## Introduction and Preface

An option gives one the right, but not the obligation, to buy or sell a security under specified terms. A call option is one that gives the right to buy, and a put option is one that gives the right to sell the security. Both types of options will have an exercise price and an exercise time. In addition, there are two standard conditions under which options operate: European options can be utilized only at the exercise time, whereas American options can be utilized at any time up to the exercise time. Thus, for instance, a European call option with exercise price K and exercise time t gives its holder the right to purchase at time t one share of the underlying security for the price K, whereas an American call option gives its holder the right to make that purchase at any time before or at time t.

A prerequisite for a strong market in options is a computationally efficient way of evaluating, at least approximately, their worth; this was accomplished for call options (of either American or European type) by the famous Black-Scholes formula. The formula assumes that prices of the underlying security follow a geometric Brownian motion. This means that if S(y) is the price of the security at time y then, for any price history up to time y, the ratio of the price at a specified future time t + y to the price at time y has a lognormal distribution with mean and variance parameters tp and to2, respectively. That is, will be a normal random variable with mean tp and variance to2. Black and Scholes showed, under the assumption that the prices follow a geometric Brownian motion, that there is a single price for a call option that does not allow an idealized trader - one who can instantaneously make trades without any transaction costs - to follow a strategy that will result in a sure profit in all cases. That is, there will be no certain profit (i.e., no arbitrage) if and only if the price of the option is as given by the Black-Scholes formula. In addition, this price depends only on the variance parameter cr of the geometric Brownian motion (as well as on the prevailing interest rate, the underlying price of the security, and the conditions of the option) and not on the parameter /x. Because the parameter a is a measure of the volatility of the security, it is often called the volatility parameter.

A risk-neutral investor is one who values an investment solely through the expected present value of its return. If such an investor models a security by a geometric Brownian motion that turns all investments involving buying and selling the security into fair bets, then this investor's valuation of a call option on this security will be precisely as given by the Black-Scholes formula. For this reason, the Black-Scholes valuation is often called a risk-neutral valuation.

Our first objective in this book is to derive and explain the Black-Scholes formula. This does require some knowledge of probability, the topic considered in the first three chapters. Chapter 1 introduces probability and the probability experiment. Random variables - numerical quantities whose values are determined by the outcome of the probability experiment - are discussed, as are the concepts of the expected value and variance of a random variable. In Chapter 2 we introduce normal random variables; these are random variables whose probabilities are determined by a bell-shaped curve. The central limit theorem is presented in this chapter. This theorem, probably the most important theoretical result in probability, states that the sum of a large number of random variables will approximately be a normal random variable. In Chapter 3 we introduce the geometric Brownian motion process; we define it, show how it can be obtained as the limit of simpler processes, and discuss the justification for its use in modeling security prices.

With the probability necessities behind us, the second part of the text begins in Chapter 4 with an introduction to the concept of interest rates and present values. A key concept underlying the Black-Scholes formula is that of arbitrage, the subject of Chapter 5. In this chapter we show how arbitrage can be used to determine prices in a variety of situations, including the single-period binomial option model. In Chapter 6 we present the arbitrage theorem and use it to find an expression for the unique nonarbitrage option cost in the multiperiod binomial model. In Chapter 7 we use the results of Chapter 6, along with the approximations of geometric Brownian motion presented in Chapter 3, to obtain a simple derivation of the Black-Scholes equation for pricing call options. In addition, we show how to utilize a multiperiod binomial model to determine an approximation of the risk-neutral price of an American put option.

In Chapter 8 we note that, in many situations, arbitrage considerations do not result in a unique cost. In such cases we show the importance of the investor's utility function as well as his or her estimates of the probabilities of the possible outcomes of the investment. Applications are given to portfolio selection problems, and the capital assets pricing model is introduced. In addition we show that, even when a security's price follows a geometric Brownian motion and call options are priced according to the Black-Scholes formula, there may still be investment opportunities that have a positive expected gain with a relatively small standard deviation. (Such opportunities arise when an investor's evaluation of the geometric Brownian motion parameter fi differs from the value that turns all investment bets into fair bets.)

In Chapter 9 we introduce some nonstandard, or "exotic," options such as barrier, Asian, and lookback options. We explain how to use Monte Carlo simulation techniques to efficiently determine the geometric Brownian motion risk-neutral valuation of such options. Our ways of exploiting variance reduction ideas to make the simulation more efficient have not previously appeared and are improvements over what is presently in the literature.

The Black-Scholes formula is useful even if one has doubts about the validity of the underlying geometric Brownian model. For as long as one accepts that this model is at least approximately valid, its use suggests the appropriate price of the option. Thus, if the actual trading option price is below the formula price then it would seem that the option is underpriced in relation to the security itself, thus leading one to consider a strategy of buying options and selling the security (with the reverse being suggested when the trading option price is above the formula price). However, one downside to the Black-Scholes formula is that its very usefulness and computational simplicity has led many to automatically assume the underlying geometric Brownian motion model; as a result, relatively little effort has gone into searching for a better model. In Chapter 10 we show that real data cannot aways be fit by a geometric Brownian motion model, and that more general models may need to be considered. For instance, one of the key assumptions of geometric Brownian motion is that the ratio of a future security price to the present price does not depend on past prices. In Chapter 10, we consider approximately 3 years of data concerning the (nearest-month) price of crude oil. Each day is characterized as being of one of four types: type 1 means that today's final crude price is down from yesterday's by more than 1%; type 2 means that the price is down by less than 1%; type 3 means that it is up by less than 1%; and type 4 that it is up by more than 1%. The following table gives the percentage of time that a type-/ day was followed by a type-j day for i, j — 1,..., 4.

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