Option pricing under the Merton model of the short rate

Previous option pricing research typically assumes that the risk-free rate or the short rate is constant during the life of the option. In this study, we incorporate the stochastic nature of the short rate in our option valuation model and derive explicit formulas for European call and put options on a stock when the short rate follows the Merton model. Using our option model as a benchmark, our numerical analysis indicates that, in general, the Black–Scholes model overvalues out-of-the-money calls, moderately overvalues at-the-money calls, and slightly overvalues in-the-money calls. Our analysis is directly extensible to American calls on non-dividend-paying stocks and to European puts by virtue of put-call parity.