Advertisement

Review of Derivatives Research

, Volume 17, Issue 2, pp 191–216 | Cite as

Efficiently pricing double barrier derivatives in stochastic volatility models

  • Marcos EscobarEmail author
  • Peter Hieber
  • Matthias Scherer
Article

Abstract

Imposing a symmetry condition on returns, Carr and Lee (Math Financ 19(4):523–560, 2009) show that (double) barrier derivatives can be replicated by a portfolio of European options and can thus be priced using fast Fourier techniques (FFT). We show that prices of barrier derivatives in stochastic volatility models can alternatively be represented by rapidly converging series, putting forward an idea by Hieber and Scherer (Stat Probab Lett 82(1):165–172, 2012). This representation turns out to be faster and more accurate than FFT. Numerical examples and a toolbox of a large variety of stochastic volatility models illustrate the practical relevance of the results.

Keywords

First-passage time Barrier options Stochastic volatility  Stochastic clock 

JEL Classification

G13 C02 C63 

References

  1. Bakshi, G., & Madan, D. (2000). Spanning and derivative security valuation. Journal of Financial Economics, 55(2), 205–238.CrossRefGoogle Scholar
  2. Ball, C. A., & Roma, A. (1994). Stochastic volatility option pricing. Journal of Financial and Quantitative Analysis, 29, 589–607.CrossRefGoogle Scholar
  3. Barndorff-Nielsen, O., & Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck based models and some of their uses in financial economics. Journal of the Royal Statistical Society: Series B, 63(2), 167–241.CrossRefGoogle Scholar
  4. Bates, D. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies, 9(1), 69–107.CrossRefGoogle Scholar
  5. Billingsley, P. (2008). Probability and measure. Wiley series in probability.Google Scholar
  6. Bingham, N., & Kiesel, R. (2004). Risk-neutral valuation. Berlin: Springer.CrossRefGoogle Scholar
  7. Black, F., & Cox, J. C. (1976). Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance, 31(2), 351–367.CrossRefGoogle Scholar
  8. Brockwell, P., Chadraa, E., & Lindner, A. (2006). Continuous-time GARCH processes. The Annals of Applied Probability, 16(2), 790–826.CrossRefGoogle Scholar
  9. Carr, P., & Crosby, J. (2010). A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options. Journal of Quantitative Finance, 10(10), 1115–1136.CrossRefGoogle Scholar
  10. Carr, P., & Lee, R. (2009). Put-call symmetry: Extensions and applications. Mathematical Finance, 19(4), 523–560.CrossRefGoogle Scholar
  11. Carr, P., & Madan, D. B. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2, 61–73.Google Scholar
  12. Carr, P., Ellis, K., & Gupta, V. (1998). Static heding of exotic derivatives. Journal of Finance, 53(3), 1165–1190.CrossRefGoogle Scholar
  13. Carr, P., Geman, H., Madan, D., & Yor, M. (2003). Stochastic volatility for Lévy processes. Mathematical Finance, 13(3), 345–382.CrossRefGoogle Scholar
  14. Carr, P., Zhang, H., & Hadjiliadis, O. (2011). Maximum drawdown insurance. International Journal of Theoretical and Applied Finance, 14(8), 1195–1230.CrossRefGoogle Scholar
  15. Christoffersen, P., Heston, S., & Jacobs, K. (2009). The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Management Science, 55, 1914–1932.CrossRefGoogle Scholar
  16. Cox, D., & Isham, V. (1980). Monographs on applied probability and statistics: Point processes. London: Chapman & Hall.Google Scholar
  17. Cox, D., & Miller, H. (1965). Theory of stochastic processes. London: Chapman & Hall.Google Scholar
  18. Cox, J., Ingersoll, J., & Ross, S. (1985). A theory of the term structure of interest rates. Econometrica, 53, 187–201.Google Scholar
  19. da Fonseca, J., Grasselli, M., & Tebaldi, C. (2007). Option pricing when correlations are stochastic: An analytical framework. Review of Derivatives Research, 10, 151–180.CrossRefGoogle Scholar
  20. Darling, D., & Siegert, A. (1953). The first passage problem for a continuous Markov process. The Annals of Mathematical Statistics, 24(4), 624–639.CrossRefGoogle Scholar
  21. Dassios, A., & Jang, J. W. (2003). Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance and Stochastics, 7, 73–95.CrossRefGoogle Scholar
  22. Derman, E., Ergener, D., & Kani, I. (1994). Forever hedged. Risk, 7, 139–145.Google Scholar
  23. Dufresne, D. (2001). The integrated square-root process. Working paper. University of Montreal.Google Scholar
  24. Dupont, D. (2002). Hedging barrier options: Current methods and alternatives. Working paper.Google Scholar
  25. Eraker, B., Johannes, M., & Polson, N. (2003). The impact of jumps in volatility and returns. Journal of Finance, 58(3), 1269–1300.CrossRefGoogle Scholar
  26. Escobar, M., Friederich, T., Seco, L., & Zagst, R. (2011). A general structural approach for credit modeling under stochastic volatility. Journal of Financial Transformation, 32, 123–132.Google Scholar
  27. Feller, W. (1951). Two singular diffusion problems. The Annals of Mathematics, 54(1), 173–182.CrossRefGoogle Scholar
  28. Geman, H., & Yor, M. (1996). Pricing and hedging double-barrier options: A probabilistic approach. Mathematical Finance, 6(4), 365–378.CrossRefGoogle Scholar
  29. Götz, B. (2011). Valuation of multi-dimensional derivatives in a stochastic covariance framework. Ph.D. Thesis, TU Munich.Google Scholar
  30. He, H., Keirstead, W., & Rebholz, J. (1998). Double lookbacks. Mathematical Finance, 8(3), 201–228.CrossRefGoogle Scholar
  31. Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Journal of Finance, 42, 327–343.Google Scholar
  32. Hieber, P., & Scherer, M. (2012). A note on first-passage times of continuously time-changed Brownian motion. Statistics & Probability Letters, 82(1), 165–172.CrossRefGoogle Scholar
  33. Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatility. Journal of Finance, 42(2), 281–300.CrossRefGoogle Scholar
  34. Hurd, T. (2009). Credit risk modeling using time-changed Brownian motion. International Journal of Theoretical and Applied Finance, 12(8), 1213–1230.CrossRefGoogle Scholar
  35. Kammer, S. (2007). A general first-passage-time model for multivariate credit spreads and a note on barrier option pricing. Ph.D. Thesis, Justus-Liebig Universität Gießen.Google Scholar
  36. Kiesel, R., & Lutz, M. (2011). Efficient pricing of constant maturity swap spread options in a stochastic volatility LIBOR market model. Journal of Computational Finance, 14(4), 37–72.Google Scholar
  37. Kilin, F. (2011). Accelerating the calibration of stochastic volatility models. Journal of Derivatives, 18(3), 7–16.CrossRefGoogle Scholar
  38. Klüppelberg, C., Lindner, A., & Ross, M. (2004). A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour. Journal of Applied Probability, 41(3), 601–622.CrossRefGoogle Scholar
  39. Lin, X. S. (1999). Laplace transform and barrier hitting time distribution. Actuarial Research Clearing House, 1, 165–178.Google Scholar
  40. Lipton, A. (2001). Mathematical methods for foreign exchange. Singapore: World Scientific.CrossRefGoogle Scholar
  41. Naik, V. (1993). Option valuation and hedging strategies with jumps in the volatility of asset returns. Journal of Finance, 48(5), 1969–1984.CrossRefGoogle Scholar
  42. Pelsser, A. (2000). Pricing double barrier options using Laplace transforms. Finance and Stochastics, 4, 95–104.CrossRefGoogle Scholar
  43. Pigorsch, C., & Stelzer, R. (2009). A multivariate Ornstein–Uhlenbeck type stochastic volatility model. Working paper.Google Scholar
  44. Raible, S. (2000). Lévy processes in finance: Theory, numerics, and empirical facts. Ph.D. Thesis, Freiburg University.Google Scholar
  45. Reiner, E., & Rubinstein, M. (1991). Breaking down the barriers. Risk 4, 8, 28–35.Google Scholar
  46. Rollin, S., Ferreiro-Castilla, A., & Utzet, F. (2011). A new look at the Heston characteristic function. Working paper.Google Scholar
  47. Schöbel, R., & Zhu, J. (1999). Stochastic volatility with an Ornstein–Uhlenbeck process: An extension. Review of Finance, 3(1), 23–46.CrossRefGoogle Scholar
  48. Sepp, A. (2006). Extended CreditGrades model with stochastic volatility and jumps. Wilmott Magazine, 50–62.Google Scholar
  49. Stein, E., & Stein, J. (1991). Stock price distributions with stochastic volatility: An analytical approach. Review of Financial Studies, 4, 727–752.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Marcos Escobar
    • 1
    Email author
  • Peter Hieber
    • 2
  • Matthias Scherer
    • 2
  1. 1.Department of MathematicsRyerson UniversityTorontoCanada
  2. 2.Lehrstuhl für Finanzmathematik (M13)Technische Universität MünchenGarching-HochbrückGermany

Personalised recommendations