Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-25T05:20:49.150Z Has data issue: false hasContentIssue false
  • English
  • Français

A SIMPLE CLOSED-FORM FORMULA FOR PRICING DISCRETELY-SAMPLED VARIANCE SWAPS UNDER THE HESTON MODEL

Part of: Mathematical finance

Published online by Cambridge University Press:  09 October 2014

SANAE RUJIVAN  and
SONG-PING ZHU
SANAE RUJIVAN*
Affiliation:
Division of Mathematics, School of Science, Walailak University, Nakhon Si Thammarat 80161, Thailand email rsanae@wu.ac.th
SONG-PING ZHU
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email spz@uow.edu.au
*
✉rsanae@wu.ac.th
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.

We develop a simplified analytical approach for pricing discretely-sampled variance swaps with the realized variance, defined in terms of the squared log return of the underlying price. The closed-form formula obtained for Heston’s two-factor stochastic volatility model is in a much simpler form than those proposed in literature. Most interestingly, we discuss the validity of our solution as well as some other previous solutions in different forms in the parameter space. We demonstrate that market practitioners need to be cautious, making sure that their model parameters extracted from market data are in the right parameter subspace, when any of these analytical pricing formulae is adopted to calculate the fair delivery price of a discretely-sampled variance swap.

Keywords

variance swapsHeston modelclosed-form exact solutionstochastic volatility

MSC classification

Secondary: 91G20: Derivative securities
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 1 , July 2014 , pp. 1 - 27
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Aït-Sahalia, Y. and Kimmel, R., “Maximum likelihood estimation of stochastic volatility models”, J. Financ. Econ. 83 (2007) 413–452; doi:10.1.1.118.1164.CrossRefGoogle Scholar
Bates, D. S., “Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options”, Rev. Financ. Stud. (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; doi:10.1086/260062.CrossRefGoogle 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
Brzez̀niak, Z. and Thomasz, Z., Basic stochastic processes (Springer-Verlag, Berlin, 2003).Google Scholar
Carr, P. and Corso, A., “Commodity covariance contracting”, Energy and Power Risk Management (2001), available at http://www.math.nyu.edu/research/carrp/papers/pdf/comprice.pdf.Google Scholar
Carr, P. and Madan, D., “Towards a theory of volatility trading”, in: Volatility (ed. Jarrow, R.), (Risk Publications, 1998), 417–427, available athttp://www.math.nyu.edu/research/carrp/papers/pdf/twrdsfig.pdf.Google Scholar
Cox, J. C., Ingersoll, J. E. Jr and Ross, S. A., “A theory of the term structure of interest rates”, Econometrica 53 (1985) 385–407; doi:10.2307/1911242.Google Scholar
Demeterfi, K., Derman, E., Kamal, M. and Zou, J., “More than you ever wanted to know about volatility swaps”, Goldman Sachs Quantitative Strategies Research Notes, 1999.Google Scholar
Garman, M., “A general theory of asset valuation under diffusion state processes”, Working paper No. 50, University of California, Berkeley, 1977, 1–54.Google Scholar
Harrison, M. and Kreps, D., “Martingales and arbitrage in multi-period securities markets”, in: Contigencies in finance (Department of Applied Economics, University of Cambridge, 1979), 381–408.Google Scholar
Harrison, M. and Pliska, R., “Martingales and stochastic integrals in the theory of continuous trading”, J. Econom. Theory 11 (1981) 215–260; doi:10.1016/0304-4149(81)90026-0.Google Scholar
Heston, S. L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Finance Stud. 6 (1993) 327–343; doi:10.1093/rfs/6.2.327.CrossRefGoogle Scholar
Jacod, J. and Protter, P., “Asymptotic error distributions for the Euler method for stochastic differential equations”, Ann. Probab. 26 (1998) 267–307; doi:10.1.1.212.7745.CrossRefGoogle Scholar
Jarrow, R., Kchia, Y., Larsson, M. and Protter, P., “Discretely sampled variance and volatility swaps versus their continuous approximations”, Financ. Stoch. 17 (2011) 305–324; doi:10.1007/s00780-012-0183-2.Google Scholar
Øksendal, B., Stochastic differential equations (Springer-Verlag, Berlin, 2003).CrossRefGoogle Scholar
Little, T. and Pant, V., “A finite-difference method for the valuation of variance swaps”, J. Comput. Financ. 5 (2001) 81–101; doi:10.1142/9789812778451_0012.Google Scholar
Merton, R. C., “Theory of rational option pricing”, Bell J. Econ. 4 (1973) 141–183; doi:10.2307/3003143.Google 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Swishchuk, A., “Modeling of variance and volatility swaps for financial markets with stochastic volatilities”, Wilmott magazine, September, Technical Article (2004) 64–72,http://people.ucalgary.ca/ aswish/StochVolatSwap.pdf.Google Scholar
Swishchuk, A. and Li, X., “Pricing variance swaps for stochastic volatilities with delay and jumps”, Int. J. Stoch. Anal. 2011 (2011); doi:10.1155/2011/435145.Google Scholar
Zhang, J. and Zhu, Y., “VIX futures”, J. Futures Markets 5 (2006) 521–531; doi:10.1002/fut.20209.Google Scholar
Zheng, W. and Kwok, Y., “Closed form pricing formulas for discretely sampled generalized variance swaps”, Math. Finance (2013); doi:10.1111/mafi.12016.Google Scholar
Zhu, S.-P. and Lian, and G., “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
Zhu, S.-P. and Lian, and G., “On the valuation of variance swaps with stochastic volatility”, Appl. Math. Comput. 219 (2012) 1654–1669; doi:10.1016/j.amc.2012.08.006.Google Scholar