The Mathematics of Derivatives Securities with Applications in MATLAB

By Mario Cerrato

Quantitative Finance is expanding rapidly. One of the aspects of the recent financial crisis is that, given the complexity of financial products, the demand for people with high numeracy skills is likely to grow and this means more recognition will be given to Quantitative Finance in existing and new course structures worldwide. Evidence has suggested that many holders of complex financial securities before the financial crisis did not have in-house experts or rely on a third-party in order to assess the risk exposure of their investments. Therefore, this experience shows the need for better understanding of risk associate with complex financial securities in the future.

The Mathematics of Derivative Securities with Applications in MATLAB provides readers with an introduction to probability theory, stochastic calculus and stochastic processes, followed by discussion on the application of that knowledge to solve complex financial problems such as pricing and hedging exotic options, pricing American derivatives, pricing and hedging under stochastic volatility and an introduction to interest rates modelling.

The book begins with an overview of MATLAB and the various components that will be used alongside it throughout the textbook. Following this, the first part of the book is an in depth introduction to Probability theory, Stochastic Processes and Ito Calculus and Ito Integral. This is essential to fully understand some of the mathematical concepts used in the following part of the book. The second part focuses on financial engineering and guides the reader through the fundamental theorem of asset pricing using the Black and Scholes Economy and Formula, Options Pricing through European and American style options, summaries of Exotic Options, Stochastic Volatility Models and Interest rate Modelling. Topics covered in this part are explained using MATLAB codes showing how the theoretical models are used practically.

Authored from an academic’s perspective, the book discusses complex analytical issues and intricate financial instruments in a way that it is accessible to postgraduate students with or without a previous background in probability theory and finance. It is written to be the ideal primary reference book or a perfect companion to other related works. The book uses clear and detailed mathematical explanation accompanied by examples involving real case scenarios throughout and provides MATLAB codes for a variety of topics.

• Language: English
• Publisher: Wiley
• Release date: Feb 24, 2012
• ISBN: 9781119973416

Book preview

Preface

This book is entirely dedicated to the Lord. ‘Love the Lord your God with all your passion and prayer and intelligence. Love others as well as you love yourself.’

Jesus Christ

I was persuaded to write this book because, in my view, the majority of books available in this area are extremely advanced for students who do not possess an excellent mathematical understanding of probability theory and stochastic calculus.

Quantitative finance has become a very popular topic in which an increasing number of students are becoming interested. However, a large number of students are ill-equipped with respect to the mathematical skills required to study this subject. This book is aimed at these students. It also assumes a sufficient understanding of general mathematics and finance, and the objective is to teach students the fundamentals of probability theory and stochastic calculus required to study quantitative finance.

The topics covered in this book are presented using both an academic approach (i.e. critically discussing the latest research in the area) and a practitioner's approach (i.e. discussing ways to implement the models practically). Thus, in my view, although the book was initially conceived for MSc/PhD students pursuing quantitative finance programmes, it is now also of use to practitioners.

The book does not attempt to cover in detail proofs of theorems and lemmas unless explicitly considered necessary, as this is not the intended goal. The objective is to use a ‘less formal’, sometimes ‘less rigorous’, approach to explain theorems, while focusing on their link with the financial models.

The book is divided into two sections. The first part is a review of the essential concepts in probability theory and stochastic calculus. It therefore does not add to this area but, unlike other books, tries to cover some topics in probability in a way that should be more accessible to those students who do not possess a strong understanding of this area. This part is also instrumental in appreciating the second part of the book. The book starts with key concepts of probability theory, such as the sigma field, and discusses different laws of convergences, using, where possible, an informal but rigorous approach.

The second part of the book discusses financial models. It does not attempt to cover every conceivable model but focuses on the main financial models and discusses ways to implement them practically. It starts discussing the Black and Scholes model and its practical limitations. It then moves on to more advanced models, such as stochastic volatility and interest rate models. These models are always critically discussed, citing the latest research in the area as well as different ways of implementing them in practice.

Generally, the chapters in this book contain MATLAB codes, which will help readers to understand the models better and tailor them to their specific cases. Computer codes are written in a simple way and with plenty of explanations. The idea is to show students and practitioners how financial models can be implemented in MATLAB.

It is still likely that this book will contain errors. I will appreciate comments from readers, which can be sent to [e-mail].

It is necessary at this point to clarify some of the mathematical notations presented in the book. Generally, if we deal with a random variable Y indexed on, say, N = 1,2,\ldots,n, the random variable is written as Yn. The same random variable, at time t, indexed on N is written as Y(t)n. Also, in the first part of the book, a Brownian motion process has been denoted by Bn. However, to avoid confusion with the numeraire Bn appearing in the second part of the book, the Brownian motion process will be denoted by Wn.

Finally, I wish to thank my students who have provided feedback on the material covered in this book. Also, I wish to thank John Crosby for his comments and support, Kostantinos Skindilias and Steve Lo for their help on Chapter and Chapter and my family for their love. A special thank you to D’Aponte Giuseppina (my mother).

Mario Cerrato

1

An Introduction to Probability Theory

1.1 THE NOTION OF A SET AND A SAMPLE SPACE

To set up an experiment we, generally, need to define the so-called outcome(s) of the experiment. We expect the outcomes to be distinct and, after collecting them, we obtain what is known as a set. Suppose that ω is a possible outcome from a random experiment, and Ω is a set. We say that , in the sense that the outcome belongs to a particular set. To clarify this, consider an example where we have two possible outcomes from a random experiment. Assume we toss a coin. There are now two possible outcomes: heads ( ) or tails ( ) so that .

This set is the sample space. Thus, a set Ω that includes all possible distinct outcomes of an experiment is called a sample set. The inclusion of distinct possible outcomes forces us to introduce the notion of an event. Indeed, since a random experiment can generate several possible outcomes, we may want to consider not just the individual outcome but also combinations of outcomes.¹

Remark 1.1

If A and Ω are sets, we say that A is a subset of Ω, where every element of A is also contained in Ω. More formally, what we call events are subsets of Ω. Consider two subsets: A and B.

1. Union : means A or B, a set of outcomes that are either in A, B or both.

2. Intersection : means A and B, a set of outcomes that are in both A and B.

3. : means not A, a set of outcomes in the universal set Ω that are not in A.

1.2 SIGMA ALGEBRAS OR FIELD

An experiment generates a series of outcomes and they all belong to the same universal set Ω. The next step is to assign a probability to each single outcome. However, given the potentially very large dimension of Ω, we are forced to first give it a certain structure. For example, consider a random experiment that can generate values anywhere on the real line. In this case, the probability that any single real number will be realized is zero. Therefore, a structure is needed that can identify which subsets of the outcome space can have probabilities assigned to them. This structure is called sigma algebra. A sigma algebra or field on Ω is a collection of subsets of Ω, say F, satisfying the following conditions:

Proposition 1.1

1. .

2. If for , the set .

3. If then .

Remark 1.2

A sigma field is a nonempty set (1) (i.e. is a field) and is closed under countable unions and intersections.² A sigma field provides the most general mathematical notion to formalize an event space.

Sigma algebra can also be interpreted as a ‘measure of information’. To understand this, consider certain ‘states of the world’ all contained in Ω. We do not know which state of the world is the true one, but we do know that the true one is contained in Ω. If we suppose the information we possess is contained in F and use proposition (1), we can reasonably assume that the true state of the world is contained in F.

1.3 PROBABILITY MEASURE AND PROBABILITY SPACE

A function is a relation where for each element, say , there exists one unique element, say , such that . Consider now the (probabilistic) function satisfying a set of assumptions. Suppose, for simplicity, that it is defined between the real numbers 0 and 1. A probability measure P, on , is a function satisfying:³

1. For any event A, and .

2. If for a countable sequence of mutually exclusive events , then

Unnumbered Display Equation

Remark 1.3

The first assumption (1) is just a matter of convention. According to Kolmogorov,⁴ we choose to measure the probability of an event with a number between 0 and 1. Equation (2) is important since we can now attach probabilities to events by using mutually exclusive events. Generally this can be split into two parts: the case, as above, of infinitely many events and the case of finitely many events.

The triple given by a set Ω, a field F of subsets of Ω and a probability measure P on is called a probability space. The probability space represents the basic mathematical structure used in probability theory to build a model.

Definition 1.1

An event A is called null if . If we say that A almost certainly occurs.

1.4 MEASURABLE MAPPING

Proposition 1.2

Let us consider a nondecreasing sequence. Let be a nondecreasing sequence of events such that ; for such a sequence in the limit , and therefore . Also, consider a nonincreasing sequence. Let be a nonincreasing sequence of events such that ; for such a sequence in limit we have , and therefore .

Theorem 1.1

Consider a sequence of events in the event space of interest. For , if , then .

Remark 1.4

The result is known as continuity properties of a sequence of events. It states that the limit of the probability of a sequence of events converges to that event. Therefore, according to this theorem, we conclude that the limit of a sequence of events is itself an event. How can we justify this?

Proof of Theorem 1.1

We provide a partial proof of Theorem 1.1 where we only consider nondecreasing events.

Let be nondecreasing, that is

and consider mutually exclusive events

Unnumbered Display Equation

For simplicity, set and consider :

Unnumbered Display Equation

Since these are mutually exclusive events and using Proposition 1.2

Unnumbered Display Equation

Therefore a sequence of events converges to the event itself (although we have proved convergence from below only, convergence from above, that is assuming nonincreasing sequence, is also needed).

1.5 CUMULATIVE DISTRIBUTION FUNCTIONS

Consider the following sequence of real numbers each indexed on the natural numbers and, for simplicity, assume that is nonstochastic. The simplest definition of convergence implies that will converge to a variable, say X, if, for any arbitrary number, say , for all , so that . Suppose now that is stochastic and defined on if , we say that converges almost certainly to X. This is the type of convergence involved in Borel's strong law of large numbers (LLN), and it implies strong convergence. This is often substituted by a weaker form of convergence. This type of convergence is, instead, associated with Bernoulli's result of a weak law of large numbers.⁵ In this case converge in probability to X if, for any , .

Remark 1.5

The former form of convergence requires as . This condition implies that it is virtually guaranteed that the values of approach the value of X as . The latter states that if you select a tolerance epsilon and choose an n that is large enough, then the value of will be guaranteed to be within the tolerance of the values of X. Thus, the values that the variable can take (which are not close to ) become increasingly restricted as n increases. However, it should be noted that may converge in probability to X, but not with probability one.⁶

1.6 CONVERGENCE IN DISTRIBUTION

Consider a sequence of random variables and its distribution function⁷ . We say that converges in distribution if as for all x for which is continuous. This means that

(1.1) Numbered Display Equation

The following relationship between different models of convergence (convergence in distribution D, in probability P and almost surely as) and different rates can be established:

Unnumbered Display Equation

Convergence in distribution (D) is the weakest form of convergence. In fact, this type of convergence is entirely defined in terms of convergence of the distribution but nothing is said about the underlying random variable. As a consequence, one can use this model of convergence even when the random variables are defined on different probability spaces. This would be impossible in the case of convergence in probability P and as.

1.7 RANDOM VARIABLES

As previously discussed, an experiment generates an outcome (for example ω, with ). The outcome takes the form of a numerical value defined over R such that and we assign a probability to every single outcome.⁸ The main problem with this approach is that the outcome set will have so many elements that it would be impossible to arrange them in a sequence so that we can count them. We can overcome this problem by using a sequence of intervals (i.e. a subset of R), for example for any . The set made of all possible subsets of the real numbers R is .⁹ Hence the pre-image of the random variable forms a mapping from the Borel field back to F such that

Unnumbered Display Equation

This will ensure that the random variable will preserve the structure of F. We can finally assign probabilities to for . Suppose that are elements of ; we have

Unnumbered Display Equation

and, assuming that ; ,

(1.2) Numbered Display Equation

1.8 DISCRETE RANDOM VARIABLES

A random variable X is called discrete if it can take on, at most, a countable infinite number of values of R, and its distribution is specified by assigning probabilities to each one such that , with and¹⁰ .

For example, consider the number of telephone calls arriving at a corporation's switchboard (say ). In this example the random variable has discontinuity at the values , and it is constant in between. Such a distribution is called atomic.

1.9 EXAMPLE OF DISCRETE RANDOM VARIABLES: THE BINOMIAL DISTRIBUTION

Suppose that we toss a coin N times, with n being the number of heads, and assume that in each trial the probability of success is and the probability of failure . Using a binomial distribution we have

(1.3)

Numbered Display Equation

In this example the random variable X has a binomial distribution that is defined as .

Example 1.1

MATLAB can be used to generate binomial random numbers. Consider a vector of length n: . Generate a matrix (z) whose rows represent all the possible combinations of the elements of c taken at the time. The MATLAB function nchoosek, z = nchoosek(c,N); Specify c = [8,10,9,11]; N = 3; z = nchoosek(c,N). The matrix z contains rows and N columns of the possible outcomes.

1.10 HYPERGEOMETRIC DISTRIBUTION

In a binomial distribution, we assume that the probability of success (i.e. tossing a coin and getting heads) is constant. This is a consequence of considering independent events (i.e. in each trial, the probability of heads and tails is independent and equal to ). Suppose instead that we have a finite population of N balls in a box. M balls are red and N – M black. Assume we draw n balls randomly (but no replacement is considered) from the population. What is the probability that p of the n balls are red?

The size of the sample M that we can draw from the population N is . Since we want exactly p balls (among n) to be red,¹¹ we have , leaving ways in which our sample balls are black. If we assume that X is the number of red balls in the sample, then X can be described by a hypergeometric distribution:

(1.4) Numbered Display Equation

Example 1.2

Consider, for example, the scenario where there are possible clients visiting a bank branch each day. Suppose that visit the local branch to open a bank account (or any other sort of operation that produces revenue) and 900 (N – M) visit the branch for general enquiries. Suppose we randomly select clients. What is the distribution of the number of clients visiting the branch in order to open a bank account? The distribution ranges from 0 to 50. Using MATLAB we can calculate the hypergeometric probability distribution function (pdf) for the given scenario. The hypergeometric pdf is obtained using the MATLAB function hygepdf. Thus, using hygepdf(p,N,M,n) given by hygepdf(0:50,1000,100,50) we obtain the distribution shown in Figure 1.1.

Figure 1.1 Example of a hypergeometric probability distribution function

ch01fig001.eps

1.11 POISSON DISTRIBUTION

There are some events that are ‘rare’, that is they do not