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APPROXIMATING LOCAL VOLATILITY FUNCTIONS OF STOCHASTIC VOLATILITY MODELS: A CLOSED-FORM EXPANSION APPROACH

Published online by Cambridge University Press:  14 July 2015

Yu An  and
Chenxu Li
Yu An
Affiliation:
Graduate School of Business, Stanford University E-mail: [email protected]
Chenxu Li
Affiliation:
Guanghua School of Management, Peking University E-mail: [email protected]
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Abstract

We propose a method for approximating equivalent local volatility functions of stochastic volatility models. Enlightened by the theory of generalized Wiener functionals proposed by Watanabe and Yoshida (1987, 1992), our key technique is to propose a closed-form expansion of conditional expectations involving marginal distributions generated by stochastic differential equations. A numerical test and an illustration of application are provided to demonstrate the efficiency of our approach.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 4 , October 2015 , pp. 547 - 563
Copyright
Copyright © Cambridge University Press 2015 

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References

1. Antonov, A. & Misirpashaev, T. (2006). Efficient calibration to FX options by Markovian projection in cross-currency LIBOR market models. SSRN eLibrary: http://ssrn.com/abstract=936087.Google Scholar
2. Antonov, A. & Misirpashaev, T. (2009). Markovian projection onto a displaced diffusion: generic formulas with applications. International Journal of Theoretical and Applied Finance 12: 507–522.Google Scholar
3. Antonov, A., Misirpashaev, T. & Piterbarg, V. (2009). Markovian projection onto a Heston model. Journal of Computational Finance 13: 23.Google Scholar
4. Avellaneda, M., Boyer-Olson, D., Busca, J., & Friz, P. (2002). Reconstructing volatility. RISK 15: 87–92.Google Scholar
5. Benhamou, E. (2002). Smart Monte Carlo: various tricks using Malliavin calculus. Quantitative Finance 2(5): 329–336.Google Scholar
6. Berestycki, H., Busca, J. & Florent, I. (2004). Computing the implied volatility in stochastic volatility models. Communications on Pure and Applied Mathematics 57: 1352–1373.Google Scholar
7. Bouchard, B., Ekeland, I. & Touzi, N. (2004). On the Malliavin approach to monte carlo approximation of conditional expectations. Finance and Stochastics 8: 45–71.Google Scholar
8. Derman, E. & Kani, I. (1994). Riding on a smile. RISK 7: 32–39.Google Scholar
9. Duffie, D., Pan, J. & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68: 1343–1376.Google Scholar
10. Dupire, B. (1994). Pricing with a smile. Risk 18–20.Google Scholar
11. Dupire, B. (1996). A unified theory of volatility. Derivatives Pricing: The Classic Collection 185–196.Google Scholar
12. Fournie, E., Lasry, J.-M., Lebuchoux, J. & Lions, P.-L. (2001). Applications of Malliavin calculus to Monte Carlo methods in finance II. Finance and Stochastics 5: 201–236.Google Scholar
13. Fournie, E., Lasry, J.-M., Lions, P.-L., Lebuchoux, J. & Touzi, N. (1999). Applications of Malliavin calculus to Monte Carlo methods in finance. Finance and Stochastics 3: 391–412.Google Scholar
14. Gatheral, J. (2006). The Volatility Surface, a Practitioner's Guide. Hoboken, NJ: John Wiley and Sons.Google Scholar
15. Gatheral, J. (2008). Consistent modeling of SPX and VIX options. Discussion Paper at The Fifth World Congress of the Bachelier Congress Society, London. Available online at: http://www.math.nyu.edu/fellows_fin_math/gatheral/Bachelier2008.pdf Google Scholar
16. Gatheral, J., Hsu, E.P., Laurence, P., Ouyang, C. & Wang, T.-H. (2012). Asymptotics of implied volatility in local volatility models. Mathematical Finance 22: 591–620.Google Scholar
17. Gatheral, J. & Wang, T.-H. (2012). The heat-kernel most-likely-path approximation. International Journal of Theoretical and Applied Finance 15:1250001/1–18.Google Scholar
18. Gobet, E., Benhamou, E. & Miri, M. (2010). Time dependent Heston model. SIAM Journal on Financial Mathematics 1: 289–325.Google Scholar
19. Guyon, J. & Henry-Labordère, P. (2011). From local to implied volatilities. Risk 79–84.Google Scholar
20. Gyöngy, I. (1986). Mimicking the one-dimensional marginal distributions of processes having an itô differential. Probability Theory and Related Fields 71: 501–516.Google Scholar
21. Hagan, P.S., Kumar, D., Lesniewski, A.S., & Woodward, D.E. (2002). Managing smile risk. Wilmott Magazine 84–108.Google Scholar
22. Henry-Labordere, P. (2005). A general asymptotic implied volatility for stochastic volatility models. SSRN eLibrary: http://ssrn.com/abstract=698601.Google Scholar
23. Henry-Labordère, P. (2008). Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing. London: CRC Press.Google Scholar
24. Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6: 327–43.Google Scholar
25. Hull, J.C. & White, A.D. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance 42: 281–300.Google Scholar
26. Kanwal, R.P. (2004). Generalized Functions: Theory and Applications. 3rd edn., Boston: Birkhäuser.Google Scholar
27. Karatzas, I. & Shreve, S.E. (1991). Brownian motion and Stochastic calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edn., New York: Springer-Verlag.Google Scholar
28. Kienitz, J. & Wittke, M. (2010). Option valuation in multivariate Sabr models-with an application to the CBS spread. Working paper.Google Scholar
29. Kunitomo, N. & Takahashi, A. (2001). The asymptotic expansion approach to the valuation of interest rate contingent claims. Mathematical Finance 11: 117–151.Google Scholar
30. Kunitomo, N. & Takahashi, A. (2003). On validity of the asymptotic expansion approach in contingent claim analysis. The Annals of Applied Probability 13: 914–952.Google Scholar
31. Li, C. (2013a). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380.Google Scholar
32. Li, C. (2013b). Supplement to “maximum-likelihood estimation for diffusion processes via closed-form density expansions”. Annals of Statistics 41: 1350–1380.Google Scholar
33. Li, C. (2014). Closed-form expansion, conditional expectation, and option valuation. Mathematics of Operations Research 39: 487–516.Google Scholar
34. Piterbarg, V. (2003). A stochastic volatility forward libor model with a term structure of volatility smiles. SSRN eLibrary: http://ssrn.com/abstract=472061 Google Scholar
35. Piterbarg, V. (2005). A multi-currency model with fx volatility skew. SSRN eLibrary: http://ssrn.com/abstract=685084 Google Scholar
36. Piterbarg, V. (2006). Smiling hybrids. RISK 19: 66–71.Google Scholar
37. Piterbarg, V. (2007). Markovian projection method for volatility calibration. Risk 85–89.Google Scholar
38. Piterbarg, V.V. (2005). Stochastic volatility model with time-dependent skew. Applied Mathematical Finance 12: 147–185.Google Scholar
39. Scott, L.O. (1987). Option pricing when the variance changes randomly: theory, estimation, and an application. Journal of Financial and Quantitative Analysis 22: 419–438.Google Scholar
40. Takahashi, A. (1995). Essays on the valuation problems of contingent claims. PhD thesis, Haas School of Business, University of California, Berkeley.Google Scholar
41. Takahashi, A. (1999). An asymptotic expansion approach to pricing contingent claims. Asia-Pacific Financial Markets 6: 115–151.Google Scholar
42. Takahashi, A., Takehara, K. & Toda, M. (2012). A general computation schemes for a high-order asymptotic expectation method. International Journal of Theoretical and Applied Finance 15:1250001/1–18.Google Scholar
43. Uchida, M. & Yoshida, N. (2004). Asymptotic expansion for small diffusions applied to option pricing. Statistical Inference for Stochastic Processes 7: 189–223.Google Scholar
44. Watanabe, S. (1987). Analysis of Wiener functionals (Malliavin calculus) and its applications to heat Kernels. Annals of Probability 15: 1–39.Google Scholar
45. Xu, G. & Zheng, H. (2009). Approximate basket options valuation for a jump-diffusion model. Insurance: Mathematics and Economics 45: 188–194.Google Scholar
46. Xu, G. & Zheng, H. (2010). Basket options valuation for a local volatility jump–diffusion model with the asymptotic expansion method. Insurance: Mathematics and Economics 47: 415–422.Google Scholar
47. Yoshida, N. (1992). Asymptotic expansions for statistics related to small diffusions. Journal of Japan Statistical Society 22: 139–159.Google Scholar
48. Yoshida, N. (1992). Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin–Watanabe. Probability Theory and Related Fields 92: 275–311.Google Scholar