exercised immediately.If the value of the second term is greater than the third term (for a certain set of call options),the call's market price will always be greater than its exercise proceeds and it will never be optimal to exercise early. To identify this set of calls,examine the conditions under which the relation Se-iT-Xe-rT>S-X Or S(e-m-l)>-X(1-er） (8) holds.Since the risk-free interest rate is positive,the expression on the right-hand side is negative.Hence,if the left-hand side is positive or zero,early exercise will never be optimal.This condition is met in cases in which is0.If is0,an American-style call will never optimally be exercised early,and the value of the American-style call is equal to the value of the European-style call,C=c.Merton(1973)was the first to identify this result and refers to the situation as the call being"worth more alive than dead." The intuition underlying the "worth more alive than dead"result can be broken down into two components-interest cost,r,and non-interest cost,i.Holding other factors constant,a call option holder prefers to defer exercise.Immediate exercise requires a cash payment of X today.On the other hand,if exercise is deferred until the call's expiration,the cash is allowed to earn interest.The present value of the exercise cost is only Xe.With respect to non-interest cost,recall that i<for physical assets that require storage.If a call on such an asset is exercised early,the asset is received immediately and storage costs begin to accrue.On the other hand,if exercise is deferred by continuing to hold the claim on the asset rather than the asset itself,storage costs are avoided.Note that,even if storage costs are zero (i.e.,with i=0),condition (8)holds because the interest cost incentive remains. For American-style call options on assets with i>(e.g.,stock index portfolio paying dividend yield and foreign currencies paying foreign interest),early exercise may be optimal.The intuition is that,while there remains the incentive to defer exercise and asset price and the exercise price.Third,you can sell it in the same marketplace.There is,after all,an 15
exercised immediately. If the value of the second term is greater than the third term (for a certain set of call options), the call’s market price will always be greater than its exercise proceeds and it will never be optimal to exercise early. To identify this set of calls, examine the conditions under which the relation iT rT Se Xe S X − − − > − or ( ) 1 (1 ) iT rT S e X e − − > − − − (8) holds. Since the risk-free interest rate is positive, the expression on the right-hand side is negative. Hence, if the left-hand side is positive or zero, early exercise will never be optimal. This condition is met in cases in which i ≤ 0 . If i ≤ 0 , an American-style call will never optimally be exercised early, and the value of the American-style call is equal to the value of the European-style call,C = c . Merton (1973) was the first to identify this result and refers to the situation as the call being “worth more alive than dead.” The intuition underlying the “worth more alive than dead” result can be broken down into two components—interest cost, r, and non-interest cost, . Holding other factors constant, a call option holder prefers to defer exercise. Immediate exercise requires a cash payment of X today. On the other hand, if exercise is deferred until the call’s expiration, the cash is allowed to earn interest. The present value of the exercise cost is only i Xe−rT . With respect to non-interest cost, recall that i < 0 for physical assets that require storage. If a call on such an asset is exercised early, the asset is received immediately and storage costs begin to accrue. On the other hand, if exercise is deferred by continuing to hold the claim on the asset rather than the asset itself, storage costs are avoided. Note that, even if storage costs are zero (i.e., with i = 0 ), condition (8) holds because the interest cost incentive remains. For American-style call options on assets with i (e.g., stock index portfolio paying dividend yield and foreign currencies paying foreign interest), early exercise may > 0 be optimal. The intuition is that, while there remains the incentive to defer exercise and asset price and the exercise price. Third, you can sell it in the same marketplace. There is, after all, an 15
earn interest on the exercise price,deferring exercise means forfeiting the income being generated on the underlying asset.The only way to capture this income is by exercising the call and taking delivery of the asset.For American-style call options on assets with i>0,early exercise may be optimal and,therefore,C>c. 2.3.2 Put options The lower price bound ofa European-style put option is p≥max(0,Xer-Ser), (9) where p is the price of a put with exercise price X and time to expiration T.The reason that the put price must exceed Xe-Se is based on a no-arbitrage portfolio involving a long position in the put,a long position of e units of the asset,and a short position of Xe in risk-free securities.If the asset price is less than or equal to the exercise price at the option's expiration,the put will be exercised.The cash proceeds from exercise are used to cover the risk-free borrowing.If the asset price is greater than the exercise price, the put expires worthless,and the asset is sold to cover the risk-free borrowing,leaving S in cash.Since the net terminal value of the portfolio is always greater than or equal to zero,its present value must be less than or equal to zero. An American-style put has an early exercise privilege,which means that the relation between the prices of American-style and European-style put options is P≥p, (10) where the upper case P represents the price of an American-style put option with the same exercise price,time to expiration and underlying asset as the European-style put. The lower price bound of an American-style put option is p>max(0,Xe-"-Se-ir,X-S) (11) This is the same as(9),except that Y-S is added within the maximum value operator. If P<X-S,a costless arbitrage profit of X-S-P can be earned by simultaneously buying the put (and exercising it)and buying the asset. active secondary market for standard calls and puts. 16
earn interest on the exercise price, deferring exercise means forfeiting the income being generated on the underlying asset. The only way to capture this income is by exercising the call and taking delivery of the asset. For American-style call options on assets with i > 0 , early exercise may be optimal and, therefore, C > c ) iT . − − rT Xe− ~ ST − , iT X − − 2.3.2 Put options The lower price bound of a European-style put option is max(0, rT p Xe Se − ≥ , (9) where p is the price of a put with exercise price X and time to expiration T. The reason that the put price must exceed rT iT Xe Se − − − is based on a no-arbitrage portfolio involving a long position in the put, a long position of e−iT units of the asset, and a short position of in risk-free securities. If the asset price is less than or equal to the exercise price at the option’s expiration, the put will be exercised. The cash proceeds from exercise are used to cover the risk-free borrowing. If the asset price is greater than the exercise price, the put expires worthless, and the asset is sold to cover the risk-free borrowing, leaving in cash. Since the net terminal value of the portfolio is always greater than or equal to zero, its present value must be less than or equal to zero. X An American-style put has an early exercise privilege, which means that the relation between the prices of American-style and European-style put options is P ≥ p , (10) where the upper case P represents the price of an American-style put option with the same exercise price, time to expiration and underlying asset as the European-style put. The lower price bound of an American-style put option is max(0, ) rT p ≥ − Xe Se − S . (11) This is the same as (9), except that X − S is added within the maximum value operator. If P < X − S , a costless arbitrage profit of X − S − P can be earned by simultaneously buying the put (and exercising it) and buying the asset. active secondary market for standard calls and puts. 16
In the case of an American-style call,early exercise is never optimal if the asset's income rate is less than or equal to zero(i.e.,is0).In the case of an American-style put, no comparable condition exists,there is always a possibility of early exercise depending on the value of S.To see this,suppose the asset price falls to 0.The put option holder will exercise immediately since (a)there is no chance that the asset price will fall further,and (b)deferring exercise means forfeiting the interest income that can be earned on the exercise proceeds.An American-style put is always worth more than the European-style put,P>p 2.3.3 Put-call parity Put-call parity uses trades in the call,the put,and the asset simultaneously to create a risk-free portfolio.Put-call parity for European-style options is given by c-p=Se-iT-Xe-rT, (12) where the call and the put have the same exercise price and time to expiration,and are written on the same underlying asset.The pricing relation is driven by a no-arbitrage argument.In this case,the no-arbitrage portfolio consists of buying eunits of the asset,buying a put,selling a call with the same exercise price,and borrowing Xe It is straightforward to show that this portfolio will be worthless when the options expire at time T regardless of the relation between the asset price and the option's exercise price. Since no one would pay a positive amount to hold such a portfolio (or a portfolio with reverse investments),the put-call parity relation(12)must hold. The set of trades used to derive put-call parity is called a conversion.If all of the trades are reversed (i.e.,sell the asset,sell the put,buy the call,and buy risk-free securities),it is called a reverse conversion.These names arise from the fact that you can create any position in the asset,options,or risk-free securities by trading (or converting) the remaining securities.The concept of conversion/reverse conversion arbitrage was introduced into the academic literature about 30 years ago.Some market participants were well aware of the concept decades earlier,however.Russell Sage,one of the great 17 In the expressionon the right-hand side of(11),the third term is greater than the second temoversome range for S,independent of the level of i. 17
In the case of an American-style call, early exercise is never optimal if the asset’s income rate is less than or equal to zero (i.e., i ≤ 0 ). In the case of an American-style put, no comparable condition exists;17 there is always a possibility of early exercise depending on the value of S. To see this, suppose the asset price falls to 0. The put option holder will exercise immediately since (a) there is no chance that the asset price will fall further, and (b) deferring exercise means forfeiting the interest income that can be earned on the exercise proceeds. An American-style put is always worth more than the European-style put, P > p . 2.3.3 Put-call parity Put-call parity uses trades in the call, the put, and the asset simultaneously to create a risk-free portfolio. Put-call parity for European-style options is given by iT rT c p Se Xe − − − = − , (12) where the call and the put have the same exercise price and time to expiration, and are written on the same underlying asset. The pricing relation is driven by a no-arbitrage argument. In this case, the no-arbitrage portfolio consists of buying e units of the asset, buying a put, selling a call with the same exercise price, and borrowing −iT rT Xe− . It is straightforward to show that this portfolio will be worthless when the options expire at time T regardless of the relation between the asset price and the option’s exercise price. Since no one would pay a positive amount to hold such a portfolio (or a portfolio with reverse investments), the put-call parity relation (12) must hold. The set of trades used to derive put-call parity is called a conversion. If all of the trades are reversed (i.e., sell the asset, sell the put, buy the call, and buy risk-free securities), it is called a reverse conversion. These names arise from the fact that you can create any position in the asset, options, or risk-free securities by trading (or converting) the remaining securities. The concept of conversion/reverse conversion arbitrage was introduced into the academic literature about 30 years ago.18 Some market participants were well aware of the concept decades earlier, however. Russell Sage, one of the great 17 In the expression on the right-hand side of (11), the third term is greater than the second term over some range for S, independent of the level of i. 17
U.S.railroad speculators of the 1800s,used conversions to circumvent usury laws.Sage extended credit to individuals under three conditions:(a)they post collateral in the form of stock(with the loan amount capped at the current stock price,S),(b)they provide a written guarantee that Sage could sell back the stock at S,and (c)they pay a cash premium to Sage for the right to buy the stock(when the loan is repaid)at S.Ignoring the cash premium,the borrower has received an interest-free loan,borrowing S and then repaying S.In reality,however,the loan is anything but interest free.The cost of the call embeds the interest cost.Conveniently,the usury laws did not apply to implicit interest rates. The early exercise feature of American-style options complicates the put-call parity relation.The specification of the relation depends on the non-interest carry cost,i. The American-style put-call parity relations are S-X≤C-P≤Se-r-Xe-rifi≤0 (13a) and Seir-X≤C-P≤S-Xe-rt if i:>0 (13b) Each inequality in (13a)and in (13b)has a separate set of no-arbitrage trades.Proofs are provided in Stoll and Whaley (1986). 2.3.4 Summary The purpose of this section was to show some of the derivatives pricing relations that can be developed under the seemingly innocuous assumption that two perfect substitutes must have the same price.Some of these relations will be used in the next section to gather intuition about the specification of option valuation formulas.The relations also serve as the basis for the empirical investigations discussed in Section 4. 3.OPTION VALUATION Valuing claims to uncertain income streams is one of the central problems in finance.The exercise is straightforward conceptually.First,the amount and the timing of The appearance of put-all parity in thecademicierre isnStol (19). 18
U.S. railroad speculators of the 1800s, used conversions to circumvent usury laws. Sage extended credit to individuals under three conditions: (a) they post collateral in the form of stock (with the loan amount capped at the current stock price, S), (b) they provide a written guarantee that Sage could sell back the stock at S, and (c) they pay a cash premium to Sage for the right to buy the stock (when the loan is repaid) at S. Ignoring the cash premium, the borrower has received an interest-free loan, borrowing S and then repaying S. In reality, however, the loan is anything but interest free. The cost of the call embeds the interest cost. Conveniently, the usury laws did not apply to implicit interest rates. The early exercise feature of American-style options complicates the put-call parity relation. The specification of the relation depends on the non-interest carry cost, i. The American-style put-call parity relations are if 0 (13a) iT rT S X C P Se Xe i − − − ≤ − ≤ − ≤ and if 0 iT rT Se X C P S Xe i − − − ≤ − ≤ − > . (13b) Each inequality in (13a) and in (13b) has a separate set of no-arbitrage trades. Proofs are provided in Stoll and Whaley (1986). 2.3.4 Summary The purpose of this section was to show some of the derivatives pricing relations that can be developed under the seemingly innocuous assumption that two perfect substitutes must have the same price. Some of these relations will be used in the next section to gather intuition about the specification of option valuation formulas. The relations also serve as the basis for the empirical investigations discussed in Section 4. 3. OPTION VALUATION Valuing claims to uncertain income streams is one of the central problems in finance. The exercise is straightforward conceptually. First, the amount and the timing of 18 The appearance of put-call parity in the academic literature is in Stoll (1969). 18
the expected cash flows from holding the claim must be identified.Next,the expected cash flows must be discounted to the present.The valuation of a European-style call option,therefore,requires the estimation of (a)the mean of the call option's payoff distribution on the day it expires,and (b)the risk-adjusted discount rate to apply to the option's expected terminal payoff. In his dissertation,Theory of Speculation,Bachelier (1900)provides the first known valuation of the European-style call option.His valuation equation,which may be written c=J(5-X)f(5)dS, (14) shows that the option's value depends its expected terminal value.Bachelier assumes that the underlying asset price follows arithmetic Brownian motion,which means f(S)is a normal density function.Unfortunately,this assumption implies that asset prices can be negative.20 To circumvent this problem,Sprenkle (1961)and Samuelson(1965)value the call under the assumption that the asset price follows geometric Brownian motion.By letting asset prices have multiplicative,rather than additive,fluctuations through time,the asset price distribution at the option's expiration is lognormal,rather than normal,and the prospect of the asset price becoming negative is eliminated.Under lognormality, Sprenkle and Samuelson show that the call option valuation formula has the form, c=e-a[Seas'N(d)-XN(d2)], (15) where d,-In(S/X)+(@s+.5o'yT ,dz=d-avT, σVT 19Many consider Bachelier to be the father of modem option pricing theory.For an interesting recount of Bachelier's life and his insights into option valuation,see Sullivan and Weithers(1991). 20 Bachelier also assumes that the asset's expected price change is zero.This implies investors are risk- neutral and money has no time value. 19
the expected cash flows from holding the claim must be identified. Next, the expected cash flows must be discounted to the present. The valuation of a European-style call option, therefore, requires the estimation of (a) the mean of the call option’s payoff distribution on the day it expires, and (b) the risk-adjusted discount rate to apply to the option’s expected terminal payoff. In his dissertation, Theory of Speculation, Bachelier (1900) provides the first known valuation of the European-style call option. His valuation equation, which may be written ( ) ( ) , (14) X c S X f S ∞ = − ∫ dS shows that the option’s value depends its expected terminal value. Bachelier assumes that the underlying asset price follows arithmetic Brownian motion,19 which means f (S) is a normal density function. Unfortunately, this assumption implies that asset prices can be negative.20 To circumvent this problem, Sprenkle (1961) and Samuelson (1965) value the call under the assumption that the asset price follows geometric Brownian motion. By letting asset prices have multiplicative, rather than additive, fluctuations through time, the asset price distribution at the option’s expiration is lognormal, rather than normal, and the prospect of the asset price becoming negative is eliminated. Under lognormality, Sprenkle and Samuelson show that the call option valuation formula has the form, c e Se N d XN d (15) c S T T = − −α α [ ( ) ( 1 )], 2 where d d T T S X S T σ σ α σ = − + + = 2 1 2 1 , ln( / ) ( .5 ) d , 19 Many consider Bachelier to be the father of modern option pricing theory. For an interesting recount of Bachelier’s life and his insights into option valuation, see Sullivan and Weithers (1991). 20 Bachelier also assumes that the asset’s expected price change is zero. This implies investors are riskneutral and money has no time value. 19