The principle of no arbitrage may be the key to understanding derivative pricing, but what kind of law is it? It is clearly not a fundamental law of nature, and is not even always obeyed by the markets. In some ways it is similar to Darwin’s theory of natural selection. An institution that does not price by arbitrage arguments the derivatives that it sells will suffer relative to institutions that do. If the price is set too high, then competitors will undercut it; if the price is too low, then the institution will be liable to market uncertainty as a hedging portfolio cannot be properly constructed.
In the competitive world of finance, such an institution would not last long.
There is a crucial point to take away from this section, and to which we shall come back again and again in the course of this book. It is that the actual probabilities of what might happen to the exchange rate (or any other underlying asset) are not important. This is because the expectation of a random variable, such as the exchange rate, may give a good idea of what the exchange rate may be in the future, but it leaves too much to chance.
What matters instead, is that we can create a trading strategy such that there is no uncertainty in the outcome. By creating a risk-free strategy that also replicates the derivative payoff function, we can uniquely determine the no arbitrage price for the derivative.
3 A Simple Casino
When it comes right down to it, putting money into the financial world can be a bit of a gamble. So there is really no better way to begin thinking about financial mathematics than by looking at betting in a Casino, which is every bit a gamble. To meet our sophisticated tastes, we will be betting in a deluxe Casino that allows not only standard wagers, but also ‘side-bets’ which we shall call derivative bets. Our Casino analogy will turn out to be a very simple, but highly effective, model for a stock market. After laying down the rules for gambling and investigating the nature of ‘ordinary’ bets, the goal will be to find a price for the derivative bets by use of the no arbitrage condition.
3.1 Rules of the Casino
Suppose that we make our way into a Casino that allows gamblers to make bets on the outcome of a coin flip. While this is probably one of the simplest Casinos imaginable, we can make it a more interesting place by increasing the complexity of the bets that can be made on the result of the coin toss.
At time 0, just before the coin toss, the initial stake for a bet is S0dollars, which you pay to the Casino. The amount that you receive back from the Casino at time t, just after the coin toss, is Stdollars, which for the ‘standard’
bet we define to be U dollars (‘up’) for heads and D dollars (‘down’) for tails.
For example, we could take
S0 = $2.00, U = $3.00, and D = $1.50. (3.1) In this case, we place $2.00 on the the table, and if the outcome is heads we get $3.00 back, while if the outcome is tails, then we only get $1.50 back. In addition to this ‘standard’ bet, we can also make a short bet. This means that at time 0 the Casino pays you S0 dollars to enter the game, but then you have to pay the Casino St dollars at time t, so the actual amount that you have to pay depends on the outcome of the coin flip. Under this naming scheme, the standard bet is actually a long bet. Since we can place both long and short bets with the same initial stake, the roles of the Casino and player are symmetric in our simple model. In a real Casino this is not, of course, the case, and the rules of the various games are designed so that the Casino will on average make money.
Since the Casino is trying to encourage gambling, it is willing to lend money at no charge. It will also hold your money for you, however no interest is earned. Thus, we can think of the Casino as having a money market account where the risk-free interest rate is zero.
Exercise 3.1 Using a simple arbitrage argument, show why we must have U > S0 > D.
The Casino is a chaotic place, but the organisers and participants are known to be honest. That is, the rules of the Casino are always obeyed. We are not told whether the coin is ‘fair’ (50-50), nor is there any implication that it is. We suspect that it isn’t fair, and after watching play for a few hours and making use of the law of averages, we conclude that the relevant probabilities are
Prob[H] = p and Prob[T ] = q, (3.2) where H, T stand for heads and tails respectively, and p + q = 1. We are also worried that these probablities may change over time.
Clearly the expected payoff from a standard bet is E[St] = pU + qD dollars. But there is no reason (a priori) to suppose that the initial stake satisfies S0 = E[St]. This is the expectation hypothesis, which we saw in the previous section is generally wrong. If S0 < E[St], then anyone willing to play this game is risk-averse, that is, they expect some profit, on average, for taking risk. If S0 > E[St], then players, on average, pay to take risk (which is typical for a Casino), and are risk-preferring. If S0 = E[St], then the players are risk-neutral, since they expect to neither gain nor lose money if they play for a long time.
3.2 Derivatives
A derivative is a kind of side-bet, with a prescribed payoff that depends on the outcome of the coin flip. The Casino is happy to allow derivative bets by special arrangement. In a typical contract, a player pays an initial bet f0
at time 0, and then receives a payoff of ft(St) dollars at time t, where ft(St) is a prescribed function of the random variable St. A derivative contract is defined by its payoff function ft(St) and its purchase price f0. It is possible, in principle, for f0 to be negative, which by convention means that the Casino pays the player to enter into the contract. Note that ft(St) can also, in
principle, be negative, in which case the player has to pay the Casino at time t.
For example, we can consider the important case of a call option, which has a payoff function
ft(St) = max[St− K, 0], (3.3) where K is a fixed number of dollars, known as the strike price, such that U > K > D. By construction, the call option pays off only when St = U . Note that many options, even if they are based on an underlying asset, do not necessarily involve the buying or selling of the underlying, but rather the cash difference between the asset value and the strike price is transferred if the terminal value of the asset exceeds the strike (assuming that the option is a call). In cases where the underlying is not transferrable, such as an option on a stock index, or the outcome of a coin flip, then a cash transfer is the only possibility. We could also consider a more complicated derivative, with a payoff function such as
ft(St) = αSt3+ βSt2+ γSt+ δ, (3.4) where α, β, γ, δ are constants. The pricing of an exotic derivative, like this one, is computationally more difficult than for a vanilla one, such as the call option above, however, mathematically they are given by the same general formula.
Now we need to determine the price f0 that someone should pay at time 0 to buy a derivative that pays off ft(St) dollars at time t. A plausible guess is
f0 = E[ft(St)]
= pft(U ) + qft(D), (3.5)
which represents the expected payoff of the derivative, that is, the probability weighted average of the possible payoffs. This guess is another typical ex-ample of the ‘expectation hypothesis’. As before, it is wrong. So how do we determine f0? Just like in the simple currency model of section 2, we want to use a no arbitrage condition to determine the correct price.