Introduction

In this chapter we consider some optimization problems involving one-time investments not necessarily tied to the movement of a publicly traded security. Section 10.2 introduces a deterministic optimization problem where the objective is to determine an efficient algorithm for finding the optimal investment strategy when a fixed amount of money is to be invested in integral amounts among n projects, each having its own return function. Section 10.2.1 presents a dynamic programming algorithm that can always be used to solve the preceding problem; Section 10.2.2 gives a more efficient algorithm that can be employed when all the project return functions are concave; and Section 10.2.3 analyzes the special case, known as the knapsack problem, where project investments are made by purchasing integral numbers of shares, with each project return being a linear function of the number of shares purchased. Models in which probability is a key factor are considered in Section 10.3. Section 10.3.1 is concerned with a gambling model having an unknown win probability, and Section 10.3.2 examines a sequential investment allocation model where the number of investment opportunities is a random quantity.

A Deterministic Optimization Model

Suppose that you have m dollars to invest among n projects and that investing x in project i yields a (present value) return of fi(x), i = 1, = n. The problem is to determine the integer amounts to invest in each project so as to maximize the sum of the returns.

Introduction

As previously noted, a key premise underlying the assumption that the prices of a security over time follow a geometric Brownian motion (and hence underlying the Black–Scholes option price formula) is that future price changes are independent of past price movements. Many investors would agree with this premise, although many others would disagree. Those accepting the premise might argue that it is a consequence of the efficient market hypothesis, which claims that the present price of a security encompasses all the presently available information – including past prices – concerning this security. However, critics of this hypothesis argue that new information is absorbed by different investors at different rates; thus, past price movements are a reflection of information that has not yet been universally recognized but will affect future prices. It is our belief that there is no a priori reason why future price movements should necessarily be independent of past movements; one should therefore look at real data to see if they are consistent with the geometric Brownian motion model. That is, rather than taking an a priori position, one should let the data decide as much as possible.

In Section 12.2 we analyze the sequence of nearest-month end-of-day prices of crude oil from 3 January 1995 to 19 November 1997 (a period right before the beginning of the Asian financial crisis that deeply affected demand and, as a result, led to lower crude prices).

Introduction

In this chapter we derive the celebrated Black–Scholes formula, which gives – under the assumption that the price of a security evolves according to a geometric Brownian motion – the unique no-arbitrage cost of a call option on this security. Section 7.2 gives the derivation of the no-arbitrage cost, which is a function of five variables, and Section 7.3 discusses some of the properties of this function. Section 7.4 gives the strategy that can, in theory, be used to obtain an arbitrage when the cost of the security is not as specified by the formula. Section 7.5, which is more theoretical than other sections of the text, presents simplified derivations of (1) the computational form of the Black–Scholes formula and (2) the partial derivatives of the no-arbitrage cost with respect to each of its five parameters.

The Black–Scholes Formula

Consider a call option having strike price K and expiration time t. That is, the option allows one to purchase a single unit of an underlying security at time t for the price K. Suppose further that the nominal interest rate is r, compounded continuously, and also that the price of the security follows a geometric Brownian motion with drift parameter μ and volatility parameter σ. Under these assumptions, we will find the unique cost of the option that does not give rise to an arbitrage.

To begin, let S(y) denote the price of the security at time y.

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 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 the 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 tμ and tσ2, respectively.

Introduction

In this chapter we look at some extensions of the basic call option model. In Section 8.2 we consider European call options on dividend-paying securities under three different scenarios for how the dividend is paid. In Section 8.2.1 we suppose that the dividend for each share owned is paid continuously in time at a rate equal to a fixed fraction of the price of the security. In Sections 8.2.2 and 8.2.3 we suppose that the dividend is to be paid at a specified time, with the amount paid equal to a fixed fraction of the price of the security (Section 8.2.2) or to a fixed amount (Section 8.2.3). In Section 8.3 we show how to determine the no-arbitrage price of an American put option. In Section 8.4 we introduce a model that allows for the possibilities of jumps in the price of a security. This model supposes that the security's price changes according to a geometric Brownian motion, with the exception that at random times the price is assumed to change by a random multiplicative factor. In Section 8.4.1 we derive an exact formula for the no-arbitrage cost of a call option when the multiplicative jumps have a lognormal probability distribution. In Section 8.4.2 we suppose that the multiplicative jumps have an arbitrary probability distribution; we show that the no-arbitrage cost is always at least as large as the Black–Scholes formula when there are no jumps, and we then present an approximation for the no-arbitrage cost.

Introduction

The options we have so far considered are sometimes called “vanilla” options to distinguish them from the more exotic options, whose prevalence has increased in recent years. Generally speaking, the value of these options at the exercise time depends not only on the security's price at that time but also on the price path leading to it. In this chapter we introduce three of these exotic-type options – barrier options, Asian options, and lookback options – and show how to use Monte Carlo simulation methods efficently to determine their geometric Brownian motion risk-neutral valuations. In the final section of this chapter we present an explicit formula for the risk-neutral valuation of a “power” call option, whose payoff when exercised is the amount by which a specified power of the security's price at that time exceeds the exercise price.

Barrier Options

To define a European barrier call option with strike price K and exercise time t, a barrier value v is specified; depending on the type of barrier option, the option either becomes alive or is killed when this barrier is crossed. A down-and-in barrier option becomes alive only if the security's price goes below v before time t, whereas a down-and-out barrier option is killed if the security's price goes below v before time t. In both cases, v is a specified value that is less than the initial price s of the security.

An Example in Options Pricing

Suppose that the nominal interest rate is r, and consider the following model for pricing an option to purchase a stock at a future time at a fixed price. Let the present price (in dollars) of the stock be 100 per share, and suppose we know that, after one time period, its price will be either 200 or 50 (see Figure 5.1). Suppose further that, for any y, at a cost of Cy you can purchase at time 0 the option to buy y shares of the stock at time 1 at a price of 150 per share. Thus, for instance, if you purchase this option and the stock rises to 200, then you would exercise the option at time 1 and realize a gain of 200 − 150 = 50 for each of the y options purchased. On the other hand, if the price of the stock at time 1 is 50 then the option would be worthless. In addition to the options, you may also purchase x shares of the stock at time 0 at a cost of 100 x, and each share would be worth either 200 or 50 at time 1.

We will suppose that both x and y can be positive, negative, or zero. That is, you can either buy or sell both the stock and the option.

Limitations of Arbitrage Pricing

Although arbitrage can be a powerful tool in determining the appropriate cost of an investment, it is more the exception than the rule that it will result in a unique cost. Indeed, as the following example indicates, a unique no-arbitrage option cost will not even result in simple one-period option problems if there are more than two possible next-period security prices.

Example 9.1a Consider the call option example given in Section 5.1. Again, let the initial price of the security be 100, but now suppose that the price at time 1 can be any of the values 50, 200, and 100. That is, we now allow for the possibility that the price of the stock at time 1 is unchanged from its initial price (see Figure 9.1). As in Section 5.1, suppose that we want to price an option to purchase the stock at time 1 for the fixed price of 150.

For simplicity, let the interest rate r equal zero. The arbitrage theorem states that there will be no guaranteed win if there are nonnegative numbers p50, p100, p200 that (a) sum to 1 and (b) are such that the expected gains if one purchases either the stock or the option are zero when pi is the probability that the stock's price at time 1 is i (i = 50, 100, 200).