The cost of carry refers to the difference between the costs and the benefits that accrue while holding an asset.Suppose a breakfast cereal producer needs 5,000 bushels of wheat for processing in two months.To lock in the price of the wheat today,he can buy it and carry it for two months.One cost of this strategy is the opportunity cost of funds.To come up with the purchase price,he must either borrow money or reduce his earning assets by that amount.Beyond interest cost,however,carry costs vary depending upon the nature of the asset.For a physical asset such as wheat,he incurs storage costs (e.g.,rent and insurance).At the same time,by storing wheat,he avoids the costs of possibly running out of his regular inventory before two months are up and having to pay extra for emergency deliveries.This benefit is called convenience yield.Thus,the cost of carry for a physical asset equals interest cost plus storage costs less convenience yield, that is, Carry costs Cost of funds storage cost convenience yield (1a) For a financial asset such as a stock or a bond,storage costs are negligible.Moreover, income (yield)accrues in the form of quarterly cash dividends or semi-annual coupon payments.The cost of carry for a financial asset is Carry costs Cost of funds income. (1b) Carry costs and benefits are modeled either as continuous rates or as discrete flows.Some costs/benefits such as the cost of funds (i.e.,the risk-free interest rate)are best modeled continuously.The dividend yield on a broadly-based stock portfolio and the interest income on a foreign currency deposit also fall into this category.Other costs/benefits like quarterly cash dividends on individual common stocks,semi-annual coupons on bonds,and warehouse rent payments for holding an inventory of grain are best modeled as discrete cash flows.In the interest of brevity,only continuous costs are considered here.13 Dividend income from holding a broadly-based stock index portfolio or interest income from holding a foreign currency is typically modeled as a constant,continuous 3 For a detailed discussion of the ways in which carrying costs can be modeled,see Whaley(2002) 10
The cost of carry refers to the difference between the costs and the benefits that accrue while holding an asset. Suppose a breakfast cereal producer needs 5,000 bushels of wheat for processing in two months. To lock in the price of the wheat today, he can buy it and carry it for two months. One cost of this strategy is the opportunity cost of funds. To come up with the purchase price, he must either borrow money or reduce his earning assets by that amount. Beyond interest cost, however, carry costs vary depending upon the nature of the asset. For a physical asset such as wheat, he incurs storage costs (e.g., rent and insurance). At the same time, by storing wheat, he avoids the costs of possibly running out of his regular inventory before two months are up and having to pay extra for emergency deliveries. This benefit is called convenience yield. Thus, the cost of carry for a physical asset equals interest cost plus storage costs less convenience yield, that is, Carry costs = Cost of funds + storage cost − convenience yield . (1a) For a financial asset such as a stock or a bond, storage costs are negligible. Moreover, income (yield) accrues in the form of quarterly cash dividends or semi-annual coupon payments. The cost of carry for a financial asset is Carry costs = Cost of funds − income . (1b) Carry costs and benefits are modeled either as continuous rates or as discrete flows. Some costs/benefits such as the cost of funds (i.e., the risk-free interest rate) are best modeled continuously. The dividend yield on a broadly-based stock portfolio and the interest income on a foreign currency deposit also fall into this category. Other costs/benefits like quarterly cash dividends on individual common stocks, semi-annual coupons on bonds, and warehouse rent payments for holding an inventory of grain are best modeled as discrete cash flows. In the interest of brevity, only continuous costs are considered here.13 Dividend income from holding a broadly-based stock index portfolio or interest income from holding a foreign currency is typically modeled as a constant, continuous 13 For a detailed discussion of the ways in which carrying costs can be modeled, see Whaley (2002). 10
rate.4 The income,as it accrues,is re-invested in more units of the asset.In this way, buying e units of a stock index portfolio today grows to exactly one unit at time T,and produces a net terminal value of S-Se.The cost of carry rate equals the difference between the risk-free rate of interest r and the dividend yield rate i for a stock index portfolio investment,and equals the difference between the domestic interest rate rand the foreign interest rate i for a foreign currency investment.The total cost of carry paid at time Tis Carry costs=Sle-1]. (2) 2.2 Valuing forward/futures using the no-arbitrage principle The value of a forward contract is inextricably linked to the cost of carry of the underlying asset.Since a forward contract requires its buyer to accept delivery of the underlying asset at time T,buying a forward contract today is a perfect substitute for buying the asset today and carrying it until time T.The present value of the payment obligation under the forward contract strategy is f,and the present value of the latter strategy is Se.Since both strategies provide exactly one unit of the asset at time T, (i.e.,S),their costs must be identical, fe-rT=Se-i (3a) If the relation(3a)does not hold,costless arbitrage profits would be possible by selling the over-priced instrument and simultaneously buying the under-priced one.The relation (3a)is the present value version of the cost of carry relation.A more familiar version is the future value form, f=Se(r-ir (3b) When the prices of the forward and the asset are such that(3a)and/or(3b)hold exactly, the forward market is said to be at full carry.Unless costless arbitrage is somehow impeded,the forward market will always be at full carry.The difference between the forward(or futures)price and the asset price is frequently referred to as the basis. 14 The carry cost rates within this framework are deterministic.In the short-run,this assumption is 11
rate.14 The income, as it accrues, is re-invested in more units of the asset. In this way, buying units of a stock index portfolio today grows to exactly one unit at time T, and produces a net terminal value of . The cost of carry rate equals the difference between the risk-free rate of interest r and the dividend yield rate i for a stock index portfolio investment, and equals the difference between the domestic interest rate r and the foreign interest rate i for a foreign currency investment. The total cost of carry paid at time T is iT e− (r i)T T S Se − − ( ) Carry costs = [ 1] r i T S e − − . (2) 2.2 Valuing forward/futures using the no-arbitrage principle The value of a forward contract is inextricably linked to the cost of carry of the underlying asset. Since a forward contract requires its buyer to accept delivery of the underlying asset at time T, buying a forward contract today is a perfect substitute for buying the asset today and carrying it until time T. The present value of the payment obligation under the forward contract strategy is rT fe− , and the present value of the latter strategy is . Since both strategies provide exactly one unit of the asset at time T, (i.e., ), their costs must be identical, iT Se− ST rT iT fe Se − − = . (3a) If the relation (3a) does not hold, costless arbitrage profits would be possible by selling the over-priced instrument and simultaneously buying the under-priced one. The relation (3a) is the present value version of the cost of carry relation. A more familiar version is the future value form, (r i)T f Se − = . (3b) When the prices of the forward and the asset are such that (3a) and/or (3b) hold exactly, the forward market is said to be at full carry. Unless costless arbitrage is somehow impeded, the forward market will always be at full carry. The difference between the forward (or futures) price and the asset price is frequently referred to as the basis. 14 The carry cost rates within this framework are deterministic. In the short-run, this assumption is 11
Futures contracts are like forward contracts,except that price movements are marked-to-market each day rather than receiving a single,once-and-for-all settlement on the contract's expiration day.5 Obviously,the sum of the daily mark-to-market price moves over the life of the futures equals the overall price movement of a forward with the same maturity.With the futures position,however,the mark-to-market profits(losses)are invested(carried)at the risk-free interest rate until the futures expires.The value of the futures position at time T,therefore,may be greater or less than the terminal value of the forward position,depending on the path that futures price follows over the life of the contract. Cox,Ingersoll and Ross(1981)(hereafter "CIR")and Jarrow and Oldfield(1981), among others,use no-arbitrage arguments to show the equivalence of forward and futures prices when interest rates are deterministic.To illustrate their argument,assume that the term structure of interest rates is flat and does not change through time.Also,assume that r is the continuously compounded interest rate on a daily basis.Now,consider a "rollover"futures position that begins,on day 0,with e futures contracts and that increases the number of futures each day by a factor e'.At the end of day 1,the position is marked-to-market,generating proceeds of e(-F).Assuming this gain/loss is carried forward until day T,the terminal gain/loss will be e(-Fe=F-F For day 2,the position is increased by a factor e'and is marked-to-market at e-(T-2(-F),generating proceeds of e-(T-2()e(-2)=F-F on day T;and so on.Because the number of futures is chosen to exactly offset the accumulated interest factor on the daily mark-to-market gain/loss,the rollover futures position has exactly the same terminal value as the long forward position.Under the no-arbitrage assumption,the valuation equation for a futures contract is the same as that of the forward,that is, F=f=Se(r-i)T (4) CIR also use no-arbitrage arguments to show the relation between forward and futures prices when interest rates are stochastic.They find that the futures price will be reasonable for most type of assets. 15 Recall that,in Section 1,we discussed the historical fact that the CBTchanged from making marketsin forward contracts to making markets in futures contracts in 1865 as a means of ensuring market integrity. 12
Futures contracts are like forward contracts, except that price movements are marked-to-market each day rather than receiving a single, once-and-for-all settlement on the contract’s expiration day.15 Obviously, the sum of the daily mark-to-market price moves over the life of the futures equals the overall price movement of a forward with the same maturity. With the futures position, however, the mark-to-market profits (losses) are invested (carried) at the risk-free interest rate until the futures expires. The value of the futures position at time T, therefore, may be greater or less than the terminal value of the forward position, depending on the path that futures price follows over the life of the contract. Cox, Ingersoll and Ross (1981) (hereafter “CIR”) and Jarrow and Oldfield (1981), among others, use no-arbitrage arguments to show the equivalence of forward and futures prices when interest rates are deterministic. To illustrate their argument, assume that the term structure of interest rates is flat and does not change through time. Also, assume that r is the continuously compounded interest rate on a daily basis. Now, consider a “rollover” futures position that begins, on day 0, with r T( 1) e− − F − F futures contracts and that increases the number of futures each day by a factor e . At the end of day 1, the position is marked-to-market, generating proceeds of e r ( − − r T( ) ~ ) 1 r T 1 . Assuming this gain/loss is carried forward until day T, the terminal gain/loss will be e F F e F − − r T− − = ( ) ( ) ( − F ~ ) 1 ~ 1 1 1 . For day 2, the position is increased by a factor e and is marked-to-market at r e F − − r T − ( ) ( F ~ ~ ) 2 2 1 , generating proceeds of e F − − r T( ) ( F e r T − F F − = − ~ ~ ( ) ) 2 ~ ~ 2 1 2 2 1 on day T, and so on. Because the number of futures is chosen to exactly offset the accumulated interest factor on the daily mark-to-market gain/loss, the rollover futures position has exactly the same terminal value as the long forward position. Under the no-arbitrage assumption, the valuation equation for a futures contract is the same as that of the forward, that is, (r i)T F f Se − = = . (4) CIR also use no-arbitrage arguments to show the relation between forward and futures prices when interest rates are stochastic. They find that the futures price will be reasonable for most type of assets. 15 Recall that, in Section 1, we discussed the historical fact that the CBT changed from making markets in forward contracts to making markets in futures contracts in 1865 as a means of ensuring market integrity. 12
less(greater)than the forward price if(a)the price changes of the futures contract and the default-free discount bond are positively(negatively)correlated and/or(b)the variance of bond price changes is less than(exceeds)the covariance between spot price changes and bond price changes.They also show that if the covariance between spot price changes and bond price changes is positive,the futures price is less than the forward price,and the futures-forward price difference is a decreasing function of the expected forward-bond covariance. 2.3.Valuing options using the no-arbitrage principle The no-arbitrage pricing results for options come in two primary forms-lower price bounds and put-call parity conditions.Each is discussed in turn. 2.3.1 Call options The lower price bound of a European-style call option is c≥max(0,Ser-XerT), (5) where c is the price of a European-style call with exercise price X and time to expiration T.The price of the call must be greater than or equal to zero since it is a privilege.The reason the call price must exceed Se-Xe-is based on the following no-arbitrage argument.Suppose a portfolio is formed by selling e units of the underlying asset and buying a European-style call.To ensure that enough cash is on hand to exercise the call at expiration,Xe in risk-free securities are also purchased.At time T,the net value of the portfolio depends on whether the asset price is above or below the exercise price.If the asset price is below the exercise price,the call expires worthless.The risk-free securities (plus accrued interest)are used to buy a unit of the asset to cover the short sale obligation.A cash balance of Y-remains.If the asset price is greater than the exercise price on day T,the call will be exercised.This requires a cash payment of which can be made exactly using the risk-free securities.The unit of the asset received upon exercising the call is used to retire the short sale obligation.Thus,if S,>Y,the net terminal value of the portfolio is certain to be 0.Considering both possible outcomes,this portfolio is certain to have a net terminal value of at least 0.This means that its initial 13
less (greater) than the forward price if (a) the price changes of the futures contract and the default-free discount bond are positively (negatively) correlated and/or (b) the variance of bond price changes is less than (exceeds) the covariance between spot price changes and bond price changes. They also show that if the covariance between spot price changes and bond price changes is positive, the futures price is less than the forward price, and the futures-forward price difference is a decreasing function of the expected forward-bond covariance. 2.3. Valuing options using the no-arbitrage principle The no-arbitrage pricing results for options come in two primary forms—lower price bounds and put-call parity conditions. Each is discussed in turn. 2.3.1 Call options The lower price bound of a European-style call option is max(0, ) iT rT c Se Xe − − ≥ − , (5) where c is the price of a European-style call with exercise price X and time to expiration T. The price of the call must be greater than or equal to zero since it is a privilege. The reason the call price must exceed iT rT Se Xe − − − is based on the following no-arbitrage argument. Suppose a portfolio is formed by selling iT e− units of the underlying asset and buying a European-style call. To ensure that enough cash is on hand to exercise the call at expiration, Xe−rT in risk-free securities are also purchased. At time T, the net value of the portfolio depends on whether the asset price is above or below the exercise price. If the asset price is below the exercise price, the call expires worthless. The risk-free securities (plus accrued interest) are used to buy a unit of the asset to cover the short sale obligation. A cash balance of X ST − ~ remains. If the asset price is greater than the exercise price on day T, the call will be exercised. This requires a cash payment of X, which can be made exactly using the risk-free securities. The unit of the asset received upon exercising the call is used to retire the short sale obligation. Thus, if , the net terminal value of the portfolio is certain to be 0. Considering both possible outcomes, this portfolio is certain to have a net terminal value of at least 0. This means that its initial T S > X 13
value must be less than or equal to 0,otherwise a costless arbitrage opportunity would exist.With Se--Ye--cs0,the lower price bound or a European-style call is c≥Ser-XerT」 In general,the lower price bound of an option is called its intrinsic value,and the difference between the option's market price and its intrinsic value is called its time value.A European-style call has an intrinsic value of max(0,Se-Ye)and a time value of c-max(0,Se-Xe).No-arbitrage principle identifies the intrinsic value of an option.The determinants of time value are the focus of Section 3. American-style options are like European-style options except that they can be exercised at any time up to and including the expiration day.Since this additional right cannot have a negative value,the relation between the prices of American-style and European-style call options is Czc, (6) where the upper case C represents the price of an American-style call option with the same exercise price and time to expiration and on the same underlying asset as the European-style call.The lower price bound of an American-style call option is C2 max(0,Se-i-Ye-T,S-X). (7) This is the same as(5),except that the term S-X is added within the maximum value operator on the right-hand side since the American-style call cannot sell for less than its early exercise proceeds,S-X.If C<S-X,a costless arbitrage profit of S-X-C can be earned by simultaneously buying the call (and exercising it)and selling the asset. The structure of the lower price bound of the American-style call (7)provides important insight regarding the motivation (or lack thereof)for early exercise.The second term in the parentheses,Se-i-e-,is the minimum price at which the call can be sold in the marketplace.6 The third term is the value of the American-style if it is 6Toexita ong position inan American-style call option,you have three ateratives.First,you can hold it to expiration,at which time you will (a)let it expire worthless if it is out of the money or(b)exercise it if it is in the money.Second,you can exercise it immediately,receiving the difference between the current 14
value must be less than or equal to 0, otherwise a costless arbitrage opportunity would exist. With , the lower price bound or a European-style call is . 0 iT rT Se Xe c − − − − ≤ rT Xe − − max(0, ) iT rT c Se X − − − − iT c S ≥ − e S − X In general, the lower price bound of an option is called its intrinsic value, and the difference between the option’s market price and its intrinsic value is called its time value. A European-style call has an intrinsic value of and a time value of . No-arbitrage principle identifies the intrinsic value of an option. The determinants of time value are the focus of Section 3. max(0, ) iT rT Se Xe − − − e American-style options are like European-style options except that they can be exercised at any time up to and including the expiration day. Since this additional right cannot have a negative value, the relation between the prices of American-style and European-style call options is C ≥ c , (6) where the upper case C represents the price of an American-style call option with the same exercise price and time to expiration and on the same underlying asset as the European-style call. The lower price bound of an American-style call option is max(0, , ) iT rT C Se Xe S − − ≥ − − X . (7) This is the same as (5), except that the term S − X is added within the maximum value operator on the right-hand side since the American-style call cannot sell for less than its early exercise proceeds, . If C < S − X , a costless arbitrage profit of S − X − C can be earned by simultaneously buying the call (and exercising it) and selling the asset. The structure of the lower price bound of the American-style call (7) provides important insight regarding the motivation (or lack thereof) for early exercise. The second term in the parentheses, Se iT rT Xe − − − , is the minimum price at which the call can be sold in the marketplace.16 The third term is the value of the American-style if it is 16 To exit a long position in an American-style call option, you have three alternatives. First, you can hold it to expiration, at which time you will (a) let it expire worthless if it is out of the money or (b) exercise it if it is in the money. Second, you can exercise it immediately, receiving the difference between the current 14