Option Pricing on Levy Based Markets Rafael Velasquez

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Option Pricing on Levy Based Markets

Rafael Velasquez

Master Thesis, 30hp Master in Mathematics, 120hp

Spring 2020

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Abstract

The development of novel methods for accurate financial market modelling has always been a significant area in financial mathematics. Therefore, this master thesis exam-ines the applicability of three Levy processes, known as CGMY, NIG and Meixner for pricing European call and put options. The theoretical modelling of Levy mar-kets will lead to the implementation of fast Fourier transform-based algorithms for simulation and option pricing. Finally, the efficiency of these named Levy processes will be compared to the traditional Black-Scholes model.

Sammanfattning

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”We have two classes of forecasters: Those who don’t know — and those who don’t know they don’t know. ”

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Acknowledgements

I would first like to thank my thesis advisors Per Åhag from the Department of Mathematics and Mathematical Statistics at Umeå University, and especially to Rasmus Leijon from Nasdaq umeå. They were always there for me whenever I ran into a trouble spot or had a question about my writing. They consistently allowed

this thesis to be my work but steered me in the right direction whenever they thought I needed it. I will always be thankful to them both for the fantastic

opportunity they gave me and the trust they put in me.

I would also like to thank my personal friends who helped me during this long process: Bo Nordlander, Max Bäckman and Oskar Nylén Nordenfors. Finally, I

would like to especially thank Lovisa Ålander for her passionate participation, input and unconditional support during this process. This work could not have

been successfully conducted without her.

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Introduction

Historically the modelling of asset returns has been of significant interest in financial mathematics since the primary goal of investing in a financial market is to make profits without taking excessive risks. This motivated Black, Scholes and Merton [6] to introduce to the world their revolutionary model for option pricing known as the Black-Scholes model in 1973. We can today claim that the Black-Scholes model is the most well-known method for modelling asset returns due to its elegance and sim-plicity. An example of this is that the Black-Scholes equation for pricing European call options can be understood for anyone with a reasonable background in mathe-matics. Unfortunately, this mathematical simplicity implies that the Black-Scholes model has several weaknesses and difficulties in describing real-life markets.

The main problem of the Black-Scholes model can be summarized in the fact that it assumes that the distribution of logarithmic asset returns to be normal distributed. This assumption implies that the Black-Scholes model expects the market to have a mild randomness and a low sensitivity to significant changes. However, empirical data from asset returns clearly show that this assumption does not hold in reality. For this reason, alternative methods for modelling asset returns have been developed throughout the years. For example in 1976 Cox and Ross [13] presented a model based on pure jump processes, and almost simultaneously Merton [42] presented a generalized version of the continuous-time diffusion process, which consisted on adding a jump term into the Black-Scholes model. These studies opened the door for the use of Levy processes for financial modelling. Recall then that Levy processes have been of relevance in several fields of study for several years, more precisely for the study of stochastic processes.

However, the Levy market models do not have the same mathematical elegance and simplicity compared with the Black-Scholes model. The reason behind this lack of mathematical elegance is that in many cases the probability density function is un-known. Therefore it is impossible to find a closed form formula for the modelling of asset returns based on Levy market assumptions. Fortunately, this problem can be solved by the use of modern numerical methods and simulation techniques. Hence, this master thesis is concerned with the development of numerical methods for pric-ing European call and put options under Levy market assumptions. We aim to convince the reader that the theory of Levy processes, in combination with modern

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2

simulation techniques, can lead to models that perform significantly better than the standard Black-Scholes model. To accomplish this, we will focus on three famous Levy processes, which are known as the CGMY, NIG and Meixner process. The mathematical background and historical development of these processes will be pre-sented in detail in Chapter 5. However, to ultimately motivate the theory of these processes, we shall start by introducing the necessary mathematical machinery. With this in mind, we shall start by presenting a brief introduction to the Black-Scholes model in Chapter 2 where we will discuss its properties, advantages but mostly expose its weaknesses to motivate the use of Levy processes for arbitrage-free option pricing. Thus, the mathematical theory for the Levy processes will be developed from fundamental results in probability and measure theory. This will be mainly covered in Chapter 3, where we will discuss the unique representation of the probability density function for Levy processes and its close relation with infinitely divisible random processes.

In Chapter 4 we will extend the theory of Levy processes by introducing the α-stable Levy processes and describe their role in the description of asset returns. We will mainly focus on the properties of this α-stable processes and the parameter that modulates its behaviour and adaptability to real-life markets. Once this theory for the Levy processes is developed, we shall continue to introduce in Chapter 6, two different numerical methods. The first one will be a simulation technique based on the work of Ballotta and Kyriakou [1] and consist of a probability density function simulation by the use of fast Fourier transform algorithms. The second numerical method will be then an option pricing algorithm based on the work of Fang [19], which consist of a novel way to apply Fourier transforms and series expansion for option pricing. The last two chapter Chapter 7 and Chapter 8 will be devoted to the presentation of our numerical results and a subsequent discussion.

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Chapter 2

Motivation

Since its introduction in 1973, the famous Black-Scholes model has been used for option pricing due to its simplicity. The Black-Scholes model is exquisite mathemat-ically speaking in the sense that it can be expressed in a closed form that can be easily understood for everyone, even without any strong background in mathematics. This simplicity implies that the Black-Scholes model does not require any massive computation capacity for the simulation of stochastic processes under its assump-tions. Therefore it has been an optimal tool for real-time option pricing. However, empirical data shows that the financial market is more complicated than what one can expect from the Black-Scholes model. For this reason, this chapter is devoted to present a brief introduction to the famous Black-Scholes model and its financial market assumptions, such that we can expose and discuss its weaknesses.

Thus the general theory for arbitrage-free pricing of European call and put options is presented in Section 2.1. The corresponding formalism to model the option price as a diffusion process is developed in Section 2.2, while the mathematical background for the Black-Scholes model is presented in Section 2.3. The chapter will then conclude in Section 2.4 with a discussion about the Black-Scholes model weaknesses and a motivation for the use of alternative stochastic processes.

We shall then remark that this chapter aims to present the classical option pricing theory and its limitations. Therefore we shall focus on the significant results from this arbitrage-free option pricing theory and shall omit several technical details. For this reason, we assume that the reader has some basic knowledge about financial mathematics, probability theory, and measure theory. Otherwise, the reader may be interested in the following for mathematical background and historical notes [5, 11, 18].

2.1 Options

Options are future financial derivatives traded on the market. There exists several types of options, each of them showing unique properties and conditions. However, in this thesis, we shall mainly focus on two types of options. These options are known as European call and put options. Hence, we say that a European call option

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4 2.2. STOCHASTIC PROCESSES

is a contract that gives the buyer of the contract the right, but not the obligation to buy a particular risky asset at a predetermined price, K, at a maturity time T. Similarly, we can define a European put option as a contract that gives the buyer of the contract the right, but not the obligation to sell a particular risky asset at a predetermined price, K, at a maturity time T.

Our aim is then to find proper prices for these options. Therefore, the following concept is of significant relevance to understanding how financial derivatives are priced.

Definition 2.1. Consider a financial market with vector price process S. Then, we say that a contingent claim with date of maturity T , is any stochastic variable X ∈ FS

T. We say that a contingent claim is called simple claim if it is of the form X = Φ(S(T )), where the function Φ is called the contract function.

Observe that for the sake of simplicity the conditionX ∈ F_TS means that it will be possible to determine the amount of money to be paid out at time T . The mathemat-ical meaning of this condition will be properly explained later in the next chapter. We can then continue by assuming that our main problem will be to establish a proper price for these claims. We will denote the price of a contingent claim as Π(t;X ) and for European options the contingent claims are given as

XCall = − K + S(T ) + , and XP ut= K − S(T ) + . Thus the arbitrage free price of the option will be given as

Π(x, t0;X ) = exp(−r∆t)EQΠ(y, T ;X ) | x.

2.2 Stochastic Processes

It is well known that a diffusion process is a solution to a stochastic differential equa-tion. Therefore, for the sake of this essay, we shall understand a stochastic process, denoted by X, as a diffusion process, if its local dynamics can be approximated by a stochastic differential equation of the form

X(t + ∆t) − X = µ t, X(t)∆t + σ t, X(t)Z(t), (2.1) where Z(t) is a normal distributed disturbance term, which is independent of the actions that has occurred before time t. The remaining terms µ, σ are given de-terministic functions known as the drift and diffusion parameters. Note that the drift parameter determines the local deterministic velocity of the process, while the diffusion parameter is a disturbance term.

The stochastic process defined in (2.1), can be modeled by considering the normal distributed disturbance term Z(t), as a so-called Wiener process.

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CHAPTER 2. MOTIVATION 5

i) ω(0) = 0;

ii) if r < s ≤ t < u, then ω(u)−ω(t), ω(s)−ω(r) are independent random variables; iii) for s < t, the stochastic variable ω(t) − ω(s) is Gaussian distributed with

expec-tation value zero and standard deviation√t − s; iv) ω has continuous trajectories.

We can now let Z(t) = ∆ω(t), where ∆ω(t) = ω(t + ∆t) − ω(t). Such that (2.1) can be rewritten into

X(t + ∆t) − X(t) = µ(t, X(t))∆t + σ(t, X(t))∆ω(t). (2.2) Note that by letting ∆t → 0 in (2.2), we obtain a stochastic differential equation of the form

dX(t) = µ(t, X(t))dt + σ(t, X(t))dω(t),

X(0) = a. (2.3)

Thus by solving the stochastic differential equation in (2.3), we obtain an integral representation for our stochastic process

X(t) = a + ˆ t 0 µ(s, X(s))ds + ˆ t 0 σ(s, X(s))dω(s). (2.4)

2.3 Black-Scholes Model

This section is devoted to a brief presentation of the main results from Black and Scholes presented in 1973 in their original article [6]. These results are famously known and there exist several books dedicated to them. Therefore we will not present any technical details for the main results presented in this section. For a detailed mathematical background, the reader may look at [5].

With this in mind we will start by considering a financial market consisting only of two assets, a risk-free asset with a price B and a stock with a price process S. In this market, the price process B is the price of the risk-free asset if it has the following dynamics:

dB(t) = r(t)B(t)dt,

where r(t) is any adapted process. Thus, it easily follows that the risk free process B takes the following form

B(t) = B(0) exp ˆ t 0 r(s)ds ,

and for the stock price S assume that it follows the dynamics shown below dS(t) = S(t)α(t, S(t))dt + S(t)σ(s, S(t))d¯ω,

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6 2.3. BLACK-SCHOLES MODEL

might be different from the probability measure defined for the stochastic process X defined on Eq. 2.4. Later on, we will show that one can choose the probability measure for the price process such that it matches the one for the stochastic process. Therefore, we shall refer as the probability measure for the price process as the P-measure, while the probability measure of X will be denoted as the Q-measure. With this distinction made we can proceed to introduce the famous Black-Scholes model. Definition 2.3. The Black-Scholes model consists of two assets with dynamics given by

dB(t) = rB(t)dt

dS(t) = αS(t)dt + σS(t)d¯ω(t). (2.5)

Where r, α, σ are deterministic constants.

We proceed to present several of the Black-Scholes results. The vast majority of them are well known, as well as the techniques used for their derivation, such as the Feynman-Kac representation formula and the Itô lemma. Therefore, we shall not present any derivation of these results in this essay. For more information about this topics the reader may look at [5, 9, 18]. The first result to be presented is then the Black-Scholes equation.

Proposition 2.4 (Black-Scholes Equation). Assume that the market is specified by the following assets dynamics: The Black-Scholes model consists of two assets with dynamics given by

dB(t) = rB(t)dt

dS(t) = α(t, S(t))S(t)dt + σ(s, S(t))S(t)d¯ω(t). (2.6) Assume as well that we want to price a contingent claim of the formX = Φ(S(T )). Then, it follows that the only pricing function of the form Π(t) = F (t, S(t)), which is consistent with the absence of arbitrage is given when the function F is a solution to

Ft+ rSFS+1₂S2σ2FSS− rF = 0

F (T, s) = Φ(S(T )). (2.7)

Remark 2.5. For simplicity we will denote the partial derivative with respect to S in Eq. 2.7 as FS. This notation will then by used every time we refer to the Feynman-Kac equation.

Proof. See e.g Theorem 7.7 in [5]

We shall continue this Black-Scholes results review by showing that Proposition. 2.4 can be applied for risk-neutral valuation. Consider then a market given by the dynamics

dB(t) = rB(t)dt

dS(t) = S(t)α t, S(t)dt + S(t)σ t, S(t)d¯ω(t), (2.8) and a contingent claimX = Φ(S(T )). Thus from Proposition 2.4 we know that the arbitrage free price is given by

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where F is a solution to (2.7). Observe then that this equation can be solved by using the stochastic representation formula of Feynman-Kac such that we obtain

F (t, s) = exp − r(T − t)Et,sφ(X(T )), where the process X is defined by the dynamics

dX(u) = rX(u)d(u) + X(u)σ u, X(u)dω(u)

X(t) = s. (2.10)

Note that the SDE in (2.10) is almost of the exactly same form of the price process S defined in (2.8). The only notable difference is that α is the local rate of returns of S, while the X process has the short rate of interest r instead. Thus, we can define a new price process under the probability measure Q as

dS(t) = rS(t)dt + S(t)σ t, S(t)dω(t). (2.11) Thus, the next major result that we shall present is the risk neutral valuation formula. which follows from (2.11)

Proposition 2.6 (Risk neutral valuation). The arbitrage free price of the claim φ(S(T )) is given by

Π(t, φ) = F (t, S(t)), where F is given by the formula

F (t, s) = exp − r(T − t)EQ

t,sφ(S(T )),

and the Q dynamics of S(t) are given by

dS(t) = rS(t)dt + S(t)σ t, S(t)dω(t).

This Q-measure is known as the Martingale measure and the reason behind this name is that under Q the normalized process B(t)S(t) turns out to be a Q-Martingale as we will show below. For more information about martingales see Chapter 3

Proposition 2.7. In the Black-Scholes model, the price process Π(t) for every traded asset has the property that the normalized price process

Z(t) = Π(t) B(t), is a Martingale under the measure Q.

Proof. We shall start by showing that under the Martingale measure Q the price process Π(t) follows the dynamics

dΠ(t) = rΠ(t)dt + g(t)dω(t).

Thus recall that from Proposition 2.4 follows that Π(t) = F (t, S(t)). Hence, by using Itô’s formula we obtain

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8 2.3. BLACK-SCHOLES MODEL

Thus recall that F satisfies the Black-Scholes equation such that we have

dΠ(t) = rΠ(t)dt + σS(t)∂F ∂Sdω(t). We can now apply Itô formula to Z(t) such that

dZ(t) = 1 B(t) dΠ(t) − Π(t) B2_(t)dB(t) = g(t) B(t)dω(t)Z(t) σS(t) Π(t) ∂F ∂Sdω(t). Therefore it follows from Definition 3.9 that Z(t) is a Martingale since EQ_{Z(T ) =}

Z(t) for all t < T .

We shall then end this chapter by postulating that the Black-Scholes formula for European options pricing can be derived by the use of geometric Brownian motion and the Risk valuation formula.

Proposition 2.8 (Black-Scholes formula). The price of a European call option with strike price K and time of maturity T is given by the formula Π(t) = F (t, S(t)), where

F (t, s) = sN [d1(t, s)] − exp − r(T − t) expKN [d2(t, s)]. (2.12)

Where N is the cumulative distribution function for the N [0, 1] distribution and

d1(t, s) = 1 σ√T − t ln(S/K) + r +1 2σ 2_{(T − t)}_, d2(t, s) = d1(t, s) − σ √ T − t.

Figure 2.1 – Adjusted close price for the NASDAQ Composite index during the

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2.4 Weaknesses of the Black-Scholes Model

To properly expose the weaknesses of the Black-Scholes model, we will focus our discussion to a brief study of the NASDAQ Composite index during the time-period between 2010−01−01 to 2019−12−31. More precisely, we will study the distribution of the adjusted close price for this index and its evolution during the time-period named above. For a better visualization of this data set, we show in Fig. 2.1 the time-based behaviour of the index.

Our aim with this section is to show that real-life markets are not normal distributed and presents a notorious heavy tail effect that the Black-Scholes model is entirely unable to take into consideration. With this in mind, we can continue to study the distribution of the NASDAQ Composite index during the last decade. Hence, we shall start by considering the logarithm of the adjusted prices, such that its distribution can be easily shown in a histogram as seen in Fig. 2.2. It is then clear that the empirical distribution seems to be more concentrated around zero, showing a notorious peak in comparison to the Gaussian distribution. This concentration around a point indicates the presence of fat tails, which demonstrates that real-life markets contains rapidly internal fluctuations that cannot be captured by the Black-Scholes model.

Figure 2.2 – A histogram for the visualization of the empirical distribution of

loga-rithm of the adjusted close prices (logaloga-rithmic returns) for the NASDAQ Composite index during the time-period between 2010 − 01 − 01 to 2019 − 12 − 31. The red line shows the traditional normal distribution.

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10 2.4. WEAKNESSES OF THE BLACK-SCHOLES MODEL

market fluctuations. To explain how this can be a problem we can consider the following example for an exotic option typically used in real-life market trading. Exemple 2.4.1. Let us consider an out-the-money option, which is an option contract that only contains intrinsic value. It is then easy to see from empirical data that the real-life market tends to price out-the-money options above their expected Black-Scholes prices. Consider now an investor using the Black-Scholes model. This investor will then believe that there exists a possibility for free profit by shorting a large number of the near-term far out-the-money options at a fraction of the market price. However, the fat tails observed in the empirical data will then imply that the short seller will be caught sooner than expected, which will finally lead the seller to

a financial crisis.

We shall then recall that financial mathematics aims to establish solid foundations to avoid any financial crisis at all levels. However, it is not an easy task to obtain a proper model that describes the real-life market at any time, because the market will change itself periodically and will present different kind of distributions during time. A possible solution to this problem can be to considering a family of the so-called α-stable distribution instead of the Gaussian distribution for the construction of the stochastic processes. More specifically, one can use a collection of Levy processes by interchanging them depending on the behaviour of the market.

Our motivation for the use of Levy processes is that they are more suitable to capture asymmetry and excess of kurtosis, which are two properties seen in the market. These family of processes and their wealth will be adequately introduced later in this essay, but for now, let us be satisfied by the behaviour of the cumulative distribution function for a specific Levy process, which can be seen in Fig. 2.4.

Figure 2.3 – A quantile-quantile plot of the adjust close price for the NASDAQ

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Figure 2.4 – A comparison between the cumulative distribution function of a

Gaus-sian distribution (green line) and the cumulative distribution function of a Levy distribution (black line).

We shall then end this section by discussing what is probably the biggest weakness of the Black-Scholes model. It is well known that asset returns present a so-called volatility clustering phenomenon, as shown in Fig 2.5. It is then evident that volatil-ity is partially predictable since it is changing over time. This feature of financial time series was first introduced in [17], and it is now an accepted fact about asset returns. We shall then recall that the Black-Scholes model assumes that the observed yields are independent and identically distributed, which is incompatible with the existence of volatility clusters.

Figure 2.5 – Logarithm of the logarithm of adjusted close price (logarithmic

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12 2.4. WEAKNESSES OF THE BLACK-SCHOLES MODEL

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Chapter 3

Probability Theory

The purpose of this master thesis is to find a reasonable price for certain kinds of European put and call options by the use of Levy processes. This chapter aims to establish the theoretical background for the construction of the α-stable theory behind the three Levy processes we will consider in this thesis, which are the CGMY, NIG and Meixner processes. Hence, the mathematical background presented in this chapter will be based on a collection of remarkable results in measure theory and probability theory. We will also consider several results published by Paul Levy and Aleksandr Khinchin [34], as well as the work of Boris Vladimirovich Gnedenko and Andrej Kolmogorov [24], for the development of this theory.

With this in mind, the Martingale theory for arbitrage-free pricing will be devel-oped in Section 3.1 from fundamental results in measure theory. The corresponding formalism to model stochastic processes as Geometric Brownian motion will be pre-sented in Section 3.2.

We shall then continue in Section 3.3 to introduce the infinitely divisible processes and show their close relation to the Levy processes by the famous Levy-Khintchine formula. While in Section 3.4, we will cover the representation of class L processes and construct the prelude for the theory of jump processes.

The chapter will then end by considering the uniqueness of the representation of the probability distribution function from the characteristic equation in Section 3.5, while the Levy processes will be then properly defined in Section 3.6. Finally for further information and historical references see [11, 12, 18, 20, 48].

3.1 Measure Theory Applied to Financial

Mathe-matics

In finance, it is necessary to keep track of the time-development of any financial derivative, as well as it is crucial to have some sense of fair play. Therefore, this section is devoted to the development of the theoretical background for Martingale theory. With this in mind, we shall proceed to introduce several results for the construction of risk-neutral Martingale measures and Itô integrals. These results

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14 3.1. MEASURE THEORY APPLIED TO FINANCIAL MATHEMATICS

will be based on fundamental results from measure theory. For this reason, it is reasonable to start this section with the following concept of significant relevance for the development of Martingale theory.

Definition 3.1. Let Y be a non-empty set, then a collectionA of subsets of Y is called a σ-algebra on Y if:

i) Y ∈A ;

ii) for each set A that belongs toA the set A{_∈_{A ;}

iii) for each {Ai}∞i=1∈A , the set S

∞

i=1Ai belongs toA .

Observe that different authors use the notation of their preference. For this reason, we shall then use the standard notation of [11]. Hence, we proceed to introduce the definition of a measure properly.

Definition 3.2. LetA be a collection of subsets of any arbitrary set Y . We then say that a measure onA is a function µ : A → [0, ∞] that satisfies A (∅) = 0 and is countably additive.

We then say that the triplet Y,A , µ is known as a measure space. This measure space is the pillar for constructing the required risk-neutral measures and stochastic integrals. Therefore we construct from this concept our theory and thus continue to introduce the concept of measurability.

Definition 3.3. Let Y,A , µ be a measure space and let A be a subset that belongs toA . Thus, we say that a function f : A → [−∞, ∞] is called measurable in A , if it fulfills one, and hence all of the following equivalent conditions:

i) For each real number t the set {y ∈ A : f (y) ≤ t} ∈A ; ii) for each real number t the set {y ∈ A : f (y) < t} ∈A ; iii) for each real number t the set {y ∈ A : f (y) ≥ t} ∈A ; iv) for each real number t the set {y ∈ A : f (y) > t} ∈A .

Now, recall that our purpose is to work with stochastic processes. Therefore it is necessary to define an adequate measure space for it. Hence, we say that a prob-ability space is given by the triplet Ω,A , P. Where P(Ω) = 1 and we say that the elements of Ω are called elementary outcomes, while the members ofA are called events. Finally note that if A ∈ A , then P(A) is the probability of the event A. Observe that from a financial point of view; we aim to work under a risk-neutral probability measure space. Therefore it is of significant relevance to properly introduce and show the existence of such a probability space. However, we still need to introduce a couple of concepts before we can guarantee the existence of such a probability space. With this in mind, we can continue by introducing the following concept of major relevance.

Definition 3.4. LetA be a family of subsets of Ω, thus we define

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CHAPTER 3. PROBABILITY THEORY 15

Note that the σ-algebra in (3.1) is the smallest σ-algebra containing A . We can then, use this definition to define random variables.

Definition 3.5. Let Ω be a topological space with open setsA . Then the Borel σ-algebra B(Ω) is the σ-algebra generated by the open sets A . In other words B(Ω) = σ(A ) and hence the sets in B(Ω) are called Borel sets. Thus we say that aA ,B-measurable map X : Ω → Rn is called a random variable.

We now have all the necessary information to define a stochastic process accurately. Observe that the definition previously presented in Chapter 2 is still correct, but unfortunately, it is not as mathematically rigorous, as the one we are going to present below.

Definition 3.6. Let Ω,A , P be a probability space and let I denote a subset of R with infinite cardinality for which we have that for each α ∈ I there is a random variable Xα: Ω → R defined on Ω, A , P. We then say that the mapping X : I × Ω → R defined by X(α, ω) is called a stochastic process.

Without any loss of generality, we can recall an important result presented in Chap-ter 2, which says that the family of stochastic processes that we consider can be seen as a diffusion process with an integral representation of the form

X(t) = a + ˆ t 0 µ(s, X(s))ds + ˆ t 0 σ(s, X(s))dω(s). (3.2)

Note that the third term in the right-hand side of (3.2), is an integral of the form ´

g(t)dω(t), where ω is a Wiener process. However, any other kind of stochastic process can easily exchange this Wiener process. Therefore, assuming no loss of generality, we shall continue this section by introducing the theory behind Itô inte-grals.

With this in mind, we can start by considering the information about the events from the stochastic process. We have that the events of the σ-algebras ofA are allocated in the so called filtration. Thus before we proceed to define it properly, recall that for any set Y and σ-algebrasA and C of Y , we say that C is a sub σ-algebra of A if and only if C ⊆ A . Thus, we can continue with the following definitions that are of major concern for the sake of this essay.

Definition 3.7. Let Ω,A , P be a probability space. We then say that a filtration is a sequence {Fn}∞n=0 of sub σ-algebras ofA that is increasing, in the sense that Fn ⊆Fn+1 holds for each n. Thus, if based upon observations of the trajectories {X(s), 0 ≤ s ≤ t}, it it possible to decide when a certain event A has occurred or not, then we write this as A ∈Ftand say that the event A isFt-measurable. Following Definition 3.7 it follows that any discrete-time stochastic process {Xt}∞t=0 is adapted to the filtration {Ft}∞t=0 if Xt is Ft-measurable for each t. Hence, we can note that the sequence {Xt}∞t=0is adapted to the filtration {Ft}∞t=0 if and only if σ(X0, ..., Xt) ⊆Ft holds for each t. This allows us to understand that the events in the σ-algebraFtare those that could be known by time t.

Our remaining problem is then to handle integrals of the type´gdω, where g is an adapted process to the filtrationFω

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16 3.1. MEASURE THEORY APPLIED TO FINANCIAL MATHEMATICS

that g ∈L2_{[a, b], where}_L2 _{denotes the set of square integrable functions. We also}

assume that g is simple. Hence it follows that there exists deterministic time steps a = t0 < t1 < t2 < ... < tn = b, such that g is constant in each sub-interval. Therefore, assuming g(s) = g(tk) for s ∈ [tk, tk+1], one can interpret the integral as a sum over discrete time steps,

ˆ b a g(s)dω(s) = n−1 X k=0 g(tk) ω(tk+1) − ω(tk) .

We can then generalize this result, for the cases when the adapted process g is not simple, by approximating g with a sequence of simple processes denoted by gn such

that _ˆ

b a

E(gn(s) − g(s))2ds → 0,

which implies that for each n, the integral´_abgn(s)dω can be seen as a well defined random variable, denoted by Zn. Therefore it is possible to prove that there exists a stochastic variable Z, such that Zn → Z (inL2) as n → ∞. Hence we can define a stochastic integral or Itô integral as

ˆ b a g(s)dω(s) = lim n→∞ ˆ b a gn(s)dω(s). (3.3)

The family of stochastic integrals defined by (3.3) have many interesting properties, but in this essay we shall just mention some of them that will be useful for us later on. For more information about the properties of stochastic integrals the reader may be interested to read [18, 33, 40, 43].

Proposition 3.8. Let g ∈L2_{[a, b] be a process adapted to the}_Fω

t -filtration. Then

the following results holds: i) ´_abE[g2_{(s)]ds < ∞;}

ii) Eh´_abg(s)dω(s)i= 0;

iii) Eh´_abg(s)dω(s)

2i

=´_abE[g2(s)]ds. Proof. See e.g Proposition 4.4 in [5]

We shall end this section by recalling that the theory of stochastic integration is strongly connected with the theory of Martingales. This is because, in some sense, we can understand Martingales as the definition of "fair play" in financial mathe-matics. The reason behind this is that we want the expected net gain or loss from further payments to be equal to zero, and especially to be independent of the process history. These Martingales will then affect the arbitrage probabilities of the market. Therefore, it is necessary to properly introduce the definition of a Martingale from a mathematical point of view.

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i) X is adapted to the filtration {Ft}t≥0; ii) E[| X(t) |] < ∞, for all t;

iii) for all s and t with s ≤ t, the following relation holds E[X(t) |Fs] = X(s). For several application in real life market studies it is necessary to relax the definition of a Martingale. Therefore we say that Definition 3.9 can be relaxed in the following way

Definition 3.10. Let s ≤ t be two numbers, then we say that any process satisfying the inequality

E[X(t) |Fs] ≤ X(s), is called a super Martingale, while a process satisfying

E[X(t) |Fs] ≥ X(s), is called a sub-Martingale.

We can then show that there exists a close relationship between the stochastic in-tegrals we will treat during this paper and the concept of Martingale previously introduced by considering the following.

Proposition 3.11. For any process g ∈L2, the process X, defined by X(t) =

ˆ t

0

g(s)dω(s),

is anFtω-Martingale. Hence, it follows that every stochastic integral is a Martingale.

Proof. Let s < t be fixed such that

E[X(t) |F_sω] = Eh ˆ t 0 g(τ )dω(τ ) |F_sωi = Eh ˆ s 0 g(τ )dω(τ ) |F_sωi+ Eh ˆ t s g(τ )dω(τ ) |F_sωi.

From Proposition 3.8 follows that the expected value Eh´₀sg(τ )dω(τ ) |Fω s

i

is mea-surable with respect to the filtrationFω

s. Therefore, it follows that Eh ˆ s 0 g(τ )dω(τ ) |F_sωi= ˆ s 0 g(τ )dω(τ ).

Similarly it follows from Proposition 3.8 that Eh´_stg(τ )dω(τ ) | F_sωi = 0. Which implies that

E[X(t) |Fsω] = ˆ s

0

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18 3.2. GEOMETRIC BROWNIAN MOTION.

3.2 Geometric Brownian Motion.

We shall properly introduce the concept of stochastic differential equations, as well as add the geometric Brownian motion as a plausible solution for a given kind of stochastic differential equation. Hence, we shall start by formally defining what is known as a SDE. Therefore, let M (n, d) denote the class of n × d matrices and consider the following conditions:

i) A d−dimensional Wiener process ω; ii) a function µ :R+×R2→Rn;

iii) a function σ :R+×Rn → M (n, d);

iv) a real vector x0∈Rn.

We then define a stochastic differential equation as

dXt= µ(t, Xt)dt + σ(t, Xt)dωt, X0= x0.

(3.4) Our goal now is to show that there exists a stochastic process X that satisfies (3.4). More precisely we want to find a process X that satisfies the following integral equa-tion Xt= x0+ ˆ t 0 µ(s, Xs)ds + ˆ t 0 σ(s, Xs)dω(s), (3.5)

for all t ≥ 0. As shown in [5], the standard method for proving the existence of a solution to the stochastic differential equation in (3.4), is to construct an iteration scheme of the Cauchy-Picard type. The idea behind this iteration scheme is to define a sequence of processes X0, X1, ... according to the recursive definition

_X0 t= x0 Xn+1_t = x0+ ´t 0µ(s, X n s)ds + ´t 0σ(s, X n s)dω(s) (3.6)

Observe that we expect the sequence {Xn}∞

n=1to converge to some limit process X, where this limit process will be a solution to the SDE. The construction of these Cauchy-Picard type scheme is outside the area of interest of this essay. How-ever, the main idea behind this construction is to use the following inequalities such that the solution to the SDE fulfils several essential properties, as postulated in the following Lemma.

Lemma 3.12. Suppose that there exists a constant K such that the following con-ditions are satisfied for all x, y and t.

kµ(t, x) − µ(t, y)k ≤ Kkx − yk; kσ(t, x) − σ(t, y)k ≤ Kkx − yk; kµ(t, x)k + kσ(t, x)k ≤ K 1 + kxk.

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i) X is Fω

t -adapted;

ii) X has continuous trajectories; iii) X is a Markov process;

iv) there exists a constant C such that EkXtk2 ≤ C exp(Ct) 1 + kx0k2;

Proof. See e.g Proposition 5.1 in [5]

We can now move on and see the geometric Brownian motion as a solution for a given type of SDE. Indeed, the use of the geometric Brownian motion is the most fundamental tool for modeling asset prices. The geometric Brownian motion (ab-breviated GBM) will be seen as an underlying process for the modeling of different kinds of stochastic processes. Therefore, we shall proceed to derive an expression for the GBM correctly. For this reason, we shall consider the following SDE

dXt= αXtdt + σXtdω(t) X0= x0.

(3.7) It is obvious that the solution to (3.7) can be seen as linear ODE, with a stochastic coefficient driven by a white noise coefficient, dω(t). Therefore, it is natural to expect the solution to the SDE in (3.7) to be an exponential function of time. For simplicity we can define the process Zt= ln(Xt), and assume that X > 0 is a solution to (3.7). Thus from Itô’s lemma follows that

dZ = 1 XdX + 1 2 − 1 X2 dX2 = αdt + σω −1 2σ 2_dt.

Which allow us to construct the following SDE instead

dZt= α −1₂σ2dt + σdω(t) Z0= ln(x0),

where the solution is given by Zt= ln(x0) + ˆ t 0 α −1 2σ 2_{dt +} ˆ t 0 σdω(t).

We can directly see that this implies that the diffusion process X is given as, X(t) = x0exp

α −1

2σ

2_{t + σω(t)}_,

and that in general the solution to the SDE in (3.7) takes the form X(t) = xtexp

α −1

2σ

2_{(T − t) + σ(ω(T ) − ω(t))}_,

which is log-normal distributed as expected from Definition 2.2. Therefore, it follows that the process Z(t) ∼ N (µZ, σ2Z) . Where µZ = ln(xt) + α −1₂σ2(T − t) and

σZ = σ √

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20 3.3. INFINITELY DIVISIBLE RANDOM PROCESSES

3.3 Infinitely Divisible Random Processes

The majority of Levy processes do not have any close form for their probability den-sity function. This is a significant concern because it implies the use of numerical methods, such as simulation techniques for the construction of this probability den-sity functions. Which as we may recall, from for example the Black-Scholes equation, are necessary to find the proper prices.

In this section, we shall introduce the theory behind the Levy-Khintchine formula for the canonical representation of the characteristic functions and will show its uniqueness for infinitely divisible random processes. With this in mind, we can say that this chapter is devoted to prove one of the essential theorems for the use of "jump processes" in financial applications, which is postulated below as.

Theorem 3.13. If the logarithm of a characteristic function, for any infinitely di-visible random variable ξ, is representable in terms of the Levy-Khintchine formula, where G(u) is a function of bounded variation. Then such a representation is unique. We shall start the proof of this theorem by introducing some necessary topics. Thus, we have that a jump process is a type of stochastic process that has discrete move-ments, called jumps, with random arrival times, rather than continuous movement. We also have that jumps with absolute values bounded away from zero cannot occur with infinity density, which is a major problem for the modeling of future financial derivatives. For this reason, we need to overcome this difficulty somehow, and one plausible solution is to introduce two function M (u) and N (u), such that we can model the probability that a jump h happens in the interval (λ, λ + dλ).

We shall then let M (u)dλ be the probability that a jump h < u < 0 occurs, and similarly, we let N (u)dλ be the probability that a jump h > u > 0 happens. We can now use these new functions in the famous formula for the logarithm representation of the characteristic function of a random variables introduced by Finetti in 1929 in [14], which is given as:

ln (fλ(t) = λ iγt −σ 2_t2 2 + c ˆ exp(iut) − 1dF (u), (3.8) where, f denotes the characteristic function, dF is the Levy measure, σ ≥ 0 and γ ∈R. Note that Eq. 3.8 can be rewritten by considering the case case when u = 0 in (3.8) and the fact that we can express the distribution function of the magnitude of the jump as P(h < u) = F (u), as shown in [48]. Hence, (3.8) takes the following form ln fλ(t) = λ iγt −σ 2_t2 2 + ˆ 0 −∞ exp(iut) − 1dM (u)) + ˆ ∞ 0 exp(iut) − 1dN (u). (3.9) However, there is another difficulty because we can note that it is still possible to have a function ξn for which the integrals

´0

−audM (u) and

´a

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for all a > 0. This should then imply that the integrals in (3.9) also are divergent. Ergo the mathematical expectation of the sum of the jumps of small magnitudes | h |≤ a can be infinite. Fortunately, this infinity can be handled in a clever way, as shown in [24]. The main idea is to introduce a term of the form λiγt into (3.9). This term will compensate for the divergent integrals, and therefore, it allows us to obtain a formula that represents the general form for the logarithm of characteristic functions for variables of finite variance

ln fλ(t) = λ iγt −σ 2_t2 2 + ˆ 0 −∞

exp(iut) − 1 − itudM (u) +

ˆ ∞ 0

exp(iut) − 1 − itudN (u). (3.10) In order to treat the infinite variance case one can follow the ideas of Paul Levy and introduce the Levy-Khintchine formula such that we obtain the following result

ln fλ(t) = λ iγt −σ 2_t2 2 + ˆ exp(itu) − 1 − itu 1 + u2 1 + u2 u2 dG(u), (3.11) where 1 + u2 u2 dG(u) = dM (u), u < 0, and 1 + u2 u2 dG(u) = dN (u), u > 0.

Please observe that the jump of G(u) at u = 0 is equal to σ2_{, and mainly that the}

continuity relation defines the integrand at zero such that exp(itu) − 1 − itu 1 + u2 1 + u2 u2 u=0 = −t 2 2.

We can now continue this section by formally introducing the concept of infinitely divisible law. This concept will be of major relevance for this essay due to the close relation between infinite divisible law and Levy processes. With this in mind we shall proceed to properly introduce the concept of infinitely divisible law as follow: Definition 3.14. Let ξ be a random variable, then we say that ξ is infinitely divisible if for every natural number n, it can be represented as

ξ = ξn1+ ξn2+ ... + ξnn,

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22 3.3. INFINITELY DIVISIBLE RANDOM PROCESSES

For simplicity, we shall, from now on, refer to the distribution functions of infinitely divisible random variables as infinitely divisible law functions. We shall now show that the uniqueness of the cumulative density function representation from the characteristic function also holds for infinitely divisible laws.

Proof of Theorem 3.13. Assume that the characteristic function f (t) has two possible representations from the Levy-Khintchine formula, with two functions G1(u) and

G2(u), as well as, two constants γ1 and γ2. Now let the positive and negative

variation of the function G1(G2) be given by G01(G02), respectevly ˜G2( ˜G2).

Thus, if we let the integrand in (3.11) be given by h(t, u) we find that

iγ1t +

ˆ

h(t, u)dG1(u) = iγ1t +

ˆ h(t, u)dG0₁(u) − ˆ h(t, u)d ˜G2(u) = iγ2t + ˆ h(t, u)dG2(u) = iγ2t + ˆ h(t, u)dG0₂(u) − ˆ h(t, u)d ˜G2(u).

thus, it follows that iγ1t +

ˆ

h(t, u)d G01(u) + ˜G2(u) = iγ2t +

ˆ

h(t, u)d ˜G1(u) + G02. (3.12)

Finally recall that the functions G0₁(u)+ ˜G2(u) and ˜G1(u)+G02(u) are non decreasing,

therefore it follows from (3.12) that for this representation to hold the following must hold: γ1 = γ2 and G01(u) + ˜G2(u) = ˜G1(u) + G02(u), which is equivalent to have

G1= G2 and so ends the proof.

We end this section by giving an example of the relevance of this kind of processes in several fields of study, such as statistics, finance, and physics.

Exemple 3.3.1. The following distributions follow an infinitely divisible law, and therefore all the processes in which such a distribution are infinitely divisible.

i) Gaussian distribution; ii) Poisson distribution;

iii) Compound Poisson distribution; iv) Geometric distribution;

v) Binomial distribution; vi) Exponential distribution;

vii) Gamma and the variance gamma distributions;

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3.4 Canonical Representation of Class L

Distribu-tions

For our purpose we will need to properly define to which kind of distribution the characteristic functions belong to. Thus, we shall follow the work of Khinchin ([24]), in order to define a class L distribution

Hence, we shall start this chapter by formally introducing the concept of a distribu-tion law. But first, we shall remark that for simplicity, the following notadistribu-tion will be used: ξ, η , ζ, and X for random variables.

Definition 3.15. A random variable X : Ω → Rn on a probability space induces a probability measure µX on R, B(Rn) by defining

µX = P X−1(A) =

ˆ

Ω

H{X (Ω)∈A}dP(Ω), (3.13)

for A ∈ B(Rn) and the Borel-set B. We then say that the measure µX is called the

distribution law of X .

Definition 3.16. Let F (x) be a distribution function, we then say that F (x) belongs to the class L, if there exists a sequence of independent random variables {ζn}∞n=1, such that for suitably chosen constants Bn> 0 and An, the distribution function of the sums ζn= 1 Bn n X k=1 ζk− An, (3.14)

converges to F (x). We also require the random variables ζnk=

ζk

Bn

1 ≤ k ≤ n, to be asymptotically constants.

We would then show that the Levy-Khintchine formula can give the characteristic function for any distribution of the class L. With this in mind we shall present, but not prove the following result.

Lemma 3.17. The distribution function F (x) belongs to the class L, if for every α between zero and one, the function F (x) is the composition of F (x/α) and some other distribution function Fα(x).

Proof. See e.g Theorem 1 of chapter 29 in [24]

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24 3.4. CANONICAL REPRESENTATION OF CLASS L DISTRIBUTIONS

Theorem 3.18. Let F (x) be a distribution function. Then F (x) belongs to the class L if the following conditions fullfields

i) The function M (u) and N (u) in (3.11) have right/left derivatives for every u; ii) The functions uM0(u), for u < 0 and uN0(u), for u > 0 are non increasing.

Proof. We shall start the proof by letting the function f (t) denote the characteristic function of a class L distribution. We shall also let α be any arbitrary number between zero and one. Thus, from (3.11) follows that

ln(f (αt)) = iγαt − σ 2_α2_t2 2 + ˆ 0 −∞ exp(iαut) − 1 − iαut 1 + u2 dM (u) + ˆ ∞ 0 exp(iαut) − 1 − iαut 1 + u2 dN (u) = iγ1t − σ2α2t2 2 + ˆ 0 −∞ exp(iut) − 1 − iut 1 + u2 dMu α + ˆ ∞ 0 exp(iut) − 1 − iut 1 + u2 dNu α , (3.15) where γ1= αγ + α ˆ 0 −∞ u3(1 − α2) (1 + u2_{)(1 + α}2_u2₎dM (u) + ˆ ∞ 0 u3(1 − α2) (1 + u2_{)(1 + α}2_u2₎dN (u)

Now consider the ratio _{f (αt)}f (t) , we know from Lemma 3.17 that this ratio generates a infinitely divisible characteristic function. Therefore it follows directly that the canonical representation is given by

ln f (t) f (αt) = i(γ − γ1)t − σ2(1 − α2)t2 2 + ˆ 0 −∞ exp(iut) − 1 − iut 1 + u2 dM (u) − Mu α + ˆ ∞ 0 exp(iut) − 1 − iut 1 + u2 dN (u) − Nu α . (3.16)

Hence we conclude from that the functions M (u) − M_αuand N (u) − Nu_αmust be non-decreasing and therefore, whatever u1 < u2 < 0 and 0 < v1 < v2, we must

have that    M (u1) − M u1 α ≤ M (u2) − M u2 α N (v1) − N v1 α ≤ N (v2) − N v2 α , then, we finally have that

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Now, by considering the inverse, we obtain that the functions M (u) and N (u) will satisfy the inequalities in (3.17) for every 0 < α < 1. Thus, the functions M (u) − Mu_αand N (v) − N_αvwill be non-decreasing functions. As a consequence of this follows that the ratio _{f (αt)}f (t) will be the characteristic function of a infinitely divisible law. Therefore the inequalities in (3.17) are sufficient for the distribution to belong the class L.

3.5 Uniqueness of the Distribution Function

We have so far found a way to find a unique representation for the characteristic function of an infinite divisible process by the Levy-Khintchine formula. Our next step is then to be able to find a unique representation of the probability density function. With this in mind, this section is devoted to the proof of the following theorem

Theorem 3.19. A distribution function is uniquely determined by its characteristic function.

Thus we have that the characteristic function of a random variable is given as Definition 3.20. The characteristic function of the Rd_{-valued random variable}

ζ is the function fζ : Rd→ R defined by

fζ (z) = E exp(izζ) = ˆ

Rd

exp(izζ)P(du). (3.18)

The reader, with a background in financial mathematics, shall recognize the relation between the higher moment derivatives of the characteristic function and the con-cepts of variance, skewness, and kurtosis. These financial concon-cepts are of significant relevance to the study of asset distributions. For more information about them, the reader may look at [5]. Now, for the reader without a background in finance, it will be enough to understand that the higher moment derivatives of a random variable are related to the derivatives at zero of its characteristic function.

Proposition 3.21. Let ζ be a random variable and fζ be a characteristic function.

i) If E | ζ |n

< ∞. Then fζ has n continuous derivatives at z = 0 and

mk= Eζk = _i1k

∂k_f ζ

∂ζk(0) for all k = 1, ..., n;

ii) If fζ has 2n continuous derivatives at z = 0, then E | ζ |2n < ∞ and mk= Eζk = _i1k ∂kfζ ∂ζk(0) for all k = 1, ..., 2n;

iii) ζ posses finite moments of all orders if and only if z → fζ(z) is C∞ at z = 0.

Then, the moments of ζ are related to the derivatives of fζ by

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26 3.5. UNIQUENESS OF THE DISTRIBUTION FUNCTION

We shall end this section by showing that the characteristic function of a random variable completely characterizes its law. In other words, we say that two random variables with the same characteristic function are identically distributed. We shall then prove that the distribution function is uniquely determined by its characteristic function, but first, we shall consider the following useful result.

Proposition 3.22. Let f (t) and F (x) be the characteristic function and distribution function of the random variable ξ. Then, if x1 and x2 are continuity points of the

function F (x) the following holds F (x2) − F (x1) = 1 2πc→∞lim ˆ c −c exp(−itx1) − exp(−itx2) it f (t)dt. (3.20)

We call (3.20) for the inversion formula. Proof. Let x1< x2 and set

Ic= 1 2π ˆ c −c exp(−itx1) − exp(−itx2) it f (t)dt, (3.21)

observe that we are not writing the limit in c explicitly; however, it is still under-statement that we will evaluate this limit at the end. Note that by substituting f (t) for its expression in terms of F (x) and changing the order of integration it follows that (3.21) can be rewritten as

Ic = 1 2π ˆ hˆ c −c

exp it(z − x1) − exp it(z − x2)

it dt i dF (z) = 1 π ˆ ˆ c 0 hsin t(z − x₁) t − sin t(z − x2) t i dtdF (z). (3.22)

The integrals in the right-hand side of (3.22) can be handled by recalling that for every α and c the following holds

| 1 π ˆ c 0 sin(αt) t dt | = | 1 π ˆ αc 0 sin(s) s ds | < 1. (3.23) Note that by letting c → ∞ in (3.23) one obtains

1 π ˆ c 0 sin(αt) t dt → 1 2, if α > 0 −1 2, if α < 0 (3.24) This approach to the limits is uniform with respect to α in every domain α > δ > 0, respectively α < −δ < 0. Therefore by choosing δ small enough for the following inequality to hold

x1+ δ < x2− δ,

thus we can rewrite (3.22) as the sum of five integrals

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where the function Ψ(c, z; x1, x2) is given by

Ψ(c, z; x1, x2) = 1 π ˆ c 0 hsin t(z − x₁) t − sin t(z − x2) t i dt.

Finally note that from (3.24) it follows that when c → ∞, the function Ψ → 0 for z < x1− δ and z > x2+ δ, and Ψ → 1 for the interval where x1+ δ < 2 < x2− δ.

Similarly as before we note that both limits are uniform with respect to z. Hence, we know that | Ψ(c, z; x1, x2) |≤ 2 in the intervals (x1− δ, x1+ δ) and (x2− δ, x2+ δ).

Thus for every δ > 0 it follows that lim

c→∞Ic= F (x2− δ) − F (x1+ δ) + R(δ, x1, x2), (3.25) where

| R(δ, x1, x2) |≤ 2F (x1+ δ) − F (x1− δ) + F (x2+ δ) − F (x2− δ).

we then note that the left hand side of (3.25) does not depend on δ and that the limit on the right hand side goes to F (x2) − F (x1) as δ → ∞.

We are now able to prove one of the significant results for the sake of this essay. But first, we shall inform the reader about the importance of this result. The im-plementations that will be presented later on, such as the COS-method and the fast Fourier transform simulation are based on the simulation of the cumulative distribu-tion funcdistribu-tion from the characteristic funcdistribu-tion. Therefore it is necessary to guarantee the uniqueness of the distribution function.

Proof of Theorem 3.19. Observe that from Proposition 3.22 it follows that for every continuity point x of the distribution function F (x), the following formula applies

F (x) = 1

2πy→∞lim c→∞lim ˆ c

−c

exp(ity) − exp(−itx)

it f (x)dt. (3.26)

Note then that the first limit in (3.26) is taken over the set of continuity points y of F (y). Hence we can conclude that this representation is unique.

3.6 Levy Processes

It is well known that the Black-Scholes model presents several difficulties to model the financial market, as discussed in Chapter 2, but especially in [10] and [48]. The main problems for the Black-Scholes model is that it cannot capture the towering peaks observed in reality as well as the fact that the market tails are significantly heavier. For this reason this simplified description of financial markets fails to explain the probability of extreme events.

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28 3.6. LEVY PROCESSES

known as a widely used class of possible discontinuous driving processes in stochastic differential equations.

The Levy process differs from the traditional GBM by the fact that the Levy processes do not have continuous paths. We instead assume that the paths of the Levy process are càdlàgs.

Definition 3.23. A stochastic process X is said to be a càdlàg if its sample paths are right continuous with left limits. Similarly a stochastic process X is said to be a càglàd if its sample paths are left continuous with right limits.

Thus we say that a Levy process is a Rd-valued stochastic processes with stationary and independent increments. More generally we say that for a filtration {Ft}t≥0 from the probability measure space Ω,A , P a Levy process is given as

Definition 3.24. An adapted process X = {Xt}t≥0with X0= 0 is a Levy Process

if

i) X has increments independent of the past, that is Xt− Xsis independent from the filtrationFsfor 0 ≤ s < t < ∞;

ii) X has stationary increments, that is Xt− Xshas the same distribution as Xt−s, for 0 ≤ s < t < ∞;

iii) Xt is stochastic continuous. That is lim h→0P | Xt+h− Xt|>  = 0, for all bigger than zero.

We shall then end this section by presenting but nor proving a compelling result for the modulation of asset returns, that connect the Levy processes to the infinitely divisible laws mentioned in the previous section.

Theorem 3.25. Let {Xt}t≥0 be a Levy process. Then for every t, Xthas infinitely

divisible distribution. Conversely if F is an infinitely divisible distribution then, there exists a Levy process {Xt} such that the distribution of X is given by F .

Proof. Cf. [24].

Thus it follows from Theorem 3.25 that a Levy process X, with distribution Xtfor

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Chapter 4

Alpha Stable Theory

The Levy alpha-stable distributions are a family of distributions that present the properties that they can be easily modified to construct processes of (in)finite vari-ation as the CGMY model. This property allows us to adapt the processes to the requirements of the real-life markets in an elegant way. Hence, the α-stable the-ory presents a simplified way to represent the distribution function based on four model parameters that control different aspects of the distributions. Thus we shall present the theory behind α-stable distribution in Section 4.1, while we will present a discussion of the effect of the model parameters in Section 4.2

For historical notes and a detailed mathematical background behind how the Levy processes that concern this thesis belong to the family of α-stable distributions, the reader, is cordially invited to read [24, 39, 45, 49].

4.1 Alpha Stable Distribution Law

Stable distributions play an increasing role as a natural generalization of the normal distribution. For this reason, this theory can be applied to financial mathematics to get a better description of the stochastic processes that describes the financial market. Therefore, for the proper classification of the stable processes, it is convenient to introduce the shorthand notation

U= V,d

to indicate that the random variables U and V have the same distribution. Thus it follows that U= aU + b means that the distributions of U and V, differ only byd location and scale parameters.

It is also necessary to introduce a random variable X , which follows the definition and properties described in Definition 3.15. With this in mind we let X1, X2, X3, ... be

independent identically distributed copies of X , with common distribution function F (x) and Sn = X1+ ... + Xn. We can then start this chapter by introducing the concept of a stable distributed random variable

Definition 4.1. The distribution F of a random variable X is stable if for each n

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30 4.1. ALPHA STABLE DISTRIBUTION LAW

there exist constants Cn > 0, Dn such that

Sn d

= CnX + Dn, (4.1)

and F is not concentrated at the origin.

We also have that a random variable is said to be strictly stable if (4.1) holds for D = 0. Similarly, we say that a random variable X is symmetric stable if, X and −X , have the same distribution. We shall now notice that there is a restriction for the norming constants, which we will present and prove in the following lemma. Lemma 4.2. Only the norming constants of the form

Cn= n

1

α, (4.2)

are valid with 0 < α ≤ 2. The constant α will be then called the characteristic exponent of the stable distribution F .

Proof. Assume that F is a stable distribution. Thus it follows that the distribution of F0ofX1−X should also be stable with the same norming constants Cn. Therefore it suffices to prove the assertion for one symmetric stable distribution F .

We shall then consider that Sm+n is the sum of the independent variables Sm and

Sm+n− Sm, which are distributed as CmX and CnX respectively. We then have that for symmetric stable distribution the following holds

Cm+nX d

= CmX1+ CnX2. (4.3)

Similarly, the sum Srk can be broken up into r independent blocks of k terms. From this we have that Crk = CrCkfor all r and k, such that, we can conclude by induction that for subscripts of the form n = rv if Cn= Crv.

We shall now prove that the sequence {Cn}∞n=1 is monotone. For this, recall that the variables in (4.3) are symmetric, which implies that for all x > 0

P Cm+nX > Cmx ≥ 1

2P CmX1> Cmx. (4.4) Thus for a fixed x we have that 1

2P CmX1> Cmx is a positive constant and hence

the ratio Cm/Cm+n must remain bounded when m → ∞ or n → ∞. We can then apply this result to m = rv _{and m + n = (r + 1)}v_{, where r is fixed and v → ∞.} Consider now an arbitrary pair of integers j, k, such that for each v there exists a unique λ for which the following holds

jλ≤ kv _{< j}λ+1_, and from the monotonicity of {Cr} follows that

C_jλ≤ Cv k < C

λ+1 j .

Hence, we have that Cj> 1, which allows us to take the logarithms such that

Option Pricing on Levy Based Markets Rafael Velasquez