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Convergence of At-The-Money Implied Volatilities to the Spot Volatility

Part of: Mathematical economics Stochastic analysis Limit theorems

Published online by Cambridge University Press:  14 July 2016

Valdo Durrleman
Valdo Durrleman*
Affiliation:
École Polytechnique
*
∗Postal address: Centre de Mathématiques Appliquées, École Polytechnique - CNRS, Route de Saclay, 91128 Palaiseau, France. Email address: vdurrleman@gmail.com
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Abstract

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We study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black-Scholes formula and of the central limit theorem for martingales.

Keywords

Implied volatilityspot volatilityrobustness formulamartingale central limit theorem

MSC classification

Secondary: 91B70: Stochastic models 60F05: Central limit and other weak theorems 60H05: Stochastic integrals
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 542 - 550
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Berestycki, H., Busca, J. and Florent, I. (2002). Asymptotics and calibration of local volatility models. Quant. Finance 2, 61–69.Google Scholar
[2] Berestycki, H., Busca, J. and Florent, I. (2004). Computing the implied volatility in stochastic volatility models. Commun. Pure Appl. Math. 57, 1352–1373.CrossRefGoogle Scholar
[3] Brace, A., Goldys, B., Klebaner, F. and Womersley, R. (2001). Market model for stochastic implied volatility with application to the BGM model. Tech. Rep., Department of Tech. Rep.Google Scholar
[4] Carmona, R. (2007). HJM: a unified approach to dynamic models for fixed income, credit and equity markets. In Paris-Princeton Lectures on Mathematical Finance 2004 (Lecture Notes Math. 1919), Springer, Berlin, pp. 1–50.Google Scholar
[5] Carr, P. (2000). A survey of preference free option valuation with stochastic volatility. Risk's 5th annual European derivatives and risk management congress, Paris.Google Scholar
[6] Carr, P. and Wu, L. (2003). What type of process underlies options? A simple robust test. J. Finance 58, 2581–2610.Google Scholar
[7] Durrleman, V. (2005). From implied to spot volatilities. Tech. Rep., Department of Tech. Rep. Available at http://math.stanford.edu/∼valdo/papers/.Google Scholar
[8] El Karoui, N., Jeanblanc-Picqué, M. and Shreve, S. (1998). Robustness of the Black and Scholes formula. Math. Finance 8, 93–126.CrossRefGoogle Scholar
[9] Jacod, J. and Shiryaev, A. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.Google Scholar
[10] Lépingle, D. and Mémin, J. (1978). Sur l'intégrabilité uniforme des martingales exponentielles. Z. Wahrscheinlichkeitsth. 42, 175–203.CrossRefGoogle Scholar
[11] Medvedev, A. and Scaillet, O. (2007). Approximation and calibration of short-term implied volatilities under Jump-diffusion stochastic volatility. Rev. Financial Studies 20, 427–459.Google Scholar
[12] Schönbucher, P. (1999). A market model for stochastic implied volatility. Philos. Trans. R. Soc. London Ser. A 357, 2071–2092.Google Scholar