Closed Solutions for Option Valuations with Stochastic Interest Rate under a Neutral Foward Risk Measure

Abstract

The nonlinear dynamics and the stochastic behaviour of the interest rate have a strong impact on the pricing of derivatives securities, particularly for long-maturity option contracts. This paper proposes closed-form solutions for the pricing of equity options when the interest rate is stochastic under the existence of an equivalent martingale measure or risk-neutral forward measure. The arbitrage-free pricing model allows to capture the dependence structure between the rates of return of the underlying stock and the default-free zero coupon bond, i. e. the two uncertainty sources can be simplified into one by means of the diffusion term of the forward price dynamics. In addition, we apply the closed solution model to the daily prices of America Movil over the period comprised between January 2nd, 2001 to May 9th, 2011. Overall, the numerical results show that in most cases, the values of the options are overestimated or underestimated with respect to the market values