No Access

TIME-CHANGED FAST MEAN-REVERTING STOCHASTIC VOLATILITY MODELS

MATTHEW LORIG

Department of Operations Research & Financial Engineering, Princeton University, Princeton, NJ 08544, USA

https://doi.org/10.1142/S0219024911006875Cited by:8 (Source: Crossref)

Abstract

We introduce a class of randomly time-changed fast mean-reverting stochastic volatility (TC-FMR-SV) models. Using spectral theory and singular perturbation techniques, we derive an approximation for the price of any European option in the TC-FMR-SV setting. Three examples of random time-changes are provided and are shown to induce distinct implied volatility surfaces. The key features of the TC-FMR-SV framework are that (i) it is able to incorporate jumps into the price process of the underlying asset (ii) it allows for the leverage effect and (iii) it can accommodate multiple factors of volatility, which operate on different time-scales.

Keywords:

References

M. Al-Gwaiz , Sturm-Liouville Theory and Its Applications ( Springer-Verlag , 2008 ) . Google Scholar
S. Alizadeh, M. Brandt and F. Diebold, The Journal of Finance 57, 1047 (2002), DOI: 10.1111/1540-6261.00454. Crossref, Web of Science, Google Scholar
W. Amrein , A. Hinz and D. Pearson , Sturm-Liouville Theory: Past and Present ( Birkhauser , 2005 ) . Crossref, Google Scholar
T. G. Andersen and T. Bollerslev, Journal of Empirical Finance 4, 115 (1997), DOI: 10.1016/S0927-5398(97)00004-2. Crossref, Google Scholar
D. Bates, Review of Financial Studies 9, 69 (1996), DOI: 10.1093/rfs/9.1.69. Crossref, Web of Science, Google Scholar
J. Bertoin, Subordinators: Examples and Applications, Lecture Notes on Probability Theory and Statistics (2004) pp. 1–91. Google Scholar
J. Bouchaud, A. Matacz and M. Potters, Physical Review Letters 87, 228701–1 (2001), DOI: 10.1103/PhysRevLett.87.228701. Google Scholar
M. Chernovet al., Journal of Econometrics 116, 225 (2003), DOI: 10.1016/S0304-4076(03)00108-8. Crossref, Web of Science, Google Scholar
D. Duffie, J. Pan and K. Singleton, Econometrica 68, 1343 (2000), DOI: 10.1111/1468-0262.00164. Crossref, Web of Science, Google Scholar
R. Engle and A. Patton, Quantitative Finance 1, 237 (2001), DOI: 10.1088/1469-7688/1/2/305. Crossref, Web of Science, Google Scholar
G. Fiorentini, A. Leon and G. Rubio, Journal of Empirical Finance 9, 225 (2002), DOI: 10.1016/S0927-5398(01)00052-4. Crossref, Web of Science, Google Scholar
J.-P. Fouque et al. , Wave Propagation and Time Reversal in Randomly Layered Media ( Springer-Verlag , 2007 ) . Google Scholar
J.-P. Fouque and M. J. Lorig, SIAM Journal on Financial Mathematics 2, 221 (2011), DOI: 10.1137/090761458. Crossref, Web of Science, Google Scholar
J.-P. Fouque , G. Papanicolaou and R. Sircar , Derivatives in Financial Markets with Stochastic Volatility ( Cambridge University Press , 2000 ) . Google Scholar
J.-P. Fouqueet al., Computational Finance 6, 1 (2003). Crossref, Web of Science, Google Scholar
J.-P. Fouqueet al., SIAM J. Applied Mathematics 63, 1648 (2003). Crossref, Web of Science, Google Scholar
J.-P. Fouqueet al., Multiscale Modeling and Simulation 2, 22 (2004), DOI: 10.1137/030600291. Crossref, Web of Science, Google Scholar
J. Gatheral , Modeling the Implied Volatility Surface , Global Derivatives and Risk Management ( Barcelona , 2003 ) . Google Scholar
S. Heston, Rev. Financ. Stud. 6, 327 (1993), DOI: 10.1093/rfs/6.2.327. Crossref, Web of Science, Google Scholar
E. Hillebrand, Journal of Econometrics 129, 121 (2005), DOI: 10.1016/j.jeconom.2004.09.005. Crossref, Web of Science, Google Scholar
E. Hillebrand, Advances in Econometrics 20, 153 (2006), DOI: 10.1016/S0731-9053(05)20025-7. Crossref, Google Scholar
D. Hinton and P. Schaefer , Spectral Theory and Computational Methods of Sturm-Liouville Problems ( CRC , 1997 ) . Google Scholar
J. Hull and A. White, The Journal of Finance 42, 281 (1987), DOI: 10.2307/2328253. Crossref, Web of Science, Google Scholar
D. Lamberton and B. Lapeyre , Introduction to Stochastic Calculus Applied to Finance ( Chapman & Hall , 1996 ) . Google Scholar
B. LeBaron, Quantitative Finance 1, 621 (2001), DOI: 10.1088/1469-7688/1/6/304. Crossref, Google Scholar
A. Melino and S. M. Turnbull, Journal of Econometrics 45, 239 (1990), DOI: 10.1016/0304-4076(90)90100-8. Crossref, Web of Science, Google Scholar
R. Mendoza-Arriaga, P. Carr and V. Linetsky, Mathematical Finance 20, 527 (2010), DOI: 10.1111/j.1467-9965.2010.00411.x. Crossref, Web of Science, Google Scholar
U. A. Mulleret al., Journal of Empirical Finance 4, 213 (1997), DOI: 10.1111/j.1467-9965.2010.00411.x. Crossref, Web of Science, Google Scholar
B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edn. (Springer-Verlag, 2005). Google Scholar
J. Perelló, J. Masoliver and J. Bouchaud, Applied Mathematical Finance 11, 27 (2004), DOI: 10.1080/1350486042000196155. Crossref, Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics. Volume I: Functional Analysis ( Academic press , 1980 ) . Google Scholar
W. Rudin , Functional Analysis ( McGraw-Hill , New York , 1973 ) . Google Scholar
K. Sato , Lévy Processes and Infinitely Divisible Distributions ( Cambridge University Press , 1999 ) . Google Scholar
L. Scott, Mathematical Finance 7, 413 (1997), DOI: 10.1111/1467-9965.00039. Crossref, Web of Science, Google Scholar
I. Stakgold , Boundary Value Problems of Mathematical Physics ( Society for Industrial Mathematics , 2000 ) . Google Scholar
J. Zhang and J. Shu, Pricing Standard & Poor's 500 index options with Heston's model, Proceedings on Computational Intelligence for Financial Engineering, IEEE International Conference (2003) pp. 85–92. Google Scholar
D. Zill and M. Cullen , Differential Equations with Boundary-Value Problems ( Brooks/Cole Pub Co , 2008 ) . Google Scholar