Arbitrage-Free Prediction of the Implied Volatility Smile

Risk Magazine, Forthcoming

18 Pages Posted: 23 Jul 2014

See all articles by Petros Dellaportas

Petros Dellaportas

Athens University of Economics and Business

Aleksandar Mijatovic

Imperial College London

Date Written: March 22, 2014

Abstract

This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to n strikes (with n large, e.g. n ≥ 40) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any n-dimensional time series model also has to satisfy these implicit restrictions at each time step, a condition that is impossible to meet since the model innovations can take arbitrary values. We solve this problem for any n ∈ N in the context of Foreign Exchange (FX) by first encoding the option prices at each time step in terms of the parameters of the corresponding risk-neutral measure and then performing the time series analysis in the parameter space. The option price predictions are obtained from the predicted risk neutral measure by effectively integrating it against the corresponding option payoffs. The non-linear transformation between option prices and the risk-neutral parameters applied here is not arbitrary: it is the standard mapping used by market makers in the FX option markets (the SABR parameterisation) and is given explicitly in closed form. Our method is not restricted to the FX asset class nor does it depend on the type of parameterisation used. Statistical analysis of FX market data illustrates that our arbitrage-free predictions outperform the naive random walk forecasts, suggesting a potential for building management strategies for portfolios of derivative products, akin to the ones widely used in the underlying equity and futures markets.

Keywords: Prediction of option prices in FX, Risk-neutral measure, Implied Volatility, trading strategy for options

Suggested Citation

Dellaportas, Petros and Mijatovic, Aleksandar, Arbitrage-Free Prediction of the Implied Volatility Smile (March 22, 2014). Risk Magazine, Forthcoming, Available at SSRN: https://ssrn.com/abstract=2469770 or http://dx.doi.org/10.2139/ssrn.2469770

Petros Dellaportas (Contact Author)

Athens University of Economics and Business ( email )

76 Patission Street
Athens, 104 34
Greece

Aleksandar Mijatovic

Imperial College London ( email )

Department of Mathematics
180 Queen's Gate
London, SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/people/a.mijatovic

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