Asia-Pacific Financial Markets

, Volume 19, Issue 3, pp 205–232 | Cite as

Pricing Discrete Barrier Options Under Stochastic Volatility

  • Kenichiro Shiraya
  • Akihiko Takahashi
  • Toshihiro Yamada


This paper proposes a new approximation method for pricing barrier options with discrete monitoring under stochastic volatility environment. In particular, the integration-by-parts formula and the duality formula in Malliavin calculus are effectively applied in pricing barrier options with discrete monitoring. To the best of our knowledge, this paper is the first one that shows an analytical approximation for pricing discrete barrier options with stochastic volatility models. Furthermore, it provides numerical examples for pricing double barrier call options with discrete monitoring under Heston and λ-SABR models.


Discrete barrier option Barrier option Knock-out option Double barrier option Stochastic volatility CEV model Heston model SABR model λ-SABR model Asymptotic expansion Malliavin calculus 


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  1. Ben Arous, G., & Laurence, P. (2009). Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-SABR model (Preprint).Google Scholar
  2. Baudoin, F. (2009). Stochastic Taylor expansions and heat Kernel asymptotics. Preprint, Department of Mathematics, Purdue University.Google Scholar
  3. Fink J. (2003) An examination of the effectiveness of static hedging in the presence of stochastic volatility. Journal of Futures Markets 23(9): 859–890CrossRefGoogle Scholar
  4. Fournié E., Lasry J.-M., Lebuchoux J., Lions P.-L., Touzi N. (1999) Applications of Malliavin calculus to Monte-Carlo methods in finance. Finance Stoch 3(4): 391–412CrossRefGoogle Scholar
  5. Fusai G., Abrahams D., Sgarra C. (2006) An exact analytical solution for discrete barrier options. Finance Stoch. 10(1): 1–26CrossRefGoogle Scholar
  6. Gatheral, J., Hsu, E. P., Laurence, P., Ouyang, C., & Wang, T.-H. (2009). Asymptotics of implied volatility in local volatility models. Forthcoming in Mathematical Finance.Google Scholar
  7. Malliavin P., Thalmaier A. (2006) Stochastic calculus of variations in mathematical finance. Springer, BerlinGoogle Scholar
  8. Malliavin P. (1997) Stochastic Analysis. Springer, New YorkGoogle Scholar
  9. Shiraya, K., Takahashi, A., & Toda, M. (2009). Pricing barrier and average options under stochastic volatility environment. Preprint, CARF-F-176, Graduate School of Economics, University of Tokyo. Forthcoming in “The Journal of Computational Finance”.Google Scholar
  10. Siopacha M., Teichmann J. (2011) Weak and strong Taylor methods for numerical solutions of stochastic differential equations. Quantitative Finance 11(4): 517–528CrossRefGoogle Scholar
  11. Takahashi A. (1999) An asymptotic expansion approach to pricing contingent claims. Asia-Pacific Financial Markets 6: 115–151CrossRefGoogle Scholar
  12. Takahashi, A. (2009). On an asymptotic expansion approach to numerical problems in finance. Selected papers on probability and statistics, Series 2 (Vol. 227 pp. 199–217). American Mathematical Society.Google Scholar
  13. Takahashi, A., Takehara, K., & Toda. M. (2009). Computation in an asymptotic expansion method. Working paper, CARF-F-149, The University of Tokyo.Google Scholar
  14. Takahashi, A., & Yamada, T. (2009). An asymptotic expansion with push-down of Malliavin weights (Preprint).Google Scholar
  15. Takahashi A., Yoshida N. (2004) An asymptotic expansion scheme for optimal investment problems. Statistical Inference for Stochastic Processes 7: 153–188CrossRefGoogle Scholar
  16. Watanabe, S. (1983). Malliavin’s calculus in terms of generalized Wiener functionals. Lecture notes in control and information science (Vol. 49). Berlin: SpringerGoogle Scholar
  17. Watanabe S. (1984) Lectures on stochastic differential equations and Malliavin calculus. Springer, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Kenichiro Shiraya
    • 1
    • 2
  • Akihiko Takahashi
    • 1
  • Toshihiro Yamada
    • 3
  1. 1.Graduate School of EconomicsThe University of TokyoTokyoJapan
  2. 2.Mizuho-DL Financial Technology Co., Ltd.TokyoJapan
  3. 3.Mitsubishi UFJ Trust Investment Technology Institute Co., Ltd. (MTEC)TokyoJapan

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