Finance and Stochastics

, Volume 15, Issue 4, pp 755–780 | Cite as

The large-maturity smile for the Heston model

Article
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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.

Keywords

Implied volatility Heston Asymptotics Large deviations 

Mathematics Subject Classification (2000)

60G44 91B70 91B25 

JEL Classification

C02 C63 G12 G13 
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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDublin City UniversityDublin 9Ireland
  2. 2.Department of MathematicsImperial College LondonLondonUK
  3. 3.Zeliade SystemsParisFrance

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