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.
Highlights
Asymptotic Solution for Call OptionsBlack and Scholes’ [1] solution for European style options assumes a geometric Brownian motion process for the underlying stock price
We propose a process for a zero-coupon bond price directly, and using that process, we derive a partial differential equation (PDE) and an asymptotic solution for a call option on the bond
While there are advantages and disadvantages to every approach, modeling bond dynamics directly is more closely aligned with the nature of trading in the asset itself; bonds, like stocks, trade in denominated currencies
Summary
Black and Scholes’ [1] solution for European style options assumes a geometric Brownian motion process for the underlying stock price. In order for spot and term structure models to provide for reasonable representation of interest rate dynamics, the models often require multiple uncertainty terms with non-constant volatility. This results in computationally complex formulations, which normally require numerical solutions. While there are advantages and disadvantages to every approach, modeling bond dynamics directly is more closely aligned with the nature of trading in the asset itself; bonds, like stocks, trade in denominated currencies We discuss the properties of our model, including the commonly used Greeks
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