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THE LARGE-MATURI TY SMILE FOR THE HESTON MODEL MARTIN FORDE AND ANTOINE JACQUIER Abstract. Using the Gärtner-Ellis theorem from large deviations 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 maturit y . Using this result, we then deriv e 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- Niels en Normal Inv erse Gaussia n model. This makes precis e the sense in which the Heston model tends to an exponential L´ evy 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-t ime, large-moneyn ess regime is needed to captu re this effec t. As a special case, we provide a rigorous proof of the well known result by Lewis [40] for the implied volatility in the usual large-time, fixed- strik e regime, at leading order. We find that there are two critical strike va lues 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. 1. Introduction In con trast to the small-time case, there is a compa ratively small liter ature on large-time asymptotics for stochastic volatility models. Lewis [40] approximated the characteristic function of the logarithmic return for the Heston model at large maturities by the leading order term of its eigenfunction expansion. Using Laplace’s method for contour integrals along a horizontal line going through the saddle point of the eigenvalue function on the imaginary axis (see the sequel paper by Forde, Jacquier & Mijatovi´ c [24] and Theorem 7.1, chapter 4 of Olver [ 43] for more details), Lewis derived an asymptotic formula for the price of a call option in the large-time, fixed-strike regime, and developed a similar asymptotic result for the Black-Scholes model. By equating the asymptotic call option formulae under the Heston model and the Black-Scholes model at zeroth order and O (1/t), Jacquier [32, 33] computed the corresponding asymptotic formula for implied vol atilit y . The leading order term in the expansion is a fixed number , indepen den t of the log-money ness, which is equal to eight times the principal eigenvalue of the Sturm-Liouville operator associated with the instantaneous varian ce process. The correct ion (or “sk ew”) term is affine in x/t, where t represents the maturity and x is the log-moneyness. From this we see that this approach captures the skew effect, but not the smile effect; we have to go to higher order to see the x 2 /t 2 term. The argument s of Lewis and Jacquier were not rigourous as they did not explicitly compute the contour of steepes t desce nt, or justif y the truncation of the eigenfunction expansion with tail estimates. The work of Forde has been supported by the European Science Foundation, AMaMeF Exchange Grant 2107 and by a SFI grant for the Edgeworth Centre for Financial Mathematics. The authors would like to thank Jin Feng, Jean-Pierre Fouque, Jim Gatheral, Archil Gulisashvili, Martin Keller-Ressel, Alan Lewis, Roger Lee and Aleksandar Mijatovi´ c for useful discussions. 1

THE LARGE-MATURITY SMILE FOR THE HESTON MODEL

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http://cermics.enpc.fr/reports/CERMICS-2004/CERMICS-2004-267.pdf

http://arxiv.org/abs/1002.3633

http://www.math.nyu.edu/fellows%20fin%20math/gatheral/madrid2004.pdf

http://arxiv.org/abs/1001.3003

http://arxiv.org/abs/0911.2992

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