Pricing Options on Foreign Currency with a Preset Exchange Rate

ABSTRACT

This paper presents a new option that can be used by agents for managing foreign exchange risk. Unlike the Garman Kolhagen model [1], (GK), this paper presents a new model with a preset exchange rate (PE), that allows the agent to take advantage of the his/her view on both the direction and magnitude of rate movement and as such provides this agent with more choices. The model has a provision for an automatic exchange of the payoff at a preset exchange rate, and upon expiration gives the agent the choice of keeping the payoff in the foreign currency or exchanging it back to the pricing currency. At the time of writing, the buyer selects the preset exchange rate. Depending on the value selected, the PE option’s price and payoff will be equal to, greater than or less than those of the GK model. A decision rule for choosing between the PE and GK models is developed by determining the expiration spot rate that equates the two models’ returns. The range of spot rates that makes the PE option’s return greater than the GK’s return is the PE preferred range. If the agent expects the expiration spot rate will be within the preferred range, the PE option is purchased. The size of the preferred range is a decreasing function of time to expiration, a decreasing function of spot rate volatility and an increasing function of the basis point spread between foreign and domestic interest rates.

This paper presents a new option that can be used by agents for managing foreign exchange risk. Unlike the Garman Kolhagen model [1], (GK), this paper presents a new model with a preset exchange rate (PE), that allows the agent to take advantage of the his/her view on both the direction and magnitude of rate movement and as such provides this agent with more choices. The model has a provision for an automatic exchange of the payoff at a preset exchange rate, and upon expiration gives the agent the choice of keeping the payoff in the foreign currency or exchanging it back to the pricing currency. At the time of writing, the buyer selects the preset exchange rate. Depending on the value selected, the PE option’s price and payoff will be equal to, greater than or less than those of the GK model. A decision rule for choosing between the PE and GK models is developed by determining the expiration spot rate that equates the two models’ returns. The range of spot rates that makes the PE option’s return greater than the GK’s return is the PE preferred range. If the agent expects the expiration spot rate will be within the preferred range, the PE option is purchased. The size of the preferred range is a decreasing function of time to expiration, a decreasing function of spot rate volatility and an increasing function of the basis point spread between foreign and domestic interest rates.

Cite this paper

A. Wolf and C. Hessel, "Pricing Options on Foreign Currency with a Preset Exchange Rate,"*Journal of Mathematical Finance*, Vol. 2 No. 3, 2012, pp. 214-224. doi: 10.4236/jmf.2012.23024.

A. Wolf and C. Hessel, "Pricing Options on Foreign Currency with a Preset Exchange Rate,"

References

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[2] Bank for International Settlements, “Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity,” 1996-2010. http://www.bis.org/publ/rpfxf10t.htm

[3] F. Black and M. Scholes, “The Valuation of Option Con- tracts and a Test of Market Efficiency,” Journal of Finance, Vol. 27, No. 2, 1972, pp. 399-417. doi:10.2307/2978484

[4] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

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

[6] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.2307/232853

[7] J. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,” Journal of International Economics, Vol. 24, No. 4, 1988, pp. 129- 145.

[8] P. Ritchken and R. Trevor, “Pricing Options under Generalized GARCH and Stochastic Volatilities,” Interna- tional Journal of Theoretical and Applied Finance, Vol. 59, No. 1, 1999, pp. 377-402. doi:10.1111/0022-1082.00109

[9] J. B. Wiggins, “Option Values under Stochastic Volatilities,” Journal of Financial Economics, Vol. 19, 1987, pp. 129-145. doi:10.1016/0304-405X(87)90009-2

[10] S. L. Heston and S. Nandi, “A Closed-Form GARCH Option Valuation Model,” The Review of Financial Studies, Vol. 13, No. 1, 2000, pp. 585-625. doi:10.1093/rfs/13.3.585

[11] P. Christoffersen, S. Jeston and K Jacobs, “Option Valuation with Conditional Skewness,” Journal of Econometrics, Vol. 131, No. 1-2, 2006, pp. 253-284. doi:10.1016/j.jeconom.2005.01.010

[12] L. Mecuri, “Option Pricing in a GARCH Model with Tempered Stable Innovations,” Finance Research Letter, Vol. 5, No. 3, 2008, pp. 172-182. doi:10.1016/j.frl.2008.05.003

[13] G. Barone-Adesi, H. Rasmussen and C. Ravanelli, “An Option Pricing Formula for the GARCH Diffusion Model,” Computational Statistics and Data Analysis, Vol. 49, No. 2, 2005, pp. 287-310. doi:10.1016/j.csda.2004.05.014

[14] A. M. Badescu and R. J. Kulperger, “GARCH Option Pricing: A Semi parametric Approach,” Insurance Mathematics and Economics, Vol. 43, No. 1, 2008, pp. 69-84. doi:10.1016/j.insmatheco.2007.09.011

[15] H. Gong, A. Thavaneswaran and J. Singh, “A Black- Scholes Model with GARCH Volatility,” The Mathematical Scientist, Vol. 35, No. 1, 2010, pp. 37-42.

[16] H. Gong, A. Thavaneswaran and J. Singh, “Stochastic Volatility Models with Application in Option Pricing,” Journal of Statistical Theory and Practice, Vol. 4, No. 4, 2010, pp. 541-557. doi:10.1080/15598608.2010.10412003

[17] R. J. Elliot, T. K. Siu and L. Chan, “Option Pricing for GARCH Models with Markov Switching,” International Journal of Theoretical and Applied Finance, Vol. 9. No. 6, 2006, pp. 825-841. doi:10.1142/SO219024906003846

[18] A. Thavaneswaran, S. Peiris and J. Singh, “Derivation of Kurtosis and Option Pricing Formula for Popular Volatility Models with Applications to Finance,” Communications in Statistics, Theory and Methods, Vol. 37, 2008, pp. 1223-1258. doi:10.1080/03610920701826435

[19] A. Thavaneswaran, J. Singh and S. S. Appadoo, “Option for Some Stochastic Volatility Models,” Journal of Risk Finance, Vol. 7, No. 4, 2006, pp. 425-445. doi:10.1108/15265940610688982

[20] E. Pillay and J. G. O’Hara, “FFT Based Option Pricing under a Mean Reverting Process with Stochastic Volatility and Jumps,” Journal of Computation and Applied Mathematics, Vol. 235, No. 12, 2011, pp. 3378-3384. doi:10.1016/j.cam.2010.10.024

[21] N. Makate and P. Sattayatham, “Stochastic Volatility Jump-Diffusion Model for Option Pricing,” Journal of Mathematical Finance, Vol. 1 No. 1, 2011, pp. 90-97. doi:10.4236/jmf.2011.13012

[22] S. Pinkham and P. Sattayatha, “European Option Pricing for a Stochastic Volatility Levy Model,” Journal of Mathematical Finance, Vol. 1 No. 1 2011, pp. 98-108. doi:10.4236/jmf.2011.13013

[23] H. Gong, A. Thavaneswaran and Y. Liang, “Recent De- velopments in Option Pricing,” Journal of Mathematical Finance, Vol. 1, No. 1, 2011, pp. 63-71. doi:10.4236/jmf.2011.13009

[24] V. G. Ivancevic, “Adaptive Wave Models for Sophisticated Option Pricing,” Journal of Mathematic Finance, Vol. 1, No. 1, 2011, pp. 41-49. doi:10.4236/jmf.2011.13006

[25] C. M. Ahn, D. C. Cho and K. Park, “The Pricing of Foreign Currency Options under Jump-Diffusion Processes,” Journal of Futures Markets, Vol. 27, No. 7, 2007, pp 669-695.

[26] I. Hart and M. Ross, “Striking Continuity,” Risk, Vol. 7, No. 6, 1994, pp. 51-56.

[1] M. B. Garman and S. W. Kohlhagen, “Foreign Currency Options Values,” Journal of International Money and Finance, Vol. 2, No. 3, 1983, pp. 231-237. doi:10.1016/S0261-5606(83)80001-1

[2] Bank for International Settlements, “Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity,” 1996-2010. http://www.bis.org/publ/rpfxf10t.htm

[3] F. Black and M. Scholes, “The Valuation of Option Con- tracts and a Test of Market Efficiency,” Journal of Finance, Vol. 27, No. 2, 1972, pp. 399-417. doi:10.2307/2978484

[4] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

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

[6] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.2307/232853

[7] J. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,” Journal of International Economics, Vol. 24, No. 4, 1988, pp. 129- 145.

[8] P. Ritchken and R. Trevor, “Pricing Options under Generalized GARCH and Stochastic Volatilities,” Interna- tional Journal of Theoretical and Applied Finance, Vol. 59, No. 1, 1999, pp. 377-402. doi:10.1111/0022-1082.00109

[9] J. B. Wiggins, “Option Values under Stochastic Volatilities,” Journal of Financial Economics, Vol. 19, 1987, pp. 129-145. doi:10.1016/0304-405X(87)90009-2

[10] S. L. Heston and S. Nandi, “A Closed-Form GARCH Option Valuation Model,” The Review of Financial Studies, Vol. 13, No. 1, 2000, pp. 585-625. doi:10.1093/rfs/13.3.585

[11] P. Christoffersen, S. Jeston and K Jacobs, “Option Valuation with Conditional Skewness,” Journal of Econometrics, Vol. 131, No. 1-2, 2006, pp. 253-284. doi:10.1016/j.jeconom.2005.01.010

[12] L. Mecuri, “Option Pricing in a GARCH Model with Tempered Stable Innovations,” Finance Research Letter, Vol. 5, No. 3, 2008, pp. 172-182. doi:10.1016/j.frl.2008.05.003

[13] G. Barone-Adesi, H. Rasmussen and C. Ravanelli, “An Option Pricing Formula for the GARCH Diffusion Model,” Computational Statistics and Data Analysis, Vol. 49, No. 2, 2005, pp. 287-310. doi:10.1016/j.csda.2004.05.014

[14] A. M. Badescu and R. J. Kulperger, “GARCH Option Pricing: A Semi parametric Approach,” Insurance Mathematics and Economics, Vol. 43, No. 1, 2008, pp. 69-84. doi:10.1016/j.insmatheco.2007.09.011

[15] H. Gong, A. Thavaneswaran and J. Singh, “A Black- Scholes Model with GARCH Volatility,” The Mathematical Scientist, Vol. 35, No. 1, 2010, pp. 37-42.

[16] H. Gong, A. Thavaneswaran and J. Singh, “Stochastic Volatility Models with Application in Option Pricing,” Journal of Statistical Theory and Practice, Vol. 4, No. 4, 2010, pp. 541-557. doi:10.1080/15598608.2010.10412003

[17] R. J. Elliot, T. K. Siu and L. Chan, “Option Pricing for GARCH Models with Markov Switching,” International Journal of Theoretical and Applied Finance, Vol. 9. No. 6, 2006, pp. 825-841. doi:10.1142/SO219024906003846

[18] A. Thavaneswaran, S. Peiris and J. Singh, “Derivation of Kurtosis and Option Pricing Formula for Popular Volatility Models with Applications to Finance,” Communications in Statistics, Theory and Methods, Vol. 37, 2008, pp. 1223-1258. doi:10.1080/03610920701826435

[19] A. Thavaneswaran, J. Singh and S. S. Appadoo, “Option for Some Stochastic Volatility Models,” Journal of Risk Finance, Vol. 7, No. 4, 2006, pp. 425-445. doi:10.1108/15265940610688982

[20] E. Pillay and J. G. O’Hara, “FFT Based Option Pricing under a Mean Reverting Process with Stochastic Volatility and Jumps,” Journal of Computation and Applied Mathematics, Vol. 235, No. 12, 2011, pp. 3378-3384. doi:10.1016/j.cam.2010.10.024

[21] N. Makate and P. Sattayatham, “Stochastic Volatility Jump-Diffusion Model for Option Pricing,” Journal of Mathematical Finance, Vol. 1 No. 1, 2011, pp. 90-97. doi:10.4236/jmf.2011.13012

[22] S. Pinkham and P. Sattayatha, “European Option Pricing for a Stochastic Volatility Levy Model,” Journal of Mathematical Finance, Vol. 1 No. 1 2011, pp. 98-108. doi:10.4236/jmf.2011.13013

[23] H. Gong, A. Thavaneswaran and Y. Liang, “Recent De- velopments in Option Pricing,” Journal of Mathematical Finance, Vol. 1, No. 1, 2011, pp. 63-71. doi:10.4236/jmf.2011.13009

[24] V. G. Ivancevic, “Adaptive Wave Models for Sophisticated Option Pricing,” Journal of Mathematic Finance, Vol. 1, No. 1, 2011, pp. 41-49. doi:10.4236/jmf.2011.13006

[25] C. M. Ahn, D. C. Cho and K. Park, “The Pricing of Foreign Currency Options under Jump-Diffusion Processes,” Journal of Futures Markets, Vol. 27, No. 7, 2007, pp 669-695.

[26] I. Hart and M. Ross, “Striking Continuity,” Risk, Vol. 7, No. 6, 1994, pp. 51-56.