BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS

A review of two financial market models: the Black–Scholes–Merton andthe Continuous–time Markov chain models

by

Haimanot Ayana and Sarah Al-Swej

Kandidatarbete i matematik / tillämpad matematik

DIVISION OF MATHEMATICS AND PHYSICSMÄLARDALEN UNIVERSITY

SE-721 23 VÄSTERÅS, SWEDEN

Bachelor thesis in Mathematics /Applied Mathematics

Date:2021-05-27

Project name:A review of two financial market models: the Black–Scholes–Merton and the Continuous–timeMarkov chain models

Author(s):Haimanot Ayana and Sarah Al-Swej

Version:5th July 2021

Supervisor(s):Anatoliy Malyarenko

Reviewer:Olha Bodnar

Examiner:Achref Bachouch

Comprising:15 ECTS credits

Abstract

The objective of this thesis is to review the two popular mathematical models of the financialderivatives market. The models are the classical Black–Scholes–Merton and the Continuous-time Markov chain (CTMC) model. We study the CTMC model which is illustrated by themathematician Ragnar Norberg. The thesis demonstrates how the fundamental results ofFinancial Engineering work in both models.

The construction of the main financial market components and the approach used for pricingthe contingent claims were considered in order to review the two models. In addition, the stepsused in solving the first–order partial differential equations in both models are explained.

The main similarity between the models are that the financial market components are thesame. Their contingent claim is similar and the driving processes for both models utilizeMarkov property.

One of the differences observed is that the driving process in the BSM model is the Brownianmotion and Markov chain in the CTMC model.

We believe that the thesis can motivate other students and researchers to do a deeper andadvanced comparative study between the two models.

Acknowledgements

We would like to thank our research supervisor Professor Anatoliy Malyarenko for his guidanceand constructive feedback at each stage of the thesis. We would also like to appreciate theMälardalens University for the Applied Mathematics program with a specialization in FinancialEngineering which is the base for the finance industry.

Contents

1 Introduction 41.1 Background of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 The content of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Background of the two market models 62.1 The Black–Scholes–Merton model . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Continuous–time Markov chain model . . . . . . . . . . . . . . . . . . 8

3 Financial market components 113.1 A Probability Space (Ω,F,P) . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Probability spaces in the BSM and CTMC model . . . . . . . . . . . 123.2 Stochastic Processes X(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Stochastic Process in the BSM and CTMC model . . . . . . . . . . . 133.3 The Filtration Ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Filtration in the BSM and CTMC model . . . . . . . . . . . . . . . . 143.4 Basic Assets Numéraire–Bank account and stocks . . . . . . . . . . . . . . . 14

3.4.1 Bank Account in the BSM and CTMC model . . . . . . . . . . . . . 153.4.2 Stocks in the BSM and CTMC model . . . . . . . . . . . . . . . . . 15

3.5 Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5.1 Portfolio in the BSM and CTMC model . . . . . . . . . . . . . . . . 16

3.6 Trading strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6.1 Trading strategy in the BSM and CTMC model . . . . . . . . . . . . 17

3.7 Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 No-arbitrtage pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.8.1 BSM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.8.2 CTMC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.9 Risk–neutral measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.10 The Fundamental Theorem of Financial Engineering . . . . . . . . . . . . . 213.11 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.12 Methods for derivative pricing . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.12.1 The Black–Scholes–Merton model . . . . . . . . . . . . . . . . . . . 223.12.2 The Continuous time Markov chain model . . . . . . . . . . . . . . 23

3.13 The method of PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2

3.13.1 Black–Scholes–Merton PDE . . . . . . . . . . . . . . . . . . . . . . 253.13.2 Norberg PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.14 Pricing formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Similarities and differences between the two models 284.1 similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Conclusions 30

A Criteria for a Bachelor Thesis 31

B An ethical description of contribution of coauthors 32

3

Chapter 1

Introduction

1.1 Background of the thesisFinancial Engineering is a multidisciplinary field relating to the creation of new financialinstruments and strategies. It is the process of employing mathematical models, financial theoryand computer programming skills to make pricing, hedging, trading and portfolio decisions.

Many models have been proposed to study the dynamics of assets price processes. But animportant discovery was achieved in the 1970s by Fisher Black, Myron Scholes and RobertMerton in the pricing of European options. This model is known as the Black–Scholes–Merton(BSM) or Black–Scholes model. In this classical model, the market is driven by Brownianmotion and it assumes constant volatility [10]. But in practice, volatility varies through time orstochastically. This drawback in the model led to the development of more complex models likethe Markov chain which have two stochastic variables, namely the stock price and volatility.

The theory of Markov chains was discovered by Andrei Markov, a Russian mathematician.Markov chain is an important mathematical tool in stochastic processes. The underlying idea isthat it exhibits the Markov property, which means the predictions about stochastic processescan be simplified by viewing the future as independent of the past given the present state of theprocess. In other words, this Markov process is a particular type of stochastic process whereonly the current value of a variable is relevant for predicting the future. The past history of thevariable and the way that the present has emerged from the past are irrelevant.

1.2 Literature reviewThe models of Financial Engineering are usually described by the system of stochastic differen-tial equations (SDE). The first attempt in this direction has been performed by L. Bachelier [1]in 1900. He modeled the movements of share prices by the Brownian motion.

The first modern model was discovered in 1973 independently by Black and Scholes [4]and Merton [13]. It will be described in more detail later. Currently, it is important that thedriving process in this model is the Brownian motion.

Alternatively, a market model can be driven by a different stochastic process. In our thesis,we consider a model developed by Ragnar Norberg [15, 14], which is driven by a continuous-

4

time Markov chain. This model has been further developed by Turra [22]. A different modeldriven by a discrete time Markov chain, has been developed by R.J. Elliott and his collaborators,see [7, 8, 9, 23].

1.3 The content of the thesisIn chapter 2, a brief description of the two market models is given. In chapter 3, the generaldefinition of the financial market and its components such as probability space, the stochasticprocess, filtration, basic assets, derivative instruments, etc is covered. In addition, examples aregiven for each financial component of the models. A review of the main technical tools of thefinancial theory like no-arbitrage pricing, numéraire, change of measure, and the FundamentalTheorem of Financial Engineering is covered in this chapter. In the end, examples that showhow the above theoretical considerations work in both models are shown.

After the comparison of the two models, in chapter 4 the similarities and differencesexhibited in both models are explained.

Then the conclusion of the thesis follows in Chapter 5.Finally, in Appendix A we show how our thesis satisfies the requirements of the Swedish

Agency for Higher Education. And at the end, the contribution of the authors is included.

5

Chapter 2

Background of the two market models

In modern financial mathematics, the theory of diffusion processes is inevitable. The Black,Scholes and Merton (BSM) model was crafted with Brownian motion but there are also othertypes of stochastic processes which are important and a better alternative in finance. A Poissonprocess,type of a Markov process, is one of them which is used to model the jumps arising inasset prices.

In modern financial mathematics, the theory of diffusion processes is inevitable. The Black,Scholes and Merton (BSM) model was crafted with Brownian motion but there are also othertypes of stochastic processes which are important and better alternative in finance. A Poissonprocess, a type of a Markov process, is one of them which is used to model the jumps arising inasset prices.

2.1 The Black–Scholes–Merton model

In the pricing of European stock options, the model developed by Fischer Black, Myron Scholesand Robert Merton in the early 1970s has achieved a major quantum leap. The model is knownas the Black–Scholes–Merton (Black–Scholes) model. It has had a huge effect on the financialmarket as it has influenced the way traders price and hedge derivatives. The model’s hugeeffect has got recognition when the scholars were awarded the Nobel prize for economics in1997. ([10])

Even though there were other researchers who have succeeded in calculating correctly theexpected payoff from a European option, knowing the correct discount rate was not easy. Toovercome this problem, Black and Scholes used the Capital Asset Pricing Model (CAPM)which relates the expected return from an asset to the risk of the return. The assumptions inCAPM were not easy and do not hold in reality. As a result, Merton preferred to use a newand different approach that involved a riskless portfolio that consists of the option and theunderlying stock. In addition, he was arguing the return on the portfolio over a short periodof time must be the risk–free return. In our thesis, we use and focus on Merton’s approachto derive the Black–Scholes–Merton model. The thesis gives emphasis on how the modelshows the risk-neutral valuation argument and how to estimate volatility from historical data orimplied from option prices.

6

The crucial part in the BSM model is it assumes that the percentage changes in the stockprice in a very short period of time are normally distributed and stock prices are log–normallydistributed, so that

lnST ∼ φ [lnS0 +(µ−σ2/2)T,σ2T ],

where

ST = stock price at future time TS0 = stock price at time 0µ = expected return on stock per yearσ = volatility of the stock price per year

The above assumption holds mathematically because the random variable ln ST is normallydistributed as a result ST has a lognormal distribution. The BSM model also assumes thecontinuously compound rate of returns are normally distributed with a mean of µ − σ2

2 andstandard deviation σ√

(T ). The expected continuously compounded return is therefore µ− σ2

2 .

The volatility which is denoted by σ is a measure of uncertainty regarding the returnsfrom the stock. The volatility of a stock can be referred to as the standard deviation of thecontinuously compounding return which is indicated in the above paragraph. Volatility canbe calculated from historical data of the stock price which is observed on a daily, weeklyand monthly basis i.e based on a fixed intervals of time. To calculate the volatility throughdetermining the standard deviation, first we need to calculate the daily return from the historicalprices of the stock by using the formula ui = ln( Si

Si−1), then we compute the expected price

(mean) of the historical prices. Then we need to work out the difference between the daily returnand the expected price and square the differences from the previous step. After determining thesum of the squared differences we divide it by the number of observations minus one. Lastly,compute the square root of the variance that is computed in the first step. The formulas for this

is given as S =√

1n−1 ∑

ni=1(ui− u)2, where u is the mean of the ui (daily return).

The Black–Scholes–Merton differential equation must be satisfied by the price of anyderivative of non–dividend-paying stock. The equation will be derived in chapter 3. A risklessportfolio consisting of a position in the stock and in the derivative is important. The argumentwhich leads to Black–Scholes–Merton differential equation is that in the absence of arbitrageopportunities, the portfolio must be the risk–free interest rate r. In a riskless portfolio, the stockprice and the derivative price are affected by the stock price which is the underlying source ofuncertainty. To remain riskless the gain or loss from the stock position always offsets the gainor loss from the derivative position, so the value of the portfolio is known with certainty.

Risk–neutral valuation is an important tool in the derivation of the Black–Scholes–Mertondifferential equation. The equation which we later discuss does not contain any variable thatis affected by the risk preferences of the investors. The model is assuming that investors arerisk neutral therefore the expected return on all underlying assets is the risk-free interest rate, r.([10])

7

2.2 The Continuous–time Markov chain modelThe Markov chain model was discovered by a Russian mathematician Andrey AndreyevichMarkov (1856–1922). Initially, he was best known for his work on stochastic processes. Laterin 1906, he becomes known for proposing Markov chains or Markov processes. In this thesis,we study a financial market driven by a continuous-time Markov chain (CTMC).

The continuous–time Markov chain model is characterized by the Markov property i.e.memorylessness, meaning given the present state, the future is independent of the past.

Definition 1. [18] A stochastic process X(t), t ≥ 0, is a continuous–time Markov chain if,for all s, t ≥ 0, and j,k,xu,0≤ u≤ s are nonnegative integers, it is true that

PX(s+t) = k|Xs = j,Xu = xu,0≤ u≤ s= PX(s+t) = k|Xs = j

This means that a continuous-time Markov chain is a stochastic process with a Markovproperty that the conditional distribution of future state k at future time s+ t, given the presentstate at j, all past states depends only on the present state and is independent of the past. ([18]).

In other words, the Markov property suggests that the current state Xs = j is enough todetermine the distribution of the future. The future is only determined by where we are at thepresent time, not anything that happens before that i.e. in this model history doesn’t matter.

Definition 2. [18] The continuous-time Markov chain is said to have homogeneous transitionprobabilities if P(Xs+t = k | Xs = j) is independent of s.

In this thesis, we will give special attention to Ragnar Norberg’s research work entitled ’AMarkov Chain Financial Market’. Ragnar Norberg, a Norwegian insurance mathematician, hasachieved and influenced well the field of insurance mathematics as well as mathematical finance.In his research, he considered a financial market driven by a continuous–time homogeneousMarkov chain. In addition, the conditions for the absence of arbitrage, completeness and thenon–arbitrage pricing of a derivative is explained in detail. Below we will define and elaborateon the terms used in his research.

Hereafter we use the notations specified below,

• Vectors and matrices are denoted in bold letters, lower and upper case respectively.

• The identity matrix is denoted by I.

Let Ytt≥0 is a continuous–time Markov chain with finite state space y = 1, ....,n, thenthe transition probabilities in the Markov chain market is

P jkt = P[Ys+t = k|Ys = j],

It is the transition probabilities of going from state j to state k at a time interval of size t.The concept of transition probabilities is associated with a random walk and it is conditionalprobability.

The transition probability on the above equation depends only on states j and k and the timedifference between s+ t and s. It does not depend on the time origin, rather it only depends on

8

how long it takes to get from the state j to k i.e. the length of the transition period. This makesthe transition probability time-homogeneous. Throughout our thesis, the transition probabilityis assumed to be time–homogeneous.

Definition 3. [15] The transition intensities from state j to state k is defined by

λjk = lim

t→0

P jkt

t, j 6= k,

exist and are constant.y j = k,λ jk ≥ 0

The transition intensities or the jump rate tells us the amount of random time that the CTMCspends in every state it visits.

Definition 4. [15] The number of directly accessible set of states from state j, can be denotedby

n j = |y j|

λj j =−λ

j =− ∑k;k∈y j

λjk

The diagonal elements in the matrix denoted by λ j j are the negative of the sum of all otheroff the diagonal matrix elements.

Definition 5. [15] The Λ jk which is equal to the λ jk is called the infinitesimal matrix or therate matrix. The rates are placed off the diagonal matrix.

The assumption here is that all states intercommunicate so P jkt > 0 for all j,k (and t > 0),

which also implies that n j > 0 for all j so there will not be absorbing states.Here after, the matrix of transitional probabilities can be denoted as Pt since Pt = (p jk

t ) andthe infinitesimal matrix, with ΛΛΛ = λ jk, such that λ jk > 0, whenever the rates are greater than 0,that means there is a positive probability that the system can have a transition from state i to j.

The Kolmogorov differential equation

Andrey Nikolayevich Kolmogorov (1903–1987), was a Russian mathematician whose workinfluenced many branches of modern mathematics including probability theory and stochasticprocesses especially Markov processes. He came up with a set of equations to describe how ina small time interval there is a probability that the state will remain unchanged, however, if itchanges the change may be radical and might lead to a jump process.

Definition 6. [15] The infinitesimal matrix is defined as Λ = (λ jk) and the matrix of transitionprobabilities as Pt = (p jk

t ). Taking the limit of the infinitesimal matrix Λ yields

Λ = limt0

1t(Pt− I),

where I is the identity matrix.

9

Taking the derivative of the matrix of transition probabilities, by the forward and backwardKolmogorov differential equations, d

dtPt = PtΛ = ΛPtwhere Λ is the infinitesimal, or the jump

rate matrix.The Kolmogorov equation relates the transition probabilities that depend on how long we

wait Pt with the derivative of the same matrix. The matrix Λ is a matrix of limits that we callthe jump rate matrix.

The exponential of the infinitesimal (jump rate) matrix multiplied by time is the solution ofthe Kolmogorov differential equation in matrix form Pt = exp(Λt).

Definition 7. [15] The indicator function relates expectation and probability. The indicator ofan event for any state Y is in state j at time t can be written as

I jt = 1Yt = j

If I jt = 1 it means that it is true the event Y is at state j at time t, and if I j

t = 0 it meansotherwise.

Definition 8. [15] The counting process can be defined as the number of direct transitions of Yfrom state j to state k ∈ y j in the time interval of (0, t]. For k /∈ y j, N jk

t ≡ 0.

N jkt = |s;0 < s≤ t,Ys−= j,Ys = k|

where∑

k;k 6= j(Nk j

t −N jkt ),

is the number of events occurred during the interval (s,T ].Therefore,

Yt = ∑j

jI jt , I

jt = I j

0 + ∑k;k 6= j

(Nk jt −N jk

t )

All the state process, the indicator processes and the counting processes carry the sameinformation, which at any time t is represented by the sigma algebra Ft

Y = σYs;0≤ s≤ t.The filtration , denoted by FY = FY

t t≥0, satisfies the conditions for right continuity, sodoes Y , I j and N jk are right–continuous. According to [11], the FY

t is a sigma algebra whichis a family of events associated with a random experiment.

Definition 9. [15] The compensated counting processes denoted as M jk, j 6= k, defined by

dM jkt = dN jk

t − I jt λ

jkdt

andM jk

0 = 0,

are zero mean, square integrable and mutually orthogonal martingales with respect to (FY ,P).

10

Chapter 3

Financial market components

In this chapter, we introduce the main object of studies in Financial Engineering, a financialmarket. Financial markets consist of several components. Each component is defined in oneof the following sections and formulated for the Black–Scholes–Merton and the continuousMarkov chain model.

3.1 A Probability Space (Ω,F,P)

A probability space contains a sample space (Ω), σ–field (F) and probability measure (P)which are collectively called the triplet.

• A sample space (Ω)

As stated by Kijima [11], a sample space is a collection of all possible outcomes in arandom experiment and it is usually denoted by Ω. In set theory, Ω is the universal set.The elementary event which is the single outcome ω is an element of Ω and the event Ais a subset of Ω. Using the notation of set theory, ω ∈Ω and A⊂Ω, respectively[11]. Asample space can be continuous or discrete. A sample space is said to be continuous if ithas uncountable sample points and discrete if it has finite or countably infinite samplepoints[16].

• σ–field (F)

The σ–field which is denoted by F contains all the family of events or outcomes underconsideration. In other words, it refers to the collection of subsets of the sample space.The family of events should satisfy the following properties to be a σ–field

1. Ω ∈ F,2. If A⊂Ω is in F then Ac = Ω\A ∈ F, and3. If An ⊂Ω,n = 1,2, ..., are in F then

⋃∞n An is in F.

• A Probability measure (P)

A Probability measure P is a real-valued function that specifies the likelihood of eachevent happening. This function P is measurable over the sample space (Ω) such that

11

1. P(Ω) = 1,

2. 0≤ P(A)≤ 1 for any event A ∈ F, and

3. For mutually exclusive events An ∈ F,n = 1,2, ..., that is, Ai∩A j =∅ for i 6= j wehave P(A1∪A2∪ ...) = ∑

∞n=1P(An). [11]

3.1.1 Probability spaces in the BSM and CTMC model

In BSM model, a stochastic process X(t,ω) : R+×Ω→ R is a function of a variable time tand an element ω in a probability space Ω, in which the events are defined. For each ω ∈Ω thestochastic process is called trajectories or sample paths which are continuous. These continuoustrajectories are an elementary event or the outcome of the experiment in this space.

Definition 10. [19] The space of elementary events for Brownian motion is the set of allcontinuous real functions.

Ω = ω(t) : R+→ R

The Brownian events consist of uncountably many Brownian trajectories. Its formulation ismore complicated than the elementary events as it contains a cylinder set which is beyond thescope of this paper.

Definition 11. [19] A σ–field in Ω is a non–empty collection of F of subsets of Ω such that

1. Ω ∈ F

2. If A ∈ F then Ac = Ω

3. If Ai ∈ F, i = (1,2, ...), then⋃

∞1 Ai ∈ F

The elements (events) of F are called measurable sets. The construction of the σ–field forBrownian motion is somehow similar to the definition that is presented in 3.1. The probabilitymeasure is defined by the events.

In CTMC model, the probability space is more complicated. The set Ω is the set of allcàdlàg functions on the interval [0,T ]. The word “càdlàg” is an abbreviation word in Frenchwhich means “right continuous with left limits”.

Definition 12. [19] A function f : [0,T ]→R is called a càdlàg function if it is right-continuouson [0,T ) and has left limits on (0,T ].

The set of all càdlàg functions on the interval [0,T ] is denoted by D([0,T ]) which is calleda Skorhod space. The σ -field F was constructed by the Ukrainian mathematician AnatoliySkorokhod in [21], see also [12]. The construction is complicated and will not be presentedhere, for further details see the textbook [2].

The σ–field in CTMC is FtY = σYs;0≤ s≤ t.

12

3.2 Stochastic Processes X(t)A Stochastic Process X(t) is a collection of random variables defined on a probability space.Its value change over time in an uncertain way, here t refers to time, and X(t) is the state ofthe process at time t. If the value of the random variable changes only at a certain fixed pointin time then X(t) is a discrete–time stochastic process. If the random variable takes any valuewithin a certain range in t, we call it a continuous–time stochastic process [10] [18].

Definition 13. The stochastic Process X(t) is a collection of random variables X(t) : t ∈T ,where the set of time epochs T is a fixed subset of the real number R. X(t) is a function oftwo variables:

x(t,ω) : T ×Ω→ R

but the stochastic process X(t) in general is entirely driven by the following multivariate randomvariables

X = (X(t1),X(t2), ....,X(tn)),

where n is a positive integer and t1, t2, ..., tn are n pairwise different time epochs.

3.2.1 Stochastic Process in the BSM and CTMC modelThe BSM model is driven by the stochastic process W (t). The stochastic process W (t), t ≥0, defined on the common probability space (Ω,F,P), is called a standard Brownian motionprocess if

1. the process starts at the origin, W (0) = 0,

2. w(t) has a continuous sample paths (trajectories),

3. it has independent increments which is also independent of time, and

4. the increment W (t+s)−W (t) is normally distributed with expected value 0 and variances.

The CTMC model is driven by the Markov process which itself is a Poisson process.A Poisson process is one of the most widely used counting process. A stochastic processNt , t ≥ 0 is said to be a counting process if Nt represents the total number of events that haveoccurred up to time t. Hence a counting process Nt , t ≥ 0 is said to be a Poisson process if

1. N(0) = 0

2. the process has independent increments i.e. if the number of events that have occurred bytime t must be independent of the number of events occurring between times t and t + s(that is N(t + s)−N(t)).

3. the increment N(t + s)−N(t) has a Poisson distribution with expected value λ (t).Theparameter λ implies the transition intensities and it is a positive number.

13

3.3 The Filtration Ft

The filtration Ft is a σ–field which contains the available information about security prices in astochastic process in the market until time t.

Definition 14. [11] the sequence of information Ft ;0 ≤ t ≤ T is called a filtration if, fors≤ t

Fs ⊆ Ft .

Given the stochastic process Xs, which is measurable for every s ∈ [0, t] the simplest andsmallest σ–field is the one generated by the process itself,

FXt = σ(Xs;0≤ s≤ t).

3.3.1 Filtration in the BSM and CTMC modelDefinition 15. [11] The filtration of Black–Scholes is denoted by Ft = σW (s);s≤ t whereFt is the σ–filed that contains all possible events of a Brownian motion W (t);0≤ t ≤ T.

Definition 16. [17] In a Markov Chain process the filtration Ft = σY (s);s ≤ t is a set ofinformation of present and past values of the stochastic process Y = Yt ;0≤ t ≤ T. Since theMarkov property indicates that the future value of the process is independent of the past, giventhe present value of the process, therefore

E[Y |Ft ] = E[Y |Yt ]

3.4 Basic Assets Numéraire–Bank account and stocks• Numéraire Numéraire can be thought of us a way of measuring the denominated price

process than the price process itself. It is a standard basis that is used to compute the realvalue of money. The numéraire here is the risk–free security S0(t).

Definition 17. [11] In a denominated price S∗i (t) = Si(t)/Z(t), the denominating positiveprocess Z(t);0≤ t ≤ T is called Numéraire.

• Bank Account

A bank account is a risk–free security and it is where our money either from saving ordebts grow at a risk free rate. The positive process denoted by B(t) is a bank account andit can also serve as a numéraire.[3]

Definition 18. [6] In a continuous time a continuous bank account, uses a continuouslycompounded interest rate r, is defined with the differential equation

dB(t) = r(t)B(t)dt, 0≤ t ≤ T.

Given the initial investment B(0) = B0 then the account balance at time t is given by theequation B(t) = B0ert .

14

3.4.1 Bank Account in the BSM and CTMC model

Definition 19. [11] Let the non-negative process r(t) be the spot rate at time t, thenthe money market account B(t) in Black–Scholes is defined by

dB(t) = r(t)B(t)dt, 0≤ t ≤ T.

Definition 20. [15] Let r j be the state–wise interest rates, where j = 1, ....,n, and let I jt

be the indicator of the event that Y is in state j at time t, then the bank account in CTMCmarket is

Bt = exp(∫ t

0rsds

)= exp

(∑

jr j∫ t

0I js ds

)

Similarly the dynamics of the bank account is defined as

dBt = Btrtdt = Bt ∑j

r jI jt dt

• Stocks

Stocks can be defined as a risky security and a claim on the company’s assets and earnings.Buying stocks is a type of investment in the form of a share of an ownership in a company.Investors buy stocks that they think will go up in value over time to get profit. The stockprice denoted by S(t) is a stochastic process.[11]

Definition 21. [11] The stochastic process S(t);0≤ t ≤ T. defined on the probabilityspace (Ω,F,P) is a positive price process in continuous time.

3.4.2 Stocks in the BSM and CTMC model

In BSM model the stock price process is given as

dS = µSdt +σSdW,0≤ t ≤ T (3.1)

where W (t) is a standard Brownian motion and µ and σ are positive constants.In CTMC model the stock price dynamics is given by

dSti = Si

t−

(∑

jα

i jI jt dt +∑

j∑

k∈y j

γi jkdN j

t k

)

where the number of assets is i = 1, ...,m, α i j is constant and γ i jk = exp(β i jk)−1.

15

3.5 PortfolioA portfolio is a position in the market where a selection of risky and risk–free securities can beheld over time to make some profit. Our choice of portfolio depends on the price processes thatis available at Ft−1.

Definition 22. [11] Let θi(t) denote the number of security i carried from time t−1 to t where0≤ t ≤ T . The vector θ(t) = (θ0(t),θ1(t), ....,θn(t))T denotes the portfolio of n securities attime t, and the portfolio process is defined as θ(t);0≤ t ≤ T.

The portfolio θ(t) is measurable with respect to Ft−1 and the portfolio process θ(t) ispredictable with respect to Ft .

3.5.1 Portfolio in the BSM and CTMC model

In BSM model portfolio is defined as a vector

h = (φ1,φ2)

where φ1 is the number of units of the stock hold in the risky security S(t) and φ2 is the numberof bonds hold in the money market account B(t). Both φ1 and φ2 are stochastic processes andthey can be positive or negative depending whether we take a long or a short position.

In CTMC model, a dynamic portfolio is made up of m+1 dimensional stochastic process

θ′t = (ηt ,ξ

′t ),

where ηt represents the number of units of the bank account held at time t, and the i–th entry in

ξt = (ξ 1t , ...,ξ

mt )′

tells us the number of units of stocks i held at time t.

3.6 Trading strategyA self-financing portfolio is a trading strategy where the value of the portfolio at any time aftera transaction is equal to its value before that transaction. Thus the portfolio does not need anyadditional investments and therefore any change in the value of the portfolio is due to changein stocks value.

Definition 23. [11] A portfolio process θ(t);0 ≤ t ≤ T is said to be self-financing if thevalue of the portfolio at time t is

V (t) = ∑i=0

θi(t +1)dGi(t), 0≤ t ≤ T,

16

The value of a self–financing portfolio is

V (t) =V (0)+n

∑i=0

∫ t

0θi(s+1)dGi(s), 0≤ t ≤ T,

where Gi is the gain obtained from security i at time t. The dynamics of the gain process is

dGi = dSi(t)+di(t)dt,

where di(t) is the dividend rate for security i at time t.

3.6.1 Trading strategy in the BSM and CTMC modelThe portfolio is defined as a vector h = (φ1,φ2) and the value process of the portfolio h isdefined by

V (t) = φ1(t)S(t)+φ2(t)B(t)

This portfolio has a deterministic value at time t = 0 and a stochastic value at t = 1 If theportfolio is not too big, then the integrals∫ T

0(φ(t))2dt, i = 1,2 (3.2)

are finite with total probability of 1. If we substitute the above equation in the value process,we get the formula

V (t) =V (0)+∫ t

0φ1(u)dS(u)+

∫ t

0φ2(u)dB(u)

The above equation is a self–financing portfolio. The economical sense is that the portfoliois created at time 0, there is no adding or withdrawal of money, so the purchase of the newasset must be financed by the sale of an old one. The variation in gain comes from the risk andnon–risky asset in the portfolio, that means there is no additional finance that will be injected.[11]

In CTMC model the value of the portfolio at time t is [15]

V θt = ηtBt +ξ

′t St = ηtBt +

m

∑i=0

ξit Si

t

whereSt = (S1

t , ..., Smt )′.

The discounted prices being marked with tilde. The value of the portfolio at time t is

V θt = ηt +ξ

′t St .

The strategy θ is self-financing (SF) if

dV θt = ηtdBt +ξ

′t dSt ,

or, equivalently,

dV θt = ξ

′t dSt =

m

∑i=1

ξit dSi

t . (3.3)

17

3.7 Contingent ClaimsAccording to Pliska [17], a contingent claim is a random variable X that represents the payoffat time T of a contract between a seller and a buyer at time t < T . The most common typesof contract are call and put options. At time T the payoff X can be positive or negative forsome states of ω ∈ Ω. For a call option the buyer will have a positive payoff if the price ofthe underlying asset at time T is greater than the price at time t. Otherwise the payoff will benegative and the buyer has the right not to exercise it.

Definition 24. A contingent claim X is attainable if the portfolio process generarte the claim.i.e. If there exist a self-financing portfolio θ(t);0≤ t ≤ T that replicate the contingent claim,such that V (T ) = X , then the value of claim X is

X =V (0)+n

∑i=0

∫ t

0θi(s)dGi(s)

The contingent claims for Black–Scholes and Markov Chain is the same for both the modelssince it’s definition does not depend on the model. An example of a contingent claim for bothmodels is the time T payoff from European call and put options. The payoff for call option is

X = max(ST −K,0),

and the payoff for put option isX = max(K−ST ,0).

3.8 No-arbitrtage pricingThe no-arbitrage pricing argument is used to determine the price of a contingent claim i.e. thevalue of the payoff X at time t = 0.

Definition 25. An arbitrage opportunity occurs when it is possible to gain a risk free profitsuch that a self financing portfolio θ(t) has no value in the beginning i.e. V (0) = 0 but laterit will have a positive value V (T ) > 0. When there is no such arbitrage opportunity in themarket the correct price of the contingent claim X is the initial cost V (0) of the replicatingportfolio θ(t).

Theorem 1. If there are no arbitrage opportunity in the market where it exists a self-financingportfolio θ(t) that replicate the claim X, the value of the portfolio at time 0, V (0), is thecorrect price of the contingent claim.

In other words, the no-arbitrage argument occurs when the guaranteed value of the portfoliois concurrent with the value of the original portfolio plus the interest earned at the risk-freerate.[24] Even though such opportunities occur infrequently in the market, two same securitieswith different price can allow arbitragers to make a risk–free profit. Through time the tradingaction of arbitragers will eliminate the price discrepancy prevailed in the market, thus the lawof no–arbitrage is self–fulfilling. [20] The initial cost between two self–financing portfoliosmust be the same in order to prevent arbitrage opportunities.

18

3.8.1 BSM modelTheorem 2. [3] In a finite time interval [0,T ], the Black–Scholes model is free of arbitrageif and only if there exists a martingale measure (risk–neutral measure) Q. And if there existsa unique probability measure Q under no–arbitrage condition then the market is completeaccording to the first and second fundamental theorem of financial engineering respectively.

In other words, a contingent claim X is attainable if there exists a self–financing portfolio hsuch that

V h(T ) = X (3.4)

h is called a replicating or a hedging portfolio. If every contingent claim is attainable we saythe market is complete. So if there is no arbitrage opportunities in the market, the initial costV (0) of the replicating portfolio will be the correct price of the contingent claim X.

3.8.2 CTMC modelFor a contingent claim H in continuous-time Markov chain, if there exists a self-financingportfolio θ ′t = (ηt ,ξ

′t ), the initial cost of the portfolio which is V θ

0 is the correct price of thecontingent claim if there are no arbitrage opportunities in the market. The absence of arbitragecan be approached from the martingale point of view.

The discounted stock price process dynamics

dSit = Si

t−

[∑

j

(α

i j− r j + ∑k∈Y j

γi jk

λjk

)I jt dt +_ j ∑

k∈Y j

γi jkdM jk

t

]

For a given information FY under the risk–neutral probability measure Q, the discountedstock prices are martingales if the drift term equals to zero, that is(

αi j− r j + ∑

k∈Y j

γi jk

λjk

)= 0. (3.5)

Therefore the discounted stock prices defined by

dSit = Si

t− ∑k∈Y j

γi jkdM jk

t ,

are martingales. The process M jkt is also a martingale under Q.

The existence of a martingale under the probability measure Q tells us that the market isarbitrage–free.

3.9 Risk–neutral measureThe risk neutral valuation principle states that the price of a derivative on an asset St is notaffected by the risk preference of the investor; so it is assumed that they have the same riskaversion. Under this assumption, the derivative price ft at time t is done as in the following:

19

1. the expected return of the asset St is the risk free rate, µ = r.

2. calculate the expected payoff of the derivative as of time t, under condition 1.

3. discount at the risk–free rate from time T to time t.

Definition 26. [17], If there exists a linear pricing measure then there cannot be any dominanttrading strategies but there can be arbitrage opportunities. There must exist a linear pricingmeasure which gives a positive mass to every state ω ∈ Ω in order to rule out arbitrageopportunities. A probability measure Q on Ω is said to be a risk neutral probability measure if

1. Q(ω)> 0, all ω ∈Ω and

2. EQ[∆S∗n] = 0,n = 1,2, ...,N

The notation of EQ[X ] means the expected value of the random variable X under the probabilitymeasure Q. Then

EQ[∆S∗n] = EQ[S∗n(1)−S∗n(0)] = EQ[S∗1]−S∗n(0),

so EQ[S∗n] = 0 is equivalent to EQ[S∗1] = S∗n(0),n = 1,2, ...,NThis is to say that under the indicated probability measure the expected time t = 1 discounted

price of each risky security is equal to its initial price. Therefore, a risk neutral probabilitymeasure is a linear pricing measure giving strictly positive mass to every ω ∈ Ω. A veryimportant result connected to this is that there are no arbitrage opportunities if and only if thereexists a risk neutral probability measure Q. [11]

The Girsanov’s theorem which is named after Igor Vladimirovich Girsanov is very crucialin modeling finance. It defines the dynamics of Brownian motion and Markov process whenthe original probability measure P changes to the risk–neutral probability measure Q. ThisTheorem which is one of the ways to the martingale approach in arbitrage theory, gives usan entire control on continuous measure transformation of the Wiener (Brownian) process [3,Chapter 11].

Theorem 3. [11] By Girsanov’s theorem the process W (t) defined by

W (t) =W (t)−∫ t

0β (u)du,0≤ t ≤ T,

is a standard Brownian motion under Q

where

β (t) =W (t)−µ(t)

σ(t)

is a stochastic process β (t) satisfying the Novikon condition

E[

exp

12

∫ T

0β

2(u)du]

< ∞

20

The Brownian motion W (t) is a standard Brownian motion under P and the new processW (t) with drift under P becomes a standard Brownian motion under the new probability measureQ by Girsanov’s theorem. The new probability measure is constructed by Q(A) = E1AY (T ),A ∈ F where Y (t) is a positive martingale with Y (0) = 1. This martingale presented as

Y (t) =dQdP

is the Radon-Nikodym density process for the change of measure from P to Q.

Theorem 4. [5] By Girsanov’s theorem a Markov chain Y under the risk–neutral probabilitymeasure Q is defined by the infinitesimal matrix Λ

Λ = (λ jk),

where the matrix Λ under Q is equivalent to the matrix Γ under P, and λ jk = 0 if and only ifλ jk = 0.

In risk–neutral measure, λ jk is a solution to equation 3.5, for all other variables strictlypositive. The existence of such solution makes the discounted stock price a martingale.

3.10 The Fundamental Theorem of Financial EngineeringThe fundamental theorem of financial Engineering (Asset Pricing Theorem) is the pillar formodern financial theory. A market model which consists asset price processes S0,S1, ...,SNand the probability measure P, is subjected to two fundamental problems. These fundamentalproblems can be tackled by the "martingale approach" to financial derivatives, which explainsthe conditions under which the market is arbitrage free and complete. The First FundamentalTheorem states that

Theorem 5 (The first fundamental theorem). There are no arbitrage opportunities in a securit-ies market if and only if there exists a risk–neutral probability measure. If this is the case, theprice of an attainable contingent claim X is given by

V (0) = EQ[

XS0(T )

](3.6)

with S0(t) = B(t) for every replicating strategy.

If S0(t) = B(t), the no-arbitrage pricing of contingent claim X can be constructed in twosteps.By

1. finding a risk–neutral probability measure Q

2. calculating the expectation of V (0) under Q

Theorem 6. Suppose that a security market admits no arbitrage opportunities then it iscomplete if and only if there exists a unique risk-neutral probability measure. "[11].

21

The second fundamental theorem of asset pricing assumes that the market is arbitrage free.And if the market is arbitrage free then it is complete if and only if the martingale measureis unique. [17] When we deal with completeness we assume that there is no arbitrage in themarket, meaning we assume that there exists a martingale measure. "A security market is saidto be complete if every contingent claim is attainable or marketable; otherwise the market issaid to be incomplete"[11].[17]

3.11 OptionsThere are two types of options, a call and a put option. A call option gives the holder of thecontract the right but not the obligation to buy an asset at a certain date (maturity date) fora certain price called a strike price. The strike price K is fixed in the contract and it is theprice at which a derivative contract can be bought or sold. And S(T ) is the market price of theunderlying asset at the maturity T. A put option gives the holder the right but not the obligationto sell an asset at a certain date for a certain price. The price specified on the contract isknown as a strike price or the exercise price. Options can be further categorized as European orAmerican option. This categorization is not geographical but rather it is based on whether theycan be exercised before or at the maturity date. Therefore, American option is an option whichcan be exercised anytime up to the expiration date and European option is an option which canbe exercised only at the expiration date.

3.12 Methods for derivative pricing

3.12.1 The Black–Scholes–Merton modelIn BSM model derivatives are priced using the risk-neutral valuation method. In order tocalculate the price of a contingent claim at time t, first we find a risk-neutral probabilitymeasure Q to define a standard Brownian motion W ∗(t) and therefore approximate the priceof the underlying asset under Q. Second we compute the expected value of the contingentclaim under the probability measure Q given the information Ft .

Let the payoff function of a European contingent claim be denoted by the function h(x),and let the price of the claim at time t be denoted by C(t). The price of a contingent claim, thatis replicated through a self-financing portfolio with b(t) units of the money market B(t) andθ(t) units of the underlying stock S(t), is determined through this formula

C(t) = b(t)B(t)+θ(t)S(t)

=C(0)+∫ t

0b(u)dB(u)+

∫ t

0θ(t)dG(u),

(3.7)

where G(t) is the gain process

G(t) = S(t)+∫ t

0δ (u)du.

22

We calculate the Ito differential of the gain process and use the stock price process S(t)defined by the stochastic differential equation

dS = [µ(t)S−δ (t)]dt +σ(t)SdW,

and getdG = S[µ(t)dt +σ(t)dW ]. (3.8)

The process W ∗ is defined by

dW ∗ = λ (t)dt +dW,

where λ is the market price of risk defined by

λ (t) =µ(t)− r(t)

σ(t).

We substitute the process W ∗ and its price of risk into the equation (3.8) and obtain

dG(t) = S[r(t)dt +σ(t)dW ∗]. (3.9)

We change the probability measure to risk-neutral measure Q using the Girsanov’s theorem tomake the process W ∗ a standard Brownian motion. Therefore, we substitute the SDE of thegain process under Q (3.9) into the formula of the price of the claim (3.7) and get

C(t) =C(0)+∫ t

0r(u)C(u)du+

∫ t

0θ(u)S(u)σ(u)dW ∗.

It follows thatdC = r(t)C(t)dt +θ(t)σ(t)S(t)dW ∗.

Choosing the numéraire B(t) for discounting the price process, such that the denominated priceS∗ = S(t)/B(t) and the denominated claim price C∗ =C(t)/B(t) gives

C∗ =C∗(0)+∫ t

0θ(u)σ(u)S∗(u)dW ∗.

We see that the denominated claim price C∗ is a martingale under the risk-neutral probability,and because C(T ) = h(S(T )) at the maturity T , we obtain by no arbitrage the price of thecontingent claim under Q

C(t) = B(t)EQt

[h(S(T ))

B(T )

], 0≤ t ≤ T

3.12.2 The Continuous time Markov chain modelLet Yt : t ≥ 0 be a continuous time homogeneous Markov chain with finite state spaceY = 1,2, . . . ,n and transition probabilities

23

p jkt = PYs+t = k |Ys = j, j,k ∈ Y .

The transition intensities

λjk = lim

t↓0

p jkt

t, j,k ∈ Y , j 6= k.

are well defined [15, p. 4.]. Let us define the set of states that are directly accessible from statej by

Y j = k ∈ Y : λjk > 0, j ∈ Y ,

and let n j be the number of elements in the set Y j. Furthermore let

I jt = 1Yt= j

be the indicator of the event Yt = j.The stochastic processes N jk

t : t ≥ 0 are now defined by

N jkt = s;∈ (0, t] : Ys− = j,Ys = k.

In other words, the number of direct transitions of the process Ys from state j ∈ Y to statek ∈ Y j in the time interval (0, t]. In our model we let the nonnegative constants r j, j ∈ Y ,denote the interest rates in state j and the locally risk-free bank account be denoted by

Bt = exp

(∑j∈Y

r j∫ t

0I js ds

).

The risky stocks

Sit = exp

(∑j∈Y

αi j∫ t

0I js ds+ ∑

j∈Y∑

k∈Y j

βi jkNik

t

), 1≤ i≤ m,

where α i j is the deterministic rate of return of the ith stock in economy state j, and β i jk

have the following sense: upon any transition of economy from state j to state k, the ith stockmakes a price jump of relative size γ i jk = exp(β i jk)−1.

Theorem 7. The above market is arbitrage-free if and only if the system of equations

αi j− r j + ∑

k∈Y j

γi jk

λjk = 0, j ∈ Y ,1≤ i≤ m

has a solution λ jk with all entries strictly positive.

Let ΓΓΓj be the matrix with m rows, n j columns, and matrix entries γ i jk, 1≤ i≤m, 1≤ k≤ n j.

Theorem 8. The above market is complete if and only if the rank of the matrix ΓΓΓj is equal to

n j.

24

Let H(YT ,SlT ) be a contingent claim of European type with maturity T written on the lth

stock.

Theorem 9. The time-t price of the contingent claim H(YT ,SlT ) is

πt = ∑j∈Y

I jt f j(t,Sl

t),

where

f j(t,s) = E∗

[exp

(−∫ T

t∑

k∈YIkurk],du

)H(Yt ,Sl

t) |Yt = j,Slt = s

]. (3.10)

3.13 The method of PDE

3.13.1 Black–Scholes–Merton PDEThe price dynamics for a European call and put options is governed by the Black–Scholesmodel which is a Partial Differential Equation (PDE). The price of the money-market accountB(t) follows the ordinary differential equation (ODE), where r in the equation is a positiveconstant.

dB(t) = rB(t)dt,0≤ t ≤ T

The price of the no dividend paying underlying asset (stocks) follows the Stochastic differentialequation (SDE) is given as

dS = µSdt +σSdW,0≤ t ≤ T (3.11)

where W (t) is a standard Brownian motion and µ and σ are positive constants.The Ito formula which solves the SDE of the stock price, equation 3.11 is

dX =

(µS

∂X∂S

+∂X∂ t

+12

σ2S2 ∂ 2X

∂S2

)dt +σS

∂V∂S

dW (3.12)

Theorem 10. [11] Let S(t) be an Ito process given by the equation 3.11. Let C(t) = f (S(t), t)represent the time t price of an option with strike price K and maturity T . The smooth functionf (S, t) is continuously differentiable in t and twice differentiable in S. For S(0) = S, we obtainfrom the Ito formula

dCC

= µc(t)dt +σc(t)dw,0≤ t ≤ T (3.13)

Where the mean rate of return of the option is

µc(t) =1C

[∂ f (S, t)

∂ t+µS

∂ f (S, t)∂S

+σ2S2

2∂ 2 f (S, t)

∂S2

],

and the volatility is

σc(t) =σSC

∂ f (S, t)∂S

.

25

Under the self–financing assumption the rate of return of the portfolio is given by

dWW

= wdSS

+(1−w)dCC

, (3.14)

Where the fraction www of the wealth W is the amount invested into the security S(t). Substitutingthe equations (3.11) and (3.13) into (3.14) yields

dWW

= (wµ +(1−w)µc(t))dt +(wσ +(1−w)σc(t))dW.

Suppose that σ 6= σc(t), and assume

w(t) =− σc(t)σ −σc(t)

,

thenw(t)σ +(1−w(t))σc(t) = 0,

indicates that the portfolio is risk–free.For such risk–free portfolio the rate of return of the portfolio, under no–arbitrage condition,

must be equal to the rate of return of a risk–free security. That is

w(t)µ +(1−w(t))µc(t) =− µσc(t)σ −σc(t)

+µc(t)σ

σ −σc(t)= r,

It follows that the mean excess return per unit of risk for the stock is equal to the mean excessreturn of the derivative, shown in equation below

−µ− rσ

=µc(t)− r

σc(t), (3.15)

By substituting the mean rate of return µc(t) and the volatility σc(t) into the equation (3.15)we will obtain the famous Black-Scholes PDE

∂X∂ t

+12

σ2S2 ∂ 2X

∂S2 + rS∂X∂S− rX = 0 (3.16)

Thus, the key boundary condition for a European call option when t = T is

X = max(S−K,0),

and for a European put option when t = T

X = max(K−S,0),

26

3.13.2 Norberg PDEFor the Norberg model, we have the following result [15]

Theorem 11. The functions (3.10) solve the boundary value problem

− r j f j(t,s)+∂ f j(t,s)

∂ t+α

l js∂ f j(t,s)

∂ s+ ∑

k∈Y j

λjk( f k(t,s(1+ γ

l jk))− f j(t,s)) = 0,

f j(T,s) = h( j,s).

(3.17)

In particular, the final condition for the European call option has the form

f j(T,s) = maxs−K,0,

where K is the strike price of the option.

3.14 Pricing formulaeThe stochastic differential equation for Black–Scholes (3.16) can be solved in closed form.This solution leads to the famous Black–Scholes formula for both call option (c) and put option(p) as described in [10]

c = S0N(d1)−Ke−rT N(d2)

p = Ke−rT N(d2)−S0N(d1)

Where

d1 =ln(S0/K)+(r+σ2/2)T

σ√

T

d2 =ln(S0/K)+(r−σ2/2)T

σ√

T= d1−σ

√T .

In contrast to the BSM model, the partial differential equations (3.17) of the Norberg modelcannot be solved in closed form even for simplest cases and require using numerical methods,which is outside the scope of this Bachelor thesis.

27

Chapter 4

Similarities and differences between thetwo models

In doing the thesis we have noticed that there are some similarities and differences between thetwo models regarding the construction of the financial market component.

4.1 similarities• The financial market components are the same for both BSM and CTMC models.

• Contingent claim is the same for both models since it is not model dependent.

• The method of PDE works for both to price European call or put options.

• The driving processes for both models have a Markov property which is the distribution ofthe future process depends only on the current state, not on the past i.e.’ memorylessness’.

• The driving forces in both models i.e. the Poisson process and the Brownian motionbelong to the Lévy process. Lévy process is a stochastic process with independent,stationary increments: it represents the motion of a point whose successive displacementsare random, in which displacements in pairwise disjoint time intervals are independent,and displacements in different time intervals of the same length have identical probabilitydistributions.

• The counting process in the CTMC model and the continuous sample paths in the BSMmodel both share the property of independent increments meaning the number of eventsthat occur in disjoint time intervals is independent.

4.2 differences• The sample space Ω in CTMC model is the set of all “càdlàg” functions. These functions

are useful for stochastic processes characterized by jumps or have a discontinuous

28

function like the Markov chain. On the contrary, the sample space in the BSM model iscontinuous sample paths or trajectories.

• In the CTMC model the finite state space is discrete and only takes a finite number ofvalues so that the random variable jumps finite steps or it is a step function, on the otherhand, the finite state space in the BSM model is continuous.

• As the Brownian motion (Wiener process) is the stochastic (driving) process in the BSMmodel, it is the Poisson process or the Markov chain in the CTMC model.

One of the drawbacks of the Brownian motion is that, even if it is the most frequentlyused model in pricing a derivative, it does not capture unexpected price changes or jumps.The CTMC model is a better one in considering the price jumps which are apparent inthe market.

• In the BSM model the market is always complete according to Theorem 2. In the CTMCmodel, the market is not always complete. The Norberg model is complete under someconditions in Theorem 8.

• The European call or put option in the BSM model has a closed analytical formula and itis mentioned in section 3.11. The PDE pricing formula in CTMC is not a closed formulaand it should be solved numerically.

• Even if it is well-known that the volatility of financial data series tends to change overtime, the volatility in the BSM model is constant but the CTMC model assumes stochasticvolatility.

29

Chapter 5

Conclusions

In this thesis, we have reviewed the two financial pricing models: Black–Scholes–Mertonand Continuous-time Markov chain model. We have defined the basic components ofa financial market and formulated the components for both models. Even though bothmodels have the same financial market components which is one of their similarities,they also have some differences. Moreover, we have learned how to price a Europeanoption using the two models. Regarding the construction of the financial components wefound the Markov chain model to be more complicated than the Black–Scholes model.But comparing the two models, we have seen that the CTMC model has more featuresthan the classical BSM model. The lack of enough literature on the Continuous-MarkovChain model was a limitation.

30

Appendix A

Criteria for a Bachelor Thesis

The Swedish National Agency for Higher Education has provided certain requirements forBachelor’s Degree thesis in mathematics, mathematical statistics, financial mathematics, andactuarial science. Hence, we have written our thesis per the requirements.

While writing the thesis, we have demonstrated knowledge and understanding of the twofinancial pricing models, namely the Black–Scholes–Merton and the Continuous-time Markovchain model. To fulfill this, we have searched and critically evaluated information from differentbooks and articles. Our ability to identify, formulate and solve problems has improved overtime and we have managed to comply with the specified time frames. Finally, we demonstratedour ability to present orally and in writing and discuss information, problems, and solutions indialogue.

31

Appendix B

An ethical description of contribution ofcoauthors

The first named author of the thesis has written Chapters 2, 4, and the CTMC model part inchapter 3. The second named author has written Chapters 1, 5, and the BSM model part inchapter 3. All the remaining parts of the thesis were written by the two authors together.

32

Bibliography

[1] L. Bachelier. Théorie de la spéculation. Ann. Sci. École Norm. Sup. (3), 17:21–86, 1900.

[2] Patrick Billingsley. Convergence of probability measures. Wiley Series in Probability andStatistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition,1999. A Wiley-Interscience Publication.

[3] Tomas Björk. Arbitrage theory in continuous time. Oxford university press, 2009.

[4] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. J.Polit. Econ., 81(3):637–654, 1973.

[5] Pierre Brémaud. Point processes and queues. Springer-Verlag, New York-Berlin, 1981.Martingale dynamics, Springer Series in Statistics.

[6] Ovidiu Calin. An introduction to stochastic calculus with applications to finance. AnnArbor, 2012.

[7] Robert J. Elliott, Chuin Ching Liew, and Tak Kuen Siu. Characteristic functions andoption valuation in a Markov chain market. Comput. Math. Appl., 62(1):65–74, 2011.

[8] Robert J. Elliott and Tak Kuen Siu. A note on differentiability in a Markov chain marketusing stochastic flows. Stoch. Anal. Appl., 33(1):110–122, 2015.

[9] Robert J. Elliott and John van der Hoek. Default times in a continuous time Markov chaineconomy. Appl. Math. Finance, 20(5):450–460, 2013.

[10] John Hull et al. Options, futures and other derivatives/John C. Hull. Upper Saddle River,NJ: Prentice Hall„ 2009.

[11] Masaaki Kijima. Stochastic processes with applications to finance. Taylor and FrancisGroup, 2013.

[12] A. N. Kolmogorov. On the Skorohod convergence. Teor. Veroyatnost. i Primenen.,1:239–247, 1956.

[13] Robert C. Merton. Theory of rational option pricing. Bell J. Econom. and ManagementSci., 4:141–183, 1973.

33

[14] Ragnar Norberg. A time-continuous markov chain interest model with applications toinsurance. Applied stochastic models and data analysis, 11(3):245–256, 1995.

[15] Ragnar Norberg. The Markov chain market. ASTIN Bulletin: The Journal of the IAA,33(2):265–287, 2003.

[16] Peyton Z Peebles. Probability, random variables, and random signal principles, volume 3.McGraw-Hill New York, NY, USA:, 1987.

[17] Stanley Pliska. Introduction to mathematical finance. Blackwell publishers Oxford, 1997.

[18] Sheldon M Ross. Stochastic processes, volume 2. Wiley, New York, 1996.

[19] Zeev Schuss. Theory and applications of stochastic processes, volume 170 of AppliedMathematical Sciences. Springer, New York, 2010. An analytical approach.

[20] Euan Sinclair. Option trading: Pricing and volatility strategies and techniques, volume445. John Wiley & Sons, 2010.

[21] A. V. Skorohod. Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen.,1:289–319, 1956.

[22] Matti Turra. A continuous-time markov chain approach to financial modelling: Analysisand applications. International Journal of Pure and Applied Mathematics, pages 1029–1054, 2016.

[23] John van der Hoek and Robert J. Elliott. Asset pricing using finite state Markov chainstochastic discount functions. Stoch. Anal. Appl., 30(5):865–894, 2012.

[24] Paul Wilmott. Paul Wilmott introduces quantitative finance. John Wiley & Sons, 2007.

34