Instrumento en MEXDER: modelo de tres factores para opciones sobre tipo de cambio
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feb 4, 2024
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https://doi.org/10.36677/ideaseningenieria.v2i1.22771
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Ma de Lourdes Nájera López
http://orcid.org/0000-0002-6824-9561
Raúl De Jesús Gutiérrez
http://orcid.org/0000-0001-6878-3038
http://orcid.org/0000-0002-6824-9561
Raúl De Jesús Gutiérrez
http://orcid.org/0000-0001-6878-3038
Resumen
A partir de las crisis cambiarias que han surgido, inversionistas, empresas y bancos centrales están en busca de instrumentos financieros que los protejan ante fluctuaciones de las variables macroeconómicas, como son el tipo de cambio y las tasas de interés. En este trabajo se presenta una extensión del modelo de Heston (1993) asociado con Hull y White (1990), resultando un modelo de tres factores para valuar el precio de las opciones sobre MXN/USD. Se estimaron los parámetros comprendiendo un periodo de 2003 a 2018 y se calcularon las primas para las opciones Call y Put. Los resultados revelan que subvalora los precios de la opción Call OTM y Put ITM y los sobrevalora para opciones Call ITM y Put OTM. La magnitud de los errores es de 11.79% para Call OTM con mayor vencimiento y de tan sólo 2.05% para el vencimiento próximo. Para las opciones Put, señala un máximo de 8.71%.
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Como citar
NÁJERA LÓPEZ, Ma de Lourdes; DE JESÚS GUTIÉRREZ, Raúl.
Instrumento en MEXDER: modelo de tres factores para opciones sobre tipo de cambio.
Ideas en Ciencias de la Ingeniería, [S.l.], v. 2, n. 1, p. 95-116, feb. 2024.
ISSN 2992-7447.
Disponible en: <https://ideasencienciasingenieria.uaemex.mx/article/view/22771>. Fecha de acceso: 12 jun. 2025
doi: https://doi.org/10.36677/ideaseningenieria.v2i1.22771.
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Artículos
Esta obra está bajo licencia internacional Creative Commons Reconocimiento-NoComercial-SinObrasDerivadas 4.0.
Citas
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2. J. Hull, and A. White. “Pricing Interest-Rate-Derivative Securities,” Review of Financial Studies.Vol. 3, No. 4, 573-592, 1990.
3. N. Bray,Report of the Presidential Task Force on Market Mechanisms. Washington, D.C. 1988.
4. Z. Ramírez, G. Vázquez y A. Bello.El casino de los derivados. Expansión. Pp. 129-140, 2008.
5. J. N. Bodurtha, and G. R. Courtadon, “The statistical distribution of exchange rates”. Journal ofInternational Economics. Vol. 22, pp. 297-319, 1987.
6. A. Melino, and S. Turnbull, The Pricing of Foreign Currency Options,Canadian Journal of Econo-mics. Vol.24, 455-480, 1991.
7. J. Hull, and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal ofFinance, 42, No. 2, 281-300, 1987.
8. E. M. Stein, and J. C. Stein, “Stock Price Distributions with Stochastic Volatility: An AnalyticApproach,” Review of Financial Studies. Vol. 4, 727-752, 1991.
9. R. Schöbel, and J. Zhu, “Stochastic volatility with an Ornstein-Uhlenbeck process: An extension,”European Finance Review, No. 4, 23–46, 1999.
10. O. Vasicek, “An equilibrium characterization of the term structure,” Journal of Financial Economics,Vol. 5, 177-188, 1997.
11. K. S. Miltersen, and D. Sondermann, “Closed-form solutions for term structure derivatives withlognormal interest rates,” Journal of Finance, Vol 52, 409–430, 1997.
12. Brace, D. Gatarek, and M. Musiela, “The market model of interest rate dynamics,” MathematicalFinance, Vol. 7, 127-155, 1997.
13. J. Cox, J. Ingersoll, and S. Ross, “A theory of the term structure of interest rates,” Econometrica,Vol. 53, 385-407, 1985.
14. D. Heath, R. Jarrow, and A. Morton, “Bond pricing and the term structure of interest rates: a newmethodology,” Econometrica, Vol. 60, 77-105, 1992.
15. J. Hull, and A. White, “One-Factor Interest-Rate Models and the Valuation of Interest-Rate Deriva-tive Securities”, Journal of Financial and Quantitative Analysis. Vol. 28, No. 2, 235-253, 1993.
16. J. K. Hunt, and A. Pelsser, “Markov functional interest rate models. Finance and Stochastics”, Vol.4, 391–408, 2000.
17. D. Brigo, and F. Mercurio, “Interest rate models. Theory and practice,” Springer Finance, 2007.
18. R. Ahlip, “Foreign exchange options under stochastic volatility and stochastic interest rates,” Inter-national Journal of Theoretical and Applied Finance, Vol. 11, No. 3, 277–294, 2008.
19. L. O. Scott, “Option pricing when the variance changes randomly: theory, estimators and, applica-tions,” Journal of Financial and Quantitative Analysis, Vol. 22, 419–438, 1997.
20. C. Bakshi, C. Cao, and Z. Chen, “Empirical performance of alternative option pricing models,” TheJournal of Finance, Vol. 52, pp. 2003-2049, 1997.
21. K. I. Amin, and V. K. Ng, “A comparison of predictable volatility models using option data,” Re-search Department working paper, International Monetary Fund, 1994.
22. T. G. Andersen, “Closed form pricing of FX options under stochastic rates and volatility,” Presen-tation at Global Derivatives Conference, Paris, 2006.
23. R. vanHaastrecht, and A. Lord, “Pricing long-dated insurance contracts with stochastic volatilityand stochastic interest rates,” Insurance: Mathematics and Economics. Vol. 45, No. 3, 2009.
24. Lord and J. Kahl, “Optimal Fourier inversion in semi-analytical option pricing,” Journal of Compu-tational Finance, Vol. 10, No. 4, 2008.
25. L. Grzelak, and C. W. Oosterlee, “On the Heston Model with stochastic Interest Rates,” MunichPersonal RePEc Archive, 2010.
26. Carr, and D.B. Madan, “Option valuation using the fast Fourier transform,” Journal of Computatio-nal Finance, Vol. 2, 61–73. 1999.
27. F. Fang, and C. W. Oosterlee, “A Novel Pricing Method for European Options Based On Fourier-Cosine Series Expansions,” Munich Personal RePEc Archive, 2008.
28. T. Haentjens, and K. J. Hout, “ADI finite difference schemes for the Heston-Hull-White PDE,” TheJournal of Computational Finance, Vol 16, pp. 83-110, 2012.
29. T. Haentjens, “ADI FD Schemes for the Numerical Solution of the Three-dimensional Heston–Cox–Ingersoll–Ross PDE,” Numerical Analysis and Applied Mathematics ICNAAM, Vol. 1479,pp. 2195-2199, 2012.
30. W. Anthonie, van del Stoep, L. Grzelak and C. Oosterlee, “A Novel Monte Carlo Approach toHybrid Local Volatility Models,” Quantitative Finance, Vol. 00, No. 00, pp. 1-31, 2016.
31. a. W. Van der Stoep, L. A. Grzelak, and C. W. Oosterlee, “The Heston Stochastic-Local Volati-lity Model: Efficient Monte Carlo Simulation,” International Journal of Theoretical and AppliedFinance, No. 17, 2014.
32. A. Cozma, and C. Reisinger, “A mixed Monte Carlo and PDE variance reduction method for foreignexchange options under the Heston-CIR model,” Working paper, arXiv:1509.01479v3 [q-fin.CP],2016.
33. C. Hendricks, M. Ehrhardt, and M. Günther, “Hybrid finite difference/pseudospectral methods forthe Heston and Heston-Hull-White PDE,” Journal of Computational Finance, Vol. 21, No 5, pp.1-33, 2018.
34. M. Z. Ullah, “Numerical Solution of Heston-Hull-White Three-Dimensional PDE with a High Or-der FD Scheme,” Mathematics, No. 7, 2019.
35. Y. Liang, and C. Xu, “An Efficient Conditional Monte Carlo Method for European Option Pricingwith Stochastic Volatility and Stochastic Interest Rate,” International Journal of Computer Mathe-matics, Vol. 97, No. 3, pp. 638-655, 2019.
36. F. Soleymani, and B. Nemati. “Pricing the financial Heston–Hull–White model with arbitrary corre-lation factors via an adaptive FDM,” Computers and Mathematics with Applications, Vol. 77, No.4, pp. 1107-1123, https://doi.org/10.1016/j.camwa.2018.10.047. 2019.
37. T. Liu, M.Z. Ullah, S. Shateyi, C. Liu, and Y. Yang. “An Efficient Localized RBF-FDMethod to Simulate the Heston-Hull-White PDE in Finance,” Mathematics. Vol. 11, pp. 833.
38. J. A. Nelder, and R. Mead, “A simplex method for function minimization,”Computer Journal, No.7, pp. 308–313, 1965.
39. F. Black, and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of PoliticalEconomy. Vol. 81, pp. 637-65, 1976.
40. S. E. Shreve, “Stochastic Calculus for Finance II,” Springer, 2004.
41. Y. M. Lin, N. Strong, and X. Xu. “Pricing FTSE 100 Index Options Under Stochastic Volatility,”Journal of Futures Markets, No. 21, pp. 197-211. 2011.
42. J. C. Hull,Options, Futures and Other Derivates, Pearson, Prentice Hall, Toronto. pp. 822. 2008.[43] M. L. Najera, “Valuación del precio de opciones europeas sobre la relación peso mexicano/dólarestadounidense para el periodo 2004-2018”, Tesis Doctoral, Repositorio Institucional UAEMéx,http://ri.uaemex.mx/handle/20.500.11799/110164. 2020.
2. J. Hull, and A. White. “Pricing Interest-Rate-Derivative Securities,” Review of Financial Studies.Vol. 3, No. 4, 573-592, 1990.
3. N. Bray,Report of the Presidential Task Force on Market Mechanisms. Washington, D.C. 1988.
4. Z. Ramírez, G. Vázquez y A. Bello.El casino de los derivados. Expansión. Pp. 129-140, 2008.
5. J. N. Bodurtha, and G. R. Courtadon, “The statistical distribution of exchange rates”. Journal ofInternational Economics. Vol. 22, pp. 297-319, 1987.
6. A. Melino, and S. Turnbull, The Pricing of Foreign Currency Options,Canadian Journal of Econo-mics. Vol.24, 455-480, 1991.
7. J. Hull, and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal ofFinance, 42, No. 2, 281-300, 1987.
8. E. M. Stein, and J. C. Stein, “Stock Price Distributions with Stochastic Volatility: An AnalyticApproach,” Review of Financial Studies. Vol. 4, 727-752, 1991.
9. R. Schöbel, and J. Zhu, “Stochastic volatility with an Ornstein-Uhlenbeck process: An extension,”European Finance Review, No. 4, 23–46, 1999.
10. O. Vasicek, “An equilibrium characterization of the term structure,” Journal of Financial Economics,Vol. 5, 177-188, 1997.
11. K. S. Miltersen, and D. Sondermann, “Closed-form solutions for term structure derivatives withlognormal interest rates,” Journal of Finance, Vol 52, 409–430, 1997.
12. Brace, D. Gatarek, and M. Musiela, “The market model of interest rate dynamics,” MathematicalFinance, Vol. 7, 127-155, 1997.
13. J. Cox, J. Ingersoll, and S. Ross, “A theory of the term structure of interest rates,” Econometrica,Vol. 53, 385-407, 1985.
14. D. Heath, R. Jarrow, and A. Morton, “Bond pricing and the term structure of interest rates: a newmethodology,” Econometrica, Vol. 60, 77-105, 1992.
15. J. Hull, and A. White, “One-Factor Interest-Rate Models and the Valuation of Interest-Rate Deriva-tive Securities”, Journal of Financial and Quantitative Analysis. Vol. 28, No. 2, 235-253, 1993.
16. J. K. Hunt, and A. Pelsser, “Markov functional interest rate models. Finance and Stochastics”, Vol.4, 391–408, 2000.
17. D. Brigo, and F. Mercurio, “Interest rate models. Theory and practice,” Springer Finance, 2007.
18. R. Ahlip, “Foreign exchange options under stochastic volatility and stochastic interest rates,” Inter-national Journal of Theoretical and Applied Finance, Vol. 11, No. 3, 277–294, 2008.
19. L. O. Scott, “Option pricing when the variance changes randomly: theory, estimators and, applica-tions,” Journal of Financial and Quantitative Analysis, Vol. 22, 419–438, 1997.
20. C. Bakshi, C. Cao, and Z. Chen, “Empirical performance of alternative option pricing models,” TheJournal of Finance, Vol. 52, pp. 2003-2049, 1997.
21. K. I. Amin, and V. K. Ng, “A comparison of predictable volatility models using option data,” Re-search Department working paper, International Monetary Fund, 1994.
22. T. G. Andersen, “Closed form pricing of FX options under stochastic rates and volatility,” Presen-tation at Global Derivatives Conference, Paris, 2006.
23. R. vanHaastrecht, and A. Lord, “Pricing long-dated insurance contracts with stochastic volatilityand stochastic interest rates,” Insurance: Mathematics and Economics. Vol. 45, No. 3, 2009.
24. Lord and J. Kahl, “Optimal Fourier inversion in semi-analytical option pricing,” Journal of Compu-tational Finance, Vol. 10, No. 4, 2008.
25. L. Grzelak, and C. W. Oosterlee, “On the Heston Model with stochastic Interest Rates,” MunichPersonal RePEc Archive, 2010.
26. Carr, and D.B. Madan, “Option valuation using the fast Fourier transform,” Journal of Computatio-nal Finance, Vol. 2, 61–73. 1999.
27. F. Fang, and C. W. Oosterlee, “A Novel Pricing Method for European Options Based On Fourier-Cosine Series Expansions,” Munich Personal RePEc Archive, 2008.
28. T. Haentjens, and K. J. Hout, “ADI finite difference schemes for the Heston-Hull-White PDE,” TheJournal of Computational Finance, Vol 16, pp. 83-110, 2012.
29. T. Haentjens, “ADI FD Schemes for the Numerical Solution of the Three-dimensional Heston–Cox–Ingersoll–Ross PDE,” Numerical Analysis and Applied Mathematics ICNAAM, Vol. 1479,pp. 2195-2199, 2012.
30. W. Anthonie, van del Stoep, L. Grzelak and C. Oosterlee, “A Novel Monte Carlo Approach toHybrid Local Volatility Models,” Quantitative Finance, Vol. 00, No. 00, pp. 1-31, 2016.
31. a. W. Van der Stoep, L. A. Grzelak, and C. W. Oosterlee, “The Heston Stochastic-Local Volati-lity Model: Efficient Monte Carlo Simulation,” International Journal of Theoretical and AppliedFinance, No. 17, 2014.
32. A. Cozma, and C. Reisinger, “A mixed Monte Carlo and PDE variance reduction method for foreignexchange options under the Heston-CIR model,” Working paper, arXiv:1509.01479v3 [q-fin.CP],2016.
33. C. Hendricks, M. Ehrhardt, and M. Günther, “Hybrid finite difference/pseudospectral methods forthe Heston and Heston-Hull-White PDE,” Journal of Computational Finance, Vol. 21, No 5, pp.1-33, 2018.
34. M. Z. Ullah, “Numerical Solution of Heston-Hull-White Three-Dimensional PDE with a High Or-der FD Scheme,” Mathematics, No. 7, 2019.
35. Y. Liang, and C. Xu, “An Efficient Conditional Monte Carlo Method for European Option Pricingwith Stochastic Volatility and Stochastic Interest Rate,” International Journal of Computer Mathe-matics, Vol. 97, No. 3, pp. 638-655, 2019.
36. F. Soleymani, and B. Nemati. “Pricing the financial Heston–Hull–White model with arbitrary corre-lation factors via an adaptive FDM,” Computers and Mathematics with Applications, Vol. 77, No.4, pp. 1107-1123, https://doi.org/10.1016/j.camwa.2018.10.047. 2019.
37. T. Liu, M.Z. Ullah, S. Shateyi, C. Liu, and Y. Yang. “An Efficient Localized RBF-FDMethod to Simulate the Heston-Hull-White PDE in Finance,” Mathematics. Vol. 11, pp. 833.
38. J. A. Nelder, and R. Mead, “A simplex method for function minimization,”Computer Journal, No.7, pp. 308–313, 1965.
39. F. Black, and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of PoliticalEconomy. Vol. 81, pp. 637-65, 1976.
40. S. E. Shreve, “Stochastic Calculus for Finance II,” Springer, 2004.
41. Y. M. Lin, N. Strong, and X. Xu. “Pricing FTSE 100 Index Options Under Stochastic Volatility,”Journal of Futures Markets, No. 21, pp. 197-211. 2011.
42. J. C. Hull,Options, Futures and Other Derivates, Pearson, Prentice Hall, Toronto. pp. 822. 2008.[43] M. L. Najera, “Valuación del precio de opciones europeas sobre la relación peso mexicano/dólarestadounidense para el periodo 2004-2018”, Tesis Doctoral, Repositorio Institucional UAEMéx,http://ri.uaemex.mx/handle/20.500.11799/110164. 2020.