APPROXIMATE FORMULAE FOR PRICING ZERO-COUPON BONDS AND THEIR ASYMPTOTIC ANALYSIS

We analyze analytic approximation formulae for pricing zero- coupon bonds in the case when the short-term interest rate is driven by a one-factor mean-reverting process with a volatility nonlinearly depending on the interest rate itself. We derive the order of accuracy of the analytical ap- proximation due to Choi and Wirjanto. We furthemore give an explicit formula for a higher order approximation and we test both approximations numerically for a class of one-factor interest rate models. Term structure models give the dependence of time to maturity of a discount bond and its present price. One-factor models are often formulated in terms of a stochastic differential equation for the instantaneous interest rate (short rate). In the theory of nonarbitrage term structure models the bond prices (yielding the interest rates) are given by a solution to a parabolic partial differential equation. The stochastic differential equation for the short rate is specified either under a real (observed) probability measure or risk-neutral one. A risk-neutral measure is an equivalent measure such that the derivative prices (bond prices in particular) can be computed as expected values. If the short rate process is considered with a real probability measure, a functiondescribing the so-called market price of risk has to be provided. The volatility part of the process is the same for both real and risk-neutral specification of the process. The changes in the drift term depend on the so called market price of risk function �. It is often assumed that the short rate evolves according to the following mean reverting stochastic differential equation