An Approximate Analytical Solution for Default-Free Zero-Coupon Bonds in a Two-Factor Vasicek Interest Rate Model with Stochastic Volatility

Abstract

To address the limitation of the traditional Vasicek interest rate model, in which volatility is assumed to be constant and therefore cannot adequately describe the stochastic evolution of interest rate volatility, this paper introduces a stochastic volatility factor into a mean-reverting short-rate framework. A two-factor Vasicek-type interest rate model is developed for the approximate analytical pricing of default-free zero-coupon bonds. By applying the integrating factor method, the short-rate process is represented in integral form, and the bond price is transformed into the conditional expectation of a discount factor generated by the future short-rate integral. Conditional on the volatility path, the conditional Gaussian property of the stochastic integral is used to derive a moment-generating-function representation. Following the stochastic-volatility approximation idea of Hull and White, the pricing expression is further approximated through the time average of the squared volatility process and a third-order Taylor expansion. The resulting formula contains correction terms determined by the variance and the third central moment of the average variance. The proposed method preserves the main effects of stochastic volatility while maintaining analytical tractability and computational efficiency, thereby providing a useful approximation framework for pricing zero-coupon bonds and related interest rate derivatives.