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

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BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS

A review of two financial market models: the Black–Scholes–Merton and

the Continuous–time Markov chain models

by

Haimanot Ayana and Sarah Al-Swej

Kandidatarbete i matematik / tillämpad matematik

DIVISION OF MATHEMATICS AND PHYSICS

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

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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–time Markov 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

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Abstract

The objective of this thesis is to review the two popular mathematical models of the financial derivatives 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 the mathematician Ragnar Norberg. The thesis demonstrates how the fundamental results of Financial Engineering work in both models.

The construction of the main financial market components and the approach used for pricing the contingent claims were considered in order to review the two models. In addition, the steps used 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 the same. Their contingent claim is similar and the driving processes for both models utilize Markov property.

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

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

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Acknowledgements

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

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Chapter 1 Introduction

1.1 Background of the thesis

Financial Engineering is a multidisciplinary field relating to the creation of new financial instruments and strategies. It is the process of employing mathematical models, financial theory and 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 an important discovery was achieved in the 1970s by Fisher Black, Myron Scholes and Robert Merton 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 Brownian motion and it assumes constant volatility [10]. But in practice, volatility varies through time or stochastically. This drawback in the model led to the development of more complex models like the 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 is that it exhibits the Markov property, which means the predictions about stochastic processes can be simplified by viewing the future as independent of the past given the present state of the process. In other words, this Markov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future. The past history of the variable and the way that the present has emerged from the past are irrelevant.

1.2 Literature review

The 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 the driving 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

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continuous-time Markov chain. This model has been further developed by Turra [22]. A different model driven 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 thesis

In chapter 2, a brief description of the two market models is given. In chapter 3, the general definition of the financial market and its components such as probability space, the stochastic process, filtration, basic assets, derivative instruments, etc is covered. In addition, examples are given for each financial component of the models. A review of the main technical tools of the financial theory like no-arbitrage pricing, numéraire, change of measure, and the Fundamental Theorem of Financial Engineering is covered in this chapter. In the end, examples that show how the above theoretical considerations work in both models are shown.

After the comparison of the two models, in chapter 4 the similarities and differences exhibited 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.

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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 other types of stochastic processes which are important and a better alternative in finance. A Poisson process,type of a Markov process, is one of them which is used to model the jumps arising in asset 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 other types of stochastic processes which are important and better alternative in finance. A Poisson process, a type of a Markov process, is one of them which is used to model the jumps arising in asset prices.

2.1 The Black–Scholes–Merton model

In the pricing of European stock options, the model developed by Fischer Black, Myron Scholes and Robert Merton in the early 1970s has achieved a major quantum leap. The model is known as the Black–Scholes–Merton (Black–Scholes) model. It has had a huge effect on the financial market as it has influenced the way traders price and hedge derivatives. The model’s huge effect has got recognition when the scholars were awarded the Nobel prize for economics in 1997. ([10])

Even though there were other researchers who have succeeded in calculating correctly the expected payoff from a European option, knowing the correct discount rate was not easy. To overcome 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 in CAPM were not easy and do not hold in reality. As a result, Merton preferred to use a new and different approach that involved a riskless portfolio that consists of the option and the underlying stock. In addition, he was arguing the return on the portfolio over a short period of time must be the risk–free return. In our thesis, we use and focus on Merton’s approach to derive the Black–Scholes–Merton model. The thesis gives emphasis on how the model shows the risk-neutral valuation argument and how to estimate volatility from historical data or implied from option prices.

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

ln ST ∼ φ [ln S0+ (µ − σ2/2)T, σ2T],

where

S_T = stock price at future time T S₀= stock price at time 0

µ = expected return on stock per year σ = volatility of the stock price per year

The above assumption holds mathematically because the random variable ln ST is normally

distributed as a result ST has a lognormal distribution. The BSM model also assumes the

continuously compound rate of returns are normally distributed with a mean of µ −σ2 2 and

standard 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 returns from the stock. The volatility of a stock can be referred to as the standard deviation of the continuously compounding return which is indicated in the above paragraph. Volatility can be calculated from historical data of the stock price which is observed on a daily, weekly and monthly basis i.e based on a fixed intervals of time. To calculate the volatility through determining the standard deviation, first we need to calculate the daily return from the historical prices of the stock by using the formula ui= ln(_SS_i−1i ), then we compute the expected price

(mean) of the historical prices. Then we need to work out the difference between the daily return and the expected price and square the differences from the previous step. After determining the sum 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 =

q

1

n−1∑ni=1(ui− ˆu)2, where ˆuis the mean of the ui(daily return).

The Black–Scholes–Merton differential equation must be satisfied by the price of any derivative of non–dividend-paying stock. The equation will be derived in chapter 3. A riskless portfolio consisting of a position in the stock and in the derivative is important. The argument which leads to Black–Scholes–Merton differential equation is that in the absence of arbitrage opportunities, the portfolio must be the risk–free interest rate r. In a riskless portfolio, the stock price and the derivative price are affected by the stock price which is the underlying source of uncertainty. To remain riskless the gain or loss from the stock position always offsets the gain or 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–Merton differential equation. The equation which we later discuss does not contain any variable that is affected by the risk preferences of the investors. The model is assuming that investors are risk neutral therefore the expected return on all underlying assets is the risk-free interest rate, r. ([10])

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2.2 The Continuous–time Markov chain model

The Markov chain model was discovered by a Russian mathematician Andrey Andreyevich Markov (1856–1922). Initially, he was best known for his work on stochastic processes. Later in 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

P{X_(s+t)= k|Xs= j, Xu= xu, 0 ≤ u ≤ s} = P{X(s+t)= k|Xs= j}

This means that a continuous-time Markov chain is a stochastic process with a Markov property that the conditional distribution of future state k at future time s + t, given the present state 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 to

determine the distribution of the future. The future is only determined by where we are at the present 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 transition probabilities 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 ’A Markov Chain Financial Market’. Ragnar Norberg, a Norwegian insurance mathematician, has achieved 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 homogeneous Markov chain. In addition, the conditions for the absence of arbitrage, completeness and the non–arbitrage pricing of a derivative is explained in detail. Below we will define and elaborate on 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 {Yt}t≥0 is a continuous–time Markov chain with finite state space y = {1, ...., n}, then

the transition probabilities in the Markov chain market is P_tjk_{= P[Y}_s+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 conditional probability.

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

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how long it takes to get from the state j to k i.e. the length of the transition period. This makes the transition probability time-homogeneous. Throughout our thesis, the transition probability is 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_tjk

t , j 6= k, exist and are constant.

yj= {k, λjk≥ 0}

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

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

nj= |yj|

λj j= −λj= −

_∑

k;k∈yj λjk

The diagonal elements in the matrix denoted by λj jare the negative of the sum of all other off the diagonal matrix elements.

Definition 5. [15] The Λjk which is equal to the λjk is called the infinitesimal matrix or the rate matrix. The rates are placed off the diagonal matrix.

The assumption here is that all states intercommunicate so P_tjk> 0 for all j, k (and t > 0), which also implies that nj> 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= (ptjk) and

the infinitesimal matrix, with ΛΛΛ = λjk, such that λjk> 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 work influenced many branches of modern mathematics including probability theory and stochastic processes especially Markov processes. He came up with a set of equations to describe how in a small time interval there is a probability that the state will remain unchanged, however, if it changes the change may be radical and might lead to a jump process.

Definition 6. [15] The infinitesimal matrix is defined as Λ = (λjk) and the matrix of transition probabilities as Pt= (ptjk). Taking the limit of the infinitesimal matrix Λ yields

Λ = lim

t0 1

t(Pt− I), where I is the identity matrix.

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Taking the derivative of the matrix of transition probabilities, by the forward and backward Kolmogorov differential equations, _dd

tPt= 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 call

the jump rate matrix.

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

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

I_tj= 1{Yt = j}

If I_tj = 1 it means that it is true the event Y is at state j at time t, and if I_tj= 0 it means otherwise.

Definition 8. [15] The counting process can be defined as the number of direct transitions of Y from state j to state k ∈ yjin the time interval of (0,t]. For k /∈ yj_{, N}jk

t ≡ 0. N_tjk= |{s; 0 < s ≤ t,Ys− = j,Ys= k}| where

∑

k;k6= j (N_tk j− N_tjk), is the number of events occurred during the interval (s, T ].

Therefore, Y_t =

_∑

j jI_tj, I_tj= I₀j+

_∑

k;k6= j (N_tk j− N_tjk)

All the state process, the indicator processes and the counting processes carry the same information, which at any time t is represented by the sigma algebra FtY = σ {Ys; 0 ≤ s ≤ t}.

The filtration , denoted by FY= {F_tY}_t≥0, satisfies the conditions for right continuity, so does Y , Ijand Njkare right–continuous. According to [11], the {F_tY} is a sigma algebra which is a family of events associated with a random experiment.

Definition 9. [15] The compensated counting processes denoted as Mjk, j 6= k, defined by dM_tjk= dN_tjk− I_tjλjkdt

and

M₀jk= 0,

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Chapter 3 Financial market components

In this chapter, we introduce the main object of studies in Financial Engineering, a financial market. Financial markets consist of several components. Each component is defined in one of the following sections and formulated for the Black–Scholes–Merton and the continuous Markov 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 a random 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 A is a subset of Ω. Using the notation of set theory, ω ∈ Ω and A ⊂ Ω, respectively[11]. A sample space can be continuous or discrete. A sample space is said to be continuous if it has uncountable sample points and discrete if it has finite or countably infinite sample points[16].

• σ –field (F)

The σ –field which is denoted by F contains all the family of events or outcomes under consideration. 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, and

3. If An⊂ Ω, n = 1, 2, ..., are in F thenS∞nAnis in F.

• A Probability measure (P)

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

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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∩ Aj= ∅ for i 6= j we

have 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 t and an element ω in a probability space Ω, in which the events are defined. For each ω ∈ Ω the stochastic process is called trajectories or sample paths which are continuous. These continuous trajectories 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 all continuous real functions.

Ω = {ω (t) : R+→ R}

The Brownian events consist of uncountably many Brownian trajectories. Its formulation is more complicated than the elementary events as it contains a cylinder set which is beyond the scope 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, ...), thenS∞1Ai∈ F

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

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

Definition 12. [19] A function f : [0, T ] → R is called a càdlàg function if it is right-continuous on [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 called a Skorhod space. The σ -field F was constructed by the Ukrainian mathematician Anatoliy Skorokhod in [21], see also [12]. The construction is complicated and will not be presented here, for further details see the textbook [2].

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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 of the process at time t. If the value of the random variable changes only at a certain fixed point in time then X (t) is a discrete–time stochastic process. If the random variable takes any value within 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 epochsT is a fixed subset of the real number R. X(t) is a function of two variables:

x(t, ω) :T × Ω → R

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

X = (X (t₁), X (t₂), ...., X (t_n)),

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

3.2.1 Stochastic Process in the BSM and CTMC model

The 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 motion process 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 variance s.

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 process {Nt,t ≥ 0} is said to be a counting process if Nt represents the total number of events that have

occurred 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 by time 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).The parameter λ implies the transition intensities and it is a positive number.

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3.3 The Filtration F

t

The filtration Ft is a σ –field which contains the available information about security prices in a

stochastic process in the market until time t.

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

s≤ t

Fs⊆ Ft.

Given the stochastic process Xs, which is measurable for every s ∈ [0,t] the simplest and

smallest σ –field is the one generated by the process itself, F_tX = σ (Xs; 0 ≤ s ≤ t).

3.3.1 Filtration in the BSM and CTMC model

Definition 15. [11] The filtration of Black–Scholes is denoted by Ft = σ {W (s); s ≤ t} where

Ft 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 = σ {Y (s); s ≤ t} is a set of

information of present and past values of the stochastic process Y = {Yt; 0 ≤ t ≤ T }. Since the

Markov propertyindicates that the future value of the process is independent of the past, given the 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 real value 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 positive

process {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 or debts grow at a risk free rate. The positive process denoted by B(t) is a bank account and it can also serve as a numéraire.[3]

Definition 18. [6] In a continuous time a continuous bank account, uses a continuously compounded 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) = B0then the account balance at time t is given by the

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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, then the 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 rjbe the state–wise interest rates, where j = 1, ...., n, and let I_tj be the indicator of the event that Y is in state j at time t, then the bank account in CTMC market is B_t = exp Z t 0 r_sds = exp

_∑

j rj Z t 0 I_sjds !

Similarly the dynamics of the bank account is defined as dB_t= Btrtdt = Bt

∑

j

rjI_tjdt

• 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 stock price denoted by S(t) is a stochastic process.[11]

Definition 21. [11] The stochastic process {S(t); 0 ≤ t ≤ T.} defined on the probability space (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 + σ SdW, 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

dSt_i = S_t−i

_∑

j αi jI_tjdt+

_∑

j k∈y

∑

j γi jkdN_tjk !

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3.5 Portfolio

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

Definition 22. [11] Let θi(t) denote the number of security i carried from time t − 1 to t where

0 ≤ t ≤ T . The vector θ (t) = (θ0(t), θ1(t), ...., θn(t))T denotes the portfolio of n securities at

time 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)} is

predictable 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 φ1is the number of units of the stock hold in the risky security S(t) and φ2is the number

of bonds hold in the money market account B(t). Both φ1and φ2are stochastic processes and

they 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_t0= (ηt, ξt0),

where ηt represents the number of units of the bank account held at time t, and the i–th entry in ξt= (ξt1, ..., ξtm)0

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

3.6 Trading strategy

A self-financing portfolio is a trading strategy where the value of the portfolio at any time after a transaction is equal to its value before that transaction. Thus the portfolio does not need any additional investments and therefore any change in the value of the portfolio is due to change in stocks value.

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

V(t) =

_∑

i=0

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The value of a self–financing portfolio is V(t) = V (0) + n

∑

i=0 Z t 0 θi(s + 1)dGi(s), 0 ≤ t ≤ T,

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

dG_i= 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 model

The portfolio is defined as a vector h = (φ1, φ2) and the value process of the portfolio h is

defined 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 the portfolio is not too big, then the integrals

Z 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) + Z t 0 φ1(u)dS(u) + Z t 0 φ2(u)dB(u)

The above equation is a self–financing portfolio. The economical sense is that the portfolio is created at time 0, there is no adding or withdrawal of money, so the purchase of the new asset must be financed by the sale of an old one. The variation in gain comes from the risk and non–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] Vθ t = ηtBt+ ξt0St = ηtBt+ m

∑

i=0 ξ_tiS_ti where ˜ St= ( ˜S1t, ..., ˜Stm)0.

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

Vθ

t = ηt+ ξt0S˜t.

The strategy θ is self-financing (SF) if dVθ t = ηtdBt+ ξt0dSt, or, equivalently, d ˜Vθ t = ξt0d ˜St= m

∑

i=1 ξ_tid ˜Si_t. (3.3)

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3.7 Contingent Claims

According to Pliska [17], a contingent claim is a random variable X that represents the payoff at time T of a contract between a seller and a buyer at time t < T . The most common types of contract are call and put options. At time T the payoff X can be positive or negative for some states of ω ∈ Ω. For a call option the buyer will have a positive payoff if the price of the underlying asset at time T is greater than the price at time t. Otherwise the payoff will be negative 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 Z t 0 θi(s)dGi(s)

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

X= max(S_T− K, 0), and the payoff for put option is

X= max(K − ST, 0).

3.8 No-arbitrtage pricing

The no-arbitrage pricing argument is used to determine the price of a contingent claim i.e. the value of the payoff X at time t = 0.

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

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

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

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3.8.1 BSM model

Theorem 2. [3] In a finite time interval [0, T ], the Black–Scholes model is free of arbitrage if and only if there exists a martingale measure (risk–neutral measure) Q. And if there exists a unique probability measure Q under no–arbitrage condition then the market is complete according 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 h such that

Vh(T ) = X (3.4)

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

3.8.2 CTMC model

For a contingent claim H in continuous-time Markov chain, if there exists a self-financing portfolio θ_t0= (ηt, ξt0), the initial cost of the portfolio which is ˜V0θ is the correct price of the

contingent claim if there are no arbitrage opportunities in the market. The absence of arbitrage can be approached from the martingale point of view.

The discounted stock price process dynamics d ˜Si_t= ˜Si_t− "

∑

j αi j− rj+

_∑

k∈Yj γi jkλ˜ jk ! I_tjdt+ _ j

_∑

k∈Yj γi jkd ˜M_tjk #

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

αi j− rj+

_∑

k∈Yj

γi jkλ˜ jk !

= 0. (3.5)

Therefore the discounted stock prices defined by d ˜Si_t= ˜Si_t−

_∑

k∈Yj

γi jkd ˜M_tjk,

are martingales. The process ˜M_tjk_{is also a martingale under Q.}

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

3.9 Risk–neutral measure

The risk neutral valuation principle states that the price of a derivative on an asset St is not

affected by the risk preference of the investor; so it is assumed that they have the same risk aversion. Under this assumption, the derivative price ft at time t is done as in the following:

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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 dominant trading strategies but there can be arbitrage opportunities. There must exist a linear pricing measure which gives a positive mass to every state ω ∈ Ω in order to rule out arbitrage opportunities. 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 probability

measure Q. Then

E_Q[∆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, ..., N

This 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 probability measure is a linear pricing measure giving strictly positive mass to every ω ∈ Ω. A very important result connected to this is that there are no arbitrage opportunities if and only if there exists a risk neutral probability measure Q. [11]

The Girsanov’s theorem which is named after Igor Vladimirovich Girsanov is very crucial in modeling finance. It defines the dynamics of Brownian motion and Markov process when the original probability measure P changes to the risk–neutral probability measure Q. This Theorem which is one of the ways to the martingale approach in arbitrage theory, gives us an 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) −

Z t

0

β (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 1 2 Z T 0 β2(u)du < ∞

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The Brownian motion W (t) is a standard Brownian motion under P and the new process ˜

W_{(t) with drift under P becomes a standard Brownian motion under the new probability measure} Q 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) = dQ

dP

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 probability measure Q is defined by the infinitesimal matrix ˜Λ

˜

Λ = ( ˜λjk),

where the matrix ˜_{Λ under Q is equivalent to the matrix Γ under P, and}λ˜ jk= 0 if and only if λjk= 0.

In risk–neutral measure, ˜λjk is a solution to equation 3.5, for all other variables strictly positive. The existence of such solution makes the discounted stock price a martingale.

3.10 The Fundamental Theorem of Financial Engineering

The fundamental theorem of financial Engineering (Asset Pricing Theorem) is the pillar for modern financial theory. A market model which consists asset price processes {S0, S1, ..., SN}

and the probability measure P, is subjected to two fundamental problems. These fundamental problems can be tackled by the "martingale approach" to financial derivatives, which explains the conditions under which the market is arbitrage free and complete. The First Fundamental Theorem 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, the price of an attainable contingent claim X is given by

V(0) = EQ X S₀(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 two

steps.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 is complete if and only if there exists a unique risk-neutral probability measure. "[11].

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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 measure is unique. [17] When we deal with completeness we assume that there is no arbitrage in the market, meaning we assume that there exists a martingale measure. "A security market is said to be complete if every contingent claim is attainable or marketable; otherwise the market is said to be incomplete"[11].[17]

3.11 Options

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

3.12 Methods for derivative pricing

3.12.1 The Black–Scholes–Merton model

In BSM model derivatives are priced using the risk-neutral valuation method. In order to calculate the price of a contingent claim at time t, first we find a risk-neutral probability measure Q to define a standard Brownian motion {W∗(t)} and therefore approximate the price of the underlying asset under Q. Second we compute the expected value of the contingent claim 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, that is replicated through a self-financing portfolio with b(t) units of the money market B(t) and θ (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) + Z t 0 b(u)dB(u) + Z t 0 θ (t)dG(u), (3.7)

where G(t) is the gain process

G(t) = S(t) +

Z t

0

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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 get

dG= 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 to make the process {W∗} a standard Brownian motion. Therefore, we substitute the SDE of the gain process under Q (3.9) into the formula of the price of the claim (3.7) and get

C(t) = C(0) + Z t 0 r(u)C(u)du + Z t 0 θ (u)S(u)σ (u)dW∗. It follows that dC= 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 price S∗= S(t)/B(t) and the denominated claim price C∗= C(t)/B(t) gives

C∗= C∗(0) +

Z t

0

θ (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 the contingent claim under Q

C(t) = B(t)E_tQ h(S(T )) B(T )

, 0 ≤ t ≤ T

3.12.2 The Continuous time Markov chain model

Let {Yt: t ≥ 0 } be a continuous time homogeneous Markov chain with finite state space

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p_tjk= P{Ys+t= k |Ys= j}, j, k ∈Y .

The transition intensities

λjk= lim

t↓0

p_tjk

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 state jby

Y j_{= { k ∈}_{Y : λ}jk_{> 0 },} _j_∈_{Y ,}

and let njbe the number of elements in the setY j. Furthermore let I_tj= 1_{Y_t_{= j}}

be the indicator of the event {Yt= j}.

The stochastic processes { N_tjk: t ≥ 0 } are now defined by N_tjk= { s; ∈ (0,t] : Ys−= j,Ys= k }.

In other words, the number of direct transitions of the process Ys from state j ∈Y to state

k∈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

B_t= exp

_∑

j∈Y rj Z t 0 I_sjds ! . The risky stocks

Si_t= exp

_∑

j∈Y αi j Z t 0 I_sjds +

_∑

j∈Y k∈Y

∑

j βi jkN_tik ! , 1 ≤ i ≤ m,

where αi j is the deterministic rate of return of the ith stock in economy state j, and βi jk have the following sense: upon any transition of economy from state j to state k, the ith stock makes a price jump of relative size γi jk= exp(βi jk) − 1.

Theorem 7. The above market is arbitrage-free if and only if the system of equations αi j− rj+

_∑

k∈Yj

γi jkλ˜ jk= 0, j∈Y ,1 ≤ i ≤ m

has a solution ˜λjk with all entries strictly positive.

Let ΓΓΓjbe the matrix with m rows, njcolumns, and matrix entries γi jk, 1 ≤ i ≤ m, 1 ≤ k ≤ nj. Theorem 8. The above market is complete if and only if the rank of the matrix ΓΓΓjis equal to nj.

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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_tjfj(t, Sl_t), where fj(t, s) = E∗ " exp − Z T t _k∈

∑

_Y I_ukrk], du ! H(Yt, Slt) |Yt = j, Stl= s # . (3.10)

3.13 The method of PDE

3.13.1 Black–Scholes–Merton PDE

The price dynamics for a European call and put options is governed by the Black–Scholes model which is a Partial Differential Equation (PDE). The price of the money-market account B(t) follows the ordinary differential equation (ODE), where r in the equation is a positive constant.

dB(t) = rB(t)dt, 0 ≤ t ≤ T

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

dS= µSdt + σ SdW, 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 = µ S∂ X ∂ S + ∂ X ∂ t + 1 2σ 2_S2∂2X ∂ S2 dt+ σ S∂V ∂ SdW (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 function f(S,t) is continuously differentiable in t and twice differentiable in S. For S(0) = S, we obtain from the Ito formula

dC

C = µc(t)dt + σc(t)dw, 0 ≤ t ≤ T (3.13) Where the mean rate of return of the option is

µc(t) = 1 C ∂ f (S, t) ∂ t + µS ∂ f (S, t) ∂ S + σ2S2 2 ∂2f(S,t) ∂ S2 , and the volatility is

σ c(t) = σ S C

∂ f (S, t) ∂ S .

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Under the self–financing assumption the rate of return of the portfolio is given by dW W = w dS S + (1 − w) dC C , (3.14)

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

dW

W = (wµ + (1 − w)µc(t))dt + (wσ + (1 − w)σ c(t))dW. Suppose that σ 6= σ c(t), and assume

w(t) = − σ c(t) σ − σ c(t), then

w(t)σ + (1 − w(t))σ c(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) = − µ σ c(t) σ − σ c(t)+

µ c(t)σ σ − σ c(t) = r,

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

−µ − r σ =

µ c(t) − r

σ c(t) , (3.15)

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

∂ X ∂ t + 1 2σ 2_S2∂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

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3.13.2 Norberg PDE

For the Norberg model, we have the following result [15]

Theorem 11. The functions (3.10) solve the boundary value problem − rjfj(t, s) +∂ f j_{(t, s)} ∂ t + α l j_s∂ fj(t, s) ∂ s +_k∈

∑

_Yj ˜ λjk( fk(t, s(1 + γl jk)) − fj(t, s)) = 0, fj(T, s) = h( j, s). (3.17)

In particular, the final condition for the European call option has the form fj(T, s) = max{s − K, 0},

where K is the strike price of the option.

3.14 Pricing formulae

The 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−rTN(d2) p= Ke−rTN(d2) − S0N(d1) Where d₁= ln(S0/K) + (r + σ 2_/2)T σ √ T d₂= ln(S0/K) + (r − σ 2_/2)T σ √ T = d1− σ √ T.

In contrast to the BSM model, the partial differential equations (3.17) of the Norberg model cannot be solved in closed form even for simplest cases and require using numerical methods, which is outside the scope of this Bachelor thesis.

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Chapter 4 Similarities and differences between the

two models

In doing the thesis we have noticed that there are some similarities and differences between the two 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 of the 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 motion

belong to the Lévy process. Lévy process is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions.

• The counting process in the CTMC model and the continuous sample paths in the BSM model both share the property of independent increments meaning the number of events that 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

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

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

• As the Brownian motion (Wiener process) is the stochastic (driving) process in the BSM model, 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 frequently used 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 in the market.

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

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

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

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Chapter 5 Conclusions

In this thesis, we have reviewed the two financial pricing models: Black–Scholes–Merton and Continuous-time Markov chain model. We have defined the basic components of a financial market and formulated the components for both models. Even though both models 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 European option using the two models. Regarding the construction of the financial components we found 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 features than the classical BSM model. The lack of enough literature on the Continuous-Markov Chain model was a limitation.

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Appendix A

Criteria for a Bachelor Thesis

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

While writing the thesis, we have demonstrated knowledge and understanding of the two financial pricing models, namely the Black–Scholes–Merton and the Continuous-time Markov chain model. To fulfill this, we have searched and critically evaluated information from different books and articles. Our ability to identify, formulate and solve problems has improved over time and we have managed to comply with the specified time frames. Finally, we demonstrated our ability to present orally and in writing and discuss information, problems, and solutions in dialogue.

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Appendix B

An ethical description of contribution of

coauthors

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

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

BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS