Interest Rates and Term Structure Models

Abstract

The assumption made in the Black-Scholes world, in particular in the Black-76 model Equation 8.10, that interest rates are non-stochastic directly contradicts the very existence of interest rate options. If interest rates were deterministic and hence predictable with certainty for all future times, we would know at time t which options will be in or out of the money upon maturity T. The options which are out of the money at maturity would be worthless at all earlier times t < T as well. The options which are in the money at maturity would be nothing other than forward transactions. Thus, the assumptions made in pricing interest rate options using the Black-76 model imply that these very options should not even exist! In spite of this fact, the option prices obtained by applying the Black-76 model are surprisingly good. The results of recent research [96][128] have shown that the effects of several “false” assumptions (in particular, the assumed equality of forward and futures prices) tend to cancel out each other.¹