Pricing And Hedging Of Derivative Securities

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Arbitrage and Risk-Neutral Pricing

By arbitrage we mean making money out of nothing without risk. For example, suppose that for a given exchange rate—say, U.S. dollars versus Japanese yen—we get two different quotes by two different traders, simultaneously, so that we can buy yen in the cheaper market and sell them in the more expensive. Even though there might be short time periods in a financial market when there is an arbitrage opportunity, such opportunities tend to disappear quickly. As other market participants observe the mismatch, the demand for the cheap yen will increase and the supply of the expensive yen will decrease, and that process will drive the quoted exchange rates to the no-arbitrage level at which they are identical. The assumption of absence of arbitrage in market models is crucial for getting any kind of sensible general results.

The formalization of this notion and its path-breaking application to finance were accomplished in the 1970s by Black and Scholes (1973), Merton (1973b, 1976, 1977), Cox and Ross (1976), Harrison and Kreps (1979), and Harrison and Pliska (1981). The revolutionary contribution of the theory of arbitrage was the realization that the absence of arbitrage implies a unique price for the claims (securities) that can be replicated in the market. Perhaps equally important for practical applications, the theory of arbitrage pricing (or pricing by no arbitrage) has developed methods for computing this unique price of a security, as well as for hedging risks of holding a position in a security. As we will see, the price of the claim that can be replicated is equal to the expected value of the payoff of the claim, discounted by the risk-free rate; however, the expected value is not taken under the "real world" probability, but under the "risk-neutral probability" or "equivalent martingale measure" (EMM).

In the following sections we show how the absence of arbitrage results in some simple bounds for the prices of standard derivative securities. We also explain the connection between arbitrage and the notion of risk-neutral probabilities. It should be remarked that the arbitrage relationships we obtain do not always hold in practice because of the idealized assumptions we make, such as the absence of transaction fees when trading.

6.1 Arbitrage Relationships for Call and Put Options; Put-Call Parity

In this chapter, as we do throughout the rest of the book, we assume that there are no arbitrage opportunities in the market. More precisely, we assume that any strategy that starts with zero investment cannot have a positive probability of producing a positive profit while at the same time having zero probability of resulting in a loss. We also assume that the bank interest rate r is positive and constant, and that there are no transaction costs in the market. (Instead of assuming a constant interest rate, we could assume that we can trade in the bond market, and by holding long or short positions in risk-free bonds, we could invest or borrow at the interest rate r.) For concreteness, we suppose continuous compounding, so that the value of K dollars at time T is worth Ke-r(T-t) at time t. Let us denote by C(t), c(t), P(t), and p(t) the time-t values of the American call option, the European call option, the American put option, and the European put option, respectively. Denote also by S the price of the stock that is the underlying security of all the options, by T the time of maturity, and by K the strike price for a given option. We now establish several comparisons among these values, which must hold, or arbitrage opportunities would exist.

We should emphasize that the results we get in this part of the chapter are model independent. More precisely, the arbitrage opportunities we will construct in the examples consist of taking a static position (one-time trade) in given securities and waiting until maturity. The positions are constructed in such a way that there is arbitrage regardless of what probability distributions those securities have.

The first of the relationships we get is that, for a European and an American call written on the same stock, and with the same maturity, we have

The first inequality says that the value of the European call is never larger than the value of the American call.

This statement is true because the holder of the American call can always wait until maturity, in which case the payoff he gets is the same as that of the European call. That is, the American call gives the same right as the European call, plus an additional possibility of exercising before maturity. More formally, suppose c(t) > C(t). Then we could sell the European call, buy the American call, and put the (positive) difference in the bank. We could then wait until maturity, when both calls have the same payoffs that would cancel each other in our portfolio, while we would have positive profit in the bank.

The second inequality says that the value of a call option is never larger than the value of the stock.

This statement is true because call options give holders the right to get one share of the stock, but they have to pay the strike price in order to get it.

For the put option we have

In other words, the value of the European put is no larger than the value of the American put, by the same argument as for the European and American calls, and the value of a put option is never larger than the strike price.

The latter statement is true because the holder of the put option gets K dollars if the option is exercised, but only upon the delivery of the stock.

In fact, since the European put payoff at maturity is at most K, the value of the European put at time t is no larger than the value that results in K dollars in the bank at time T, when c(t) < C(t) < S(t)

invested in the bank at time t:

Otherwise, we would write the put and deposit p(t) > Ke-r(T-t) in the bank. At maturity, we would have more than K in the bank. This amount is enough to pay the put holder if the put is exercised, and in exchange we would get the stock, with a value of at least zero. Therefore, we have a positive profit (arbitrage opportunity).

The next result says that the value of the European call of a stock that pays no dividends is no smaller than the difference between the stock price and the discounted strike price:

Suppose that this statement is not true, so that c(t) + Ke-r(T-t) < S(t). In other words, it looks like the call and the cash might be cheap while the stock might be expensive. We take advantage of this difference by constructing an arbitrage portfolio. More precisely, consider the strategy consisting of the following two positions at time t:

1. Sell the stock short.

2. Buy the call and deposit Ke-r(T-t) dollars in the bank.

Because we assumed c(t) + Ke-r(T-t) < S(t), when we take these positions, we have extra money to deposit in the bank at time t. However, at maturity time T, we have - S(T) in position 1; as for position 2, if the option is in the money, that is, S(T) > K, our payoff will be S(T) - K + K = S(T), and if the option is out of the money, we will have K > S(T). Therefore, in any case we have no loss at time T [and maybe a profit, if K > S(T)], and we made a positive profit (money in the bank) at time t. We point out that this strategy might not work if the stock pays dividends before maturity. The reason is that, in this case, the short position in the stock will involve the obligation to pay the dividends. Then we cannot guarantee that there will be no loss.

For the put option we have the following: The value of the European put on a stock that pays no dividends until maturity is never smaller than the discounted strike price minus the stock price:

We argue as before. If the previous inequality does not hold, implement the strategy consisting of the following two positions at time t:

2. Buy the put and the stock.

This strategy yields a profit at time t, and it can be verified that it yields a zero profit at maturity, at worst.

Thus far, we have only obtained bounds (inequalities) on option prices. The following classical result gives precise information (as an equality) on the relationship between the European and the American call prices:

The price of an American call on a stock that pays no dividends until maturity is equal to the price of the corresponding European call (that is, written on the same stock and with the same maturity).

Suppose that this is not the case. We only show that we cannot have C(t) > c(t), because we already know that C(t) > c(t) [we showed how to construct an arbitrage portfolio if C(t) < c(t)]. If C(t) > c(t), we could sell the American call and buy the European call, while depositing the extra money in the bank. If at time t < T the American option is exercised, we need to pay S(t) - K. We can cover that expense by selling the call option, because we know from expression (6.4) that c(t) > S(t) - Ke-r(T-t) > S(t) - K. If the American option is never exercised, there is no obligation to cover. This is an arbitrage opportunity, so we cannot have C (t) > c(t).

A corollary of previous arguments is the fact that an American call written on a stock that does not pay dividends until maturity should not be exercised before maturity.

Indeed, if the American option is exercised early at t < T, its payoff is S(t) - K, which is lower than the price of the option C(t), because we know that we have C(t) > c(t) > S(t) - Ke-r(T-t) > S(t) - K. Thus, if the option holder wants to get out of the option position, it is more profitable to sell it than to exercise it.

When the underlying stock pays dividends before maturity, the previous statement is not necessarily true, and it may be better to exercise the American option earlier, in order to be able to collect the dividends. Our argument might not hold anymore because, as we observed previously, inequality (6.4) might not be satisfied if the stock pays a dividend and we cannot conclude that C (t) > S(t) - K. For the American put option, even if the stock pays no dividends, it may be optimal to exercise early. As an extreme example, suppose that the stock is worth zero (or very close to zero) at some time t < T (the company went bankrupt, or almost bankrupt). If the option is exercised at t, its payoff is K (or very close to K), and when we deposit it in the bank we will have more than K dollars at time T. This action is definitely better than waiting until time T to exercise, when the highest possible payoff is K dollars. Thus it is better to exercise at time t.

Let us also mention that for an American call option on a stock that pays dividends, if the option is exercised before maturity, it should be exercised only immediately before one of the times the dividend is paid. We have shown that in the absence of dividends it is optimal to wait until maturity. If there is a dividend to be paid, before the payment date the option behaves as an option on a non-dividend-paying stock, so it should not be exercised until that date.

Assuming that the stock pays no dividends, we finish this section by discussing another classical result that relates the prices of the European call and the European put, called put-call parity:

In words, the call price plus the present value of K dollars in the bank is equal to the put price plus the stock price.

This result follows from looking at the payoffs of these two positions at maturity: holding a call plus discounted K dollars in the bank results in a payoff of one share S(T) at maturity if S(T) > K, and it results in K dollars if S(T) < K. Holding a put and a share of the stock results in the same payoff at maturity. Consequently, since there is no possibility of early exercise, the prices of the two positions have to be the same. In the following example we show how to extract arbitrage profit when the put-call parity does not hold.

Example 6.1 (Deviation from the Put-Call Parity) Suppose that the today's price of stock S is S(0) = $48. The European call and put on S with strike price 45 and maturity T of three months are trading at $4.95 and $0.70, respectively. Three-month T-bills (nominal value $100) are trading at $98.50. We want to see whether there is arbitrage and take advantage if there is. We can find the discount factor, call it d, from the price of the T-bill:

98.5 = 100d from which d = 0.985. We now compute the value of the left-hand side of the put-call parity, c(t) + Kd = 49.275

and the value of the right-hand side, p(t) + S(t) = 48.7

The left- and the right-hand sides are not equal; therefore, there is an arbitrage opportunity. We sell overvalued securities; that is, we write one call and sell short T-bills with a nominal value of $45, from which we receive $49.275. We buy the undervalued securities using this money; that is, we buy one put and one share of the stock at the cost of $48.7. We still have left 49.275 - 48.7 = $0.575 at our disposal. We hold our positions until maturity. If S(T) > 45, then we have to deliver one share of the stock that we hold to cover the call position, and we use $45 that we get for that share to cover our short position in the T-bill. If S(T) < 45, the call we wrote is out of the money, while we get $45 for the share of the stock we have, by exercising the put. We use $45 to cover our short position in the T-bill. In both cases the net cash flow of liquidating the portfolio is zero. This outcome is arbitrage, because the amount $0.575 that we had at our disposal was not used to liquidate the portfolio.

Put-call parity does not apply to American options. The most we can say, if the stock pays no dividends, is

This result follows from the put-call parity for the European options and from the fact that P(t) > p(t) and C(t) = c(t). In fact, we have

The right-hand-side inequality follows from expression (6.7). As for the left-hand side, suppose that it is not true, that is, S(t) + P (t) > C(t) + K. Then we could sell short one share of the stock, sell the put, buy the call, and deposit more than K dollars in the bank. If the put is exercised at time t , we get one share of the stock to cover our short position, and we have to pay for it K dollars, which is less than what we already have in the bank. If the put is never exercised, we are short one share of the stock at maturity, which can be covered by exercising the call option we have, that is, by buying the share for K dollars from the bank. In any case, we make money out of nothing; therefore, there is an arbitrage opportunity.

6.2 Arbitrage Pricing of Forwards and Futures 6.2.1 Forward Prices

Let us first consider the problem of pricing a forward on an asset S that pays no dividends (or coupons). For concreteness, we call that asset a stock. We claim that, at least in theory, the no-arbitrage forward price F (t), agreed upon at time t, has to satisfy

In other words, the forward price is equal to the time-T value (that is, compounded at the appropriate interest rate) of one share's worth deposited in the bank at time t.

In order to see this point, suppose it is not true. For example, suppose F(t) > S(t)er(T-t). We can then, at time t, take a short position in the forward contract and buy the stock by borrowing S(t) from the bank. At time T, we have to return S(t)er(T-t) to the bank, and we can do so even while making a positive profit, because we deliver the stock and get in exchange F(t) dollars, with F(t) > S(t)er(T-t). This outcome is arbitrage. Conversely, we can construct an arbitrage portfolio if F(t) < S(t)er(T-t) by selling the stock short, depositing the proceeds in the bank, and taking a long position in the forward contract.

Next, suppose the stock pays known dividends during the time interval [t, T], whose present time-t value (the discounted future payments) is denoted D (t). We can then use a similar argument (see Problem 12) to show

The intuition for this result is that the dividend makes a long position in the stock cheaper, since the holder of the stock will later receive the dividends. Thus the present value of dividends is subtracted from the stock value on the right-hand side of equation (6.9). Similarly, if the stock pays the dividends continuously at the constant rate q, the present value of the stock net of dividends becomes S(t )e—q(T—t), and we have

Example 6.2 (Forward Pricing with Dividends) A given stock trades at $100, and in six months it will pay a dividend of $5.65. The one-year continuous interest rate is 10%, and the six-month continuous interest rate is 7.41%. According to the analogue of expression (6.9) with different annualized interest rates for six months and for one year, we find the forward price as

Suppose that the price is instead 104; that is, the forward contract is undervalued. We now construct an arbitrage strategy. We take a long position in the forward contract (at zero initial cost) and sell the stock short for $100. We buy the six-month bond in the amount of 5.65e—00741x0'5 = 5.4445. We invest the remaining balance, 100 — 5.4445 = 94.5555, in the one-year bond. In six months, we have to pay the dividend of $5.65 and we can do it by using the proceeds of our investment in the six-month bond. In one year, we have bonds maturing for a nominal rate of 94.5555e01 = 104.5. We pay $104 for the stock (according to the conditions of the forward contract), deliver the stock to cover the short position, and keep 104.5 — 104 = 0.5, which is an arbitrage profit.

Consider now a forward contract on a foreign currency. Here, S(t) denotes the current price in dollars of one unit of the foreign currency. We denote by rf the foreign risk-free interest rate (with continuous compounding). We observe that the foreign (continuously paid) interest rate is equivalent to a (continuously paid) dividend: the holder of the foreign currency will receive the interest rate very much as the holder of the stock will receive the dividend. We then have

If, for example, we had F(t) < S(t)e(r —rf )(T—1), we could take a long position in the forward contract, borrow e—rf(T—t) units of foreign currency, and deposit its value in dollars of

S(t)e—rf (T—t) in the domestic bank. At time T we would have to pay F(t) to get one unit of foreign currency; this unit would cover our debt to the foreign bank (remember that we had borrowed e—rf (T—t), and this amount, compounded at the foreign rate rf, becomes equal to one), and we would still have S(t)e(r—rf)(T—t) in the domestic bank to more than cover the expense F(t). We can similarly construct an arbitrage portfolio if F(t) > S(t)e(r—rf)(T—t).

6.2.2 Futures Prices

We recall that the main difference between a futures contract and a forward contract is that the position in the futures contract is updated ("marked to market") periodically (typically, daily): the changes in the futures price for a given contract are credited or charged to the account of the investor with a position in the contract. The main result affecting the futures prices is the futures-forward price equivalence:

If the interest rate is deterministic, the futures price is equal to the forward price. In order to show this point, assume for simplicity that the length of the period between two resettlement dates (when the contract is marked to market) for the futures contract is equal to one unit of time (say, one day) and that the continuously compounded interest rate for every period is equal to the constant r. Consider the following strategy:

1. At time 0 take a long position in e—r (T-1) futures contracts with price F (0).

2. At time 1 increase the position to e-r(T—2) contracts, and so on, until time T — 1, when the position is increased to one contract. During each period invest the profit or loss in the bank at rate r.

This strategy is not possible if the interest rate is stochastic because we need to know the value of the interest rate in the future. The profit or loss in the period (k, k + 1) is

which, being invested in the bank, increases to

at time T. The total profit or loss is therefore equal to

]T[ F (k + 1) — F (k)] = F (T) — F (0) = S(T) — F (0)

However, the profit of the form S(T) — G(0) is also the profit of the forward contract with price G(0). Since these strategies both require zero investment, we need to have F (0) = G (0), or, otherwise, there would be an arbitrage opportunity. Indeed, if, for example,

G (0) > F (0), we could use the futures strategy we have described with zero initial investment, while taking a short position in the forward contract, therefore having a profit equal to S(T) — F(0) — [S(T) — G(0)] = G(0) — F(0) > 0 at time T.

In general, if the interest rates are random, then the theoretical futures price is not equal to the theoretical forward price. We say "theoretical" because in practice there may be other factors influencing these prices in different ways, like taxes or liquidity considerations. One general rule of thumb for their relation is as follows: if the underlying asset S is strongly positively correlated with the interest rate, then the futures price will tend to be higher than the forward price, because increases in S tend to happen together with increases in the interest rate, so that the profit of holding a futures contract may typically be invested in the bank at a high rate of interest. Also, a decrease in S will tend to happen together with a decrease in the interest rate, so that the loss of holding a futures contract can typically be financed by borrowing from the bank at a low rate of interest. These two effects make the futures contract more valuable, while they do not affect the forward contract, for which there is no marking to market. However, if the underlying asset S is strongly negatively correlated with the interest rate, the futures price will tend to be lower than the forward price, by similar reasoning (see Problem 16).

We discuss the theoretical relationship between the forward price and the futures price more carefully in a later section.

6.2.3 Futures on Commodities

We assume in this section that the futures price is equal to the forward price. Thus far we have considered futures on financial assets. If a futures contract is written on a commodity (gold, silver, corn, oil, and so on), the preceding formulas may not be valid, even in theory. For example, it may be costly to hold a commodity because of storage costs. First, we suppose that there is no advantage in holding a commodity (we will explore this issue later). Storage costs can be interpreted as negative dividends (an additional payment the holder of the commodity has to make). So, by analogy with equation (6.9), if the total storage cost is known, denoting its present value by U(t), we have

Similarly, by analogy with equation (6.10), if the storage cost is proportional to the price of the commodity and is paid continuously in time at the constant rate u, we get

Recall that these relations are obtained with the assumption that there is no advantage in holding a commodity. However, for some commodities it may be the case that holding them is preferable to taking a position in the futures contract and getting them only in the future. For example, there may be benefits to owning the commodity in the case of shortages, or to keep a production process running, or because of their consumption value. In such situations, the value of the futures contract becomes smaller than in the case in which there are no benefits in holding the commodity, so we conclude

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