Fast Computation of Vanilla Prices in Time-Changed Models and Implied Volatilities Using Rational Approximations

30 Pages Posted: 25 Oct 2011 Last revised: 4 Nov 2011

See all articles by Martijn Pistorius

Martijn Pistorius

Imperial College London

Johannes Stolte

Imperial College London

Date Written: October 25, 2011

Abstract

We present a new numerical method to price vanilla options in time-changed Brownian motion models quickly. 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 (FFT) 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.

Keywords: Rational approximations, Time-changed Brownian motion models, Inverse Black-Scholes formula

JEL Classification: G12, G13, C63

Suggested Citation

Pistorius, Martijn and Stolte, Johannes, Fast Computation of Vanilla Prices in Time-Changed Models and Implied Volatilities Using Rational Approximations (October 25, 2011). Available at SSRN: https://ssrn.com/abstract=1946786 or http://dx.doi.org/10.2139/ssrn.1946786

Martijn Pistorius

Imperial College London ( email )

South Kensington Campus
Exhibition Road
London, Greater London SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/people/m.pistorius

Johannes Stolte (Contact Author)

Imperial College London ( email )

South Kensington Campus
Exhibition Road
London, Greater London SW7 2AZ
United Kingdom

Do you have negative results from your research you’d like to share?

Paper statistics

Downloads
140
Abstract Views
1,554
Rank
372,809
PlumX Metrics