Pricing average options under time-changed Lévy processes
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- 作者：Akira Yamazaki (1)
- 关键词：Average options ; Time ; changed Lévy processes ; Gram–Charlier expansion ; Affine processes ; Quadratic Gaussian processes ; G13 ; C63
- 刊名：Review of Derivatives Research
- 出版年：2014
- 出版时间：April 2014
- 年：2014
- 卷：17
- 期：1
- 页码：79-111
- 全文大小：455 KB
- 参考文献：1. Albrecher, H. (2004). The valuation of Asian options for market models of exponential Lévy type. In A. De Schepper, J. Dhaene, H. Reynaerts, W. Schoutens, P. Van Goethem, & M. Vanmaele (Eds.), / Proceedings of the second day of actuarial and financial mathematics. Brussels: Royal Flemish Academy of Arts and Sciences.
2. Albrecher, H., & Predota, M. (2002). Bounds and approximations for discrete Asian options in a variance-gamma model. / Grazer Mathematische Berichte, / 345, 35-7.
3. Albrecher, H., & Predota, M. (2004). On Asian option pricing for NIG Lévy processes. / Journal of Computational and Applied Mathematics, / 172(1), 153-68. CrossRef
4. Albrecher, H., & Schoutens, W. (2005). Static hedging of Asian options under stochastic volatility models using fast Fourier transform. In A. Kyprianou, W. Schoutens, & P. Wilmott (Eds.), / Exotic option pricing and advanced Lévy models. New York: Wiley.
5. Albrecher, H., Dhaene, J., Goovaerts, M., & Schoutens, W. (2003). / Static hedging of Asian options under Lévy models: The comonotonicity approach. Working Paper, Katholieke Universiteit Leuven.
6. Barndorff-Nielsen, O. (1998). Process of normal inverse Gaussian type. / Finance and Stochastics, / 2, 41-8. CrossRef
7. Bayraktar, E., & Xing, H. (2011). Pricing Asian options for jump diffusion. / Mathematical Finance, / 21(1), 117-43. CrossRef
8. Carr, P., & Wu, L. (2004). Time-changed Lévy processes and option pricing. / Journal of Financial Economics, / 71, 113-41. CrossRef
9. Carr, P., & Wu, L. (2007). Stochastic skew in currency options. / Journal of Financial Economics, / 86(1), 213-47. CrossRef
10. Carr, P., Geman, H., Madan, D., & Yor, M. (2003). Stochastic volatility for Lévy processes. / Mathematical Finance, / 13(3), 345-82. CrossRef
11. Carr, P., Lee, R., & Wu, L. (2012) Variance swaps on time-changed Lévy processes. / Finance and Stochastics, 16(2), 335-55.
12. Collin-Dufresne, P., & Goldstein, R. (2002). Pricing swaptions within an affine framework. / Journal of Derivatives, / 10, 1-8. CrossRef
13. Cox, J., Ingersoll, J., & Ross, S. (1985). A theory of the term structure of interest rates. / Econometrica, / 53, 385-07. CrossRef
14. Curran, M. (1994). Valuing Asian and portfolio options by conditioning on the geometric mean price. / Management Science, / 40(12), 1705-711. CrossRef
15. Duffie, D., & Kan, R. (1996). A yield-factor model of interest rates. / Mathematical Finance, / 6, 379-06. CrossRef
16. Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. / Econometrica, / 68, 1343-376. CrossRef
17. Figueroa-López, J. (2009). Nonparametric estimation of time-changed Lévy models under high-frequency data. / Advances in Applied Probability, / 41(4), 1161-188. CrossRef
18. Filipovi?, D., & Mayerhofer, E. (2009). / Affine diffusion processes: Theory and applications. Working Paper, arXiv.
19. Forde, M., & Jacquier, A. (2010). Robust approximations for pricing Asian options and volatility swaps under stochastic volatility. / Applied Mathematical Finance, / 17(3), 241-59. CrossRef
20. Foschi, P., Pagliarani, S., & Pascucci, A. (2011). / Black–Scholes formulae for Asian options in local volatility models. Working Paper.
21. Fouque, J., & Han, C. (2003). Pricing Asian options with stochastic volatility. / Quantitative Finance, / 3(5), 352-62.
22. Fusai, G., & Meucci, A. (2008). Pricing discretely monitored Asian options under Lévy processes. / Journal of Banking and Finance, / 32, 2076-088. CrossRef
23. Geman, H., & Yor, M. (1993). Bessel processes, Asian options and perpetuities. / Mathematical Finance, / 4, 345-71.
24. Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. / Review of Financial Studies, / 6, 327-43. CrossRef
25. Huang, J., & Wu, L. (2004). Specification analysis of option pricing models based on time-changed Lévy processes. / Journal of Finance, / 59, 1405-439. CrossRef
26. Itkin, A., & Carr, P. (2010). Pricing swaps and options on quadratic variation under stochastic time change models -discrete observations case. / Review of Derivatives Research, / 13, 141-76. CrossRef
27. Jarrow, R., & Rudd, A. (1982). Approximate option valuation for arbitrary stochastic processes. / Journal of Financial Economics, / 10, 347-69. CrossRef
28. Kallsen, J., & Pauwels, A. (2011). Variance-optimal hedging for time-changed Lévy processes. / Applied Mathematical Finance, / 18(1), 1-8. CrossRef
29. Leippold, M., & Wu, L. (2002). Asset pricing under the quadratic class. / Journal of Financial and Quantitative Analysis, / 37, 271-95. CrossRef
30. Levy, E. (1992). Pricing European average rate currency options. / Journal of International Money and Finance, / 6, 474-91. CrossRef
31. Linetsky, V. (2004). Spectral expansions for Asian (average price) options. / Operations Research, / 52(6), 856-67. CrossRef
32. Madan, D., & Milne, F. (1991). Option pricing with VG martingale components. / Mathematical Finance, / 1, 39-6. CrossRef
33. Rogers, L. C. G., & Shi, Z. (1995). The value of Asian option. / Journal of Applied Probability, / 32, 1077-088. CrossRef
34. Shiraya, K., & Takahashi, A. (2011). Pricing average options on commodities. / Journal of Futures Markets, / 31(5), 407-39. CrossRef
35. Tanaka, K., Yamada, T., & Watanabe, T. (2005). / Approximation of interest rate derivatives-prices by Gram–Charlier expansion and bond moments. Working Paper, Bank of Japan.
36. Umezawa, Y., & Yamazaki, A. (2012). / Pricing path-dependent options with discretely monitoring under time-changed Lévy processes. Working Paper, Social Science Research Network.
37. Vecer, J. (2001). A new PDE approach for pricing arithmetic average Asian options. / Journal of Computational Finance, / 4(4), 105-13.
38. Vecer, J., & Xu, M. (2004). Pricing Asian options in a semimartingale model. / Quantitative Finance, / 4(2), 170-75. CrossRef
39. Wong, H., & Cheung, Y. (2004). Geometric Asian options: Valuation and calibration with stochastic volatility. / Quantitative Finance, / 4(3), 301-14. CrossRef - 作者单位：Akira Yamazaki (1)
1. Graduate School of Business Administration, Hosei University, 2-16, Tamachi, Ichigaya, Sinjuku-ku, Tokyo, 162-0843, Japan - ISSN：1573-7144
文摘
This paper presents an approximate formula for pricing average options when the underlying asset price is driven by time-changed Lévy processes. Time-changed Lévy processes are attractive to use for a driving factor of underlying prices because the processes provide a flexible framework for generating jumps, capturing stochastic volatility as the random time change, and introducing the leverage effect. There have been very few studies dealing with pricing problems of exotic derivatives on time-changed Lévy processes in contrast to standard European derivatives. Our pricing formula is based on the Gram–Charlier expansion and the key of the formula is to find analytic treatments for computing the moments of the normalized average asset price. In numerical examples, we demonstrate that our formula give accurate values of average call options when adopting Heston’s stochastic volatility model, VG-CIR, and NIG-CIR models.