Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-25T10:16:34.170Z Has data issue: false hasContentIssue false
  • English
  • Français

A CLOSED-FORM PRICING FORMULA FOR VARIANCE SWAPS WITH MEAN-REVERTING GAUSSIAN VOLATILITY

Part of: Mathematical finance

Published online by Cambridge University Press:  10 September 2014

LI-WEI ZHANG
LI-WEI ZHANG*
Affiliation:
School of Mathematics, Jilin University, Changchun, 130012, China email jluzlw@163.com
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.

Although variance swaps have become an important financial derivative to hedge against volatility risks, closed-form formulae have been developed only recently, when the realized variance is defined on discrete sample points and no continuous approximation is adopted to alleviate the mathematical difficulties associated with dealing with the discreteness of the sample data. In this paper, a new closed-form pricing formula for the value of a discretely sampled variance swap is presented under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model. With the newly found analytical formula, not only can all the hedging ratios of a variance swap be analytically derived, the numerical values of the swap price can be efficiently computed as well.

Keywords

variance swapsmean-reverting Gaussian volatility modelclosed-form solutiondiscrete sampling

MSC classification

Secondary: 91G20: Derivative securities
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 4 , April 2014 , pp. 362 - 382
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Aonuma, K. and Nakagawa, H., “Valuation of credit default swap and parameter estimation for Vasicek-type hazard rate model”, Working Paper, University of Tokyo, 1998.Google Scholar
Bates, D. S., “Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options”, Rev. Financ. Stud. 9 (1996) 69–107; doi:10.1093/rfs/9.1.69.Google Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Polit. Econ. 81 (1973) 637–654.CrossRefGoogle Scholar
Brenner, M., Ou, E. Y. and Zhang, J. E., “Hedging volatility risk”, J. Banking Finance 30 (2006) 811–821; doi:10.1016/j.jbankfin.2005.07.015.Google Scholar
Broadie, M. and Jain, A., “The effect of jumps and discrete sampling on volatility and variance swaps”, Int. J. Theor. Appl. Finance 11 (2008) 761–797; doi:10.1142/S0219024908005032.CrossRefGoogle Scholar
Carr, P. and Corso, A., “Commodity covariance contracting”, Energy Risk–April (2001) 42–45.Google Scholar
Carr, P. and Lee, R., “Volatility derivatives”, Annu. Rev. Financ. Econom. 1 (2009) 319–339 ; doi:10.1146/annurev.financial.050808.114304.CrossRefGoogle Scholar
Carr, P. and Madan, D., “Towards a theory of volatility trading”, in: Volatility: new estimation techniques for pricing derivatives (Risk Publications, London, 1998), 417–427.Google Scholar
Demeterfi, K., Derman, E., Kamal, M. and Zou, J., “More than you ever wanted to know about volatility swaps”, Technical Report, Goldman Sachs Quantitative Strategies Research Notes, 1999.Google Scholar
Dupire, B., “Exploring volatility derivatives: new advances in modelling”, Presentation at New York University, 2005.Google Scholar
Elliott, R., Siu, T. and Chan, L., “Pricing volatility swaps under Heston’s stochastic volatility model with regime switching”, Appl. Math. Finance 14 (2007) 41–62; doi:10.1080/13504860600659222.Google Scholar
Elliott, R. J., Van Der Hoek, J. and Malcolm, W. P., “Pairs trading”, Quant. Finance 5 (2005) 271–276; doi:10.1080/14697680500149370.Google Scholar
Garman, M. B., “A general theory of asset valuation under diffusion state processes”, Technical Report, University of California at Berkeley, 1976.Google Scholar
Gorovoi, V. and Linetsky, V., “Black’s model of interest rates as options, eigenfunction expansions and Japanese interest rates”, Math. Finance 14 (2004) 49–78 ; doi:10.1111/j.0960-1627.2004.00181.x.CrossRefGoogle Scholar
Grunbichler, A. and Longstaff, F., “Valuing futures and options on volatility”, J. Banking Finance 20 (1996) 985–1001; doi:10.1016/0378-4266(95)00034-8.Google Scholar
Heston, S. L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Financ. Stud. 6 (1993) 327–343; doi:10.1093/rfs/6.2.327.Google Scholar
Heston, S. L. and Nandi, S., “A closed-form GARCH option valuation model”, Rev. Financ. Stud. 13 (2000) 585–625; doi:10.1093/rfs/13.3.585.Google Scholar
Heston, S. L. and Nandi, S., “Derivatives on volatility: some simple solutions based on observables”, Federal Reserve Bank of Atlanta, 2000.Google Scholar
Howison, S., Rafailidis, A. and Rasmussen, H., “On the pricing and hedging of volatility derivatives”, Appl. Math. Finance 11 (2004) 317–346; doi:10.1080/1350486042000254024.Google Scholar
Little, T. and Pant, V., “A finite-difference method for the valuation of variance swaps”, Quantitative Analysis in Financial Markets: Collected Papers of the New York University Mathematical Finance Seminar, 2001.Google Scholar
Lucia, J. J. and Schwartz, E. S., “Electricity prices and power derivatives: evidence from the Nordic power exchange”, Rev. Deriv. Res. 5 (2002) 5–50; doi:10.1023/A:1013846631785.Google Scholar
Merton, R., “The theory of rational option pricing”, Bell J. Econom. Manage. Sci. 1 (1973) 141–183; doi:10.2307/3003143.Google Scholar
Nelson, D. B., “ARCH models as diffusion approximations”, J. Econom. 45 (1990) 7–38 ; doi:10.1016/0304-4076(90)90092-8.Google Scholar
Øksendal, B., Stochastic differential equations (Springer, Berlin Heidelberg, 2003).CrossRefGoogle Scholar
Rujivan, S. and Zhu, S. P., “A simplified analytical approach for pricing discretely sampled variance swaps with stochastic volatility”, Appl. Math. Lett. 25 (2012) 1644–1650 ; doi:10.1016/j.aml.2012.01.029.Google Scholar
Schöbel, R. and Zhu, J., “Stochastic volatility with an Ornstein–Uhlenbeck process: an extension”, Eur. Financ. Rev. 3 (1999) 23–46; doi:10.1023/A:1009803506170.Google Scholar
Schoutens, W., “Moment swaps”, Quant. Finance 5 (2005) 525–530 ; doi:10.1080/14697680500401490.Google Scholar
Schwartz, E. S., “The stochastic behavior of commodity prices: implications for valuation and hedging”, J. Finance 52 (1997) 923–973; doi:10.1111/j.1540-6261.1997.tb02721.x.Google Scholar
Scott, L. O., “Option pricing when the variance changes randomly: theory, estimation, and an application”, J. Financ. Quant. Anal. 22 (1987) 419–438; doi:10.2307/2330793.Google Scholar
Scott, L. O., “Pricing stock options in a jump diffusion model with stochastic volatility and interest rates: applications of Fourier inversion methods”, Math. Finance 7 (1997) 413–426 doi: 10.1111/1467-9965.00039.Google Scholar
Stein, E. and Stein, J., “Stock price distributions with stochastic volatility: an analytic approach”, Rev. Financ. Stud. 4 (1991) 727–752; doi:10.1093/rfs/4.4.727.CrossRefGoogle Scholar
Vasicek, O., “An equilibrium characterization of the term structure”, J. Financ. Econom. 5 (1977) 177–188; doi:10.1016/0304-405X(77)90016-2.Google Scholar
Williams, D., Probability with martingales (Cambridge University Press, Cambridge, 1991).Google Scholar
Windcliff, H., Forsyth, P. and Vetzal, K., “Pricing methods and hedging strategies for volatility derivatives”, J. Banking Finance 30 (2006) 409–431; doi:10.1016/j.jbankfin.2005.04.025.Google Scholar
Zheng, W. and Kwok, Y. K., “Closed form pricing formulas for discretely sampled generalized variance swaps”, Math. Finance (2012) in press; doi: 10.1111/mafi.12016.Google Scholar
Zhu, S. P. and Lian, G. H., “A closed-form exact solution for pricing variance swaps with stochastic volatility”, Math. Finance 21 (2011) 233–256; doi:10.1111/j.1467-9965.2010.00436.x.Google Scholar