Approximate Formulae for Pricing Zero-coupon Bonds and Their Asymptotic Analysis

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 function λ describing 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 λ. It is often assumed that the short rate evolves according to the following mean reverting stochastic differential equation