Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-19T03:46:09.027Z Has data issue: false hasContentIssue false

APPROXIMATE PRICING OF DERIVATIVES UNDER FRACTIONAL STOCHASTIC VOLATILITY MODEL

Published online by Cambridge University Press:  15 January 2024

Y. HAN*
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, China; e-mail: zxd22@mails.jlu.edu.cn
X. ZHENG
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, China; e-mail: zxd22@mails.jlu.edu.cn
*

Abstract

This paper examines the issue of derivative pricing within the framework of a fractional stochastic volatility model. We present a deterministic partial differential equation system to derive an approximate expression for the derivative price. The proposed approach allows for the stochastic volatility to be expressed as a composition of deterministic functions of time and a fractional Ornstein–Uhlenbeck process. We apply this method to the European option pricing under the fractional Stein–Stein volatility model, demonstrating its feasibility and reliability through numerical simulations. Our numerical simulations also illustrate the impact of the parameters in the fractional stochastic volatility model on the option price.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alòs, E., Mazet, O. and Nualart, D., “Stochastic calculus with respect to Gaussian processes”, Ann. Probab. 29 (2001) 766801; doi:10.1214/aop/1008956692.CrossRefGoogle Scholar
Bayer, C., Friz, P. and Gatheral, J., “Pricing under rough volatility”, Quant. Finance 16 (2016) 887904; doi:10.1080/14697688.2015.1099717.CrossRefGoogle Scholar
Beben, M. and Orłowski, A., “Correlations in financial time series: established versus emerging markets”, Eur. Phys. J. B 20 (2001) 527530; doi:10.1007/s100510170233.CrossRefGoogle Scholar
Biagini, F., Øksendal, B., Sulem, A. and Wallner, N., “An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion”, Proc. Roy. Soc. Lond. Ser. A 460 (2004) 347372; doi:10.1098/rspa.2003.1246.CrossRefGoogle Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Polit. Econ. 81 (1973) 637657; doi:10.1086/260062.CrossRefGoogle Scholar
Cheridito, P., Kawaguchi, H. and Maejima, M., “Fractional Ornstein–Uhlenbeck processes”, Electron. J. Probab. 8 (2003) 114; doi:10.1214/EJP.v8-125.CrossRefGoogle Scholar
Chernov, M., Gallant, A. R., Ghysels, E. and Tauchen, G., “Alternative models for stock price dynamics”, J. Econom. 116 (2003) 225257; doi:10.1016/S0304-4076(03)00108-8.CrossRefGoogle Scholar
Decreusefond, L. and Ustunel, A. S., “Stochastic analysis of the fractional Brownian motion”, Potential Anal. 10 (1999) 177214; doi:10.1023/A:1008634027843.CrossRefGoogle Scholar
Duncan, T. E., Hu, Y. and Pasik-Duncan, B., “Stochastic calculus for fractional Brownian motion. I. Theory”, SIAM J. Control Optim. 38 (2000) 582612; doi:10.1137/S036301299834171X.CrossRefGoogle Scholar
Elliott, R. J., Siu, T. and Chan, L., “Pricing volatility swaps under Heston’s stochastic volatility model with regime switching”, Appl. Math. Finance 14 (2007) 4162; doi:10.1080/13504860600659222.CrossRefGoogle Scholar
Elliott, R. J. and Van Der Hoek, J., “A general fractional white noise theory and applications to finance”, Math. Finance 13 (2003) 301330; doi:10.1111/1467-9965.00018.CrossRefGoogle Scholar
Garnier, J. and Sølna, K., “Correction to Black–Scholes formula due to fractional stochastic volatility”, SIAM J. Financial Math. 8 (2017) 560588; doi:10.1137/15M1036749.CrossRefGoogle Scholar
Gatheral, J., Jaisson, T. and Rosenbaum, M., “Volatility is rough”, Quant. Finance 18 (2018) 933949; doi:10.1080/14697688.2017.1393551.CrossRefGoogle Scholar
Gillespie, D. T., “The chemical Langevin equation”, J. Chem. Phys. 113 (2000) 297306; doi:10.1063/1.481811.CrossRefGoogle Scholar
Gulisashvili, A., Viens, F. and Zhang, X., “Extreme-strike asymptotics for general Gaussian stochastic volatility models”, Ann. Finance 15 (2019) 59101; doi:10.1007/s10436-018-0338-z.CrossRefGoogle Scholar
Hu, Y. and Øksendal, B., “Fractional white noise calculus and applications to finance”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003) 132; doi:10.1142/S0219025703001110.CrossRefGoogle Scholar
Huang, B. and Yang, C. W., “The fractal structure in multinational stock returns”, Appl. Econ. Lett. 2 (1995) 6771; doi:10.1080/135048595357591.CrossRefGoogle Scholar
Johnson, H. and Shanno, D., “Option pricing when the variance is changing”, J. Financ. Quant. Anal. 22 (1987) 143151; doi:10.2307/2330709.CrossRefGoogle Scholar
Mandelbrot, B. B. and Van Ness, J. W., “Fractional Brownian motions, fractional noises and applications”, SIAM Rev. 10 (1968) 422437; doi:10.1137/1010093.CrossRefGoogle Scholar
Merton, R. C., “Theory of rational option pricing”, Bell J. Econom. Manag. Sci. 4 (1973) 141183; doi:10.2307/3003143.CrossRefGoogle Scholar
Necula, C., “Option pricing in a fractional Brownian motion environment”, Math. Rep. (Bucur.) 6 (2004) 259273; doi:10.2139/ssrn.1286833.Google Scholar
Neuberger, A., “The log contract”, J. Portf. Manag. 20 (1994) 7480; doi:10.3905/jpm.1994.409478.CrossRefGoogle Scholar
Rujivan, S. and Zhu, S., “A simple closed-form formula for pricing discretely-sampled variance swaps under the Heston model”, ANZIAM J. 56 (2014) 127; doi:10.1017/S1446181114000236.Google Scholar
Sepp, A., “Pricing options on realized variance in the Heston model with jumps in returns and volatility Part II. An approximate distribution of discrete variance”, J. Comput. Finance 16 (2012) 332; doi:10.21314/JCF.2012.240.CrossRefGoogle Scholar
Stein, E. M. and Stein, J. C., “Stock-price distributions with stochastic volatility – an analytic approach”, Rev. Financ. Stud. 4 (1991) 727752; doi:10.1093/rfs/4.4.727.CrossRefGoogle Scholar
Wiggins, J. B., “Option values under stochastic volatility: theory and empirical estimates”, J. Financ. Econ. 19 (1987) 351372; doi:10.1016/0304-405X(87)90009-2.CrossRefGoogle Scholar
Zhu, S. and Lian, G., “A closed-form exact solution for pricing variance swaps with stochastic volatility”, Math. Finance 21 (2011) 233256; doi:10.1111/j.1467-9965.2010.00436.x.CrossRefGoogle Scholar