Asymptotic Analysis of Implied Volatility
The implied volatility was first introduced by H.A. Latané and R.J. Rendleman under the name “the implied standard deviation”. Latané and Rendleman studied standard deviations of asset returns, which are implied in actual call option prices when investors price options according to the Black-Scholes model. Chapter 9 mainly is concerned with the asymptotics of the implied volatility at extreme strikes. It presents sharp model-free asymptotic formulas for the implied volatility established by the author and the higher order extensions of these formulas found by K. Gao and R. Lee. The chapter also provides a characterization of implied volatility models free of static arbitrage, and discusses certain symmetries hidden in stochastic asset price models. These symmetries can be used to analyze the asymptotic behavior of the implied volatility at small strikes knowing how the volatility behaves at large strikes.
KeywordsAsymptotic Formula Implied Volatility Symmetric Model Static Arbitrage Strike Price
- [CL10]Carr, P., Lee, R., Implied volatility in stochastic volatility models, in: Encyclopedia of Quantitative Finance, 2010. Google Scholar
- [DM10]De Marco, S., On probability distributions of diffusions and financial models with non-globally smooth coefficients, Ph.D. Dissertation, Université Paris-Est Marne-la-Vallée and Scuola Normale Superiore di Pisa, 2010. Google Scholar
- [DMM10]De Marco, S., Martini, C., The term structure of implied volatility in symmetric models with applications to Heston, preprint, 2010; available at papers.ssrn.com/sol3/papers.cfm?abstract_id=1622828.
- [Dur04]Durrleman, V., From implied to spot volatilities, Ph.D. Dissertation, Princeton University, 2004. Google Scholar
- [Fri10]Friz, P., Implied volatility: large strike asymptotics, in: Cont, R. (Ed.), Encyclopedia of Quantitative Finance, pp. 909–913, Wiley, Chichester, 2010. Google Scholar
- [GL11]Gao, K., Lee, R., Asymptotics of implied volatility to arbitrary order, preprint, 2011; available at http://ssrn.com/abstract=1768383.
- [Gat06]Gatheral, J., The Volatility Surface: A Practitioner’s Guide, Wiley, Hoboken, 2006. Google Scholar
- [LR76]Latané, H. A., Rendleman, R. J. Jr., Standard deviations of stock price ratios implied in option prices, Journal of Finance 31 (1976), pp. 369–381. Google Scholar
- [Lee04a]Lee, R., Implied volatility: statics, dynamics, and probabilistic interpretation, in: Baeza-Yates, R., Glaz, J., Ghyz, H., et al. (Eds.), Recent Advances in Applied Probability, Springer, Berlin, 2004. Google Scholar
- [Reb04]Rebonato, R., Volatility and Correlation: The Perfect Hedger and the Fox, 2nd ed., Wiley Finance, New York, 2004. Google Scholar
- [Rop09]Roper, M., Implied volatility: general properties and asymptotics, Ph.D. thesis, The University of New South Wales, 2009. Google Scholar
- [Rop10]Roper, M., Arbitrage free implied volatility surfaces, preprint, 2010. Google Scholar