Pricing Stock and Bond Options when the Default-Free Rate is Stochastic

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Abstract
We derive formulas for the valuation of call options on stocks and bonds when the default-free rate is stochastic. The formulas highlight the role of the correlation between the unanticipated returns on the underlying security and the changes in the short-term rate in determining the options value. Our numerical analysis indicates that option prices predicted by the proposed formula differ from those predicted by the Black and Scholes formula when this correlation is relatively large and the short-term rate's instantaneous variance is relatively large as well. Moreover, the proposed formula predicts higher (lower) stock option prices than those predicted by the Black and Scholes formula for correlation values that are lower (higher) than some positive critical value.

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