The problem of fast pricing, hedging, and calibrating of derivatives is considered when the underlying does not follow the standard Black–Scholes–Merton model but rather a mean-reverting and deterministic volatility model. Mean-reverting models are often used for volatility, commodities, and interest-rate derivatives, while the deterministic volatility accounts for the nonconstant implied volatility. Trading desks often use numerical methods for real-time pricing, hedging, and calibration when implementing such models. A more efficient alternative is to use an analytic formula, even if only approximate. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations to the density function that can be used to derive simple formulas for pricing derivatives. Such approximations are usually only valid away from any boundaries, yet for some derivatives the values of the underlying near the boundaries are needed such as when interest rates are very low or for pricing put options. Hence, the ray approximation may not yield acceptable results. A new asymptotic approximation near boundaries is derived, which is shown to be of value for pricing certain derivatives. The results are illustrated by deriving new analytic approximations for European derivatives and their high accuracy is demonstrated numerically.
A Comparative Analysis of the Black-Scholes- Merton Model and the Heston Stochastic Volatility Model