Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T17:21:39.860Z Has data issue: false hasContentIssue false
  • English
  • Français

From characteristic functions to implied volatility expansions

Part of: Mathematical economics Limit theorems

Published online by Cambridge University Press:  21 March 2016

Antoine Jacquier  and
Matthew Lorig
Antoine Jacquier*
Affiliation:
Imperial College London
Matthew Lorig*
Affiliation:
University of Washington
*
∗ Postal address: Department of Mathematics, Imperial College London, London SW7 2AZ, UK. Email address: a.jacquier@imperial.ac.uk
∗∗ Postal address: Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For any strictly positive martingale S = eX for which X has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log-strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Lévy model, Merton (1976), one infinite activity exponential Lévy model (variance gamma), and one stochastic volatility model, Heston (1993). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

Keywords

Implied volatility expansionexponential Lévyaffine classHestonadditive process

MSC classification

Primary: 91B70: Stochastic models
Secondary: 60F99: None of the above, but in this section 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 837 - 857
Copyright
Copyright © Applied Probability Trust 2015 

References

Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, DC.Google Scholar
Benaim, S. and Friz, P. (2009). Regular variation and smile asymptotics. Math. Finance 19, 1–12.Google Scholar
Berestycki, H., Busca, J. and Florent, I. (2004). Computing the implied volatility in stochastic volatility models. Commun. Pure Appl. Math. 57, 1352–1373.Google Scholar
Breeden, D. T. and Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. J. Business 51, 621–651.Google Scholar
Brown, J. W. and Churchill, R. V. (1996). Complex Variables and Applications, 6th edn. McGraw-Hill, New York.Google Scholar
Deuschel, J. D., Friz, P. K., Jacquier, A. and Violante, S. (2014). Marginal density expansions for diffusions and stochastic volatility II: Applications. Commun. Pure Appl. Math. 67, 321–350.Google Scholar
Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.CrossRefGoogle Scholar
Duffie, D., Pan, J. and Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376.Google Scholar
Fouque, J.-P., Papanicolaou, G., Sircar, R. and S⊘lna, K. (2003). Singular perturbations in option pricing. SIAM J. Appl. Math. 63, 1648–1665.Google Scholar
Fouque, J.-P., Papanicolaou, G., Sircar, R. and S⊘lna, K. (2011). Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press.Google Scholar
Friz, P., Gerhold, S., Gulisashvili, A. and Sturm, S. (2011). On refined volatility smile expansion in the Heston model. Quant. Finance 11, 1151–1164.CrossRefGoogle Scholar
Fukasawa, M. (2011). Asymptotic analysis for stochastic volatility: Edgeworth expansion. Electron. J. Prob. 16, 764–791.CrossRefGoogle Scholar
Gatheral, J. (2004). A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives. Presentation at Global Derivatives & Risk Management 2004, Madrid.Google Scholar
Gatheral, J. and Jacquier, A. (2014). Arbitrage-free SVI volatility surfaces. Quant. Finance 14, 59–71.Google Scholar
Gatheral, J. et al. (2012). Asymptotics of implied volatility in local volatility models. Math. Finance 22, 591–620.CrossRefGoogle Scholar
Gulisashvili, A. and Stein, E. M. (2010). Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models. Appl. Math. Optimization 61, 287–315.Google Scholar
Guo, G., Jacquier, A., Martini, C. and Neufcourt, L. (2012). Generalised arbitrage-free SVI volatility surfaces. Preprint. Available at http://arxiv.org/abs/1210.7111.Google Scholar
Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward, D. E. (2002). Managing smile risk. Wilmott Magazine 2002, 84–108.Google Scholar
Henry-Labordère, P. (2009). Analysis, Geometry, and Modeling in Finance. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
Jacquier, A. and Lorig, M. (2013). The smile of certain Lévy-type models. SIAM J. Financial Math. 4, 804–830.Google Scholar
Jacquier, A., Keller-Ressel, M. and Mijatović, A. (2013). Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models. Stochastics 85, 321–345.CrossRefGoogle Scholar
Lee, R. W. (2004). The moment formula for implied volatility at extreme strikes. Math. Finance 14, 469–480.CrossRefGoogle Scholar
Lewis, A. L. (2001). A simple option formula for general jump-diffusion and other exponential Lévy processes. Preprint. Available at http://ssrn.com/abstract=282110.Google Scholar
Lipton, A. (2002). The vol smile problem. Risk (February 2002), 61–65.Google Scholar
Lorig, M. (2013). The exact smile of some local volatility models. Quant. Finance 13, 897–905.Google Scholar
Lorig, M., Pagliarani, S. and Pascucci, A. (2014). Explicit implied volatilities for multifactor local-stochastic volatility models. Preprint. Available at http://arxiv.org/abs/1306.5447.Google Scholar
Lukacs, E. (1970). Characteristic Functions, 2nd edn. Hafner, New York.Google Scholar
Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The variance gamma process and option pricing. Europ. Finance Rev. 2, 79–105.Google Scholar
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3, 125–144.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Takahashi, A. and Toda, M. (2013). Note on an extension of an asymptotic expansion scheme. Internat. J. Theoret. Appl. Finance 16, 1350031.CrossRefGoogle Scholar
Tankov, P. (2011). Pricing and hedging in exponential Lévy models: review of recent results. In Paris-Princeton Lecture Notes on Mathematical Finance, Springer, Berlin, pp. 319–359.Google Scholar