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Contingent Claims Pricing Part III Contingent Claims Pricing 191 Chapter 7 Basics of Derivative Pricing Chapter 4 showed how general pricing relationships for contingent claims could be derived in terms of an equilibrium stochastic discount factor or in terms of elementary securities. This chapter takes a more detailed look at this important area of asset pricing.1 The field of contingent claims pricing experienced explo- sive growth following the seminal work on option pricing by Fischer Black and Myron Scholes (Black and Scholes 1973) and by Robert Merton (Merton 1973b). Research on contingent claims valuation and hedging continues to expand, with significant contributions coming from both academics and finance practitioners. This research is driving and is being driven by innovations in financial markets. Because research has given new insights into how potential contingent securi- ties might be priced and hedged, financial service providers are more willing to introduce such securities to the market. In addition, existing contingent securities motivate further research by academics and practitioners whose goal is to improve the pricing and hedging of these securities. 1 The topics in this chapter are covered in greater detail in undergraduate and masters- level financial derivatives texts such as (McDonald 2002) and (Hull 2000). Readers with a background in derivatives at this level may wish to skip this chapter. For others without this knowledge, this chapter is meant to present some fundamentals of derivatives that provide a foundation for more advanced topics covered in later chapters. 193 194 CHAPTER 7. BASICS OF DERIVATIVE PRICING We begin by considering two major categories of contingent claims, namely, forward contracts and option contracts. These securities are called derivatives because their cashflows derive from another “underlying” variable, such as an asset price, interest rate, or exchange rate.2 For the case of a derivative whose underlying is an asset price, we will show that the absence of arbitrage op- portunities places restrictions on the derivative’s value relative to that of its underlying asset.3 In the case of forward contracts, arbitrage considerations alone may lead to an exact pricing formula. However, in the case of options, these no-arbitrage restrictions cannot determine an exact price for the deriva- tive, but only bounds on the option’s price. An exact option pricing formula requires additional assumptions regarding the probability distribution of the underlying asset’s returns. The second section of this chapter illustrates how options can be priced using the well-known binomial option pricing technique. This is followed by a section covering different binomial model applications. The next section begins with a reexamination of forward contracts and how they are priced. We then compare them to option contracts and analyze how the absence of arbitrage opportunities restricts option values. 7.1 Forward and Option Contracts Chapter 3’s discussion of arbitrage derived the link between spot and forward contracts for foreign exchange. Now we show how that result can be generalized to valuing forward contracts on any dividend-paying asset. Following this, we compare option contracts to forward contracts and see how arbitrage places limits on option prices. 2 Derivatives have been written on a wide assortment of other variables, including commod- ity prices, weather conditions, catastrophic insurance losses, and credit (default) losses. 3 Thus, our approach is in the spirit of considering the underlying asset as an elementary security and using no-arbitrage restrictions to derive implications for the derivative’s price. 7.1. FORWARD AND OPTION CONTRACTS 195 7.1.1 Forward Contracts on Assets Paying Dividends Similar to the notation introduced previously, let F0 be the current date 0 forward price for exchanging one share of an underlying asset periods in the future. Recall that this forward price represents the price agreed to at date 0 but to be paid at future date > 0 for delivery at date of one share of the asset. The long (short) party in a forward contract agrees to purchase (deliver) the underlying asset in return for paying (receiving) the forward price. Hence, the date > 0 payoff to the long party in this forward contract is S F0 , where S is the spot price of one share of the underlying asset at the maturity date of the contract.4 The short party’spayoff is simply the negative of the long party’spayoff. When the forward contract is initiated at date 0, the parties set the forward price, F0 , to make the value of the contract equal zero. That is, by setting F0 at date 0, the parties agree to the contract without one of them needing to make an initial payment to the other. Let Rf > 1 be one plus the per-period risk-free rate for borrowing or lending over the time interval from date 0 to date . Also, let us allow for the possibility that the underlying asset might pay dividends during the life of the forward contract, and use the notation D to denote the date 0 present value of dividends paid by the underlying asset over the period from date 0 to date .5 The asset’sdividends over the life of the forward contract are assumed to be known at the initial date 0, so that D can be computed by discounting each dividend payment at the appropriate date 0 risk-free rate corresponding to the time until the dividend payment is made. In the analysis that follows, we also assume that risk-free interest rates are nonrandom, though most of our results in this section and the next continue to hold when interest rates are assumed to change 4 Obviously S is, in general, random as of date 0 while F0 is known as of date 0. 5 In our context, "dividends" refer to any cashflows paid by the asset. For the case of a coupon-paying bond, the cashflows would be its coupon payments. 196 CHAPTER 7. BASICS OF DERIVATIVE PRICING randomly through time.6 Now we can derive the equilibrium forward price, F0 , to which the long and short parties must agree in order for there to be no arbitrage opportunities. This is done by showing that the long forward contract’s date payoffs can be exactly replicated by trading in the underlying asset and the risk-free asset. Then we argue that in the absence of arbitrage, the date 0 values of the forward contract and the replicating trades must be the same. The following table outlines the cashflows of a long forward contract as well as the trades that would exactly replicate its date payoffs. Date 0 Trade Date 0 Cashflow Date Cashflow Long Forward Contract 0 S F0 Replicating Trades 1) Buy Asset and Sell Dividends S0 + DS 2) Borrow R F0 F0 f Net Cashflow S0 + D + R F0 S F0 f Note that the payoff of the long forward party involves two cashflows: a positive cashflow of S , which is random as of date 0, and a negative cashflow equal to F0 , which is certain as of date 0. The former cashflow can be replicated by purchasing one share of the underlying asset but selling ownership of the dividends paid by the asset between dates 0 and .7 This would cost S0 D, where S0 is the date 0 spot price of one share of the underlying asset. 6 This is especially true for cases in which the underlying asset pays no dividends over the life of the contract, that is, D = 0. Also, some results can generalize to cases where the underlying asset pays dividends that are random, such as the case when dividend payments are proportional to the asset’s value. 7 In the absence of an explict market for selling the assets’dividends, the individual could borrow the present value of dividends, D, and repay this loan at the future dates when the dividends are received. This will generate a date 0 cashflow of D, and net future cashflows of zero since the dividend payments exactly cover the loan repayments. 7.1. FORWARD AND OPTION CONTRACTS 197 The latter cashflow can be replicated by borrowing the discounted value of F0 . This would generate current revenue of Rf F0 . Therefore, the net cost of replicating the long party’s cashflow is S0 D R F0 . In the absence of f arbitrage, this cost must be the same as the cost of initiating the long position in the forward contract, which is zero.8 Hence, we obtain the no-arbitrage condition S0 D R F0 = 0 (7.1) f or F0 = (S0 D) R (7.2) f Equation (7.2) determines the equilibrium forward price of the contract. Note that if this contract had been initiated at a previous date, say, date 1, at the forward price F 1 = X, then the date 0 value (replacement cost) of the long party’spayoff, which we denote as f0, would still be the cost of replicating the two cashflows: f0 = S0 D R X (7.3) f 8 If S0 D R F0 < 0, the arbitrage would be to perform the following trades at f date 0: 1) purchase one share of the stock and sell ownership of the dividends; 2) borrow Rf F0 ; 3) take a short position in the forward contract. The date 0 net cashflow of these three transactions is (S0 D) + R F0 + 0 > 0, by assumption. At date the individual f would: 1) deliver the one share of the stock to satisfy the short forward position; 2) receive F0 as payment for delivering this one share of stock; 3) repay borrowing equal to F0 . The date net cashflow of these three transactions is 0 + F0 F0 = 0.
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