1 +12µ
2
1 µ2−3µ
2
2 −4µ1µ3+µ4)+. . .

where µn, n = 1, 2, 3, . . . are raw moments.
Note: The function Ψ(ω) is the cumulant generating function ( characteristic exponent) of the char-
acteristic function φS(t) of the Merton’s log-return density function.
2.3.5 Remark.
A moment µn of a probability function P(x) taken about the origin, is given by
µn = xn
− 0 = xn
F(x)dx.
Now the ﬁrst four cumulants of the characteristic exponent of equation (2.3.10) are
k1 = α −
σ2
2
− λk + λµ
k2 = σ2
+ λδ2
+ λµ2
k3 = λ(3δ2
µ + µ3
)
k4 = λ(3δ4
+ 6µ2
δ2
+ µ4
).
The mean, variance, skewness and excess kurtosis annualized per unit of time of Merton’s log-return
density are as follows;
• The mean of the log-return st is the ﬁrst moment of the cumulant generating function Ψ(ω), that
is
E [st] = k1 = α −
σ2
2
− λk + λµ
= α −
σ2
2
− λ eµ+1
2
δ2
− 1 + λµ
• The variance of the log-return st is the second moment of the cumulant generating function Ψ(ω),
that is
V ar [st] = k2 = σ2
+ λδ2
+ λµ2
• The skewness of the log-return st is the third moment of the cumulant generating function Ψ(ω),
that is
skew [st] = E
(S − E [st])3
( V ar [st])3
,
from the linearity of expectation we have
skew [st] =
E s3
t − 3k1E s2
t + 3k2
1E [st] − k3
1
k
3
2
2
=
k3
k
3
2
2

Hence, for the Merton jump-diffusion model we have;
skew [st] =
λ(3δ2µ + µ3)
(σ2 + λδ2 + λµ2)
3
2
.
Note: If µ < 0 the log-return density is negatively skewed and if µ > 0, then the log-return
density is positively skewed. It is symmetry if µ = 0.
• The excess kurtosis of the log-return st is the fourth moment of the cumulant generating function
Ψ(ω), that is
kurt [st] =
E (S − E [st])4
( V ar [st])4
,
from the linearity of expectation we have
kurt [st] =
E s4
t − 4k1E s3
t + 6k2
1E s2
t − 3k4
1
k2
2
=
k4
k2
2
Hence, for the Merton jump-diffusion model we have;
kurt [st] =
λ(3δ4 + 6µ2δ2 + µ4)
(σ2 + λδ2 + λµ2)2
.
Let us observe these properties of Merton’s log-return density. First is the skewness property; Figure
2.1 below shows that, the Merton jump-diffusion model incorporates skewness. The log-return density
Figure 2.1: Different values of µ for Merton’s log-return density (2.3.8). The values of µ are µ = 0 in
black, µ = 0.7 in red and µ = −0.7 in green. Parameters α = 0.05, σ = 0.3, λ = 0.2 and δ = 0.4 are
fixed.
for the Merton jump-diffusion model is symmetric at µ = 0 in black, negatively skewed for µ = −0.7 in
green and positively skewed for µ = 0.7 in red. Figure 2.2 below also shows Merton’s log-return density
superimposed by the Black-Scholes log-return for reference. Clearly, we see that, the Merton jump-
diffusion model(MJD) incorporates skewness whereas the Black-Scholes model(BS) does not. Finally,

Figure 2.2: Merton log-return density (MJD) (2.3.8) versus Black-Scholes log-return density (BS)
(2.3.9) for µ = 0.9. Parameters α = 0.05, σ = 0.3, λ = 0.2 and δ = 0.4 are fixed.
Figure 2.3 below illustrates the comparison of the peaks for Merton’s log-return density and Black-
Scholes density, that is, leptokurtic features. The log-return density for the Merton Jump-diffusion
model has a higher peak (in black) than that of the Black-Scholes model (in red). This comparison can
also be observed from Figure 2.2 above. In addition, Merton’s density has a heavier (flatter) tail than
that of Black-Scholes.
Figure 2.3: Merton’s log-return density (2.3.8) versus Black-Scholes log-return density (2.3.9). Fixed
parameters are µ = 0, λ = 0.2, α = 0.05, σ = 0.3, δ = 0.4
2.3.6 Merton Jump-Diffusion Measure.
The measure for the Merton jump-diffusion model depends on the measures for compound Poisson and
Poisson processes. The product of the measure of the Poisson process, λ, and the total mass (density)
jump size f(dx) is the measure for the compound Poisson process l(dx);
l(dx) = λf(dx).

This means that, the measure l(dx) of the compound Poisson process is a measure of the average
number of jumps size per unit of time. The measure l(dx) > 0 on the real line However, is not a
probability measure since its average number of jump sizes per unit of time is not equal to 1, thus
λ = l(dx), λ ∈ R+
.
Both Poisson and compound Poisson processes are examples of finite Lévy processes since their measures
are finite.
Now, since the log-return jump size (dx) is normally distributed with mean µ and variance δ2, that is,
f(dx) =
1
δ
√
2π
exp −
(dx − µ)2
2δ2
,
then, the Merton jump-diffusion measure is given by
l(dx) = λ f(x) =
λ
δ
√
2π
exp −
(dx − µ)2
2δ2
.
Large values of the Poisson intensity λ means that frequent occurrence of jumps are not expected and
the Merton’s log-return density becomes flatter-tailed as shown below in Figure 2.4. The larger the
value of λ the more the Merton’s log-return density approaches the Black-Scholes density. We can also
Figure 2.4: Different values of intensity, λ, for Merton’s log-return density (2.3.8).
λ = 1 in black, λ in red and λ in green. Fixed parameters are µ = 0, α = 0.05, σ = 0.3, λ = 1 and
δ = 0.4.
observe that excess kurtosis in the case of smaller values of intensity, that is, λ = 1 or λ = 3 are much
larger than the case of large values of intensity , λ = 5. This is due to the fact that excess kurtosis is
standardised by standard deviation.

Section 2.4. Option Pricing using Martingale Approach Page 17
2.4 Option Pricing using Martingale Approach
The focus for this section is to study option pricing using martingale approach. Therefore, it will be
useful to start with some basic definitions in finance.
2.4.1 Definition (Financial market).
A financial market is a market in which financial instruments such as bonds, stocks and commodities
are traded. Examples include the bond market, stock market (such as the New York Stock Exchange
and the Dar es Salaam Stock Exchange), foreign exchange market, futures and options markets. (Cont
and Tankov, 2004)
2.4.2 Definition (Contingent claim).
Contingent claim (Derivative security) is a financial instrument whose value depends on the price of
some underlying products. For instance, option prices depend on stock prices. (Cont and Tankov, 2004)
2.4.3 Definition (option).
An option is a contract giving the holder the right, but not the obligation, to buy or sell at or within a
specified (maturity) date T at a strike (exercise or predetermined) price K. (Cont and Tankov, 2004)
2.4.4 Remark.
• the right to buy an option is referred to as Call option whereas the right to sell an option is Put
option.
• types of options are
- European options: can be exercised only at the maturity or expiration date T only.
- American options: can be exercised at any time from initial time t = 0, up to the maturity date
T.
- Asian options depend on the average price over a period of time.
- Lookback options: depend on the maximum or minimum price over a period of time.
- Barrier options depend on some price level being attained or not.
For our case, we will look at the European option pricing.
2.4.5 Martingale Approach for Option Pricing.
The pricing rule says that, the value of an European option with terminal pay-off H(S(T)) = max [S(T) − K, 0]
at maturity date T is expressed as a discounted conditional expectation of its terminal pay-off under
risk-neutral probability Q :
Ct = E e−r(T−t)
H(S(T))|Ft .
Applying the Markov property in proposition 2.1.7 we have
Ct = C(t, S).
Then
C(t, S) = E e−r(T−t)
H(S(T))|S(t) = S .
Let us assume that the presence of a money market account ert where r is risk-free interest rate. Then
we would require the discounted option process Ht(S(t))
ert to be martingale under the martingale mea-
sure Q. It is well-known that martingale measure Q is not unique due to jumps. From the beginning of

chapter one, we assumed that our market is complete under the probability measure P but, the Merton
jump-diffusion model in our case is incomplete. Hence, we need to change the measure from complete
probability measure P to an incomplete martingale measure Q.
2.4.6 Definition (Equivalence Measure).
Define two probability measures Q and P on Ω with σ− algebra on Ft. Then, measure P is absolutely
continuous with respect to measure Q, that is P Q;
if Q(A) = 0 P(A) = 0, ∀A ⊂ Ft.
If Q P and P Q, then both P and Q measures are equivalent (Q P).
The definition above can be found in (Cont and Tankov, 2004).
2.4.7 Theorem.
Let P and Q be two measures on a probability space such that Q is absolutely continuous with respect
to P, that is , Q P. Then there is a unique non-negative function Z : Ω → (0, 1) such that
1) Z is F−measurable .
2) Q(A) = A Z(x)dP(x) < ∞, ∀A ⊂ Ft and ∀x ∈ Ω.
In this case
dQ = ZdP,
⇒ Z =
dQ
dP
.
The function Z is the Randon-Nikodým derivative of Q with respect to P.
Theorem 2.4.7 can be found in (Cont and Tankov, 2004). For the proof of Theorem 2.4.7 see Appendix
5.1.
We already know that the process (B(t))t≥0 is a standard Brownian motion on the complete probability
space (Ω, F, P), and the Merton jump-diffusion asset price process, under the actual probability
measure P of the complete space, is given in integral form as
L(t) = ln
S(t)
S(0)
= α −
σ2
2
− λk t + σB(t) +
N(t)
m=1
Ym.
So
S(t) = S(0)eL(t)
= S(0) exp



α −
σ2
2
− λk t + σB(t) +
N(t)
m=1
Ym



.
We will use index i as the number of jumps: N(T − t) ≡ i = 0, 1, . . .
Also, process
N(T−t)
m=1 Ym is normally distributed with mean iµ and variance iδ2.
Now, due to the several equivalent martingale risk-neutral measures Q ∼ P under which the discounted
price process (e−rtS(t))t≥0 becomes a martingale, the Merton jump-diffusion process is an incomplete
process or model. Due to this, we will use Merton’s martingale risk-neutral measure Q ∼ P equivalent
to Q ∼ P by changing the drift part of the Brownian motion process and keep all other parts unchanged,
that is
S(t) = S(0) exp



αQ
t + σB(t)Q
+
N(t)
m=1
Ym



. (2.4.1)

where B(t)Q is a standard Brownian motion under the risk-neutral measure Q and αQ is chosen such
that ˆS(t) = S(t)e−rt is a martingale under Q.
So
αQ
= r −
σ2
2
− λk
where r is the risk-free interest rate. Hence,
S(t) = S(0) exp



r −
σ2
2
− λk t + σB(t)Q
+
N(t)
m=1
Ym



. (2.4.2)
2.4.8 Remark.
The processes B(t)Q Normal r − σ2
2 − λk , σ2t ,
N(t)=i
m=1 Ym Normal(iµ, iδ2) and
Ym iidNormal(µ, δ2). Since we assumed that the processes B(t)Q, N(t) and Ym are independent,
then σB(t)Q +
N(t)=i
m=1 Ym Normal r − σ2
2 − λk , σ2t + iδ2 .
We will now work under the martingale risk-neutral measure Q. Hence, the value of the European call
option of the Merton jump-diﬀusion model (MJD) under the measure Q with terminal pay-oﬀ H(S(T))
is expressed as
VMJD(t, S(t)) = VMJD = EQ
e−r(T−t)
H(S(T))|Ft .
Applying the Markov property in proposition 2.1.7 and using τ = T − t, we have
VMJD = EQ

e−rτ
H

S(t) exp



r −
σ2
2
− λk τ + σB(τ)QM
+
N(τ)
m=1
Ym






 ,
VMJD = e−rτ
EQ

H

S(t) exp



r −
σ2
2
− λk τ + σB(τ)Q
+
N(τ)
m=1
Ym






 .
Since the index i denotes the number of jumps per unit of time, then VMJD(t, S(t)) can be conditioned
by it as follows
VMJD = e−rτ
∞
i=0
Q(N(τ) = i)EQ
H S(t) exp r −
σ2
2
− λk τ + σB(τ)Q
+
i
m=1
Ym .
We recall that k = eµ+δ2
2 − 1, so
VMJD = e−rτ
∞
i=0
e−λτ (λτ)i
i!
EQ
× (2.4.3)
H S(t) exp r −
σ2
2
− λ eµ+δ2
2 − 1 τ + σB(τ)Q
+
i
m=1
Ym .
From equation (2.4.3) above, we see that the exponent of the exponential function
exp r −
σ2
2
− λ eµ+δ2
2 − 1 τ + σB(τ)Q
+
i
m=1
Ym

is normally distributed with mean r − σ2
2 − λ eµ+σ2
2 − 1 τ + iµ and variance σ2τ + iδ2.
With the same distribution we can rewrite the exponent of equation (2.4.3) as
r −
σ2
2
− λ eµ+δ2
2 − 1 τ + iµ + σ2
+
iδ2
τ
1
2
B(τ)Q
.
The reason for the above operation is that a normal density is uniquely determined by its mean and
variance. Therefore, we can write VMJD(t, S(t)) as follows;
VMJD = e−rτ
∞
i=0
e−λτ (λτ)i
i!
EQ
×
H S(t) exp r −
σ2
2
− λ eµ+δ2
2 − 1 τ + iµ + σ2
+
iδ2
τ
1
2
B(τ)Q
.
Introducing the terms iδ2
2τ and −iδ2
2τ , we have
VMJD = e−rτ
∞
i=0
e−λτ (λτ)i
i!
EQ
×
H S(t) exp r −
1
2
σ2
+
iδ2
τ
+
iδ2
2τ
− λ eµ+δ2
2 − 1 τ + iµ + σ2
+
iδ2
τ
1
2
B(τ)Q
.
Let σi = σ2 + iδ2
τ , then we have
VMJD = e−rτ
∞
i=0
e−λτ (λτ)i
i!
EQ
×
H S(t) exp r −
1
2
σ2
i +
iδ2
2τ
− λ eµ+δ2
2 − 1 τ + iµ + σiB(τ)Q
.
= e−rτ
∞
i=0
e−λτ (λτ)i
i!
EQM
×
H S(t) exp i µ +
δ2
2
− λ eµ+δ2
2 − 1 τ exp r −
1
2
σ2
i τ + σiB(τ)Q
.
The Black-Scholes price for the option is given by
VBS(τ, S(t); σ) = e−rτ
∞
i=0
e−λτ (λτ)i
i!
EQ
H S(t) exp r −
1
2
σ2
i τ + σiB(τ)QM
.
Hence, the formula for pricing the option for the Merton jump-diﬀusion model is expressed as the
discounted weighted average of the Black-Scholes price conditioned by i (number of jumps per unit

of time), is given by
VMJD = e−rτ
∞
i=0
e−λτ (λτ)i
i!
VBS(τ, S(t), σ)
= e−rτ
∞
i=0
e−λτ (λτ)i
i!
EQ
×
H S(t) exp r −
1
2
σ2
i − λ eµ+δ2
2 − 1 +
2iµ + iδ2
2τ
τ + σiB(τ)Q
=
∞
i=0
e−βτ (βτ)i
i!
EQ
×
H S(t) exp r −
1
2
σ2
i − λ eµ+δ2
2 − 1 +
2iµ + iδ2
2τ
τ + σiB(τ)Q
,
where β = λ(1 + k) = eµ+δ2
2 .

3. Model Calibration and Results
This chapter focuses on estimating the parameters of the Merton jump-diffusion model in equation
(2.3.6) using daily closing price data from the Dar es Salaam Stock Exchange. Also we will compare the
European call option prices of the Merton jump-diffusion and the Black-Scholes Models by assuming
that Dar es Salaam Stock Exchange price options. We will first discretize the Merton jump-diffusion
model (2.3.6) and then consider the Expectation Maximisation (EM) algorithm to obtain the estimation
of the model parameters. Lastly, we will compare the option prices from the Merton jump-diffusion and
the Black-Scholes Models.
3.1 Discrete -Time Process
The computation of the stock return can be done using the continuously compound (log-returns) returns
or relative returns. The relative return is defined as Rn = S(n)−S(n−1)
S(n−1) where as, the log-returns is
defined as
Rn = ln
S(n)
S(n − 1)
= ln S(n) − ln S(n − 1) (3.1.1)
over the unit time interval (n − 1, n] . The log-return is often used in theoretical modelling, however,
measurement of the underlying stock prices in continuous form are rarely. Hence, based on sampled
values of the stock prices S(t), inference must be made which will lead to the discrete-time process Rn
in equation (3.1.1). According to (Kou, 2008) “the difference between simple and log returns for daily
data is quite small, although it could be substantial for monthly and yearly data”.
Now, the discrete-time process of equation (2.3.6) is given by
Rn = ln S(n) − ln S(n − 1)
= ln S(0) + (α −
σ2
2
− λk)t + σB(n) +
N(n)
m=1
Ym −



ln S(0) + (α −
σ2
2
− λk)(n − 1) + σB(n − 1) +
N(n−1)
m=1
Ym



= (α −
σ2
2
− λk) + σ [B(n) − B(n − 1)] +
Nn
m=1
Ym,
where Nn = N(n) − N(n − 1).
Then
Rn = Gn +
Nn
m=1
Ynm, (n = 1, 2, . . .) (3.1.2)
where Gn = (α − σ2
2 − λk) + σ [B(n) − B(n − 1)] and Nn = N(n) − N(n − 1). Gn is an independent
random variables (Gaussian) with mean (α − σ2
2 − λk) and variance σ2. Also, Nn is an independent
random variable (Poisson process) with mean λ. Finally, Ynm (the number of jumps that occurs in the
time interval (n − 1, n] ) is an independent Gaussian process with mean µ and variance δ2.
22

Section 3.2. Estimation Page 23
3.1.1 Remark.
Gn, Nn, and Ynm are mutually independent discrete-time processes. Gn (incomplete or latent data) is
used to estimate the parameters α, σ2, δ2, µ and λ of the underlying stock price process.
Since Gn is a linear combination of the independent Gaussian random variables Gn and Nn, it has
mean (α − σ2
2 − λk) + µNn and variance σ2 + δ2Nn. In effect, Gn are identically and independently
distributed. The probability density function of Gn is
f(g) =
∞
i=0
φ(g; (α −
σ2
2
− λk) + iµ, σ2
+ iδ2
), (−∞ < g < ∞). (3.1.3)
We see that, Gn has an infinite Gaussian distribution with mixing coefficients. The mixing coefficients
are given by a Poisson process Nn with intensity λ.
3.2 Estimation
Many methods of parameter estimation of jump-diffusion models have been proposed and used by
various authors. Ball and Torous (1985) used direct maximum likelihood estimation and assumed that
µ = 0. They applied this procedure to the probability density (3.1.3 ). Also, Beckers (1981) and
Press (1967) used the method of matching sample, moments and population cumulants in fitting the
restricted form of the jump-diffusion model by assuming that α = 0 by (Press, 1967) and µ = 0 by
(Beckers, 1981). Their methods were not satisfactory due to the negative estimates of the variances
σ2 and δ2. Pickard et al. (1986) were the first to develop Expectation Maximization (EM) algorithm
to calculate the maximum likelihood estimates for jump-diffusion models, however, it was Jiang et al.
(1998) who made indirect inference in fitting the parameters of some jump-diffusion models.
As stated in chapter one, maximum likelihood estimation has it own properties and is efficient and con-
sistent with asymptotic Gaussian distributions under it is general condition of sufficiently large number
of observations. However, in our case the number of observations is not enough to implement the maxi-
mum likelihood estimation. In the next section we will study a new version of Expectation Maximization
(EM) procedure based on Pickard et al. (1986) version to estimate the Merton jump-diffusion model
parameters in equation (2.3.6).
3.3 Expectation Maximisation (EM) Procedure
The main objective in this section is to build on the EM algorithm developed by Pickard et al. (1986)
which will be more efficient than the direct maximum likelihood estimates under condition where the
latter is unsatisfactory.
To implement the EM procedure we will need two sets of data, that is, complete data, Rn and latent
(incomplete data), in our case is Gn, all in equation (3.1.2). The complete data is defined as independent
random vector as follows;
Rn =
(Gn, Nn) for Nn = 0
(Gn, Nn, Yn1, Yn2, . . . , YnNn ) for Nn = i > 0
where n = 1, 2, , . . . , T.
Since we have an estimation problem consisting of complete data (R1, R2, . . . , RT ) which is unobserved,

Section 3.3. Expectation Maximisation (EM) Procedure Page 24
we need to ﬁt the parameters of the Merton jump-diﬀusion model with probability density in equation
(2.3.8) to the data.
Before we move onto the EM algorithm, we need the log-likelihood of the complete data Rn. The
likelihood of the complete data is given by
Lc
(θ|θ0
) =
T
n=1
f (Rn; θ|Nn = i)
where θ = α, σ2, θ, µ, λ
T
.
Hence, the log-likelihood of the complete data Rn is given by
ln Lc
(θ|θ0
) = ln
T
n=1
f(Rn; θ|Nn = i)
= ln
T
n=1
f(Gn; θ|Nn = 0) or
T
n=1
f(Ynm; θ|Nn = i > 0)
=
T
n=1
ln f(Gn; θ|Nn = 0) +
T
n=1
ln f(Ynm; θ|Nn = i > 0)
=
T
n=1
ln
1
√
2πσ2
exp −
(Gn − α )2
2σ2
+
T
n=1
ln
e−λλi
i!
1
2π(iδ)2
exp −
Nn
m=1
(Ym − iµ)2
2(iδ)2
=
T
n=1
−
1
2
ln(2π) −
1
2
ln(σ2
) −
(Gn − α )2
2σ2
+
T
n=1
−λ + i ln(λ) − ln(i!) −
1
2
ln(2πi) −
1
2
ln(iδ2
) −
1
2δ2
Nn
m=1
Ym
Nn
− µ
2
=
T
n=1
−
1
2
ln(2π) −
1
2
ln(σ2
) −
(Gn − α )2
2σ2
+
T
n=1
−λ + i ln(λ) − ln(i!) −
1
2
ln(2πi) −
1
2
ln(iδ2
) −
1
2δ2
Nn
m=1
(Ynm − µ)2
ln Lc
(θ|θ0
) = −
T
2
ln(2π) −
T
2
ln(σ2
) −
1
2σ2
T
n=1
(Gn − α )2
− λT + ln(λ)
T
n=1
Nn −
T
n=1
ln(Nn!) −
1
2
T
n=1
ln(2πNn) −
1
2
T
n=1
ln(Nnδ2
) −
1
2σ2
T
n=1
Nn
m=1
(Ynm − µ)2

Now, given the observations (G1, G2, . . . , GT ) the best predictor of ln(Lc(θ|θ0)) is
Q(θ|θ0
) = E0
[ln(Lc
(θ))|Rn] (3.3.1)
where E0 is the expectation with respect to the time distribution of the initial parameter vector θ0 =
(α0, δ2
0, µ0, σ2
0, λ0)T . Equation (3.3.1) is the starting point for the EM procedure. Fixing initial
parameters θ0, we estimate new parameters ˆθ = (ˆα, ˆδ2, ˆµ, ˆσ2, ˆλ)T by
(i) ﬁrst evaluating the conditional expectation on the right hand side of equation (3.3.1) given θ0.
This process is referred to as the Expectation-step (E-step).
(ii) maximizing Q(θ|θ0) subject to θ and this process is also referred to as Maximization-step (M-step).
The EM criterion for E-step is
Q(θ|θ0
) = E0
ln(Lc
(θ|θ0
))|Rn
= −
T
2
ln(2π) −
T
2
ln(σ2
) −
1
2σ2
T
n=1
E0
(Gn − α )2
|Rn − λT + ln(λ)
T
n=1
E0
[Nn|Rn] −
T
n=1
E0
[ln(Nn!)|Rn] −
1
2
T
n=1
E0
ln(Nnδ2
)|Rn −
1
2σ2
T
n=1
E0
Nn
m=1
(Ynm − µ)2
|Rn
= −
T
2
ln(2π) −
T
2
ln(σ2
) −
1
2σ2
T
n=1
E0
(Gn − α )2
|Rn − λT + ln(λ)
T
n=1
E0
[Nn|Rn] −
T
n=1
E0
[ln(Nn!)|Rn] −
1
2
T
n=1
E0
ln(Nnδ2
)|Rn −
1
2σ2
T
n=1
E0
Nn(Ynm − µ)2
.
Next, the EM procedure for M-step is as follows; we maximise Q(θ|θ0) subject to θ.
Taking partial derivative of Q(θ|θ0) with respect to λ, we have
∂Q(θ|θ0)
∂λ
= −T +
1
λ
T
n=1
E0
[Nn|Rn] = 0
∴ ˆλ =
1
T
T
n=1
E0
[Nn|Rn] .
Also, taking partial derivative of Q(θ|θ0) with respect to α , we get
∂Q(θ|θ0)
∂α
= −
1
2σ
∂
∂α
T
n=1
E0
G2
n|Rn − 2α
T
n=1
E0
[Gn|Rn] + α
2
T = 0
⇒ 0 = −
1
σ2
T
n=1
E0
[Gn|Rn] + α T
∴ ˆα =
1
T
T
n=1
E0
[Gn|Rn] .

But ˆα = ˆα − ˆσ2
2 − ˆλˆk, where ˆk = eˆµ+ ˆσ2
2 − 1.
So
ˆα =
ˆσ2
2
+ ˆλˆk +
1
T
T
n=1
E0
[Gn|Rn] .
Next, taking partial derivative of Q(θ|θ0) with respect to σ2, we obtain
∂Q(θ|θ0)
∂σ2
= −
T
2
1
σ2
+
1
2(σ2)2
T
n=1
E0
(Gn − α )2
|Rn = 0
∴ ˆσ2
=
1
T
T
n=1
E0
(Gn − α )2
|Rn .
Also, taking partial derivative of Q(θ|θ0) with respect to δ2 , we obtain
∂Q(θ|θ0)
∂δ2
= −
1
2δ2
T
n=1
E0
[Gn|Rn] +
1
2(σ2)2
T
n=1
E0
Nn(Ynm − µ)2
|Rn = 0
⇒ ˆδ2
=
T
n=1 E0 Nn(Ynm − µ)2|Rn
T
n=1 E0 [Gn|Rn]
∴ ˆδ2
=
1
ˆλT
T
n=1
E0
Nn(Ynm − µ)2
|Rn .
Finally, taking partial derivative of Q(θ|θ0) with respect to µ we get
∂Q(θ|θ0)
∂µ
=
∂
∂µ
−
1
2δ2
T
n=1
E0
Nn(Ynm − µ)2
|Rn = 0
0 = 2
T
n=1
E0
[NnYnm|Rn] + 2µ
T
n=1
E0
[Nn|Rn]
ˆµ =
T
n=1 E0 [NnYnm|Rn]
T
n=1 E0 [Nn|Rn]
∴ ˆµ =
1
ˆλT
T
n=1
E0
[NnYnm|Rn] .
For us to implement the above formulae for ˆλ, ˆδ2, ˆµ, ˆσ2, ˆα , we need to evaluate all the conditional
expectations with respect to the initial parameters θ0 = (α0, σ2
0, µ, δ2
0, λ0)T .

3.3.1 Remark.
From covariance and correlation, the best linear predictor of Y given X is
L(Y |X) = E(Y ) +
Cov(X, Y )
V ar(X)
[X − E(X)] .
3.3.2 Lemma.
Suppose E(Y |X) is a linear function of X, thus E(Y |X) = a + bX for constants a and b, then
E(Y |X) = L(Y |X) = E(Y ) +
Cov(X, Y )
V ar(X)
[X − E(X)] ,
∴ a = E(Y ) − E(X)
Cov(X, Y )
V ar(X)
and
b =
Cov(X, Y )
V ar(X)
.
3.3.3 Theorem.
Suppose (X, Y ) has a bivariate normal distribution with mean E [Y |X = x] = E(Y )+Cov(X,Y )
V ar(X) [X − E(X)],
then for x ∈ R conditional variance of Y given X = x is normal given by
V ar [Y |X = x] = V ar(Y )

1 −
Cov(X, Y )
V ar(X)V ar(Y )
2

 .
For the proofs of Lemma 3.3.2 and Theorem 3.3.3 see appendix 5.2.
From Lemma 3.3.2 and Theorem 3.3.3 with δ = bσ, b ∈ R+, we have
E0
[Gn|Rn, Nn] = α0 − µ0
σ2
δ2
+
σ2
σ2 + Nnδ2
Rn − α + µ0
σ2
δ2
= α −
µ0
b2
0
+
1
1 + Nnb2
0
Rn − α +
µ0
b2
0
(3.3.2)
and
V ar0
[Gn|Rn, Nn] = σ2
0 1 −
σ2
σ2 + Nnδ2
= σ2
0 1 −
1
1 + Nnb2
0
. (3.3.3)
Now, using the results in equations (3.3.2) and (3.3.3), we have
E0
[Gn|Rn] = α0 −
µ0
b2
0
+ E0 1
1 + Nnb2
0
|Rn Rn − α0 +
µ0
b2
0
= (α0 −
σ2
0
2
− λ0k0) −
µ0
b2
0
+ γn(b0) Rn − α0 +
µ0
b2
0

and
E0
(Gn − α )2
|Rn = E0
[Gn|Rn] − ˆα
2
+ σ2
0 1 − E0 1
1 + Nnb2
0
|Rn +
V ar0 1
1 + Nnb2
0
|Rn Rn − α0 +
µ0
b2
0
2
= E0
[Gn|Rn] − ˆα
2
+ σ2
0 [1 − γn(b0)] + νn(b0) Rn − α0 +
µ0
b2
0
2
= E0
[Gn|Rn] − ˆα
2
+ σ2
0 [1 − γn(b0)] + νn(b0) Rn − (α0 −
σ2
0
2
− λ0k0) +
µ0
b2
0
2
,
where γn(b0) = E0 1
1+Nnb2
0
|Rn and νn(b0) = V ar0 1
1+Nnb2
0
|Rn .
The quantities γn(b0) and νn(b0) help to explicitly determine the parameters ˆα and ˆσ2.
Also, to be able to determine ˆµ and ˆδ2, we use the following results which holds for all Nn;
NnE0
[Ynm|Rn, Nn] = 1 −
1
1 + Nnb2
0
Rn − α0 +
µ0
b2
0
(3.3.4)
and
NnV ar0
[Ynm|Rn, Nn] = δ2
0 Nn − 1 +
1
1 + Nnb2
0
(3.3.5)
Applying the results in equations (3.3.4) and (3.3.5), we obtain
E0
[NnYnm|Rn] = 1 − E0 1
1 + NnB2
0
|Rn Rn − α0 +
µ0
b2
0
= [1 − γn(b0)] Rn − (α0 −
σ2
0
2
− λ0k0) +
µ0
b2
0
.
Also,
E0
Nn(Ynm − ˆµ)2
|Rn = δ2
0 E0
[Nn|Rn] − 1 + γn(b0) + ˆµ −
E0 [NnYnm|Nn]
E0 [Nn|Rn]
2
+
b2
0γn(b0) (1 − γn(b0)) − b2
0νn(b0) −
(1 − γn(b0))2
E0 [Nn|Rn]
Rn − (α0 −
σ2
0
2
− λ0k0) +
µ0
b2
0
Finally, we have to evaluate γn(b0), νn(b0) and E0 [Nn|Rn] . We can see that γn(b0) and νn(b0) are
functions of b0, Rn and θ0.
Now the conditional probability of Nn = i condition on Rn is given by
P0(Nn = i|Rn) ∝ φ(Rn; α0 + iµ, σ2
0 + iδ2
0)
λi
0
i!
.
Thus,
P0(Nn = i|Rn) =
φ(Rn; α0 + iµ, σ2
0 + iδ2
0)
λi
0
i!
∞
i=1 φ(Rn; α0 + iµ, σ2
0 + iδ2
0)
λi
0
i!
. (3.3.6)

Note: ˆα0 = α0 − σ2
2 − λ0k0 and k0 = eµ0+σ2
2 − 1.
The quantities γn(b0), νn(b0) and E0 [Nn|Rn] can be evaluated using equation (3.3.6).
Now,
γn(b0) = E0 1
1 + Nnb0
|Rn =
∞
i=0
1
1 + ib0
P0(Nn = i|Rn)
=
∞
i=0
1
1 + ib0
λi
i! φ(Rn ; ˆα0 + iˆµ0, σ2 + iˆδ2
0)
∞
i=0
λi
0)
.
Also,
νn(b0) = V ar
1
1 + Nnb0
|Rn
= E0 1
1 + Nnb0
2
|Rn − E0 1
1 + Nnb0
|Rn
2
=
∞
i=0
1
1 + ib0
2
P0(Nn = i|Rn) − (γn(b0))2
=
∞
i=0
1
1 + ib0
2 λi
0)
∞
i=0
λi
0)
−
∞
i=0
1
1 + ib0
λi
0)
∞
i=0
λi
0)
2
.
Lastly,
E0
[Nn|Rn] =
∞
i=0
e−λλi
i!
φ(Rn ; ˆα0 + iˆµ0, σ2
+ iˆδ2
0).
Substituting γn(b0), νn(b0), E0 [Nn|Rn] and the evaluated expectations into the formulae for the
estimated parameters ˆλ, ˆσ2, ˆα
2
, ˆδ2 , ˆµ, we obtain:
ˆλ =
1
T
T
n=1
∞
i=0
e−λλi
i!
φ(Rn ; ˆα0 + iˆµ0, σ2
+ iˆδ2
0)
=
1
T
T
n=1



∞
i=0
e−λλi
i!
1
2π(σ2
0 + iδ2
0)
exp



−
Rn − (α0 −
σ2
0
2 − λ0k0) − iµ0
2
2(σ2
0 + iδ2
0)






.

Also,
ˆσ2
=
1
T
T
n=1
E0
(Gn − ˆα )|Rn
=
1
T
T
n=1
E0
[Gn|Rn] − ˆα
2
+ σ2
0 1 −
∞
i=0
1
1 + ib0
λi
0)
∞
i=0
λi
0)
+
1
T
T
n=1
νn(b0) Rn − α0 −
σ2
0
2
− λ0k0 +
µ0
b2
0
2
=
1
T
T
n=1



α0 −
µ0
b2
0
+
∞
i=0
1
1 + ib0
λi
0)
∞
i=0
λi
0)
− ˆα
2



+
1
T
T
n=1
σ2
0 1 −
∞
i=0
1
1 + ib0
λi
0)
∞
i=0
λi
0)
+
1
T
T
n=1
νn(b0) Rn − α0 −
σ2
0
2
− λ0k0 +
µ0
b2
0
2
.
Next,
ˆα =
1
T
T
n=1
E0
[Gn|Rn]
=
1
T
T
n=1
ˆα0 −
µ0
b2
0
+ γn(b2
0) Rn − ˆα0 +
µ0
b2
0
= ˆα0 −
µ0
b2
0
+
1
T
T
n=1
∞
i=0
1
1 + ib0
λi
0)
∞
i=0
λi
0)
Rn − ˆα0 +
µ0
b2
0
.
So,
ˆα =
ˆσ2
2
+ ˆλˆk + α0 −
σ2
0
2
− λ0k0 −
µ0
b2
0
+
1
T
T
n=1
∞
i=0
1
1 + ib0
λi
0)
∞
i=0
λi
0)
Rn − α0 −
σ2
0
2
− λ0k0 +
µ0
b2
0
.

Also,
ˆδ2
=
1
ˆλT
T
n=1
δ2
0 E0
[Nn|Rn] − 1 + γn(b0) +
1
ˆλT
T
n=1
E0
[Nn|Rn] ˆµ −
E0 [NnYnm|Rn]
E0 [Nn|Rn]
2
1
ˆλT
T
n=1
b2
0γn(b0)(1 − γn(b0)) − b2
0νn(b0) −
(1 − γn(b0))2
E0 [Nn|Rn]
Rn − ˆα0 +
µ0
b2
0
2
.
Finally,
ˆµ =
1
ˆλT
T
n=1
E0
[NnYnm|Rn]
=
1
ˆλT
T
n=1
[1 − γn(b0)] Rn − α0 +
µ0
b2
0
=
1
ˆλT
T
n=1
1 −
∞
i=0
1
1 + ib0
λi
0)
∞
i=0
λi
0)
×
Rn − α0 −
σ2
0
2
− λ0k0 +
µ0
b2
0
.
3.3.4 Estimation of Merton Jump-Diffusion Model.
In this subsection we will estimate the Merton jump-diffusion model for a range of stocks. The estimation
method for the maximum likelihood estimation is the one discussed in the previous section and it is
implemented in Python programming language.
We look at six (6) actively Stocks of the Dar es Salaam Stock Exchange , each with daily closing prices
in the period January 2, 2007 to June 26, 2014.
The density function function for t period log-returns, st, has the form of equation (2.3.8), thus
F(st) =
∞
i=1
e−λt(λt)i
i!
φ st ; (α −
σ2
2
t + iδ2
(3.3.7)
where k = eµ+δ2
2 − 1.
The log-likelihood function of the density (3.3.7) the one is presented in the previous section. Many
authors usually estimate the parameters of equation (3.3.7) by maximising with respect to θ. However,
it is invalid to use direct maximum likelihood estimation in the Merton jump-diffusion model. The
reason is that, equation (3.3.7) is unbounded and also, based on the analogy of (Kiefer, 1978), if the
estimated mean (ˆα − ˆσ2
2 − ˆλk)t + iˆµ is chosen so that log-returns st exactly equal to the true mean
(α− σ2
2 −λk)t+iµ for any i then, as the estimated variance ˆσ2 +iˆδ2 goes to zero, the density function
F(st) increases without bound. This is due to the mixture of Gaussian distributions with different
means and variances of the log-return of Merton jump-diffusion model. In addition, the weight eλt(λt)i
i!
in equation (3.3.7) is not known and that, it is difficult to identify from the various distributions the
observation comes from. Therefore, the variances of the various distributions are different which makes

Section 3.4. Results and Discussion Page 32
the direct maximum likelihood estimation not applicable. Beckers (1981) and Ball and Torous (1985)
used the direct maximum likelihood in their estimation, and in some instances it was not surprising
that they got negative variances and other parameters outside the feasible parameter region. Following
Matsuda (2004), the implementation of the proposed procedure in the previous section for parameter
estimation, we need to make parametrisation of both µ and δ to help obtain the estimates of θ. So for
a fixed a, b ∈ D (D is a compact set on R+ ), we set δ = bσ and µ = aσ. This helps to make a
good initial guess for the EM algorithm.
Now, we approximate the density function of the Merton jump-diffusion model in equation (3.3.7) by
the first M terms of the sum for the estimation. The reason is that, the first M terms corresponds
to the discrete mixture of M normally distributed. To obtain consistent and asymptotically normally
distributed estimates, the evaluation of the log-likelihood will be in one dimension, because the variances
are described by σ and δ only. Lastly, due to the error which will results from the approximation of
equation (3.3.7) by first M terms, we will use sufficiently large M so that such error will be minimised.
Using daily observations, empirical results have shown that, there is no significant difference in the
estimates from M = 20. However, for practical purpose and implementation we will use M = 150.
For the initialisation of the EM algorithm, we use the values assumed by (Press, 1967) (α = 0) and
(Matsuda, 2004) (σ = 1, and λ = 0.2). Therefore, from the parametrisation of δ, µ and the assumed
value of σ by (Matsuda, 2004)(σ = 1), we have a = 0, b = 1 . Hence, our initial parameter vector for
the EM algorithm is θ0 = (α0 = 0, µ0 = 0, σ0 = 1, δ0 = 1, λ0 = 0.2).
3.4 Results and Discussion
The results from the Merton jump-diffusion model are as follows: The estimated parameters of the
Merton jump-diffusion model for the 6 stocks of the Dar es Salaam Stock Exchange is presented in
Table 3.1 below. For comparison purpose we also estimate the parameters of the Black-Scholes model
for the 6 Stocks in Table 3.2. The most desired feature of the Merton jump-diffusion model is the arrival
of the jump which is determined by the parameter λ (the expected jump amplitude).
Stock ˆα ˆµ ˆσ ˆδ ˆλ
SWISSPORT 0.0013 (0.045) 0.0031 (0.007) 0.396 (0.0073) 0.028 (0.0008) 20.493 (0.014)
TBL 0.00032 (0.055) 0.0007 (0.001) 0.2491 (0.0068) 0.0121 (0.0007) 31.152 (0.019)
SIMBA 0.0121 (0.005) 0.0067 (0.002) 0.284 (0.007) 0.005 (0.0001) 26.035 (0.014)
TWIGA 0.0017 (0.044) 0.00098 (0.003) 0.50172 (0.008) 0.0301 (0.0009) 27.818 (0.0143)
NMB -0.14175 (0.072) 0.00014 (0.0001) 0.12494 (0.0039) 0.0157 (0.0007) 29.09 (0.016)
DCB -0.1835 (0.109) 0.00011 (0.0001) 0.08791 (0.0009) 0.0090 (0.0001) 11.930 (0.007)
The estimations are based on the daily log-returns. The values in parentheses are standard errors.
Table 3.1: Merton Jump-Diffusion Model.
The empirical result presented in Table 3.1 shows that the expected jump amplitudes for all the 6
Stocks were high. Although, the values for λ does not really support theoretical (literature) claim that,
λ must be low enough to describe extreme events, it is beyond the scope of this report to make more
investigation into the improvement of including a jump component in the Black-Scholes model for such
purpose. One main interest of this report is to seek whether there are jumps in the stock prices. The
fact that values of λ are not zero show that, there are some jumps in the stocks. Comparing the result
in Table 3.1 and that of (Honore, 1998), it is clear that, the results presented in this report gives more

low values of λ but, not as low as to support theory and literature. Hence, the Merton jump-diffusion
model presented in Chapter two can be used to describe extreme events.
Also, the values of the volatilities ˆσ and ˆδ are non-negative which confirms that, the proposed Expectation
Maximisation (EM) algorithm is more efficient and robust than the direct Maximum Likelihood method.
From the descriptive statistics of the data from the Dar es Salaam Stock Exchange in appendix 5.4 of
Table 5.2 two of the Stocks are positively skewed and the other four negatively skewed. The skewness
of the 6 Stocks can also be seen from the histograms in Figures 3.7, 3.8 and 3.9 about their respective
means provided in the descriptive statistics of the data in Table 5.2. In addition, all the Stocks have
negative kurtosis which means that they have flatter-tailed distributions. Therefore, the Merton jump-
diffusion model can be used to describe the extreme events of the Stocks since it has been shown in
Chapter two that Merton model incorporates such features which are not found in Black-Scholes model.
STOCKS ˆα ˆσ
SWISSPORT 0.0273 (0.03) 0.14231 (0.0023)
TBL 0.038171 (0.032) 0.12163 (0.0022)
SIMBA 0.16261 (0.041) 0.14935 (0.004)
TWIGA 0.21713 (0.06) 0.15073 (0.0042)
NMB 0.3433 (0.075) 0.48517 (0.006)
DCB 0.25183 (0.067) 0.54489 (0.0062)
The estimations are based on the daily log-returns. The values in parentheses are standard errors.
Table 3.2: Black-Scholes Model.
To evaluate the price of an option (contingent claim), we need a dynamical model which describes the
value of the underlying stock S(t) at all times t between when the contingent claim (option contract)
is entered into the expiration data T. Such a dynamical model is the one presented in chapter two (that
is, Merton jump-diffusion model) with solution (2.3.7).
In part (a) of Figures 3.1, 3.2, 3.3, 3.4, 3.5 and 3.6 we plotted the daily closing prices of the Dar es
Salaam Stock Exchange for 6 Stocks (all on the left). Observe how the prices of all the Stocks have
increased dramatically in the last few years up to 2014. Also, part (b) of the same figures exhibit one
stimulated path of equation (2.4.2) which has been fitted to the daily closing prices from from Dar es
Salaam Stock Exchange. Following theory and literature we used low value for λ (that is, λ = 0.2, a
value assumed by Matsuda (2004)) in the stimulated paths. All the paths start at their respective last
closing prices in our time series data. These last closing prices are specified under each figure. The
stimulated path for each Stock represents possible future evolvements of the stock prices for the next 8
years corresponding to 2032 trading days. For the purpose of equal time scaling, note that, the time for
all stimulated path figures are in years and need to be scaled to trading days which corresponds to that
of the daily closing prices figures.The Dar es Salaam Stock Exchange has 254 trading days in a year
then, with change in time of dt = 0.001, the number of time steps for a year is 1001 and approximately
4 time steps for 1 trading day. Hence, stimulation for 8 years corresponds to 2032 trading days with
8008 time steps.

(a) Daily price levels of SWISSPORT. (b) Stimulated price levels of SWISS-
PORT.
Figure 3.1: Plots of SWISSPORT: The stimulated path of the Merton jump-diffusion (2.4.2) is fitted to
the daily closing prices of the Dar es Salaam Stock Exchange. Last closing price is 680 and parameters
are α = 0.15, σ = 0.396, λ = 0.2, µ = 0.007 and δ = 0.028.
(a) Daily price levels of TBL. (b) Stimulated price levels of TBL.
Figure 3.2: Plots of TBL: The stimulated path of the Merton jump-diffusion (2.4.2) is fitted to the
daily closing prices of the Dar es Salaam Stock Exchange. Last closing price is 1580 and parameters are
α = 0.15, σ = 0.249, λ = 0.2, µ = 0.0007 and δ = 0.012.
(a) Stimulated price levels of SIMBA. (b) Stimulated price levels of SIMBA.
Figure 3.3: Plots of SIMBA (a): The stimulated path of the Merton jump-diffusion (2.4.2) is fitted to
the daily closing prices of the Dar es Salaam Stock Exchange. Last closing price is 960 and parameters
are α = 0.15, σ = 0.284, λ = 0.2, µ = 0.0067 and δ = 0.005.

(a) Daily price levels of TWIGA. (b) Stimulated price levels of TWIGA.
Figure 3.4: Plots of TWIGA: The stimulated path of the Merton jump-diffusion (2.4.2) is fitted to the
α = 0.15, σ = 0.502, λ = 0.2, µ = 0.001 and δ = 0.03.
(a) Daily price levels of DCB. (b) Stimulated price levels of DCB.
Figure 3.5: Plots of DCB: The stimulated path of the Merton jump-diffusion (2.4.2) is fitted to the
α = 0.15, σ = 0.088, λ = 0.2, µ = 0.0001 and δ = 0.009.
(a) Daily price levels of NMB. (b) Stimulated price levels of NMB.
Figure 3.6: Plots of DCB: The stimulated path of the Merton jump-diffusion (2.4.2) is fitted to the
α = 0.15, σ = 0.125, λ = 0.2, µ = 0.0014 and δ = 0.016.

3.4.1 Option Pricing.
The goal of this subsection is to compare European call option prices from Merton jump-diffusion model
and the Black-Scholes model. The reason is to investigate the difference in the prices.
Although, Dar es Salaam Stock Exchange do not price options, assuming they do price options using
Black-Scholes model (since most Stock Exchange price options by Black-Scholes model) then, the
difference in the prices are presented in Table 3.3. In the risk-neutral world, the Martingale approach for
pricing European call option under st = ln S(t)
S(0) studied in chapter two is given by equation (3.3.8).
We approximate equation (3.3.8) by the first M terms (we take M = 150), that is
VMJD =
M
i=0
e−βτ (βτ)i
i!
EQ
H S(t) exp r −
1
2
σ2
i − λ eµ+δ2
2 − 1 +
2iµ + iδ2
2τ
τ + σiB(τ)QM
,
(3.3.8)
where β = λ(1 + k) = eµ+δ2
2 and τ = T − t. We assume that the riskless interest rate ,r, is constant
over the period 20 − January − 2007 to 26 − June − 2014 (in our case we use the average interest
rate over the selected period of time horizon, thus r = 15%).
STOCKS Initial price Strike Price BS MJD
SWISSPORT 680 600 361.57 200.84
TBL 1580 1500 434.36 334.42
SIMBA 960 900 322.82 217.92
TWIGA 650 600 266.46 198.88
NMB 970 900 232.53 199.26
DCB 350 300 97.23 91.78
Table 3.3: European Call option prices from Merton Jump-Diffusion (MJD) and Black-Scholes (BS)
Models.
There is a significant difference in the prices of the European Call option in Table 3.3 for the first four
Stocks. The Merton jump-diffusion model gives less price difference than often expected due to the
high value of the expected jump amplitude or jump intensity (λ). Theory and modern literature such
Kou (2008) and Matsuda (2004) assume a very low jump intensity.
(a) (b)
Figure 3.7: Histograms of SWISSPORT and TBL.

(a) (b)
Figure 3.8: Histograms of SIMBA and TWIGA.
(a) (b)
Figure 3.9: Histograms of NMB and DCB.

4. Conclusion
In this report, we have shown that, the Merton jump-diffusion model incorporates features such as
higher peak corresponding to fat tails (kurtosis) and skewness (asymmetric features) than found in the
Black-Scholes normal distribution. In addition, we have also shown that, the log-return in the Merton
jump-diffusion model is equivalent to a discrete mixture of M normally distributed variables, as M
goes to infinity. From literature (Kiefer, 1978), it is known that the likelihood function is unbounded
which leads to inconsistency in parameter estimation. Thus, care must be taken in estimating Merton
jump-diffusion model, since the direct maximum likelihood estimate is invalid. For consistent and
asymptotically normally distributed estimator, we proposed a new method based on the EM algorithm
developed by Pickard et al. (1986) in which the log-likelihood is bounded with respect to variances
between the jump and diffusion parts.
Based on the proposed method of parameter estimation, the empirical results show that, stock prices are
significantly described to some extent by the Merton jump-diffusion model, as the estimated expected
jump amplitude is of such a size that adding a jump component looks like introducing a Brownian motion
process. However, according to Honore (1998), “there is empirical evidence that adding a second jump
component with a constant jump amplitude improves the results”.
In addition, examining the European call option prices to the effect of moving away from the classical
Black-Scholes model to the Merton jump-diffusion model was made. There was significant differences
in the prices of the Black-Scholes and the Merton jump-diffusion models. The Merton-diffusion model
gives less price than often expected, since theory and literature mostly assumes a very low jump intensity.
Finally, adding a second jump component with a constant jump amplitude is a possible extension
of the Merton jump-diffusion model presented in this report. Also, the report can be extended to
include hedging options in incomplete markets since, the market for the Merton jump-diffusion model
is incomplete.
38

5. APPENDIX.
5.1 Appendix A
Proof of Theorem (2.4.7):
Proof.
Suppose P and Q are two probability measures on a probability space (Ω, F) such that Q is absolutely
continuous with respect to P, that is Q P.
1) We need to show that there exist a non-negative function Z : Ω → (0, 1) such that Z is
F−measurable, that is , if Z = E [X|F] then (Z > b) as a set in F for each real b.
Now, from deﬁnition of measurable set
Z = E [X|F] =
[X; A]
P(Ai)
1A1
Z =
i
ai1A1 .
If we set
ai =
[X; A]
P(Ai)
,
then the set (Z ≥ b) is a union of some of the Ai’s, namely, those Ai for which ai ≥ b. But the
union of any collection of the Ai is in F. Hence, the function Z is an F−measurable.
2) We now show that, Q(A) = A ZdP < ∞.
Suppose C is the class of all non-negative function Z, integrable with respect to P, such that
A
ZdP ≤ Q(A), ∀A ∈ Ω
and we write α = sup ZdP : Z ∈ C , C = 0 since 0 ∈ C.
Moreover,
0 ≤ ZdP ≤ Q(A) < ∞ for every Z ∈ C.
0 ≤ α < ∞.
Since α is the accumulation (cluster) point of the set ZdP : Z ∈ C , there exist a sequence
of functions in C such that α = limn→∞ ZndP.
Let n be positive integer and deﬁne a function gn : Ω → [0, 1] by
gn = max {Z1, Z2, . . . , Zn} .
Since gn is the maximum of non-negative measurable functions it is non-negative measurable
function.
Let
B = Ai ∩ ∩n
j=1,j=i(Zi − Zj)−1
([0, 1]) , for i = 1, 2, 3, . . . , n
39

Section 5.1. Appendix A Page 40
Set A1 = B1, A2 = B2 − A1, . . . , An = Bn − ∪n−1
i=1 Bi .
That is, A = A1 ∩ . . . ∩ An is a disjoint union such that gn(x) = Zi(x) for x ∈ Ai. Since
gn =
n
i=1
ZiχAi
it is integrable. In eﬀect we have
A
gndP =
A
n
i=1
ZiχAi dP =
n
i=1 A
ZiχAi dP
A
gndP =
n
i=1 A
ZiχAi dP ≤
n
i=1
Q(Ai) = Q(A)
Since Q is a measure and Ai’s are disjoint, therefore we have that gn ∈ C.
Now, let Z0 ; X → [0, ∞] be the function
Z0(x) = sup {Zn(x) : n = 1, 2, 3, . . .} .
We have
Z0(x) = lim
n→∞
gn(x).
Then {gn} is a non-decreasing sequence of non-negative measurable functions that converges to
Z0.
By applying the Monotone Convergence Theorem we have
X
Z0dP = lim
n→∞ X
ZndP.
In a parallel development, X ZndP is a non-decreasing sequence in X ZdP : Z ∈ C , since
each gn ∈ C.
Then X gndP ≤ α for every n and thus,
X
Z0dP = lim
n→∞ X
gndP ≤ α. (5.1.1)
Also, because Zn ≤ gn for all n, we have
X
ZndP ≤
X
gndP, ∀n.
Then
α = lim
n→∞ X
ZndP ≤ lim
n→∞ X
gndP = lim
n→∞ X
Z0dP. (5.1.2)
From equations (5.1.1) and (5.1.2 )we have
X
Z0dP = α.

Section 5.1. Appendix A Page 41
Therefore
A
Z0dP = lim
n→∞ A
gndP
A
Z0dP ≤ lim
n→∞
Q(A) = Q(A).
So Z0 ∈ C.
Since Z0 is integrable function, there exist a ﬁnite valued-function Z such that
Z = Z0 a.e(P).
Let Q0 : Ω → [0, 1] be the function given by
Q0 = Q(A) −
A
ZdP.
Q ≥ 0 since A ZdP = A Z0dP ≥ Q(A).
Hence Q0 is a measure. Also, Q0 is ﬁnite since Q is.
Now if g = Z + εχB for some ε > 0 then g is integrable and
A
gdP =
A
ZdP +
A
εχBdP by linearity of integral.
A
gdP =
A
ZdP + εP(A ∩ B)
A
ZdP =
A∩B
ZdP +
A−B
ZdP + εP(A ∩ B) since the integral is additive.
A
ZdP ≤
A∩B
ZdP +
A−B
ZdP + Q(A ∩ B) −
A∩B
ZdP
=
A−B
ZdP + Q(A ∩ B)
= Q(A − B) + Q(A ∩ B), since Z0 ∈ C
= Q(A)
∴
A
gdP ≤ Q(A), since Q is additive.
However,
X
gdP =
X
ZdP + εP(B) > α since εP(B) and
X
ZdP =
X
Z0dP = α.

Section 5.2. Appendix B. Page 42
We get a contradiction since α is the supremum of the set
X
ZdP : Z ∈ C
and
X
gdP ∈
X
ZdP : Z ∈ C ,
so g ∈ R.
Hence Q0(A) = 0 for every measurable set A, that is
Q(A) =
A
ZdP < ∞.
Let now show the uniqueness of the function Z .
Let g be another non-negative measurable function satisfying Q(A) = A gdP. Since Q(A) < ∞
for every A ∈ Ω, we have that g is integrable on every A ∈ Ω.
By linearity of the Lebesque integral for integrable functions, we have
0 = Q(A) − Q(A)
0 =
A
gdP −
A
ZdP
0 =
A
(g − Z)dP, by linearity
for every A ∈ ω. Then we have
Z = g.
5.2 Appendix B.
Proof of Lemma (3.3.2) is as follows:
Proof.
Firstly. the unconditional expectation of Y given that E(Y |X) = a + bX for some constant a and b is
E(Y ) = E [E(Y |X)]
= E [a + bX]
= a + bE(X)
∴ a = E(Y ) − bE(X).

Next,
Cov(X, Y ) = Cov(X, E(Y |X))
= Cov(X, a + bX)
= aE(X) + bE(X2
) − E(X).E(a + bX)
= bE(X2
) − b(E(X))2
= b E(X2
) − (E(X))2
Cov(X, Y ) = bV ar(X)
∴ b =
Cov(X, Y )
V ar(X)
.
Hence,
E(Y |X) = E(Y ) − bE(X) + bX
= E(Y ) + b [X − E(X)]
= E(Y ) +
Cov(X, Y )
V ar(X)
[X − E(X)] .
So, E(Y |X) is the best predictor of Y among all the functions of X.
Proof of Theorem (3.3.3) is as follows:
Proof.
Since Y is a continuous random variable, using the deﬁnition of the conditional variance of Y given
X = x for continuous random variables we have
V ar(Y |X) =
∞
−∞
(y − E(Y |X))2
h(y|x)dy
From Lemma (3.3.2) we have
V ar(Y |X) =
∞
−∞
y − E(Y ) −
Cov(X, Y )
V ar(X)
[X − E(X)]
2
h(y|x)dy. (5.2.1)
Multiplying both sides of equation (5.2.1) by fX(x) and integrating over the range x (that is R), we
get
∞
−∞
V ar(Y |X)fX(x)dx =
∞
−∞
∞
−∞
y − E(Y ) −
Cov(X, Y )
V ar(X)
[X − E(X)]
2
h(y|x)fX(x)dydx
(5.2.2)
Since V ar(Y |X) is constant of x and from the R.H.S of equation (5.2.2) f(x, y) = h(y|x)fX(x) then
we have unconditional expectation on the R.H.S. Thus, equation (5.2.2) becomes;

V ar(Y |X)
∞
−∞
fX(x)dx = E y − E(Y ) −
Cov(X, Y )
V ar(X)
[X − E(X)]
2
.
Now, by deﬁnition of a valid probability density function,
∞
−∞
fX(x)dx = 1.
Then
V ar(Y |X) = E (y − E(Y )) −
Cov(X, Y )
V ar(X)
[X − E(X)]
2
.
Expanding the R.H.S and distributing the expectation we obtain;
V ar(Y |X) = E (y − E(Y ))2
− 2
Cov(X, Y )
V ar(X)
(X − E(X))(y − E(Y )) +
Cov(X, Y )
V ar(X)
2
(x − E(X))2
.
So identifying the various terms we have;
V ar(Y |X) = V ar(Y ) − 2
Cov(X, Y )
V ar(X)
Cov(X, Y ) +
Cov(X, Y )
V ar(X)
2
V ar(X)
= V ar(Y ) − 2
Cov(X, Y )
V ar(X)V ar(Y )
V ar(Y )
V ar(X)
Cov(X, Y ) +
Cov(X, Y )
V ar(X)V ar(Y )
V ar(Y )
V ar(X)
2
V ar(X)
= V ar(Y ) − 2
Cov(X, Y )
V ar(X)V ar(Y )
V ar(Y )
V ar(X)
Cov(X, Y )
V ar(Y )
V ar(Y ) +
Cov(X, Y )
V ar(X)V ar(Y )
V ar(Y )
V ar(X)
2
V ar(X)
= V ar(Y ) − 2
Cov(X, Y )
V ar(X)V ar(Y )
Cov(X, Y )
V ar(X)V ar(Y )
V ar(Y ) +
Cov(X, Y )
V ar(X)V ar(Y )
2
V ar(Y )
= V ar(Y ) −
Cov(X, Y )
V ar(X)V ar(Y )
2
V ar(Y )

Section 5.3. Appendix C. Page 45
∴ V ar(Y |X) = V ar(Y )

1 −
Cov(X, Y )
V ar(X)V ar(Y )
2

 .
5.3 Appendix C.
Sample Securities of Dar es Salaam Stock Exchange.
SYMBOLS NAME OF SECURITY SAMPLE PERIOD
DCB Dar es Salaam Community Bank Jan 2, 2009- June 26, 20014
NMB National Microﬁnance Bank Jan 2, 2009- June 26, 20014
SIMBA Simba Cement Company Jan 2, 2007- June 26, 20014
SWISSPORT Swissport International Limited Jan 2, 2007- June 26, 20014
TBL Tanzania Breweries Limited Jan 2, 2007- June 26, 20014
TWIGA Twiga Cement Company Jan 2, 2007- June 26, 20014
Table 5.1: List of Symbols.
5.4 Appendix D.
Descriptive Statistics.
Stocks Min. 1st Q. Mean Med. 3rd Q. Max. Std. Skew. Kurt. Obs.
DCB 300 350 655.44 740 830 860 206.24 -0.708 -1.212 1524
NMB 700 900 2000.28 2200 2600 3140 840.78 -0.319 -1.231 1524
SIMBA 960 1700 1771.21 1820 1920 2380 358.86 -0.389 -0.444 2032
SWISSPORT 540 600 657.65 640 710 820 76.16 -0.498 -0.840 2032
TBL 1560 1700 1761.72 1780 1820 2080 120.40 0.232 -0.044 2032
TWIGA 640 1400 1549.91 1640 1820 2100 426.64 -0.829 -0.269 2032
Table 5.2: Descriptive statistics of the daily stock levels.

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