Abstract
This paper is devoted to studying the difference between the fair strike of a volatility swap and the at-the-money implied volatility (ATMI) of a European call option. It is well known that the difference between these two quantities converges to zero as the time to maturity decreases. In this paper, we make use of a Malliavin calculus approach to derive an exact expression for this difference. This representation allows us to establish that the order of convergence is different in the correlated and uncorrelated cases, and that it depends on the behavior of the Malliavin derivative of the volatility process. In particular, we see that for volatilities driven by a fractional Brownian motion, this order depends on the corresponding Hurst parameter \(H\). Moreover, in the case \(H\geq 1/2\), we develop a model-free approximation formula for the volatility swap in terms of the ATMI and its skew.
Similar content being viewed by others
References
Akahori, J., Song, X., Wang, T.H.: Probability density of lognormal fractional SABR model. Working paper (2017). Available online at: arXiv:1702.08081
Alòs, E.: A generalization of the Hull and White formula with applications to option pricing approximation. Finance Stoch. 10, 353–365 (2006)
Alòs, E., León, J.A.: On the curvature of the smile in stochastic volatility models. SIAM J. Financ. Math. 8, 373–399 (2017)
Alòs, E., León, J.A., Vives, J.: On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch. 11, 571–589 (2007)
Bergomi, L., Guyon, J.: The smile in stochastic volatility models. Working paper (2011). Available online at: https://ssrn.com/abstract=1967470
Carr, P., Lee, R.: Robust replication of volatility derivatives, PRMIA award for Best Paper in Derivatives. In: MFA 2008 Annual Meeting (2008). Available online at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.378.189&rep=rep1&type=pdf
Carr, P., Lee, R.: Volatility derivatives. Annu. Rev. Finance Econ. 1, 319–339 (2009)
Comte, F., Renault, E.: Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291–323 (1998)
Demeterfi, K., Derman, E., Kamal, M., Zou, J.: A guide to volatility and variance swaps. J. Deriv. 6(4), 9–32 (1999)
El Euch, O., Fukasawa, M., Gatheral, J., Rosenbaum, M.: Short-term at-the-money asymptotics under stochastic volatility models. Working paper (2018). Available online at: https://ssrn.com/abstract=3111471
Feinstein, S.: Forecasting stock market volatility using options on index futures. Econ. Rev. 74(3), 12–30 (1989)
Friz, P., Gatheral, J.: Valuation of volatility derivatives as an inverse problem. Quant. Finance 5, 531–542 (2005)
Fukasawa, M.: Asymptotic analysis for stochastic volatility: martingale expansion. Finance Stoch. 15, 635–654 (2011)
Fukasawa, M.: Volatility derivatives and model-free implied leverage. Int. J. Theor. Appl. Finance 17, 1450002 (2014)
Gatheral, J., Jaisson, T., Rosenbaum, M.: Volatility is rough. Quant. Finance 18, 933–949 (2018)
Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)
Nualart, D., Răşcanu, A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53, 55–81 (2002)
Romano, M., Touzi, N.: Contingent claims and market completeness in a stochastic volatility model. Math. Finance 7, 399–410 (1997)
Willard, G.A.: Calculating prices and sensitivities for path-independent securities in multifactor models. J. Deriv. 5, 45–61 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
E.A. supported by grants ECO2014-59885-P and MTM2016-76420-P.
K.S. supported by CARF (Center for Advanced Research in Finance).
Rights and permissions
About this article
Cite this article
Alòs, E., Shiraya, K. Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach. Finance Stoch 23, 423–447 (2019). https://doi.org/10.1007/s00780-019-00384-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00780-019-00384-5