Fast computation of vanilla prices in time-changed models and implied volatilities using rational approximations

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier transform method with respect to accuracy and speed appears to favour the newly developed method in the cases considered. We present error estimates for the option prices. Additionally, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. To achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than one can by the use of other common methods. Error estimates are presented for a wide range of parameters.