The Pricing of Bond Futures Options

We derive closed-form solutions for bond futures and forward prices and bond spot options, bond futures and forward options, when the instantaneous interest rates follow a mean-reverting diffusion process. The pricing formulas are consistent with the Black-Scholes option formula so that they can be easily applied. These formulas include the present value operator which makes them look different from the Black-Scholes formula. An important conclusion is that as long as the bond forward price is greater than the bond futures price, the bond forward call option price is greater than the bond futures call option price. Since the marking to market from the futures contract lowers the futures price compared to the forward price, the bond futures call option price is strictly lower than the corresponding bond forward call option price. This result is contrast to Ahn (1996a) which shows the numerical analysis that the currency futures option price is strictly greater than the corresponding currency forward option price since the currency futures price at the expiration date is greater than the corresponding currency forward price.