The Large Maturity Smile for the Heston Model

Posted: 27 Nov 2009 Last revised: 4 Jun 2012

See all articles by Martin Forde

Martin Forde

Dublin City University - Department of Mathematical Sciences

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Date Written: November 26, 2009

Abstract

Using the Gartner-Ellis theorem from large deviation theory, we characterize the leading-order behaviour of call option prices under the Heston model, in a new regime where the maturity is large and the log-moneyness is also proportional to the maturity. Using this result, we then derive the implied volatility in the large-time limit in the new regime, and we find that the large-time smile mimics the large-time smile for the Barndorff-Nielsen Normal Inverse Gaussian model. This makes precise the sense in which the Heston model tends to an exponential Levy process for large times. We find that the implied volatility smile does not flatten out as the maturity increases, but rather it spreads out, and the large-time, large-moneyness regime is needed to capture this effect. As a special case, we are able to rigourously prove the well known result by Lewis for the implied volatility in the usual large-time, fixed-strike regime, at leading order. We find that there are two critical strike values where there is a qualitative change of behaviour for the call option price, and we use a limiting argument to compute the asymptotic implied volatility in these two cases.

Keywords: Heston model, asymptotics, smile, large deviations, calibration

JEL Classification: G12, G13, C6

Suggested Citation

Forde, Martin and Jacquier, Antoine, The Large Maturity Smile for the Heston Model (November 26, 2009). Available at SSRN: https://ssrn.com/abstract=1514080 or http://dx.doi.org/10.2139/ssrn.1514080

Martin Forde

Dublin City University - Department of Mathematical Sciences ( email )

Dublin
Ireland

Antoine Jacquier (Contact Author)

Imperial College London ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://wwwf.imperial.ac.uk/~ajacquie/

The Alan Turing Institute ( email )

British Library, 96 Euston Road
London, NW12DB
United Kingdom

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