Abstract
We investigate the effect of stochastic fluctuations of an interest rate on the value of a derivative. We derive the modified Black-Scholes equation that describes evolution of the value of a derivative averaged over an ensemble of stochastic fluctuations of the rate of interest and depends on the "renormalized" values of volatility and rate of interest. We present the explicit expressions for the renormalized volatility and interest rate that incorporate the corrections owing to the short-term stochastic variations of the interest rate. The stochastic component of the interest rate tends to enhance the effective volatility and reduce the effective interest rate that determine an evolution of the option pricing "smoothed out" over the stochastic variations. The results of numerical solution of the modified Black-Scholes equation with the renormalized coefficients are illustrated for an American put option on non-dividend-paying stock.
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