Computation of Deterministic Volatility Surfaces

The 'volatility smile' is one of the well-known biases of Black-Scholes models for pricing options. In this paper, we introduce a robust method of reducing this bias by pricing subject to a deterministic functional volatility $\sigma = \sigma (S,t)$. This instantaneous volatility is chosen as a spline whose weights are determined by a regularised numerical strategy that approximately minimises the difference between Black-Scholes vanilla prices and known market vanilla prices over a range of strikes and maturities; these Black-Scholes prices are calculated by solving the relevant partial differential equation numerically using finite element methods. The instantaneous volatility generated from vanilla options can be used to price exotic options where the skew and term-structure of volatility are important, and we illustrate the application to barrier options