Pricing of Double Barrier Options by Spectral Theory

Dell'Era Mario, M.D. (2008): Pricing of Double Barrier Options by Spectral Theory.

Abstract

We propose to discuss the efficiency of the spectral method for computing the value of Double Barrier Options. Using this method, one may write the option price as a Fourier series, with suitable coefficients. We propose a simple approach for its computing. One consider the general case, in which the volatility is time dependent, but it is immediate extend our methodology also in the case of constant volatility. The advantage to write the arbitrage price of the Double Barrier Options as Fourier series, is matter of computation complexity. The methods used to evaluate options of this kind have a high value of computation complexity, furthermore, them have not the capacity to manage it, while using our method, one can define, through an easy analytical report, the computation complexity of the problem, and also one can choice its accuracy.

• Item Type: MPRA Paper
• Language: English
• Keywords: Options Pricing, Computation Complexity.
• Subjects: G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

References

Abate, J., and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA Journal of Computing, 7, 36–43.

Akahori, J. (1995). Some formulae for a new type of path-dependent option. Annals of Applied Probability, 5, 383–8.

Andersen, L., Andreasen, J., and Eliezer, D. (2000). Static replication of barrier options: some general results. Working paper, General Re Financial Products.

Black, F., and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–59.

Borodin, A. N., and Salminen, P. (1996). Handbook of Brownian Motion. Birkhauser, Boston.

Boyle, P. P., and Tian, Y. (1999). Pricing lookback and barrier options under the CEV process. Journal of Financial and Quantitative Analysis, 34 (Correction: P. P. Boyle, Y. Tian, and J. Imai, Lookback options under the CEV process: a correction, JFQA web site http://depts.washington.edu/jfqa/ in “Notes, Comments, and Corrections”).

Broadie, M., Cvitanic, J., and Soner, H. M. (1998). Optimal replication of contingent claims under portfolio constraints. Review of Financial Studies, 11, 59–79.

Broadie, M., Glasserman, P., and Kou, S. (1997). A continuity correction for discrete barrier options. Mathematical Finance, 7, 325–49.

Carr, P., Ellis, K. and Gupta, V. (1998). Static hedging of exotic options. Journal of Finance, 53, 1165–90.

Chesney, M., Jeanblanc-Picque, M., and Yor, M. (1997). Brownian excursions and Parisian barrier options. Annals of Applied Probability, 29, 165–84.

Chesney, M., Cornwall, J., Jeanblanc-Picque, M., Kentwell, G., and Yor, M. (1997). Parisian pricing, RISK, January, 77–9.

Cheuk, T. H. F., and Vorst, T. C. F. (1996). Complex barrier options. Journal of Derivatives, 4 (Fall), 8–22.

Choudhury, G., Lucantoni, D., and Whitt, W. (1994). Multidimensional transform inversion with applications to the transient M/G/1 queue. Annals of Applied Probability, 4, 719–40.

Cox, J. (1975). Notes on Option Pricing I: Constant Elasticity of Variance Diffusions. Working Paper, Stanford University (reprinted in Journal of Portfolio Management, 1996, 22, 15–17).

Cox, D. R., and Miller, H. D. (1965). The Theory of Stochastic Processes. Wiley, New York.

Cox, J., and Ross, S. (1976). The valuation of options for alternative stochastic processes. Structuring, pricing and hedging double-barrier step options Journal of Financial Economics, 3, 145–66.

Dassios, A. (1995). The distribution of the quantile of a Brownian motion with drift and the pricing of path-dependent options. Annals of Applied Probability, 5, 389–98.

Davydov, D., a nd Linetsky, V. (2000). Pricing options on scalar diffusions: an Eigenfunction expansion approach. Submitted for publication.

Davydov, D., and Linetsky, V. (2001). Pricing and Hedging Path-Dependent Options under the CEV Process. Management Science, 47, 949–65.

Derman, E., Ergener, D., and Kani, I. (1995). Static options replication. Journal of Derivatives, 2(4), 78–95.

Derman, E., and Kani, I. (1996). The ins and outs of barrier options. Derivatives Quarterly, Winter, 55–67 (Part I) and Spring, 73–80 (Part II).

Douady, R. (1998). Closed-form formulas for exotic options and their lifetime distribution. International Journal of Theoretical and Applied Finance, 2, 17–42.

Duffie, D. (1996). Dynamic Asset Pricing, 2nd Edition, Princeton University Press, Princeton, New Jersey.

Feller, W. (1971). Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York.

Fitzsimmons, P. J., and Pitman, J. (1997). Kac’s moment formula and the Feynman–Kac formula for additive functionals of a Markov process. Working paper, Berkeley.

Fu, M., Madan D., and Wang, T. (1997). Pricing Asian options: a comparison of analytical and Monte Carlo methods. Computational Finance, 2, 49–74.

Fusai, G. (1999). Corridor options and arc–sine law. Preprint, University of Florence.

Fusai, G. and Tagliani, A. (2001). Pricing of occupation-time derivatives: continuous and discrete monitoring. Journal of Computational Finance, 5(1), 1–37.

Gallus, C. (1999). Exploding hedging errors for digital options. Finance and Stochastics, 3. 187–201.

Geman, H., and Eydeland, A. (1995). Domino effect: inverting the Laplace transform. RISK, April, 65–7.

Geman, H., and Yor, M. (1993). Bessel processes, Asian options and perpetuities. Mathematical Finance, 3, 349–75.

Geman, H., and Yor, M. (1996). Pricing and hedging double barrier options: a probabilistic approach. Mathematical Finance, 6, 365–78.

Hart, I., and Ross, M. (1994). Striking continuity. RISK, June, 51.

He, H., Keirstead, W., and Rebholz, J. (1998). Double lookbacks. Mathematical Finance, 8, 201–28.

Hsu, H. (1997). Surprised parties. RISK, April.

Hui, H. C. (1997). Time-dependent barrier option values. The Journal of Futures Markets, 17,667–88.

Hui, H. C. Lo, C. F., and Yuen, P. H. (1997). Comment on “pricing double-barrier options using Laplace transforms”, Finance and Stochastics, 4, 105–7.

Hugonnier, J.-N. (1999). The Feynman–Kac formula and pricing occupation time derivatives. International Journal of Theoretical and Applied Finance, 2, 153–78.

Jamshidian, F. (1997). A note on analytical valuation of double barrier options. Working Paper, Sakura Global Capital.

Jeanblanc-Picque, M., Pitman, J., and Yor, M. (1997). The Feynman–Kac formula and decomposition of Brownian paths. Computational and Applied Mathematics, 16, 27–52.

Karatzas, I., and Shreve, S. (1991). Brownian Motion and Stochastic Calculus, 2nd Edition. Springer-Verlag, New York.

Kunitomo, N., and Ikeda, M. (1992). Pricing options with curved boundaries. Mathematical Finance, 4, 275–98.

Linetsky, V. (1998). Steps to the barrier. RISK, April, 62–5.

Linetsky, V. (1999). Step options. Mathematical Finance, 9, 55–96.

Merton, R. C. (1973). Theory of rational options pricing. Bell Journal of Economics and Management Science, 2, 275–98.

Miura, R. (1992). A note on look-back options based on order statistics. Hitotsubashi Journal of Commerce Management, 27, 15–28.

Pechtl, A. (1995). Classified information. RISK, June, 59–61.

Pelsser, A. (2000). Pricing double barrier options using analytical inversion of Laplace transforms. Finance and Stochastics, 4, 95–104.

Revuz, D., and Yor, M. (1999). Continuous Martingales and Brownian Motion, Third edition, Springer, Berlin.

Rogers, L. C. G., and Zane, O. (1995). Valuing moving barrier options. Computational Finance, 1, 5–12.

Rubinstein, M., and Reiner, E. (1991). Breaking down the barriers. RISK, September, 28–35.

Schroder, M. (2000). On the valuation of double-barrier options: computational aspects. Journal of Computational Finance, 3 (4).

Schmock, U., Shreve, S., and Wystup, U. (1999). Valuation of exotic options under short-selling constraints. Working paper, Carnegie Mellon University.

Sidenius, J. (1998). Double barrier options: valuation by path counting. Computational Finance, 1, 63–79.

Snyder, G. L. (1969). Alternative forms of options. Financial Analysts Journal, 25, 93–9.

Taleb N. (1997). Dynamic Hedging. J. Wiley Sons, New York.

Toft, K., and Xuan, C. (1998). How well can barrier options be hedged by a static portfolio of standard options? Journal of Financial Engineering, 7, 147–75.

Tompkins, R. (1997). Static versus dynamic hedging of exotic options: an evaluation of hedge performance via simulation. Net Exposure, 2, November, 1–37.

Vetzal, K., and Forsyth, P. (1999). Discrete Parisian and delayed barrier options: a general numerical approach. Advances in Futures and Options Research, 10, 1–15.

Wystup, U. (1998). Valuation of exotic options under short-selling constraints as a singular stochastic control problem. PhD dissertation, Carnegie Mellon University.

Wystup, U. (1999). Dealing with dangerous digitals. Working paper, http://www.mathfinance.de

Zhang, P. (1997). Exotic Options. World Scientific, Singapore.