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The large-maturity smile for the Heston model

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Abstract

Using the Gärtner–Ellis theorem from large deviations theory, we characterise 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 Lévy 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 provide a rigorous proof of the well-known result by Lewis (Option Valuation Under Stochastic Volatility, Finance Press, Newport Beach, 2000) 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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

    Martin Forde

  2. Department of Mathematics, Imperial College London, London, UK

    Antoine Jacquier

  3. Zeliade Systems, 56 Rue Jean-Jacques Rousseau, Paris, France

    Antoine Jacquier

Authors
  1. Martin Forde
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  2. Antoine Jacquier
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Correspondence to Martin Forde.

Additional information

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.

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Forde, M., Jacquier, A. The large-maturity smile for the Heston model. Finance Stoch 15, 755–780 (2011). https://doi.org/10.1007/s00780-010-0147-3

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  • DOI: https://doi.org/10.1007/s00780-010-0147-3

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