An Asymptotic Solution for Call Options on Zero-Coupon Bonds

Abstract

We present an asymptotic solution for call options on zero-coupon bonds, assuming a stochastic process for the price of the bond, rather than for interest rates in general. The stochastic process for the bond price incorporates dampening of the price return volatility based on the maturity of the bond. We derive the PDE in a similar way to Black and Scholes. Using a perturbation approach, we derive an asymptotic solution for the value of a call option. The result is interesting, as the leading order terms are equivalent to the Black–Scholes model and the additional next order terms provide an adjustment to Black–Scholes that results from the stochastic process for the price of the bond. In addition, based on the asymptotic solution, we derive delta, gamma, vega and theta solutions. We present some comparison values for the solution and the Greeks.