Asymptotics and calibration of local volatility models

We derive a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation. Using this equation we establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in- and out-of-the-money options. This in turn leads us to propose a new formulation near expiry of the calibration problem for the local volatility model, which we show to be well posed. In the Black–Scholes–Merton model [4,24], it is assumed that the price of a non-dividend paying stock St follows the log-normal stochastic differential equation dSt = St (µ dt + σ dWt), (1) where t is time, µ and σ are constants and Wt is a standard Brownian motion. The parameter σ is called the volatility of the stock St. It is well-known that the priceC(St, t;K, T) of a European call option written on St with strike K and maturity T satisfies the linear parabolic partial differential equatio