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 abitrage 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.

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).

Introduction

In this chapter we consider some optimization problems involving onetime investments not necessarily tied to the movement of a publicly traded security. Section 11.2 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 11.2.1 presents a dynamic programming algorithm that can always be used to solve the preceding problem; Section 11.2.2 gives a more efficient algorithm that can be employed when all the project return functions are concave; and Section 11.2.3 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 11.3. Section 11.3.1 is concerned with a gambling model having an unknown win probability, and Section 11.3.2 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.

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

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 14.2 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).

At the end of the last chapter, we looked beyond abortion to the issue of infanticide, thus confirming the suspicions of supporters of the sanctity of human life that once abortion is accepted, euthanasia lurks around the corner. For them, that is an added reason for opposing abortion. Euthanasia has, they point out, been rejected by doctors since the fifth century BC, when physicians first took the Oath of Hippocrates and swore ‘to give no deadly medicine to anyone if asked, nor suggest any such counsel’. Moreover, they argue, the Nazi extermination programme is a terrible modern example of what can happen once we give the state the power to kill innocent human beings.

It is true that if one accepts abortion on the grounds provided in the preceding chapter, the case for killing other human beings, in certain circumstances, is strong. As I shall try to show in this chapter, however, this is not something to be regarded with horror, and the use of the Nazi analogy is utterly misleading. On the contrary, once we abandon those doctrines about the sanctity of human life that – as we saw in Chapter 4 – collapse as soon as they are questioned, it is the refusal to accept killing that, in some cases, is horrific.

When the first edition of this book appeared in 1979, no country had legalized euthanasia, although in Switzerland a physician could prescribe lethal drugs to patients seeking aid in dying.

In the previous chapter, we briefly considered the argument that the only obligation we have to strangers is not to harm them. For most of human existence, that view would have been easy to live by. Our ancestors lived in groups of no more than a few hundred people, and those on the other side of a river or mountain range might as well have been living in a separate world. We developed ethical principles to help us to deal with problems within our community, not to help those outside it. The harms that it was considered wrong to cause were generally clear and well defined. We developed inhibitions against, and emotional responses to, such actions, and these instinctive or emotional reactions still form the basis for much of our moral thinking.

Today, we are connected to people all over the world in ways our ancestors could not have imagined. The discovery that human activities are changing the climate of our planet has brought with it knowledge of new ways in which we can harm one another. When you drive your car, you burn fossil fuel that releases carbon dioxide into the atmosphere. You are changing the chemical composition of the atmosphere and, hence, the climate. What does this do to others?

Previous chapters of this book have discussed what we ought, morally, to do about several practical issues and what means we are justified in adopting to achieve our ethical goals. The nature of our conclusions about these issues – the demands they make on us – raises a further, more fundamental question: why should we act morally?

Take our conclusions about the use of animals for food, or the aid the rich should give the poor. Some readers may accept these conclusions, become vegetarians, and do what they can to reduce absolute poverty. Others may disagree with our conclusions, maintaining that there is nothing wrong with eating animals and that they are under no moral obligation to do anything about reducing absolute poverty. There is also, however, likely to be a third group: readers who find no fault with the ethical arguments of these chapters yet do not change their diets or their contributions to aid for the poor. Of this third group, some may just be weak-willed, but others may want an answer to a further practical question: if the conclusions of ethics require so much of us, they may ask, why should we bother about ethics at all?

UNDERSTANDING THE QUESTION

‘Why should I act morally?’ is a different type of question from those that we have been discussing up to now. Questions like ‘Why should I treat people of different ethnic groups equally?’ or ‘Why is abortion justifiable?’ seek ethical reasons for acting in a certain way.

A river tumbles through forested ravines and rocky gorges towards the sea. The state hydro-electricity commission sees the falling water as untapped energy. Building a dam across one of the gorges would provide three years of employment for a thousand people, and longer-term employment for twenty or thirty. The dam would store enough water to ensure that the state could economically meet its energy needs for the next decade. This would encourage the establishment of energy-intensive industry thus further contributing to employment and economic growth.

The rough terrain of the river valley makes it accessible only to the reasonably fit, but it is nevertheless a favoured spot for bushwalking. The river itself attracts the more daring whitewater rafters. Deep in the sheltered valleys are stands of rare Huon pine, many of the trees being more than a thousand years old. The valleys and gorges are home to many birds and animals, including an endangered species of marsupial mouse that has seldom been found outside the valley. There may be other rare plants and animals as well, but no one knows, for scientists are yet to investigate the region fully.

Should the dam be built? This is one example of a situation in which we must choose between very different sets of values. The description is loosely based on a proposed dam on the Franklin River, in the south-west of Australia's island state, Tasmania.

THE BASIS OF EQUALITY

The period since the end of World War II has seen dramatic shifts in moral attitudes on issues like abortion, sex outside marriage, same-sex relationships, pornography, euthanasia and suicide. Great as the changes have been, no new consensus has been reached. The issues remain controversial, and the traditional views still have respected defenders.

Equality seems to be different. The change in attitudes towards inequality – especially racial inequality – has been no less sudden and dramatic than the change in attitudes towards sex, but it has been more complete. Racist assumptions shared by most Europeans at the beginning of the twentieth century have become totally unacceptable, at least in public life. A poet could not now write of ‘lesser breeds without the law’, and retain – indeed enhance – his reputation, as Rudyard Kipling did in 1897. This does not mean that there are no longer any racists, but only that they must disguise their racism if their views and policies are to have any chance of general acceptance. The principle that all humans are equal is now part of the prevailing political and ethical orthodoxy. But what, exactly, does it mean and why do we accept it?

Once we go beyond the agreement that blatant forms of racial discrimination are wrong and raise questions about the basis of the principle that all humans are equal, the consensus starts to weaken.

RACISM AND SPECIESISM

In the previous chapter, I gave reasons for believing that the fundamental principle of equality, on which the idea that humans are equal rests, is the principle of equal consideration of interests. Only a basic moral principle of this kind can allow us to defend a form of equality that embraces almost all human beings, despite the differences that exist between them. (The exceptions are human beings who are not and have never been conscious and therefore have no interests to be considered – a topic to be discussed in Chapters 6 and 7.) Although the principle of equal consideration of interests provides the best possible basis for human equality, its scope is not limited to humans. When we accept the principle of equality for humans, we are also committed to accepting that it extends to some nonhuman animals.

When I wrote the first edition of this book, in 1979, I warned the reader that the suggestion I was making here might seem bizarre. It was then generally accepted that discrimination against members of racial minorities and against women ranked among the most important moral and political issues. Questions about animal welfare, however, were widely regarded as matters of no real significance, except for people who are dotty about dogs and cats. Issues about humans, it was commonly assumed, should always take precedence over issues about animals.