Abstract
In this paper we discuss a new approach to extend a class of solvable stochastic volatility models (SVM). Usually, classical SVM adopt a CEV process for instantaneous variance where the CEV parameter γ takes just few values: 0—the Ornstein–Uhlenbeck process, 1/2—the Heston (or square root) process, 1—GARCH, and 3/2—the 3/2 model. Some other models, e.g. with γ = 2 were discovered in Henry-Labordére (Analysis, geometry, and modeling in finance: advanced methods in option pricing. Chapman & Hall/CRC Financial Mathematics Series, London, 2009) by making connection between stochastic volatility and solvable diffusion processes in quantum mechanics. In particular, he used to build a bridge between solvable superpotentials (the Natanzon superpotentials, which allow reduction of a Schrödinger equation to a Gauss confluent hypergeometric equation) and existing SVM. Here we propose some new models with \({\gamma \in \mathbb{R}}\) and demonstrate that using Lie’s symmetries they could be priced in closed form in terms of hypergeometric functions. Thus obtained new models could be useful for pricing volatility derivatives (variance and volatility swaps, moment swaps).
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References
Albanese C., Campolieti G., Carr P., Lipton A. (2001) Black–Scholes goes hypergeometric. Risk Magazine 14: 99–103
Bakshi, G., Ju, N., & Yang, H. (2004). Estimation of continuous time models with an application to equity volatility. Technical report, University of Maryland working paper.
Brockhaus, O., & Long, D. (2000). Volatility swaps made simple. Risk, 92–93.
Carr P., Linetsky V. (2006) A jump to default extended CEV model: An application of Bessel processes. Finance and Stochastics 10: 303–330
Carr P., Sun J. (2007) A new approach for option pricing under stochastic volatility. Review of Derivatives Research 10: 87–250
Chacko G., Viceira L. (1999) Spectral gmm estimation of continuous-time processes. Harvard University, Technical report
Craddock M. (2009) Fundamental solutions, transition densities and the integration of Lie symmetries. Journal of Differential Equations 246: 2538–2560
Craddock M., Lennox K. (2007) Lie group symmetries as integral transforms of fundamental solutions. Journal of Differential Equations 232: 652–674
Davydov D., Linetsky V. (2001) Pricing and hedging path-dependent options under the CEV process. Management Science 47(7): 949–965
Gatheral, J. (2004). A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives. In: Global derivatives and risk management 2004, Madrid, May 26, 2004. Available online at http://faculty.baruch.cuny.edu/jgatheral/madrid2004.pdf.
Gatheral, J. (2008) Consistent Modeling of SPX and VIX options. In: The Fifth World Congress of the Bachelier Finance Society, London, July 18, 2008. Available online at http://www.math.nyu.edu/fellows_fin_math/gatheral/Bachelier2008.pdf.
Goldenberg D. (1991) A unified method for pricing options on diffusion-processes. Journal of Financial Economics 29: 3–34
Henry-Labordére P. (2009) Analysis, geometry, and modeling in finance: Advanced methods in option pricing. Chapman & Hall/CRC Financial Mathematics Series, London
Heston S. (1993) Closed-form solution for options with stochastic volatility, with applicationto bond and currency options. Review of Financial Studies 6(2): 327–343
In’t Hout K. J., Welfert B. D. (2007) Stability of ADI schemes applied to convection-diffusion equations with mixed derivative terms. Applied Numerical Mathematics 57: 19–35
Ishida, I., & Engle, R. (2002). Modelling variance of variance: The square root, the affine, and the CEV GARCH models. Technical report, NYU.
Itkin A., Carr P. (2010) Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case. Review Derivatives Research 13: 141–176
Javaheri, A. (2004). The volatility process: A study of stock market dynamics via parametric stochastic volatility models and a comparison to the information embedded in option prices. PhD thesis.
Jones C. (2003) The dynamics of stochastic volatility: Evidence from underlying and options markets. Journal of Econometrics 116: 118–224
Lavoie J., Osler T., Tremblay R. (1976) Fractional derivatives and special functions. SIAM Review 18: 240–268
Lee R. (2004) The moment formula for implied volatility at extreme strikes. Mathematical Finance 14(3): 469–480
Lee R. (2008) Gamma swaps. University of Chicago, Technical report
Lewis A. L. (2000) Option valuation under stochastic volatility. Finance Press, Newport Beach, CA
Lin Y.-N. (2007) Pricing VIX futures: Evidence from integrated physical and risk-neutral probability measures. The Journal of Futures Markets 27: 1175–1217
Olver P. (1993) Applications of Lie groups to differential equations, Vol. 107 of graduate texts in mathematics (2nd ed.). Springer, New York
Revuz D., Yor M. (1999) Continuous martingales and Brownian motion (3rd ed.). Springer, Berlin
Romano M., Touzi N. (1997) Contingent claims and market completeness in a stochastic volatility model. Mathematical Finance 7(4): 279–302
Schoutens W. (2005) Moment swaps. Quantitative Finance 5(6): 525–530
Sepp A. (2008) Pricing options on realized variance in Heston model with jumps in returns and volatility. Journal of Computational Finance 11(4): 33–70
Swishchuk A. (2004) Modeling of variance and volatility swaps for financial markets with stochastic volatilities. WILMOTT Magazine 2: 64–72
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I thank Peter Carr, Mark Craddock, Alexander Antonov, Igor Halperin, Michael Spector, Roza Galeeva and an anonymous referee for useful comments and discussion. I assume full responsibility for any remaining errors.
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Itkin, A. New solvable stochastic volatility models for pricing volatility derivatives. Rev Deriv Res 16, 111–134 (2013). https://doi.org/10.1007/s11147-012-9082-0
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DOI: https://doi.org/10.1007/s11147-012-9082-0
Keywords
- Volatility derivatives
- Variance swap
- Options
- Stochastic volatility model
- Lie symmetry
- Closed-form solution
- Pricing