An Undergraduate Introduction to Financial Mathematics

The following sections are included:

• Preface

• Preface to the Second Edition

• Preface to the First Edition

• Contents

One of the first types of investments that people learn about is some variation on the savings account. In exchange for the temporary use of an investor’s money, a bank or other financial institution agrees to pay interest, a percentage of the amount invested, to the investor. There are many different schemes for paying interest. In this chapter we will describe some of the most common types of interest and contrast their differences. Along the way the reader will have the opportunity to renew their acquaintanceship with exponential functions and the geometric series. Since an amount of capital can be invested and earn interest and thus numerically increase in value in the future, the concept of present value will be introduced. Present value provides a way of comparing values of investments made at different times in the past, present, and future. As an application of present value, several examples of saving for retirement and calculation of mortgages will be presented. Sometimes investments pay the investor varying amounts of money which change over time. The concept of rate of return can be used to convert these payments in effective interest rates, making comparison of investments easier.

Since the number and interactions of forces driving the values of investments are so large and complex, development of a deterministic mathematical model of a market is likely to be impossible. In this book a probabilistic or stochastic model of a market will be developed instead. This chapter presents some elementary concepts of probability and statistics. Here the reader will find explanations of discrete events and their outcomes. A discrete outcome can take on only one value from a list of a finite (or countable) number of values. For example the outcome of a roll of a fair die can be only one of the six values in the set {1, 2, 3, 4, 5, 6}. No one ever rolls a die and discovers the outcome to be π for example. Basic methods for determining the probabilities of outcomes will be presented. The concept of the random variable, a numerical quantity whose value is not known until an experiment is conducted, will be explained. There are many different kinds of discrete random variables, but one that frequently arises in financial mathematics is the binomial random variable. While statistics is a field of study unto itself, two important descriptive statistics will be introduced in this chapter, expected value (or mean) and standard deviation. The expected value provides a number which is representative of typical values of a random variable. The standard deviation is a number which provides a measure related to the width of an interval centered at the mean into which values of the random variable are likely to fall. As will be seen when discussing specific experiments, the standard deviation measures the degree to which values of the random variable are “spread out” around the mean.

Whereas in Chapter 2 random variables could take on only a finite number of values taken from a set with gaps between the values, in the present chapter, continuous random variables will be described. A continuous random variable can take on an infinite number of different values from a range without gaps. Calculus-based methods for determining the expected value and standard deviation of a continuous random variable will be described. A very useful type of continuous random variable for the study of financial mathematics is the normal random variable. We will see that when many random factors and influences come together (as in the complex situation of a financial market), the sum of the influences can be modeled by a normal random variable. Finally we will examine some stock market data and detect the presence of normal randomness in the fluctuations of stock prices.

The concept known as arbitrage is subtle and can seem counter-intuitive. Formally there are two types of arbitrage (creatively named type A and type B). Type A arbitrage is a trading strategy that results in an initial positive cash flow (the investor receives a positive amount of income initially) and there is no risk of loss in the future. Type B arbitrage is a trading strategy in which the investor needs no initial cash investment, has no risk of future loss, and has a positive probability of profit in the future [Cornuejols and Tütüncü (2007)]. Informally arbitrage exists whenever two financial instruments are mis-priced relative to one another. Due to the mis-pricing, it becomes possible to make a financial gain. For example, suppose bank A issues loans at a 5% interest rate and bank B offers a savings account which pays 6% interest. A person could take out a loan from bank A and place the loan into savings with bank B. When it becomes time to repay bank A for the loan, the person closes the savings account with bank B, repays the principal and interest and still has 1% of the loaned amount as profit. The assumption that financial markets are efficient prevents such obvious arbitrage opportunities from being commonplace. When arbitrage opportunities arise, investors wanting to make a profit flock to the mis-priced instruments and the financial market reacts by correcting the pricing of the instruments.

In this chapter we will introduce and explain some of the concepts surrounding the probabilistic models used to capture the behavior of stock, security, option, and index prices. Random walks are related to properties of discrete random variables studied in Chap. 2. Brownian motion can be thought of as a continuous random process which is the limiting case of a random walk as the time and space steps become infinitesimally small. Topics covered here could be expanded into an entire book of their own. For our purposes we will explore just enough of the stochastic calculus to provide some justification for Itô’s Lemma, the main result of this chapter.

This chapter introduces some of the concepts and terminology associated with the buying and selling of securities such as stocks. The main issue discussed will be the pricing of forward contracts and futures. The present chapter will give the reader the opportunity to apply the theory of interest, the arbitrage principle, and some elementary stochastic processes to the problem of pricing two commonly traded financial derivatives. The term “derivative” is used because the values of these financial instruments is “derived” from underlying securities or commodities. The material of this chapter can also be treated as a “warm-up” exercise for the later chapters on options and the development of the Black-Scholes option pricing formula.

In the present world of finance there are many types of financial instruments which go by the name of options. At its simplest, an option is the right, but not the obligation, to buy or sell a security such as a stock for an agreed upon price at some time in the future. The agreed upon price for buying or selling the security is known as the strike price. Options come with a time limit at which (or prior to) they must be exercised or else they expire and become worthless. The deadline by which they must be exercised is known as the exercise time, strike time, or expiry date. Often the exercise time will simply be called expiry. In the remainder of the text the three terms will be used interchangeably. An option to buy a security in the future is called a call option. An option to sell is known as a put option…

In Chapter 7 the Black-Scholes partial differential equation was derived and summarized in Eq. (7.15). Every European style option satisfies this PDE. The differences in the options are due to different boundary and payoff conditions. In the present chapter we will solve the Black-Scholes equation with boundary and final conditions appropriate for a European style call option. Several methods are available to solve this equation. Some take the limit of a discrete time binomial model of security prices (see [Ross (1999)]). In [Wilmott et al. (1995)] the Black-Scholes PDE is transformed through a sequence of changes of variables to an ordinary differential equation which is solved by elementary means. Their change of variable approach will be mimicked here, but the Fourier Transform will be used to solve the equation. For readers unfamiliar with the Fourier Transform, a brief introduction to the relevant operations are included in this chapter. The Fourier Transform is a standard mathematical device used to convert certain types of partial differential equations into ordinary differential equations. A more complete introduction to Fourier Transforms can be found in either [Greenberg (1998)] or [Jeffrey (2002)]…

Now that the solution to the Black-Scholes equation is known we can investigate the sensitivity of the solution to changes in the independent variables. Knowledge of these sensitivities allows a portfolio manager to minimize changes in the value of a portfolio when underlying variables such as the risk-free interest rate change. The material of the present chapter will be used extensively in Chapter 10. An understanding of the sensitivity of an option’s value to changes in its independent variables will also provide a new way of interpreting the Black-Scholes equation itself…

Hedging is the practice of making a portfolio of investments less sensitive to changes in market variables such as the prices of securities and interest rates. If F(S, t) is a solution to the Black-Scholes partial differential equation Eq. (7.15), the quantity Δ = F_S is central to hedging a portfolio. In fact this same quantity Δ was used to derive the Black-Scholes PDE. The term “Delta” may be used in two senses in this chapter. In some cases Δ will refer to the instantaneous rate of change of a financial derivative with respect to the value of the underlying security. This is the mathematical Delta. The term may also be used as part of the phrase “Delta neutral” used to describe a portfolio consisting of the underlying security and financial derivatives. The portfolio $\mathcal{P}$ made up of one option F and Δ shares of the underlying security S is Delta neutral if $\mathcal{P}_S = 0$. This is the quantitative analytical sense of “Delta”. The Black-Scholes equation can be thought of as the statement that when the Δ of a portfolio is zero, the rate of return from a portfolio consisting of ownership of the underlying security and sale of the option (or the opposite positions) should be the same as the rate of return at the risk-free interest rate on the same net amount of cash…

The work done to develop the Black-Scholes option pricing formulas in Chapters 7-8 can readily be extended to other types of options that generalize the European-style options on non-dividend paying stocks. In this chapter we will develop variations on the Black-Scholes model to price options on stocks which pay either discrete or continuous dividends and to options with payoffs other than the magnitude of the difference between the stock value and the strike price at expiry. Unless otherwise specified all options discussed in this chapter will still have the European style of exercise.

There are several notions of the idea of optimizing a portfolio of securities, options, bonds, cash, etc. An optimal portfolio could be defined as one with a maximal rate of return or it could mean a portfolio for which the probability of a large fluctuation (usually in the downward direction) in the value of the portfolio is minimized. Alternatively an optimal portfolio for some investors could combine the two notions allowing the investor to specify for example an acceptable level of return and then designing the portfolio with the minimum chance of deviating from that return. The probability of deviating from a desired rate of return will serve as our definition of risk associated with a portfolio. Thus in this chapter we will explore portfolio optimality in the sense of maximizing the rate of return while minimizing the variance in the rate of return, and hence minimizing the risk to the investor. We will also introduce the Capital Assets Pricing Model which attempts to relate the rate of return of a specific investment to the rate of return for the entire market of investments. We will see in this chapter that the difference in the expected rates of return for a specific security and the risk-free interest rate is proportional to the difference in the expected rates of return for the market and the risk-free interest rate. In order to explain this proportionality relationship we will introduce two additional statistical measures, covariance and correlation. We will also make use of these concepts in developing additional hedging strategies.

For the most part the preceding chapters have been concerned with European-style options, their properties, and their uses in hedging. For the reader who has come this far, the final chapter will explore the pricing and properties of American-style options. Recall that American-style options may be exercised at any time up to expiry. In this chapter we will see there are important differences between the European- and American-style options as well as nearly paradoxical similarities. We will develop a means of approximating the values of American puts and calls using the binomial approach outlined in Section 8.5. Before tackling the pricing procedure for American-style options, readers should understand the binomial model presented in the earlier section.

The following sections are included:

• Appendix A Sample Stock Market Data

• Appendix B Solutions to Chapter Exercises
  • The Theory of Interest
  • Discrete Probability
  • Normal Random Variables and Probability
  • The Arbitrage Theorem
  • Random Walks and Brownian Motion
  • Forwards and Futures
  • Options
  • Solution of the Black-Scholes Equation
  • Derivatives of Black-Scholes Option Prices
  • Hedging
  • Extensions of the Black-Scholes Model
  • Optimizing Portfolios
  • American Options

• Bibliography

• Index