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**Pricing options under stochastic interest rates: a new approach.**
*(English)*
Zbl 1157.91363

Summary: We generalize the Black-Scholes option pricing formula by incorporating stochastic interest rates. Although the existing literature has obtained some formulae for stock options under stochastic interest rates, the closed-form solutions have been known only under the Gaussian (Merton type) interest rate processes. We show that an explicit solution, which is an extended Black-Scholes formula under stochastic interest rates in certain asymptotic sense, can be obtained by extending the asymptotic expansion approach when the interest rate volatility is small. This method, called the small-disturbance asymptotics for Itō processes, has recently been developed by N. Kunitomo and A. Takahashi [Math. Finance 11, No. 1, 117–151 (2001; Zbl 0994.91023); Ann. Appl. Probab. 13, No. 3, 914–952 (2003; Zbl 1091.91037)] and A. Takahashi [Asia-Pac. Financ. Mark. 6, No. 2, 115–151 (1999; Zbl 1153.91568)]. We found that the extended Black-Scholes formula is decomposed into the original Black-Scholes formula under the deterministic interest rates and the adjustment term driven by the volatility of interest rates. We illustrate the numerical accuracy of our new formula by using the Cox-Ingersoll-Ross model for the interest rates.

### MSC:

91G20 | Derivative securities (option pricing, hedging, etc.) |

91G30 | Interest rates, asset pricing, etc. (stochastic models) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |