Closed-Form Approximate Solutions of Window Barrier Options with Term-Structure Volatility and Interest Rates Using the Boundary Integral Method

ABSTRACT

In this study we propose an approach to solve a partial differential equation (PDE), the boundary integral method, for the valuation of both discrete and continuous window barrier options, as well as multi-window barrier options within a deterministic term structure of volatility and interest rates. Numerical results reveal that the proposed method yields rapid and highly accurate closed-form approximate solutions. In addition, the term structure will have a significant impact on the valuation.

In this study we propose an approach to solve a partial differential equation (PDE), the boundary integral method, for the valuation of both discrete and continuous window barrier options, as well as multi-window barrier options within a deterministic term structure of volatility and interest rates. Numerical results reveal that the proposed method yields rapid and highly accurate closed-form approximate solutions. In addition, the term structure will have a significant impact on the valuation.

KEYWORDS

Window Barrier Option; Integral Method; Integral Representation; Green’s Function; PDE Approach

Window Barrier Option; Integral Method; Integral Representation; Green’s Function; PDE Approach

Cite this paper

Y. Hsiao, "Closed-Form Approximate Solutions of Window Barrier Options with Term-Structure Volatility and Interest Rates Using the Boundary Integral Method,"*Journal of Mathematical Finance*, Vol. 2 No. 4, 2012, pp. 291-302. doi: 10.4236/jmf.2012.24032.

Y. Hsiao, "Closed-Form Approximate Solutions of Window Barrier Options with Term-Structure Volatility and Interest Rates Using the Boundary Integral Method,"

References

[1] R. Heynen and H. Kat, “Partial barrier options,” Journal of Financial Engineering, Vol. 3, No. 4, 1994, pp. 253-274.

[2] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143

[3] J. C. Cox and M. Rubinstein, “Options Markets,” Prentice-Hall, Upper Saddle River, 1985.

[4] M. Rubinstein and E. Reiner, “Breaking Down the Barriers,” RISK, Vol. 4, No. 8, 1991, pp. 28-35.

[5] E. Haug, “The Complete Guide to Option Pricing Formulas,” McGraw Hill, New York, 1997.

[6] N. Kunitomo and M. Ikeda, “Pricing Options with Curved Boundaries,” Mathematical Finance, Vol. 2, No. 4, 1992, pp. 275-298. doi:10.1111/j.1467-9965.1992.tb00033.x

[7] H. Geman and M. Yor, “Pricing and Hedging Double-Barrier Options: A Probabilistic Approach,” Mathematical Finance, Vol. 6, No. 4, 1996, pp. 365-378. doi:10.1111/j.1467-9965.1996.tb00122.x

[8] J. C. Cox, S. A. Ross and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of Financial Economics, Vol. 7, No. 3, 1979, pp. 229-264. doi:10.1016/0304-405X(79)90015-1

[9] P. P. Boyle, “Option Valuation Using a Three-Jump Process,” International Options Journal, Vol. 3, 1986, pp. 7-12.

[10] P. P. Boyle, “A Lattice Framework for Option Pricing with Two State Variables,” Journal of Financial and Quantitative Analysis, Vol. 23, No. 1, 1988, pp. 1-12. doi:10.2307/2331019

[11] P. P. Boyle and S. H. Lau, “Bumping Up against the Barrier with the Binomial Method,” Journal of Derivatives, Vol. 1, No. 4, 1994, pp. 6-14. doi:10.3905/jod.1994.407891

[12] P. Ritchken, “On Pricing Barrier Options,” Journal of Derivatives, Vol. 3, No. 2, 1995, pp. 19-28. doi:10.3905/jod.1995.407939

[13] M. L. Wang, Y. H. Liu and Y. L. Hsiao, “Barrier Option Pricing: A Hybrid Method Approach,” Quantitative Finance, Vol. 9, No. 3, 2009, pp. 341-352. doi:10.1080/14697680802595593

[14] S. Figlewski and B. Gao, “The Adaptive Mesh Model: A New Approach to Efficient Option Pricing,” Journal of Financial Economics, Vol. 53, No. 3, 1999, pp. 313-351. doi:10.1016/S0304-405X(99)00024-0

[15] M. Albert, J. Fink and K. E. Fink, “Adaptive Mesh Modeling and Barrier Option Pricing under a Jump-Diffusion Process,” Journal of Financial Research, Vol. 31, No. 4, 2008, pp. 381-408. doi:10.1111/j.1475-6803.2008.00244.x

[16] M. Broadie, P. Glasserman and S. Kuo, “Connecting Discrete and Continuous Path-Dependent Options,” Finance and Stochastics, Vol. 3, No. 1, 1999, pp. 55-82. doi:10.1007/s007800050052

[17] P. H?rfelt, “Extension of the Corrected Barrier Approximation by Broadie, Glassman, and Kou,” Finance and Stochastics, Vol. 7, 2003, pp. 231-243. doi:10.1007/s007800200077

[18] P. P. Boyle and Y. Tian, “An Explicit Finite Difference Approach to the Pricing of Barrier Options,” Journal Applied Mathematical Finance, Vol. 5, No. 1, 1998, pp. 17-43. doi:10.1080/135048698334718

[19] D. Ahn, S. Figlewski and B. Gao, “Pricing Discrete Barrier Options with an Adaptive Mesh Model,” Journal of Derivatives, Vol. 6, No. 4, 1999, pp. 33-43. doi:10.3905/jod.1999.319127

[20] S. G. Kou, “On Pricing of Discrete Barrier Options,” Statistica Sinica, Vol. 13, No. 4, 2003, pp. 955-964.

[21] G. K. Mitov, S. T. Rachev, Y. S. Kim and F. J. Fabozzi, “Barrier Option Pricing by Branching Processes,” International Journal of Theoretical and Applied Finance, Vol. 12, No. 7, 2009, pp. 1055-1073. doi:10.1142/S0219024909005555

[22] F. Hu and C. Knessl, “Asymptotics of Barrier Option Pricing under the CEV Process,” Applied Mathematical Finance, Vol. 17, No. 3, 2010, pp. 261-300. doi:10.1080/13504860903335355

[23] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062

[24] R. Heynen and H. Kat, “Discrete Partial Barrier Options with a Moving Barrier,” Journal of Financial Engineering, Vol. 5, No. 3, 1996, pp. 199-209.

[25] G. F. Armstrong, “Valuation Formulae for Window Barrier Options,” Applied Mathematical Finance, Vol. 8, No. 4, 2001, pp. 197-208. doi:10.1080/13504860210124607

[26] P. Carr, “Two Extensions to Barrier Option Valuation,” Applied Mathematical Finance, Vol. 2, No. 3, 1995, pp. 173-209. doi:10.1080/13504869500000010

[27] D. Chance, “The Pricing and Hedging of Limited Exercise Caps and Spreads,” Journal of Financial Research, Vol. 17, No. 4, 1994, pp. 561-584.

[28] H. Kat and L. Verdonk, “Tree Surgery,” RISK, Vol. 8, No. 2, 1995, pp. 53-56.

[29] M. L. Wang and S. Y. Shen, “On Pricing of the Up-and-Out Call—A Boundary Integral Method Approach,” Asia Pacific Management Review, Vol. 10, No. 3, 2005, pp. 205-213.

[30] M. L. Wang and Y. L. Hsiao, “A PDE Approach to Valuation of Discrete Barrier Option,” working paper, 2012.

[1] R. Heynen and H. Kat, “Partial barrier options,” Journal of Financial Engineering, Vol. 3, No. 4, 1994, pp. 253-274.

[2] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143

[3] J. C. Cox and M. Rubinstein, “Options Markets,” Prentice-Hall, Upper Saddle River, 1985.

[4] M. Rubinstein and E. Reiner, “Breaking Down the Barriers,” RISK, Vol. 4, No. 8, 1991, pp. 28-35.

[5] E. Haug, “The Complete Guide to Option Pricing Formulas,” McGraw Hill, New York, 1997.

[6] N. Kunitomo and M. Ikeda, “Pricing Options with Curved Boundaries,” Mathematical Finance, Vol. 2, No. 4, 1992, pp. 275-298. doi:10.1111/j.1467-9965.1992.tb00033.x

[7] H. Geman and M. Yor, “Pricing and Hedging Double-Barrier Options: A Probabilistic Approach,” Mathematical Finance, Vol. 6, No. 4, 1996, pp. 365-378. doi:10.1111/j.1467-9965.1996.tb00122.x

[8] J. C. Cox, S. A. Ross and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of Financial Economics, Vol. 7, No. 3, 1979, pp. 229-264. doi:10.1016/0304-405X(79)90015-1

[9] P. P. Boyle, “Option Valuation Using a Three-Jump Process,” International Options Journal, Vol. 3, 1986, pp. 7-12.

[10] P. P. Boyle, “A Lattice Framework for Option Pricing with Two State Variables,” Journal of Financial and Quantitative Analysis, Vol. 23, No. 1, 1988, pp. 1-12. doi:10.2307/2331019

[11] P. P. Boyle and S. H. Lau, “Bumping Up against the Barrier with the Binomial Method,” Journal of Derivatives, Vol. 1, No. 4, 1994, pp. 6-14. doi:10.3905/jod.1994.407891

[12] P. Ritchken, “On Pricing Barrier Options,” Journal of Derivatives, Vol. 3, No. 2, 1995, pp. 19-28. doi:10.3905/jod.1995.407939

[13] M. L. Wang, Y. H. Liu and Y. L. Hsiao, “Barrier Option Pricing: A Hybrid Method Approach,” Quantitative Finance, Vol. 9, No. 3, 2009, pp. 341-352. doi:10.1080/14697680802595593

[14] S. Figlewski and B. Gao, “The Adaptive Mesh Model: A New Approach to Efficient Option Pricing,” Journal of Financial Economics, Vol. 53, No. 3, 1999, pp. 313-351. doi:10.1016/S0304-405X(99)00024-0

[15] M. Albert, J. Fink and K. E. Fink, “Adaptive Mesh Modeling and Barrier Option Pricing under a Jump-Diffusion Process,” Journal of Financial Research, Vol. 31, No. 4, 2008, pp. 381-408. doi:10.1111/j.1475-6803.2008.00244.x

[16] M. Broadie, P. Glasserman and S. Kuo, “Connecting Discrete and Continuous Path-Dependent Options,” Finance and Stochastics, Vol. 3, No. 1, 1999, pp. 55-82. doi:10.1007/s007800050052

[17] P. H?rfelt, “Extension of the Corrected Barrier Approximation by Broadie, Glassman, and Kou,” Finance and Stochastics, Vol. 7, 2003, pp. 231-243. doi:10.1007/s007800200077

[18] P. P. Boyle and Y. Tian, “An Explicit Finite Difference Approach to the Pricing of Barrier Options,” Journal Applied Mathematical Finance, Vol. 5, No. 1, 1998, pp. 17-43. doi:10.1080/135048698334718

[19] D. Ahn, S. Figlewski and B. Gao, “Pricing Discrete Barrier Options with an Adaptive Mesh Model,” Journal of Derivatives, Vol. 6, No. 4, 1999, pp. 33-43. doi:10.3905/jod.1999.319127

[20] S. G. Kou, “On Pricing of Discrete Barrier Options,” Statistica Sinica, Vol. 13, No. 4, 2003, pp. 955-964.

[21] G. K. Mitov, S. T. Rachev, Y. S. Kim and F. J. Fabozzi, “Barrier Option Pricing by Branching Processes,” International Journal of Theoretical and Applied Finance, Vol. 12, No. 7, 2009, pp. 1055-1073. doi:10.1142/S0219024909005555

[22] F. Hu and C. Knessl, “Asymptotics of Barrier Option Pricing under the CEV Process,” Applied Mathematical Finance, Vol. 17, No. 3, 2010, pp. 261-300. doi:10.1080/13504860903335355

[23] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062

[24] R. Heynen and H. Kat, “Discrete Partial Barrier Options with a Moving Barrier,” Journal of Financial Engineering, Vol. 5, No. 3, 1996, pp. 199-209.

[25] G. F. Armstrong, “Valuation Formulae for Window Barrier Options,” Applied Mathematical Finance, Vol. 8, No. 4, 2001, pp. 197-208. doi:10.1080/13504860210124607

[26] P. Carr, “Two Extensions to Barrier Option Valuation,” Applied Mathematical Finance, Vol. 2, No. 3, 1995, pp. 173-209. doi:10.1080/13504869500000010

[27] D. Chance, “The Pricing and Hedging of Limited Exercise Caps and Spreads,” Journal of Financial Research, Vol. 17, No. 4, 1994, pp. 561-584.

[28] H. Kat and L. Verdonk, “Tree Surgery,” RISK, Vol. 8, No. 2, 1995, pp. 53-56.

[29] M. L. Wang and S. Y. Shen, “On Pricing of the Up-and-Out Call—A Boundary Integral Method Approach,” Asia Pacific Management Review, Vol. 10, No. 3, 2005, pp. 205-213.

[30] M. L. Wang and Y. L. Hsiao, “A PDE Approach to Valuation of Discrete Barrier Option,” working paper, 2012.