Derivative securities Forwards A forward contract is a sale transaction, which is consummated in the future, but with all details of the transaction specified in the present. The time at which the contract settles is called the expiration date. The asset (often a commodity) on which the forward is based is called the underlying. A forward contract generally requires no upfront payment. It should be noted that both sides of a forward are exposed to a significant risk of nonperformance of the other party (i.e., credit risk), and for that reason, forwards are rarely used in practice, and even if they are used, they are private transactions, not traded on an exchange. Forwards are negotiated and customized for specific parties. They are created on commodities, financial assets, indices, or currency exchange rates. They are arranged by brokers or dealers, who make money on the spread. Note that the price paid in the market for a given asset is called the spot price (as opposed to the price named in the forward contract, the forward price). The person buying the underlying in a forward contract is said to be long forward, and the person selling the underlying is said to be short forward. We have the following Payoff to long forward = Spot price at expiration Forward price Payoff to short forward = Forward price Spot price at expiration By analyzing the payoffs of a forward we can also conclude that Long forward = Underlying PV(Forward Price) Forward price (price at which the contract is agreed upon) depends on the spot price of the underlying, and converges to it as the contract nears maturity (otherwise arbitrage opportunities would be present). Consider a forward on an n-year zero-coupon bond, for the time k in the future. Let us write F( 0, k) for the forward price. Let P( n + k) be the price of a risk-free zero-coupon bond maturing at time n + k, and let P( k) be the price of a risk-free zero coupon bond maturing at time k. This case does not seem to allow for a straightforward application of the formula above. However, we have the following: PV(Forward Price) = PV( F( 0,k) ) = F( 0, k)! P( k). The underlying in this case is a zero-coupon bond maturing at time n + k, issued at time n, which effectively is a unit monetary amount paid at time n + k, so that its present value is P( n + k). Because we assume that a long forward, with Long forward = Underlying PV(Forward Price), is costless to enter, we must have 0 = P( n + k)! F( 0,k) " P( k), so that ( ) = P( n + k) P( k) F 0, k. You also can show this by using a no-arbitrage argument (assume all maturity values are units):
- Buy a k-year zero-coupon bond and then buy a forward contract to purchase an n-year bond k years from now; you are thus entitled to a unit payment at time n + k. - Purchase an (n + k)-year zero coupon bond now with the same unit terminal value. - Since these strategies would produce identical payoffs, they must cost the same: P( k)! F( 0,k) = P( n + k), or F 0,k Note that the key formula ( ) = P( n + k) P( k) F 0, k ( ) = P( n + k) P( k) effectively says that the forward price is the current price of the underlying accumulated to maturity, at a risk free interest force of interest r (credit risk is ignored in this model). General relationship between spot and forward prices Let S be the current spot price at time. Then the relationship between spot and forward prices is: F( t) = S! e rt " D( t), where F( t) is the futures price to be paid at time t agreed upon at time 0, D( t) is the cash flows produced by the asset from time 0 to time t, accumulated to time t, and r is the risk-free force of interest. This relationship is known as the forward-spot parity (it also applies to futures). If the cash flow is a dividend, and the dividend is payable at a continuous fashion, with the force of interest! for the rate of dividend payment, then F( t) = ( S! e "#t )! e rt. For commodities, the underlying does not produce income, but it requires the cost of carry (the total cost of carrying, i.e. storing, transporting etc.), and if that cost is expressed as a continuously compounded annualized rate c, we obtain this relationship F t ( )! e rt. ( ) = S! e ct This relationship is known as the cost of carry relation between forward (also applies to futures, see below) and spot prices. A futures contract is basically like a forward, but with a lot of things added in order to make it tradable on the exchange, and eliminate the counterparty credit risk. Futures are exchange-traded and are standardized with regard to maturity, size, and the underlying asset. Futures are also marked-to-market each day (with cash flows required from all parties): this means that each side of the trade must deposit a certain required margin and the margin must be sufficient for the position held not just at the beginning, but each day the position is held. Also, the exchange may impose a price limit on a daily movement in the futures price (once that limit is reached, trading is halted for a specified period of time). While there may be some slight differences between prices of forwards and futures in practice, due to margin requirement and intermediate cash flows in future contracts, we will generally assume that pricing formulas for futures are the same as pricing formulas.
for forwards that we developed above. We will discuss possible divergence between forward price and futures price later. Forwards versus futures overview: Security feature Forward Futures Type of market Dealer or broker Exchange Liquidity Low (almost zero) Very high Contract form Custom Standard Performance guarantee Creditworthiness Mark-to-market Transaction costs Bid-ask spread Fees or commission Because futures are usually traded for hedging or speculation, without actual intent of buying or selling the underlying, many futures contracts are settled in cash, without the delivery of the underlying. Stock index futures, for example for the S&P 500 index, are always settled in cash. Hedging spot prices with futures Assume a company knows now that it will want to issue bonds at a later date and it is afraid that rates will rise between now and that date, forcing it into higher cost of debt. The company can hedge its position by holding a short position in Treasury-Bond futures: - If rates go up, it will be forced to issue debt at a higher rate but this loss will be offset by a gain from the short futures position. - If rates go down, it can issue debt at a lower rate, but this gain will be offset by a loss from the short futures position. - This kind of a hedge is called a cross hedge, because the underlying of the hedge (futures) is different than the position hedged. As a result the company may still be assuming basis risk: the risk of divergence between the hedge underlying and the security hedged; even if Treasury rates remain level, the company may be forced into higher rates if corporate bonds spreads widen. Stock index futures No futures are written on the DJIA (Dow Jones Industrial Average), but there are futures contracts on the S&P 500, the NYSE composite, the MMI (Major Market Index, closely correlated with DJIA), and the Nikkei (Tokyo market index). MMI futures contracts sell for 250 times the index value. Suppose you want to hedge a portfolio of X dollars of a diversified portfolio of stocks of large companies (commonly called blue chips). You can t use DJIA futures, but MMI is available. You can hedge the portfolio by shorting n futures contracts, where n is obtained from n! 250! S = X where S is the spot index level. This creates a hedge: if the index rises, the loss on the futures contracts is offset by a gain in the portfolio, and if the index falls, the loss on the portfolio is offset by a gain in the futures contracts. In general, for index futures, F( t) = S! e ( r "# )t, where! is the dividend yield of the index. This shows us that when the dividend yield is greater than the risk-free rate, the futures price will be less than the current index price. Exercise 2.1
Assume it is now June 30, 2002, and a Treasury Bill maturing September 30, 2002 with $10,000 face value is selling for $9,955.20. Current spot price for gold is $315 per ounce. What is the price of the futures contract for gold with September 30, 2002 delivery? Assume that there is neither any cost of storage for gold, nor any convenience yield to owning it. Solution. The information about the Treasury Bill gives us information about the risk-free interest rate. Since gold does not pay any dividends, all we have is this relationship between spot and futures price (spot-futures parity): F = Se r! = $315 " 10,000.00 9955.20 = $316.42. Options In general, an option is a contract in which one side acquires the right to buy (or sell, but only one of these two) the underlying at a predetermined price (might be a function of something, but the conditions are stated in advance) at a predetermined time or by predetermined time. In hedging with forwards and futures, upside potential must be sacrificed in order to receive downside protection. When hedging with options, this is not the case. Options have asymmetrical payoff patterns such that they only pay off when the index/security/commodity price moves in a specific direction. In an option contract, the long side had the right, but not the obligation, to purchase or sell (depending on the option) a security at a specified price. The other (short) side must be the counterparty and provide the market for the right of the long side. This sounds good for the long side, bad for the short side, but the long side must pay an up-front premium to the short side. Option terminology - American options are exercisable at any time prior to expiration. - European options can be exercised only at maturity. - Bermuda options can only be exercised during prescribed periods before maturity, but not the entire period from now till maturity. - A call option gives the long side the right to buy the underlying at a fixed price, called the strike price, or exercise price. - A put option gives the long side the right to sell the underlying at a fixed price, called the strike price, or exercise price. Remember that every right (to buy or to sell) of the long side originates from the obligation of the short side to accommodate the right granted to the long side. The process of creation an option by the short side is called option writing. Let us now assume that the underlying is a stock. If S t is the stock price at the time of expiration of a European option, and K is the strike price, then the following table summarizes the long side payoffs of call and put options: S t < K S t = K S t > K Call Option Payoff 0 0 S t! K
Put Option Payoff K! S t 0 0 Equivalently (for the long side), Call option payoff = max S t! K,0 Put option payoff = max K! S t,0 ( ) = ( S t! K ) +. ( ) = ( K! S t ) +. The above are true for American options, but the stock price need not be the price at expiration of the option; it can be the price at any time until the expiration. The value that an option has if it were exercised instantly is known as its intrinsic value. Before its expiration, the option may have a price higher than intrinsic value, and the difference between the two is called the time value of the option. An option whose intrinsic value is positive is said to be in-the-money. If the underlying is trading at exactly the exercise price, then the option is said to be at-the-money. An option that is at-the-money has intrinsic value of zero, but the opposite is not necessarily true. If an option has intrinsic value of zero, but it is not at-the-money, we say that the option is out-of-the-money. Options generally have positive time value before expiration. This is, in fact, a must for an American option. The price of American options can never be less than their intrinsic value, since arbitrage opportunities would exist otherwise (someone could buy the American option and immediately exercise it, profiting a gain). What happens if an option is issued on a stock that pays a dividend? In a sense, nothing. The dividend has no effectively no influence on the situation of the parties involved in the option trade. The only influence of the dividend is that affected on the stock price itself. When a dividend is paid, stock price is reduced by the amount of the dividend. Effectively, before the time of the payment, the stock trades with a dividend, and after that, without a dividend. But the price that applies to the option contract is the price of the stock, either the one with the dividend, or without it. Exercise 2.2 Mr. Romuald Carcosheek writes a six-month put option on the MIDWIG (an index of the Warsaw Stock Exchange in Warsaw, Poland) index with the exercise price of 2750. In order to be able to write options, he must put down a margin deposit of 1000. He does not receive interest on his margin deposit. The put option he writes on MIDWIG sells for a premium of 75. In six months, MIDWIG index stands at 2700. Assuming that Mr. Carcosheek earns interest on the option premium at a nominal annual rate of 4% compounded semiannually, and assuming that at option expiration he buys the index at 2750 from the long side of the option contract and sells it immediately in the market, calculate the effective annual rate of return Mr. Carcosheek will have earned over the six month period. Solution. Mr. Carcosheek puts down an investment in the amount of 1000. He receives the option premium of 75, and that premium, with interest, after six months is worth
" 75! 1+ 0.04 % 2!1 2 # $ 2 & ' = 75!1.02 = 76.50. When he buys MIDWIG for 2750 and sells it for 2700, he suffers a loss of 50. Thus his net cash flow at the end of the six-month period is 26.50, and his effective annual rate of return earned is 1+ 26.50 2! $ " # 1000 % & ' 1 ( 5.37%. Exercise 2.3 Mr. Romuald Carcosheek owns a home that costs 20,000,000 PLN. The house is so well built and so well protected that it can only suffer damage from a fire or an earthquake. Mr. Carcosheek purchases an insurance policy that will pay for the damage to his house due to fire or earthquake any time during the next year, with a 1,000,000 PLN deductible, for a premium of 100,000 PLN. Assuming that the value of the house does not change for any other reason than a fire or an earthquake, describe an option contract that Mr. Carcosheek can use to obtain the same protection as that given by the insurance policy. Solution. Let us write S for the value of the house. The payoff (not counting the insurance premium) of the insurance policy is # 0, if S! 19,000,000, & $ ' = max ( 19000000 " S,0 % 19,000,000 " S, if S < 19,000,000, ( ). This is the same as the payoff of an American put option on the house, with exercise price of K = 19,000,000 and expiration date equal to the last day the insurance policy is valid. Exercise 2.4 Countrybank is entering the Polish market and has decided to offer a new attractive Certificate of Deposit, tied to the WIG index (an index of the Warsaw Stock Exchange in Warsaw, Poland). The certificate is issued for two years, and it promises to pay the full amount deposited plus 70% of the performance of the WIG index over that period. Express the payoff of the certificate in terms of an appropriate option contract. Solution. Assume for simplicity that the amount of the initial deposit K in the certificate is the current value of WIG. Let S stand for the value of the index in two years. In two years, the certificate will be worth $ K + 0.7!( S " K ), if S > K, ' % ( = K + 0.7! max( S " K,0). & K, if S # K, ) Therefore, Countrybank certificate simply pays 70% of the payoff of a European call option on WIG with exercise price of K, which expires when the certificate expires, as the return on the deposit.
3. Insurance Created With Options, Collars, Floors, Caps, Spreads, Straddles As we had already pointed out in an exercise in Section 2, some options positions result in the same payoffs as certain insurance policies. For example, a put option with the exercise price of $50 on a stock selling now for $60 provides insurance against the fall of the price below $50. This insurance provided by a put is sometimes called a floor, as it puts a floor under losses that the long position can suffer. On the other hand, there are situations when an opposite form of insurance is needed. Suppose that an insurance company promises to pay its customer the rate of return on a stock market index, and the customer holds an account valued at $1,000,000, and the current value of the index is 1000. Thus the customer holds 1000 units of the stock market index. In order to protect itself (i.e., insure itself) against a sharp increase in the stock market index (which would result in a corresponding sharp increase in the liability to the customer), the insurance company can buy 1000 calls on a unit of the stock market with the exercise price of 1000. Then any increase in the stock market index above the current value of customer s account would be fully covered by the increase in the value of these calls, and the company s liability would be fully insured. This form of insurance is sometimes called a cap, as it caps the value of the liability of the company. Caps and Floors Caps and floors, however, are most commonly used in the context of insuring against interest rate risk. They provide one-sided protection against movements in a floating interest rate. They are commonly embedded in adjustable rate mortgages in the United States. The cost can either be paid for up-front or embedded in the interest rate paid. An interest rate cap consists of a series of caplets, and an interest rate floor consists of a series of floorlets. An interest rate caplet is analogous to a call option on the level of interest rates: at expiration, if the interest rate is above the strike rate, the caplet pays ( i! k) times the notional, and zero otherwise (i is the index rate, k is the strike rate). An interest rate floorlet is analogous to a put option on the level of interest rates; at expiration, if the interest rate is below the strike rate, the caplet pays ( k! i) times the notional, and zero otherwise (i is the index rate, k is the strike rate). An interest rate collar is a long position in a cap plus a short position in a floor; it effectively makes a payment to its holder whenever the index rate is outside the boundaries set by the strike rates (the floor rate is below the cap rate, and the range between them leaves the floating rate alone). Covered and naked option writing If you own the underlying and write an option, this is called covered writing. If you do not own the underlying, and write an option, this is called naked writing. Exercise 3.1
Mr. Romuald Carcosheek owns 100,000 shares of Megabank, a Polish bank whose shares are traded on the Warsaw Stock Exchange. Mr. Carcosheek is concerned that the current price of PLN 50 of the Megabank stock is too high, and it is likely to decline to PLN 45 range. He decides to protect himself, at least partially, by writing at-the-money calls on his shares, which currently sell for 1 PLN per share. He is able to invest the premium received from writing these calls in a risk-free one-year Polish government bond earning 6%. In one year, the shares of Megabank are selling for PLN 47. Calculate Mr. Carcosheek s total rate of return over that one-year period, including his unrealized loss on the shares, premiums received for options written, and interest earned on the premiums invested in a risk-free one-year Polish government bond. Solution. Mr. Carcosheek begins the year with 100,000 shares of Megabank worth PLN 50 each, for a total initial investment of PLN 5,000,000. By writing covered calls, he earns PLN 100,000, which accumulate to PLN 106,000 by the end of the year by being invested in a government bond. But at the end of the year, his 100,000 shares are worth only PLN 4,700,000, so that his total amount at the end of the year is PLN 4,806,000. His rate of return is 4,806,000 5,000,000! 1 =!3.88%. While this may look bad, note that the shares declined by 6%, so was able to cushion his loss by utilizing the strategy of writing covered calls. An important observation to make concerning European calls and puts is that if a European call and a European put have the same underlying, same maturity date, and the same exercise price, then Call! Put = Forward. In other words, a investor who buys a European call with exercise price K and writes a European put on the same underlying, same maturity date, and the same exercise price, will experience exactly the same payoff as an investor who enters into a long forward position with the same maturity date and the same exercise price K. Short forward position is replicated by a portfolio of a long put and short call (both European). Put-Call Parity Let us make the following assumptions: - No dividends payable on the underlying. - All investors may borrow and lend at the risk-free rate. - There are no transaction fees or taxes. - Short selling and borrowing are allowed, fractional shares may be traded. - There are no arbitrage opportunities. Under these assumptions, consider the following two portfolios: Portfolio 1: Long European call option with maturity date t years from now, plus Ke!rt invested in the risk-free asset. Portfolio 2: Long European put and one share of the underlying. We assume that the call and put have the same exercise price and maturity.
Then the value of these two portfolios is as follows: S T < K S T > K Long call 0 S T! K Ke!rt in risk-free asset K K Total Portfolio 1 K S T Long put K! S T 0 One share of stock S T S T Total Portfolio 2 K S T We see that these two portfolios produce identical payoffs regardless of the price of the underlying. Therefore, these two portfolios must sell for the same price, i.e., c + Ke!rt = S + p where c is the call price, and p is the put price. This relationship is called the put-call parity. Exercise 3.2 The current price, as of June 30, 2002, of a $325 call on September 30, 2002, gold is $12, with the spot price being $315 per ounce, and a three-month Treasury-Bill maturing on September 30, 2002 with $10,000 maturity value selling now for $9,955.20. Find the price of a September 30, 2002, $325 (exercise price) gold put as of June 30, 2002. Assume that all conditions for put-call parity to hold are satisfied. Solution. We use the put-call parity formula c! p = S! PV(K). In this case, c = 12, S = 315, and the present value of a cash flow paid on September 30, 2002, as of June 30, 2002, is established by multiplying that cash flow by 9955.20 10000. We get 12! p = 315! 9955.20 " 325 #!8.54. 10000 This gives us p! 20.54. Exercise 3.3 You are given the following information: - An option market satisfies the condition for put-call parity. - The current underlying security price is 100. - A call option with a strike price of 105 and maturity one year from now has a current price of 4. - A put option with a strike price of 105 and maturity one year from now has a current price of 6. Determine the one-year risk-free interest rate.
Solution. We use the put-call parity relationship c! p = S! PV ( K ). Substituting the data given in the problem we get 4! 6 = 100! 105 1+ i, or 102 = 105 1+ i. The solution is i = 3 102! 2.94%. Spreads, Collars, Straddles An option spread is a position of only calls or only puts, in which some options are purchased and some are written. Here are some of these strategies: A bull spread consists of a long call with a lower exercise price and short call with higher exercise price, both calls expiring at the same time. A bear spread consists of a long call with a higher exercise price and a short call with a lower exercise price, both calls expiring at the same time. A box spread consists of a long call and short put with the same exercise price, and another position of a long put and a short call with the same exercise price, but different than the previous one. This amounts to being long one forward and short another forward. The strategy is purely a means of borrowing or lending money. A ratio spread is constructed by buying by buying m calls at one exercise price and selling n calls at a different exercise price, with same maturity and same underlying. A collar is created by purchasing a put option and writing a call option. A reverse position is a short collar. A collar width is the difference between the call and put strike prices. A zero-cost collar is created by adjusting strike prices so that there is no cost or income to the transaction. A straddle is a position consisting of a long call and a long put with the same exercise price. This position benefits from high volatility. Short straddle benefits from low volatility. A strangle consist of a long out-of-the-money call and a long out-of-the-money put. This is a strategy similar to a straddle, but at a lower cost. A butterfly spread consists of a short straddle combined with a long out-of-the-money put and a long out-of-the-money call. The position could be symmetric or asymmetric. 4. Managing Risk With Derivatives Hedging with forwards or futures A long position in stocks or commodities can be hedged by holding a short position in a forward or in futures. Portfolio insurance You can insurance a minimum value of a portfolio by buying a put option. Insurance by selling a call
Used to hedge the risk of a long portfolio of stocks or commodities. This does not provide complete protection, but reduces risk with premium income. But recall put-call parity: c! p = S! PV ( K ), so that S + p = c + PV ( K ). Thus the long position insured with a put can be replicated by buying a bond and a call option. Why do firms manage risk? Financial risk management should be viewed from the perspective of the Modigliani- Miller Theorem: financing mode affects value only if it increases firm s output (without increasing costs, or at least not increasing them above the benefit created), reduces taxes, reduces bankruptcy costs, or reduces agency costs. But there may be good reasons not to hedge: Derivatives involve high transaction costs. Hedging requires costly expertise. Hedging increases agency costs (e.g., rogue traders). Hedging increases taxes or other government costs (e.g., regulatory). Cross-Hedging There are situations when hedging is achieved not by a position in a security identical or directly related to the originally position held, but a more remotely related security. This means practically that the movements of prices of the hedged security will not be mimicked by the hedge, but will only be related somehow. Such a situation is best modeled with a probability-based model, and probability is officially not used in the Course FM/2, but since the reasoning involved in this model and the resulting formula are simple, you should learn the formula. Suppose that an investor holds Q units of a security whose price is S and hedges it with a short position in H units of a security whose price is F, with! S being the standard deviation of the changes in price of a unit of the hedged instrument,! F being the standard deviation in the changes in price of the hedging instrument, and! being the correlation of the price changes of the two instruments. Then the investor s total position is! = QS " HF. The variance of this position is Var! ( ) = Var ( QS " HF) = Q 2 # 2 S + H 2 # 2 F " 2QH $# S # F. We want to derive the value of H, which minimizes this variance. Note that the value of Q is given. We take the derivative of Var! Note that ( )!H!Var " = 2H# F 2 $ 2Q%# S # F = 0. ( ) with respect to H and set it equal to 0
! 2 Var (") = 2# 2!H 2 F > 0. The value of H at which the derivative of Var! H = Q!" S " F, produces the lowest variance portfolio. Also, ( ) with respect to H is equal to 0, i.e., h = H Q =!" S " F is called the optimal hedge ratio or minimum variance hedge ratio. Consider a company that is hired to repair roads in a small town every year. The materials used in the process of road repair are not traded in financial markets, but they are produced from crude oil, an asset for which futures contracts are readily available. Suppose that this company is paid P r per kilometer of a road repair, and that N c barrels of crude oil are needed to repair N r kilometers of roads. Let P c be the crude oil price per barrel. Then this company s profit (we ignore other possible expenses and sources of profits for now) is! = N r " P r # N c " P c. Now suppose that this company enters into H futures contracts on crude oil, assuming for simplicity that each contract covers one barrel. If F is the futures price, the profit on such a hedged position is! H = N r " P r # N c " P c + H ( P c # F). Now suppose that the objective is to minimize the variance of the hedged position. The variance of the hedged position is (we make a simplifying assumption that F does not vary, as we are doing this calculation for this moment in time, based on historical estimates of volatility and correlations) 2! " H = N 2 r #! 2 Pr + ( H $ N c ) 2 #! 2 Pc + 2( H $ N c ) # N r # Cov( P r, P c ). The variance-minimizing hedge ratio is ( ) H * = N c! N r " Cov P r,p c. 2 # c In this hedge ratio, we can interpret the first term as hedging costs, and the second term as hedging revenue. The coefficient that N r is multiplied by is the result of regression of the price of a kilometer of a road on the price of a barrel of oil. The resulting minimum variance is 2! " H* = N 2 r #! 2 2 Pr #( 1$ % Pr,P c ). This positive variance indicates that there is risk left in this position, due to possible divergence of the revenue earned from the road and the price of oil. This kind of risk due to divergence of the value of the portfolio hedged and the value of the hedge used is called the basis risk. 5. More on Forwards and Futures
Alternative ways to buy a stock Buy the stock directly, for cash. Fully leveraged purchase: borrow all money used to buy the stock. Prepaid forward contract: pay for the stock today, receive it at time t in the future. Forward contract. Prepaid forward contract In the absence of dividends, the price on the prepaid forward contract is today s stock price S. The reason is simple: you will own the stock anyway, so you should pay its market price. We will denote the prepaid forward price by F P 0,t. If we use a subscript in the stock price notation to denote the time of the price, we have F P 0,t = S 0. If the prepaid forward is on a stock that pays a dividend, then the forward contract holder does not receive that dividend, and the prepaid forward price is the price of the stock without any of the dividends paid through time t. Forward contract Let us denote by F 0,t the forward price for transaction to occur at time t. As we had already noted before, F 0,t = S 0! e rt, where r is the risk-free force of interest between time 0 and time t. If the stock pays a dividend, the price of the stock should be used without any of the dividends payable between time 0 and time t. The forward premium is defined as the ratio F 0,t. The annualized forward premium is 1 S 0 t! ln " F 0,t % # $ S 0 & ', and, in the absence of arbitrage opportunities, it equals the risk-free force of interest (or the difference between the risk-free force and the force of dividend). Note that the payoff of a long forward position is S t! F 0,t. This payoff can be replicated by buying 1 share of stock with borrowed funds of e!rt " F 0,t. This shows again that F 0,t = S 0! e rt. It also illustrates the fact that Forward = Stock Risk-Free Zero-Coupon Bond in the amount e!rt " F 0,t, and Stock = Forward + Risk-Free Zero-Coupon Bond in the amount e!rt " F 0,t, and Risk-Free Zero-Coupon Bond in the amount e!rt " F 0,t = Stock Forward. Note that given the market prices of a stock and of a forward on that stock, we get the implied risk-free interest rate from the last identity. That implied interest rate is called the implied repo rate. A transaction in which you buy the underlying and short the offsetting forward is called a cash-and-carry. As we see from the above, this is equivalent to a purchase of a risk-free zero-coupon bond. Market makers in forwards often offset their position by buying the underlying, and creating the cash-and-carry position. A reverse cash-and-carry is created by being short underlying and long forward, and is equivalent to borrowing at the risk-
free rate. We should note that the rate on cash-and-carry is the risk-free rate only if the forward is prices by the familiar equation F 0,t = S 0! e rt. Otherwise, arbitrage can be utilized to earn riskless profit. In practice, however, an attempt to arbitrage the difference between F 0,t and S 0! e rt involve transaction costs. Suppose that the spot bid and ask prices are S b < S a and that the forward bid and ask prices are F b < F a. Assume also that there is a transaction cost k for a transaction in the stock or its forward, and that the force of interest for borrowing and lending differs, with r b > r l. We assume all transaction costs to occur at time 0, and none at time t. Consider an arbitrageur who sells the forward and buys the stock (assume no dividends for simplicity, or consider the stock without the dividend). The arbitrageur will have an upfront cash cost of k plus ( S a + k). We assume he/she borrows to finance that position. At time t the payoff is ( ) " e rb t S a + 2k!## "## $ + F 0,t! S!# " $# + S t t = F 0,t! S a + 2k!## "## $. Repayment of borrowing ( ) Value of short forward ( ) " e rb t Denoted by F + The quantity F + is the upper bound for current forward price so that the arbitrage is not profitable. In the same fashion, the lower bound F! below which arbitrage is profitable is given by the formula F! = S b! 2k actually underestimate all costs involved. ( ) " e r l "t. It should be noted that this analysis may Quasi-arbitrage is a substitution of a lower-yielding position by a higher-yielding position. If a company can borrow at 8.5% and lend at 7.5%, clearly there is no arbitrage possible. But if a company is already lending at 7.5%, and it is possible to arrange for a cash-and-carry with implied repo rate of 8%, an arbitrage becomes possible. Does the forward price predict the future price? When you invest in stock, you expect to earn the risk-free rate plus the risk premium. But when buy enter into a forward, you put down no money, so you should not get the riskfree rate, just the risk premium (as you are still exposed to the risk of the underlying). Consider a one-year forward. Let r be the risk-free force of interest for that year. Let! be the expected force of return of the stock. Then we have F 0,1 = S 0! e r and E( S 1 ) = S 0! e ". This tells us that ( ) = F 0,1! e " #r. E S 1 The expression! " r is the risk-premium for the underlying. This tells us that the price of a forward is a downward (assuming positive risk premium) biased predictor of the future price of the underlying, and the degree of the bias is determined by the riskpremium of the asset. Recall that a forward can be replicating by a long position in the underlying (earning the risk-free rate plus the risk-premium) and a short position in a zero-coupon risk-free bond (earning the risk-free rate). As the risk-free rate is paid, only the risk-premium accrues to the forward.
More on Futures Forward and futures prices may differ. The reason is that with the futures contracts, interest is earned on any mark-to-market proceeds, and lost on any required margin deposits (although sometimes interest may be paid on the margin balance). The required margin during the life of a contract (called the maintenance margin) is generally lower than the initial margin, but is nevertheless required. We will illustrate this in an exercise. Exercise 5.1. Consider a futures contract on the stock market index in a hypothetical country called Cuba Libre, such index being called Viva Cuba Libre, or VCL for short. Suppose that you enter into a long position in this contract with a notional value of 1 million libretas (currency of Cuba Libre). The margin required is 7%, or 70,000 libretas. Each contract corresponds to 1000 units of VCL index, and that 1000 is used as the contract multiplier. Futures are initially trading at 1000 libretas. Calculate the dollar-weighted rate of return over a week on this contract, assuming the following: The maintenance margin requirement is 5%. Risk-free interest rate is 0.02% per day. Money held with broker earns this interest. When additional margin deposit is required, you pay it. When funds become available to withdraw, you take them out immediately. Prices at the close of the days of the week for the futures on the VCL index are: Monday: 990 libretas; Tuesday: 1025 libretas; Wednesday: 1075 libretas; Thursday: 920 libretas; Friday: 1000 libretas. Position is costless to enter into and to close (other than the margin deposit), and it is closed at the end of the week. Compare the net cash flow from the futures contract to that for a costless forward entered at the beginning of the week, for the purchase at the end of the week. Solution. We have the following history of this futures position Futures Price Margin price change balance Week beginning 1000 --- 70000 Monday close 990 10 70000 1.0002 1000 10 = 60014 As of Monday close, maintenance margin required is 5% of 990,000, i.e., 49,500, so that 10,514 libretas are withdrawn and 49,500 libretas remain in the margin account. Tuesday close 1025 35 49500 1.0002 + 1000 35 = 84509.9 As of Tuesday close, maintenance margin required is 5% of 1025,000, i.e., 51,250, so that 33,259.90 libretas are withdrawn and 51,250 libretas remain in the margin account.
Wednesday close 1075 50 51250 1.0002 + 1000 50 = 101260.25 As of Wednesday close, maintenance margin required is 5% of 1075,000, i.e., 53,750, so that 47,510.25 libretas are withdrawn and 53,750 libretas remain in the margin account. Thursday close 920 155 53750 1.0002 1000 155 = 101239.25 As of Thursday close, maintenance margin required is 5% of 920,000, i.e., 46,000, so that 147,239.25 libretas are deposited and 46,000 libretas are in the margin account at the end of the day Thursday. Friday close 1000 80 46000 1.0002 + 1000 80 = 126009.20 As of Friday close, the position is liquidated and the margin balance of 126009.20 libretas is collected. Therefore, the dollar-weigthed rate of return over the trading week (five days) was: 70000 + 10514 + 33259.90 + 47510.25 147239.25 + 126009.20 70000 5 5 10514 4 5 33259.90 3 5 47510.25 2 5 + 147239.25 1 0.1039%. 5 This is a weekly rate of return, corresponding to annualized rate of return of approximately 5.5485%. Let us now compare this to a forward contract. We enter a forward at the beginning of the week. Assuming that transaction is costless, the price at which we set the purchase is the same as the initial futures price, i.e., 1000, if we take the futures-spot parity F = Se δt and the forward-spot parity formula F = Se δt to hold identically for both contracts. Then at the end of the week the price is 1000, and the transaction results in a zero return. This is quite interesting: we have a net positive cash flow of 54.10 from the futures contract, but a zero return from the forward. But the forward return makes sense: over the week, the underlying increased in value by the rate of return equal to the risk-free rate, and by holding a forward you had to give up that return, as you did not have to tie up the money by investing it in the underlying. The futures part is more interesting: you had several cash flows for the week. Those cash flows end up netting to a positive number, 54.10, and this is what created your return. The structure of the futures contract and changes in the price in this problem forces you to adopt the strategy of buying when the price is down, and selling when the price is up, creating a positive return, in a sense, out of nothing. One important practical implication of the fact that in general futures tend to benefit from volatility of the underlying, as in the long run most volatility in on the upside, is that when we hedge with futures, fewer futures contracts than equivalent forwards are commonly used. Nevetheless, most of the time we will assume the same pricing formula for forwards and futures. This is illustrated by the following exercises. Exercise 5.2 Consider a futures contract that calls for delivery of 1000 ounces of gold on July 1, 2006, priced on May 1 of the same year. Suppose that the current quoted spot price of gold is
$680 per ounce, and the current annual risk-free continuously compounded interest rate is 4.75%. Assume that gold carrying cost and convenience yield are zero. What is the current futures price assuming that the fundamental relation between cash and futures prices holds? Solution. The fundamental relation is F = Se rt. We are given that r = 0.0475, t = 2 12, S = $680,000 (1000 oz.) so that F = Se rt 12 = 680,000 e 0.0475 685, 404.70. 2 Exercise 14.5 Suppose that the value of the S&P 500 stock index is currently $1,300. If one year Treasury Bill (zero coupon bond) interest rate is 6%, and the expected dividend yield on the S&P 500 Index is 2%, both of these interest rates expressed as effective annual rates of return, what should the one-year maturity futures price be? Solution. The futures-spot parity relationship, with adjustment for dividends, assuming annual effective risk-free interest rate, and annual effective dividend yield, is (approximately): F 0 = S 0 1 + d ( 1 + r F ). where d is the dividend yield of the underlying. Substituting data given we get: $1, 300 F 0 = 1.06 $1, 350.98. 1.02 You might wonder whether we can use the expected dividend yield in this exercise, if the actual dividend yield is uncertain. If you do so wonder, you are most certainly raising a valid point. This formula assumes that the dividend yield is known with certainty, and in real life applications that would not be the case. One more question about the above is whether the Treasury-Bill rate is appropriate for the risk-free rate used in the calculations above. The problem with the Treasury-Bill rate is that it tends to be relatively lower than other short-term interest rates, and some believe it to be unnaturally so. One possible explanation is that the margin in transactions such as buying on margin, short-selling, option writing, or purchases of futures, can be posted in cash, in which case it generally does not earn interest, or in Treasury-Bills, which earn interest automatically, as they are discounted zero-coupon bonds of short maturity (up to one year). Thus traders may buy up Treasury-Bills, bidding up their prices, and bidding down their yields. For that reason, some prefer using another index of short-term interest rates: London Interbank Offered Rate (LIBOR). LIBOR is calculated by British Bankers Association as an average of rates at which leading banks borrow from each other (or, equivalently, deposit money with each other) in U.S. dollars (on this exam, we only discuss dollar LIBOR, but the index is also calculated for other currencies). The index is calculated separately for maturities of 1 day, 1 week, 2 weeks, 1 month, 2 months, 3 months, and for every number of months until 12 (i.e., one year).
Quanto index futures The Nikkei 225 futures contract traded on the Chicago Mercantile Exchange is quite peculiar. Its values are derived from the Nikkei 225 index, but the currency in which it is expressed is the U.D. dollar, not the Japanese yen. The size of that futures contract is $5 times the numerical reading of the Nikkei 225 index. It is cash-settled based on the opening Osaka quotation of the Nikkei 225 index on the second Friday of the expiration month. A contract of that nature is called a quanto contract: referring to an index in one currency, but traded in another currency. What are the uses of stock index futures? Asset allocation. This can mean switching from stocks to Treasury-Bills or the other way around. But using longer term Treasury-Bond futures together with stock index futures we can also construct allocations between stock index and a bond portfolio. If we own a diversified stock portfolio and short S&P 500 futures ot reduce stock allocation, while going long Treasury-Bond futures, the first transaction converts stocks into T-Bills, and the second one converts T-Bills into T-Bonds. This combined use of futures is called futures overlay. You can also use futures to convert a bond-portfolio manager into a stock portfolio manager. Suppose that you have found a bond portfolio manager who can beat the Treasury-Bond index consistently, but you want to invest in stocks. You can hire that manager, while simultaneously shorting Treasury Bond futures (thus converting the manager to T-Bill plus this manager s outperformance of the index) and going long S&P 500 futures (thus converting T-Bill return into stock return). Cross-hedging. This refers to using a futures contract on a different security than the one we are hedging. It is quite common to hedge diversified stock portfolios with S&P 500 futures, after adjusting for the size of the portfolio in relation to the size of the contract, and for the beta of the portfolio. Currency futures and forwards These instruments are used to hedge against changes in currency exchange rates. The simplest instrument is a currency prepaid forward. Suppose that in one year you want to have 1 liberta. Suppose that the risk-free force of interest in libertas is r For and the riskfree force of interest in U.S. dollars is r Dom. To obtain 1 liberta in one year, we must have e!r For today. Suppose that the exchange rate at time t is E t dollars needed to purchase one liberta. We conclude that in order to assure a purchase of one liberta a year from now we need F P 0,1 = E 0! e "r For dollars. This is the price of a prepaid forward. The currency forward price is therefore F 0,1 = E 0! e "r For! e r Dom. In general, the currency forward price (assuming a costless contract to be entered an time 0 and realized at time t) is ( ). t F 0,t = E 0! e "r For!t! e r Dom!t = E 0! e r Dom "r For Covered interest arbitrage is a transaction consisting of borrowing in domestic currency, lending in a foreign currency, and entering into a forward transaction to purchase the foreign currency. The principle behind this is that a position in foreign risk-free bonds, with the currency risk hedged, pays the same return as domestic risk-free bond.
Eurodollar futures The Eurodollar contract is based on a $1 million 3-month deposit earning LIBOR. There is also a 1-month contract, handled similarly. Suppose that the current LIBOR is 1.25% over 3 months. By convention, this is annualized by multiplying by 4, to the quoted annualized LIBOR rate of 5.0%. The Eurodollar futures price at expiration of the contract is 100 Annualized 3-month LIBOR. The settlement is based on then current LIBOR (i.e., for the future three months counting from the date of settlement). This futures contract is used for hedging interest rate risk. 6. Swaps Swaps are generally derivatives that trade cash flows between counterparties based on two pieces of underlying. Such a contract is written for a specific term, called the swap term, or swap tenor. An unusual, and simplest type of a swap is an exchange of a single payment for multiple payments (or, possibly, multiple deliveries of some form of underlying) in the future. This is called a prepaid swap. Swaps are commonly settled financially, i.e., currency payments are exchanged. But it is also possible to create a physical delivery swap. For example, one could arrange for a delivery of an ounce of gold every year for the next twenty years in exchange for a fixed payment of $600 every year. Note that this transaction can be actually decomposed into a series of forwards (and that is not a coincidence). Such physical delivery swap can also be settled financially, of course, by the side delivering the gold paying the other party the difference between $600 and the price of an ounce of gold at the time of delivery. It is rare that the two counterparties of a swap are both market participants. Swaps are arranged as private transactions and not traded on exchanges. They are typically arranged by a dealer, for a client of that dealer. The dealer ends up holding one side of the swap deal, but usually seeks to make another swap arrangement that would at least partially offset that position. That second transaction is commonly called back-to-back transaction, or matched book transaction. The dealer can also seek to hedge the exposure using traded derivative instruments. It is generally difficult, however, to find an exact hedge in the market, and the dealer may have to look at the dealer s entire portfolio exposure and hedge pieces of it, combining exposures from various transactions. The market value of a swap When a swap is entered into, it is standard that no payments change hands and future payments committed to by each party have the same market value. As time passes, the value of each counterparty s position changes. Because what one party pays, the other one receives, the total of the values remains zero. This may create credit risk for the party, which has a positive value of the swap. Interest rate swaps A plain vanilla (i.e., the simplest kind, as in the plain vanilla ice cream) interest rate swap is a contract between two counterparties, requiring them to make them interest
payments to each other over the term of the contract, based on different types of underlying bonds or interest rate indices. The long party (fixed-rate payer) pays interest to the second at a fixed rate, while the short party (floating-rate payer) pays interest to the first at a rate that changes ( floats ) according to a specified index. This kind of a swap is also called a pay-fixed swap, because the long party pays fixed. The actual cash payments are determined by multiplying the relevant rate of interest by a face amount, or principal, which is called the notional principal, or just the notional. For example, consider a semiannual swap with a notional of $10 million: Long: Pay 4% and receive 6-month LIBOR, Short: Pay 6-month LIBOR and receive 4%. If the 6-month LIBOR on the first payment date were 3%, then: Fixed-rate payer pays floating-rate payer $10M! 0.04 2 = $200,000, Floating-rate payer pays fixed-rate payer $10M! 0.03 2 = $150,000. This would be actually settled by the fixed-rate payer paying the floating-rate payer $50,000 rather than the two parties paying offsetting amounts. Swaps are commonly used to manage interest rate risk exposure. An insurance company can use swaps to produce an income stream that better matches its liability structure. Suppose that you are managing a company that issued fixed-rate liabilities that credit 5.50% (e.g., GIC or SPDA), and backs them with a bond that pays 3-month LIBOR + 1%. There is a mismatch creating interest rate risk exposure. Now suppose that this company enters into a swap where it pays 3-month LIBOR and receives 5.50%. This would result in a net 1% profit to the company, without any interest rate risk. This, of course, is an idealized example, but you should understand how it works. The company used a swap to convert its floating-rate asset to a fixed-rate asset that better matched its liabilities; in doing so, it locked in a spread of 100 bps. In practice, most commonly, life insurance companies seek to convert their fixed coupon bonds income into floating rate income. A swap in which a party receives a floating rate in exchange for fixed payments on bonds that this party already holds is called an asset swap. Swap rate A long (pay-fixed) swap position is equivalent to buying a fixed coupon bond with funds borrowed at the swap s floating rate. If an interest rate swap is arranged in such a way that no payment is made upfront, and in exchange for fixed rate payment, the long side receives the market floating rate, the resulting fixed interest rate is called the swap rate. The swap rate turns out to be simply the coupon rate on a par coupon bond, with that bond s maturity equal to the swap term. The swap curve Thanks to Eurodollar futures on 3-month LIBOR, 3-month LIBOR forward rates can be found for up to 10 years. The swap rates established against the floating rates equal to those LIBOR forward rates constitute the swap curve. The swap spread is the difference between a given swap rate and a Treasury Bond yield for the corresponding maturity.