Asymptotic analysis for stochastic volatility: Martingale expansion

Finance and Stochastics

Volumn 15, Issue 4, 2011, Pages 635-654

  Fukasawa, Masaaki ^a  

^a OSAKA UNIVERSITY   (Japan)

Author keywords

Asymptotic expansion; Fast mean reversion; Fractional Brownian motion; Jump diffusion; Partial Malliavin calculus; Yoshida's formula

Indexed keywords

EID: 82355170197     PISSN: 09492984     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00780-010-0136-6     Document Type: Article

References (26)

1. Alós, E., León, J. A., Vives, J.: On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch. 11, 571-589 (2007).
2. Conlon, J. G., Sullivan, M. G.: Convergence to Black-Scholes for ergodic volatility models. Eur. J. Appl. Math. 16, 385-409 (2005).
3. Fouque, J. P., Papanicolaou, G., Sircar, K. R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000).
4. Fouque, J. P., Papanicolaou, G., Sircar, R., Solna, K.: Singular perturbations in option pricing. SIAM J. Appl. Math. 63, 1648-1665 (2003).
5. Fouque, J. P., Papanicolaou, G., Sircar, R., Solna, K.: Multiscale stochastic volatility asymptotics. Multiscale Model. Simul. 2, 22-42 (2003).
6. Fouque, J. P., Papanicolaou, G., Sircar, R., Solna, K.: Maturity cycles in implied volatility. Finance Stoch. 8, 451-477 (2004).
7. Fouque, J. P., Papanicolaou, G., Sircar, R., Solna, K.: Timing the smile. Wilmott Mag. Mar/Apr, 59-65 (2004).
8. Fouque, J. P., Sircar, R., Solna, K.: Stochastic volatility effects on defaultable bonds. Appl. Math. Finance 13, 215-244 (2006).
9. Fukasawa, M.: Edgeworth expansion for ergodic diffusions. Probab. Theory Relat. Fields 142, 1-20 (2008).
10. Fukasawa, M.: Asymptotic analysis for stochastic volatility: edgeworth expansion (2008, submitted). http://arxiv. org/abs/1004. 2106.
11. Hagan, P. S., Kumar, D., Lesniewski, S., Woodward, D. E.: Managing smile risk. Wilmott Mag. 18(11), 84-108 (2002).
12. Heston, S. L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327-343 (1993).
13. Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Finance 42, 281-300 (1987).
14. Jacod, J., Shiryaev, A. N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2002).
15. Khasminskii, R. Z., Yin, G.: Uniform asymptotic expansions for pricing European options. Appl. Math. Optim. 52, 279-296 (2005).
16. Lee, R. W.: Implied and local volatilities under stochastic volatility. Int. J. Theor. Appl. Finance 4, 45-89 (2001).
17. Lewis, A. L.: Option Valuation under Stochastic Volatility, with Mathematica Code. Finance Press, Newport Beach (2000).
18. Masuda, H., Yoshida, N.: Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model. Stoch. Process. Appl. 115, 1167-1186 (2005).
19. Nicolato, E., Venardos, E.: Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type. Math. Finance 13, 445-466 (2003).
20. Nelson, D. B.: ARCH models as diffusion approximations. J. Econom. 45, 7-38 (1990).
21. Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Probability and its Applications (New York). Springer, Berlin (2006).
22. Osajima, Y.: The asymptotic expansion formula of implied volatility for dynamic SABR model and FX hybrid model. UTMS preprint series. The University of Tokyo (2006). http://faculty. ms. u-tokyo. ac. jp/users/preprint/preprint2006. html.
23. Renault, E., Touzi, N.: Option hedging and implicit volatilities. Math. Finance 6, 279-302 (1996).
24. Yoshida, N.: Asymptotic expansion for statistics related to small diffusions. J. Jpn. Stat. Soc. 22, 139-159 (1992).
25. Yoshida, N.: Malliavin calculus and asymptotic expansion for martingales. Probab. Theory Relat. Fields 109, 301-342 (1997).
26. Yoshida, N.: Malliavin calculus and martingale expansion. Bull. Sci. Math. 125, 431-456 (2001).