Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T07:01:33.641Z Has data issue: false hasContentIssue false
  • English
  • Français

AN ANALYTICAL APPROACH FOR VARIANCE SWAPS WITH AN ORNSTEIN–UHLENBECK PROCESS

Part of: Mathematical economics

Published online by Cambridge University Press:  19 July 2017

JIAN-PENG CAO Open the ORCID record for JIAN-PENG CAO [Opens in a new window]  and
YAN-BING FANG
JIAN-PENG CAO
Affiliation:
School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China email [email protected], [email protected]
YAN-BING FANG*
Affiliation:
School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China email [email protected], [email protected]
*
✉[email protected]
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.

Pricing variance swaps have become a popular subject recently, and most research of this type come under Heston’s two-factor model. This paper is an extension of some recent research which used the dimension-reduction technique based on the Heston model. A new closed-form pricing formula focusing on a log-return variance swap is presented here, under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model (Ornstein–Uhlenbeck process). Numerical tests in two respects using the Monte Carlo (MC) simulation are included. Moreover, we discuss a procedure of solving a quadratic differential equation with one variable. Our method can avoid the previously encountered limitations, but requires more time for calculation than other recent analytical discrete models.

Keywords

variance swapsOrnstein–Uhlenbeck processclosed-form solutionstochastic volatility

MSC classification

Primary: 91B70: Stochastic models
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 1 , July 2017 , pp. 83 - 102
Copyright
© 2017 Australian Mathematical Society 

References

Bakshi, G., Cao, C. and Chen, Z., “Empirical performance of alternative option pricing models”, J. Finance 52 (1997) 2003–2049 ; doi:10.1111/j.1540-6261.1997.tb02749.x.Google Scholar
Bates, D. S., “Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options”, Rev. Finac. 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 ; doi:10.1086/260062.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.Google Scholar
Brockhaus, O. and Long, D., “Volatility swaps made simple”, Risk 19 (2000) 92–95, https://www.researchgate.net/publication/248350031_Volatility_Swaps_Made_Simple_Risk.Google Scholar
Carr, P. and Corso, A., “Commodity covariance contracting”, Energy Risk (2001) 42–45, http://engineering.nyu.edu/files/twrdsfig.pdf.Google Scholar
Carr, P. and Lee, R., “Robust replication of volatility derivatives”, in: Mathematics in finance working paper series (Courant Institute of Mathematical Sciences, New York University, NY, 2008) 1–48; https://pdfs.semanticscholar.org/2ea8/6c38b34913eedd05a79c880198a27b824217.pdf.Google Scholar
Carr, P. and Lee, R., “Realised volatility and variance: options via swaps”, Risk 5 (2007) 76–83, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.463.7161&rep=rep1&type=pdf.Google 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; doi:10.1017/CBO9780511569708.013.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, 1–52, http://www.emanuelderman.com/writing/entry/more-than-you-ever-wanted- to-know-about-volatility-swaps-the-journal-of-der.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(1) (2007) 41–62 ; doi:10.1080/13504860600659222.Google Scholar
Garman, M. B., “A general theory of asset valuation under diffusion state processes”, in: Research program in finance working paper (Center for Research in Management Science, Berkley, CA, 1976) , http://EconPapers.repec.org/RePEc:ucb:calbrf:50.Google 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.CrossRefGoogle Scholar
Heston, S. L. and Nandi, S., “Derivatives on volatility: some simple solutions based on observables”, in: Working paper series (Federal Reserve Bank of Atlanta, GA, 2000) 1–19; https://pdfs.semanticscholar.org/d2db/dfa9b5957efb5f78657c6aded9da1bcbe57b.pdf.Google Scholar
Javaheri, A., Wilmott, P. and Haug, E., “GARCH and volatility swaps”, Quant. Finance 4 (2004) 589–595 ; doi:10.1080/14697680400000040.Google Scholar
Jia, Z. L., Bi, X. C. and Zhang, S. G., “Pricing variance swaps under stochastic volatility with an Ornstein–Uhlenbeck process”, J. Syst. Complex 28 (2015) 1412–1425 ; doi:10.1007/s11424-015-3165-6.Google Scholar
Lian, G. H., Chiarella, C. and Kalev, P. S., “Volatility swaps and volatility options on discretely sampled realized variance”, J. Dynamics Control 47 (2014) 239–262 ; doi:10.1016/j.jedc.2014.08.014.Google Scholar
Little, T. and Pant, V., “A finite-difference method for the valuation of variance swaps”, J. Comput. Finance 5 (2001) 81–103 ; doi:10.21314/JCF.2001.057.CrossRefGoogle Scholar
Merton, R., “The theory of rational option pricing”, Bell J. Econom. Manage Sci. 1 (1973) 141–183 ; doi:10.2307/3003143.Google Scholar
Poularikas, A. D., The transforms and applications handbook, 2nd edn The electrical engineering handbook series (CRC Press LLC, Boca Raton, 2000).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.Google Scholar
Rujivan, S. and Zhu, S. P., “A simple closed-form formula for pricing discretely-sampled variance swaps under the Heston model”, ANZIAM J. 56 (2014) 1–27 ; doi:10.21914/anziamj.v56i0.7455.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
Sepp, A., “Pricing options on realized variance in the Heston model with jumps in returns and volatility”, J. Comput. Finance 11 (2008) 33–70 ; doi:10.21314/JCF.2008.185.Google Scholar
Stein, E. M. and Stein, J. E., “Stock price distributions with stochastic volatility and interest rates”, Rev. Financ. Stud. 4 (1991) 727–752 ; doi:10.1093/rfs/4.4.727.Google Scholar
Swishchuk, A., “Modeling of variance and volatility swaps for financial markets with stochastic volatilities, Wilmott magazine September issue”, Technical Article 2 (2004) 64–72 ;http://math.ucalgary.ca/files/finlab/StochVolatSwap.pdf.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
Zhang, L.-W., “A closed-form pricing formula for variance swaps with mean-reverting Gaussian volatility”, ANZIAM J. 55 (2014) 362–382 ; doi:10.1017/S144618111400011X.Google Scholar
Zheng, W. D. and Kwok, Y. K., “Closed form pricing formulas for discretely sampled generalized variance swaps”, Math. Finance 24 (2014) 855–881 ; 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
Zhu, S. P. and Lian, G. H., “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
Zhu, S. P. and Lian, G. H., “Pricing forward-start variance swaps with stochastic volatility”, Appl. Math. Comput. 250 (2015) 920–933 ; doi:10.1016/j.amc.2014.10.050.Google Scholar