Pricing Interest Rate Options with the Black Futures Option Model

Transcription

1 Bond Evaluation, Selection, and Management, Second Edition by R. Stafford Johnson Copyright 2010 R. Stafford Johnson APPENDIX I Pricing Interest Rate Options with the Black Futures Option Model I.1 BLACK FUTURES MODEL An extension of the B-S OPM that is sometimes used to price interest rate options is the Black futures option model. The model is defined as follows: C 0 = [ f 0N(d 1 ) XN(d 2 )] e R f T P 0 = [X(1 N(d 2)) f 0 (1 N(d 1 ))] e R f T ln( f 0/ X) + (σ 2 f /2)T σ f T d 2 = d 1 σ f T σ f 2 = variance of the logarithmic return of futures prices = V(ln(f n /f 0 ) T = time to expiration expressed as a proportion of a year R f = continuously compounded annual risk-free rate [if simple annual rate is R, the continuously compounded rate is ln(1+r)] N(d) = cumulative normal probability; this probability can be looked up in a standard normal probability table or by using the following formula: N(d) = 1 n(d), for d < 0 N(d) = n(d), for d > 0, n(d) = 1.5[ ( d ) ( d ) ( d ) ( d ) 4 ] 4 d =absolute value of d Example: T-Bill Futures Consider the European futures T-bill call options we priced in Appendix H in which the futures option had an exercise price of and expiration of one year and the current futures price was f 0 = If the simple risk-free rate is 5%, implying a 763

2 764 APPENDIX I: PRICING INTEREST RATE OPTIONS WITH THE BLACK FUTURES OPTION MODEL continuously compounded rate of 4.879% [= ln(1.05)], and the annualized standard deviation of the futures logarithmic return, σ (ln(f n /f 0 )), is.00158, then using the Black futures model the price of the T-bill futures call would be C 0 = [ ( ) 98.75( )] e (.04879)(1) = ln( /98.75) + (.00158)2 /2)(1) d 2 = = N(.24175) = N(.24017) = = Example: T-Bond Futures As a second example, consider one-year put and call options on a CBOT T-bond futures contract, with each option having an exercise price of $100,000. Suppose the current futures price is $96,115, the futures volatility is σ (ln(f n /f 0 )) =.10, and the continuously compounded risk-free rate is.065. Using the Black futures option model, the price of the call option would be $2,137 and the price of the put would be $5,777: C 0 = [$96,115 (.36447) $100,000 ( )] e (.065)(1) = $2,137 P 0 = [$100,000( ) $96,115( )] e (.065)(1) = $5,777 ln(96115/100,000) + (.01/2)(1).10 = d 2 = = N(.34625) = N(.44625) = It should be noted that the call and futures prices are also consistent with put-call futures parity: P 0 C 0 = PV(X f 0) P0 = (X f 0) e R f T + C0 P0 = ($100,000 $96,115) e (.065)(1) + $2,137 = $5,777

3 Appendix I: Pricing Interest Rate Options with the Black Futures Option Model 765 Also, note that the Black model can be used to price a spot option. In this case, the current futures price, f 0, is set equal to its equilibrium price as determined by the carrying cost model: f 0 = S 0 (1+R f ) T (Accrued interest at T). If the carrying cost model holds, the price obtained using the Black model will be equal to the price obtained using the B-S OPM. Pricing Caplets and Floorlets with the Black Futures Option Model The Black futures option model also can be extended to pricing caplets and floorlets by (1) substituting T* fort in the equation for C* (for a caplet) or P* (for a floorlet), T* is the time to expiration on the option plus the time period applied to the interest rate payoff time period, φ: T* = T + φ; (2) using an annual continuously compounded risk-free rate for period T* instead of T; (3) multiplying the Black adjusted-futures option model by the notional principal times the time period: (NP) φ. C 0 = φ(np) [RN(d 1) R X N(d 2 )] e R f T P 0 = φ(np) [R X(1 N(d 2 )) R(1 N(d 1 ))] e R f T ln(r/r X) + (σ 2 /2)T σ σ T d 2 = d 1 σ T Example: Pricing a Caplet Consider a caplet with an exercise rate of X = 7%, NP = $100,000, φ =.25, expiration = T =.25 year, and reference rate = LIBOR. If the current LIBOR were R = 6%, the estimated annualized standard deviation of the LIBOR s logarithmic return were.2, and the continuously compounded riskfree rate were %, then using the Black model, the price of the caplet would be C 0 =.25($100,000) [.06( ).07 ( )] e ( )(.5) = 4.34 ln(.06/.07) + (.04/2)(.25).2 = d 2 = d = N( ) = N( ) = Example: Pricing a Cap Suppose the caplet represented part of a contract that caps a two-year floating-rate loan of $100,000 at 7% for a three-month period. The cap consists of seven caplets, with expirations of T =.25 years,.5,.75, 1, 1.25, 1.5, and 1.75.

4 766 APPENDIX I: PRICING INTEREST RATE OPTIONS WITH THE BLACK FUTURES OPTION MODEL The value of the cap is equal to the sum of the values of the caplets comprising the cap. If we assume a flat yield curve such that the continuous rate of % applies, and we use the same volatility of.2 for each caplet, then the value of the cap would be $254.38: Expiration Price of Caplet In practice, different volatilities for each caplet are used in valuing a cap or floor. The different volatilities are referred to as spot volatilities. They are often estimated by calculating the implied volatility on a comparable Eurodollar futures option. Note Even though the B-S OPM and the Black model can be used to estimate the equilibrium price of interest rate options and futures options, there are at least two problems. First, the OPM is based on the assumption that the variance of the underlying asset is constant. In the case of a bond, though, its variability tends to decrease as its maturity becomes shorter. Second, the OPM assumes the interest rate is constant. This assumption does not hold for options on interest-sensitive securities. In spite of these problems, the B-S OPM and the Black futures model are still used to value interest rate options. PROBLEMS AND QUESTIONS Note: The appendix problems can be done using the B-S OPM Excel program available on the Web site. 1. Suppose a T-bill futures is priced at f 0 = 99 and has an annualized standard deviation of.00175, and that the continuously compounded annual risk-free rate is 4%. a. Using the Black futures option model, calculate the equilibrium price for a three-month T-bill futures call option with an exercise price of b. Using the Black futures option model, calculate the equilibrium price for a three-month T-bill futures put option with an exercise price of

5 Appendix I: Pricing Interest Rate Options with the Black Futures Option Model Suppose a T-bond futures expiring in six months is priced at f 0 = 95,000 and has an annualized standard deviation of.10, and that the continuously compounded annual risk-free rate is 5%. a. Using the Black futures option model, calculate the equilibrium price for a six-month T-bond futures call option with an exercise price of 100,000. b. Using the Black futures option model, calculate the equilibrium price for a six-month T-bond futures put option with an exercise price of 100,000.