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as the case may be. As a measure of how much smaller the futures price becomes, we define the convenience yield (which represents the value of owning the commodity) as the value y for which

In general, any type of cost or benefit associated with a futures contract (whether it is a storage cost, a dividend, or a convenience yield) is called cost of carry. We define it as the value c for which

We have hinted before that the absence of arbitrage implies that the contingent claims that can be replicated by a trading strategy could be priced by using expectations under a special, risk-neutral probability measure. In the present section we explain why this is the case. The main results of this section are summarized in figure 6.1.

6.3.1 Martingale Measures; Cox-Ross-Rubinstein (CRR) Model

The modern approach to pricing financial contracts, as well as to solving portfolio-optimization problems, is intimately related to the notion of martingale probability measures. As we shall see, prices are expected values, but not under the "real-world" or "true" probability; rather, they are expected values under an "artificial" probability, called risk-neutral probability or equivalent martingale measure (EMM).

Expected value Solution to a PDE

Figure 6.1

Risk-neutral pricing: no arbitrage, completeness, and pricing in financial markets.

Expected value Solution to a PDE

Figure 6.1

Risk-neutral pricing: no arbitrage, completeness, and pricing in financial markets.

We first recall the notion of a martingale: Consider a process X whose values on the interval [0, s] provide information up to time s. Denote by Es the conditional expectation given that information. We say that a process X is a martingale if

(We implicitly assume that the expected values are well defined.) This equation can be interpreted as saying that the best possible prediction for the future value X (t) of the process X is the present value X (s). Or, in profit/loss terminology, a martingale process, on average, makes neither profits nor losses. In particular, taking unconditional expected values in equation (6.14), we see that E[X(t)] = E[X(s)]. In other words, expected values of a martingale process do not change with time.

Recall our notation A that we use for any value A discounted at the risk-free rate. We say that a probability measure is a martingale measure for a financial-market model if the discounted stock prices Si are martingales.

Let us see what happens in the Cox-Ross-Rubinstein model with one stock. Recall that in this model the price of the stock at period t +1 can take only one of the two values, S(t )u or S(t)d, with u and d constants such that u > 1 + r > d, where r is the constant risk-free rate, and we usually assume d < 1. At every point in time t, the probability that the stock takes the value S(t)u is p, and, therefore, q := 1 - p is the probability that the stock will take the value S(t)d. Consider first a single-period setting. A martingale measure will be given by probabilities p* and q* := 1 - p* of up and down moves, such that the discounted stock price is a martingale:

Here, E * = E0 denotes the (unconditional, time t = 0) expectation under the probabilities p*, 1 - p*. Solving for p* we obtain

We see that the assumption d < 1 + r < u guarantees that these numbers are indeed positive probabilities. Moreover, these equations define the only martingale measure with positive probabilities. Furthermore, p* and q* are strictly positive, so that events that have zero probability under the "real-world" probability measure also have zero probability under the martingale measure, and vice versa. We say that the two probability measures are equivalent and that the probabilities p*, q* form an equivalent martingale measure or EMM.

In order to make a comparison between the actual, real probabilities p, 1 - p and the risk-neutral probabilities p*, 1 - p*, introduce the mean return rate f of the stock as determined from

Then a calculation similar to the preceding implies that we get expressions analogous to equations (6.15):

Thus we can say that the risk-neutral world is the world in which there is no compensation for holding the risky assets, hence in which the expected return rate of the risky assets is equal to the risk-free rate r.

We want to make a connection between the price of a contingent claim and the possibility of replicating the claim by trading in other securities, as discussed in chapter 3. Denote now by 8 the number of shares of stock held in the portfolio and by x the initial level of wealth, X (0) = x. The rest of the portfolio, x - 8 S(0), is invested in the bank at the risk-free rate r. Therefore, from the budget constraint of the individual, the discounted wealth XX (1) at time 1 is given by

Since the discounted stock price is a martingale under probabilities p*, 1 - p*, we have E*[S(1)] = S(0), hence

In other words, the discounted wealth process is also a martingale under the equivalent martingale measure.

We say that the expected value of the discounted future wealth in the risk-neutral world is equal to the initial wealth.

In fact, we have the following result on the possibility of replicating a contingent claim C (that is, a security with a payoff C at moment 1) starting with initial amount x. We denote by Cu the value of the contingent claim in the up state [if the stock goes up to uS(0)] and by Cd the value of the contingent claim in the down state.

THEOREM 6.1 A claim C can be replicated starting with initial wealth C (0) if and only if E *[C] = C (0).

Proof We have already shown that if X(1) = C, then E*[XX(1)] is equal to the initial wealth x = C(0). Conversely, suppose that E*[C] = C(0); that is, p*Cu + q *Cd = C (0) (6.18)

However, a strategy 5C with initial cost D(0) replicates C if

Cu = 5S(0)u + [D(0) - 5S(0)](1 + r), Cd = 5S(0)d + [D(0) - 5S(0)](1 + r)

These last two equations determine the unique strategy 5 and the cost D(0) of replicating C. Using equations (6.15) we can easily verify that this amount D(0) is equal to C(0) given by equation (6.18). Therefore, C(0) is indeed the initial amount needed to replicate C. ■

For the sake of completeness, we also mention another way of studying replication, without using risk-neutral probabilities. Introduce the following random variable, called risk-neutral density:

I q */q with probability q

We have

In other words, a different way to write equation (6.17), that is, the budget constraint for the wealth process, is

Results similar to the preceding results for the single-period model are also valid in the multiperiod Cox-Ross-Rubinstein model. First, it can be verified that the martingale probabilities are still given by the same expressions as in equations (6.15). {Verification is accomplished by solving the equation EJ[S(t + 1)] = S(t).} Next, we define the risk-neutral density Z(T) as the product of T independent copies of the random variable Z(1) in definition (6.19). We then have the budget constraint

Moreover, Theorem 6.1 is still valid (see Problem 19). 6.3.2 State Prices in Single-Period Models

Consider now a more general single-period model with N stocks. For stock j, the value Sj (1) at the end of the period is a random variable that can take K possible values, s ],..., sf. We say that there are K possible states of the world at time 1. A vector d = (d\,..., dK) is called a state-price vector or a vector of Arrow-Debreu prices, if di > 0 for all i = 1,..., K, and we have for all stocks j = 1,..., N. This equation means that the stock price today is obtained as a weighted average of its future values, with the weights being the state prices.

We observe that the elements di of the vector d "measure" the value at moment 0 of one unit of currency at moment 1, in state i. That is, di is the price of a security with possible values si = 1, sk = 0 for all k = i. If we assume, as usual, that there is a risk-free asset S0 such that S0(0) = 1, S0(1) = 1 + r, where r > 0 is a constant, and that equation (6.22) is also valid for S0, we get

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