Asymptotic Analysis for Stochastic Volatility: Edgeworth Expansion

Masaaki Fukasawa (ETH Zürich)

Abstract


The validity of an approximation formula for European option prices under a general stochastic volatility model is proved in the light of the Edgeworth expansion for ergodic diffusions. The asymptotic expansion is around the Black-Scholes price and is uniform in bounded payoff functions. The result provides a validation of an existing singular perturbation expansion formula for the fast mean reverting stochastic volatility model.

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Pages: 764-791

Publication Date: April 18, 2011

DOI: 10.1214/EJP.v16-879

References

  1. AlÚs, E. A generalization of the Hull and White formula with applications to option pricing approximation. Finance Stoch. 10 (2006), no. 3, 353ñ365. Math. Review 2007k:60197
  2. Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Labys, P. The distribution of realized exchange rate volatility. J. Amer. Statist. Assoc. 96 (2001), no. 453, 42ñ55. Math. Review number not available.
  3. Bhattacharya, R. N.; Ranga Rao, R. Normal approximation and asymptotic expansions. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-London-Sydney, 1976. Math. Review 436272
  4. Borisov, I. S. Estimation of the rate of convergence of distributions of additive functionals of a sequence of sums of independent random variables. Sibirsk. Mat. é. 19 (1978), no. 3, 530ñ546, 716. Math. Review 0501274
  5. Conlon, J. G.; Sullivan, M. G. Convergence to Black-Scholes for ergodic volatility models. European J. Appl. Math. 16 (2005), no. 3, 385ñ409. Math. Review 2006g:91082
  6. Feller, W. An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons, Inc., New York-London-Sydney 1971. Math. Review 270403
  7. Fitzsimmons, P. J.; Pitman, J. Kac's moment formula and the Feynman-Kac formula for additive functionals of a Markov process. Stochastic Process. Appl. 79> (1999), no. 1, 117ñ134. Math. Review 2000a:60136
  8. Fouque, J. P.; Papanicolaou, G.; Sircar, K. R. Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge, 2000. Math. Review 2002g:91082
  9. Fouque, J. P.; Papanicolaou, G.; Sircar, R.; Solna, K. Singular perturbations in option pricing. SIAM J. Appl. Math. 63 (2003), no. 5, 1648ñ1665. Math. Review 2004g:91070
  10. Fouque, J. P.; Sircar, R.; Solna, K. Stochastic Volatility Effects on Defaultable Bonds. Applied Mathematical Finance, 13 (2006), no. 3, 215-244. Math. Review number not available.
  11. Fukasawa, M. Edgeworth expansion for ergodic diffusions. Probab. Theory Related Fields 142 (2008), no. 1-2, 1ñ20. Math. Review 2010a:62043
  12. Fukasawa, M. Asymptotic Analysis for stochastic volatility: martingale expansion. Finance Stoch. forthcoming.
  13. Gatheral, J. The Volatility Surface: A Practitionerís Guide. John Wiley and Sons: Hoboken, NJ. 2006. Math. Review number not available.
  14. Hall, P. The bootstrap and Edgeworth expansion. Springer Series in Statistics. Springer-Verlag, New York, 1992. Math. Review 93h:62029
  15. Khasminskii, R. Z.; Yin, G. Uniform asymptotic expansions for pricing European options. Appl. Math. Optim. 52 (2005), no. 3, 279ñ296. Math. Review 2006f:91099
  16. Malinovskiĭ, V. K. Limit theorems for Harris Markov chains. I. Teor. Veroyatnost. i Primenen. 31 (1986), no. 2, 315ñ332. Math. Review 88b:60157
  17. Petrov, V. V. Sums of independent random variables, Springer-Verlag, New York-Heidelberg. 1975. Math. Review 322927
  18. Skorokhod, A. V. Asymptotic methods in the theory of stochastic differential equations. Translations of Mathematical Monographs, 78. American Mathematical Society, Providence, RI. 1989. Math. Review 913305
  19. Veretennikov, A. Y. On lower bounds for mixing coefficients of Markov diffusions. From stochastic calculus to mathematical finance, 623ñ633, Springer, Berlin, 2006. Math. Review 2007f:60069
  20. Yoshida, N. Malliavin calculus and asymptotic expansion for martingales. Probab. Theory Related Fields 109 (1997), no. 3, 301ñ342. Math. Review 99k:60068
  21. Yoshida, N. Partial mixing and Edgeworth expansion. Probab. Theory Related Fields 129 (2004), no. 4, 559ñ624. Math. Review 2005g:62035
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