Valoración de opciones dependientes de trayectoria usando la transformada de Mellin
Valoración de opciones dependientes de trayectoria usando la transformada de Mellin
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León Nieto, D.I. 2016. Valoración de opciones dependientes de trayectoria usando la transformada de Mellin. ODEON. 10 (oct. 2016), 117–156. DOI:https://doi.org/10.18601/17941113.n10.06.
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En muchos casos no existe, o es muy difícil, encontrar una solución analítica para la valoración de opciones con perfiles de pago complejos. Con tal motivación se presenta un marco general para la valoración de opciones dependientes de trayectoria, y los casos más relevantes como opciones asiáticas y opciones lookback. Dicho análisis está soportado en la deducción de la ecuación diferencial parcial Black-Scholes para el caso general y los casos específicos. Posteriormente se abordará la aplicación de la transformada de Mellin a la valoración de opciones dependientes de trayectoria. Se encuentra que este método tiene un gran potencial por desarrollar, sencillo y fácil de implementar, porque reduce el problema de valoración bajo la perspectiva de la ecuación diferencial parcial de Black-Scholes a la solución numérica de una integral.
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AitSahlia, F. y Lai, T. L. (1997). Valuation of discrete barrier and hindsight options. Journal of Financial Engineering, 6 (2), 169-177.
Babbs, S. (2000). Binomial valuation of lookback options. Journal of Economic Dynamics and Control, 24 (11), 1499-1525.
Beckers, S. (1980). The constant elasticity of variance model and its implications for option pricing. The Journal of Finance, 35 (3), 661-673.
Black, F. y Scholes, M. (1973). The pricing of options and corporate liabilities. The journal of political economy, 637-654.
Borovkov, K. y Novikov, A. (2002). On a new approach to calculating expectations for option pricing. Journal of Applied Probability, 39 (4), 889-895.
Boyle, P. P. (1999). Pricing lookback and barrier options under the CEV process. Journal of financial and quantitative analysis, 34 (02), 241-264.
Broadie, M., Glasserman, P. y Kou, S. G. (1999). Connecting discrete and continuous path-dependent options. Finance and Stochastics, 3 (1), 55-82.
Brychkov, Y. A., Glaeske, H. J., Prudnikov, A. P. y Tuan, V. K. (1992). Multidimensional Integral Transformations. Gordon and Breach Science Publishers.
Cahen, E. (1894). Sur la fonction ζ(s) de Riemann et sur des functions analogues, In Annales scientifiques de l’École Normale Supérieure (vol. 11, pp. 75-164). Société mathématique de France.
Cai, N. y Kou, S. G. (2011). Option pricing under a mixed-exponential jump diffusion model. Management Science, 57 (11), 2067-2081.
Chandra, S. R. y Mukherjee, D. (2016). Barrier Option Under Lévy Model: A PIDE and Mellin Transform Approach. Mathematics, 4 (1), 2.
Chandra, S. R., Mukherjee, D. y SenGupta, I. (2014). Mellin Transform Approach for Pricing of Lookback Option Under Levy Process with Minimal Martingale Measure. Available at SSRN 2232509.
Conze, A. y R. Vishwanathan (1991). Path-dependent options: The case of Look- back options. Journal Finance, 46, 1893-1907.
Debnath, L. y Bhatta, D. (2014). Integral Transforms and Their Applications (2 ed.). Chapman and Hall/CRC Press.
Dewynne, J. N. y Shaw, W. T. (2008). Differential equations and asymptotic solutions for arithmetic Asian options: ‘Black–Scholes formulae’ for Asian rate calls. European Journal of Applied Mathematics, 19 (04), 353-391.
Elshegmani, Z. A. y Ahmed, R. R. (2011). Analytical solution for an arithmetic Asian option using Mellin transforms. International Journal of Mathematical Analysis, 5 (25-28), 1259-1265.
Elshegmani, Z. A., Ahmad, R. R. y Zakaria, R. H. (2011). New pricing formula for arithmetic Asian options using PDE approach. Appl. Math. Sci., Ruse, 5 (77-80), 3801-3809.
Eltayeb, H. y Kilicman, A. (2007). A note on Mellin transform and partial differential equations. International Journal of Pure and Applied Mathematics, 34 (4), 457.
Fabozzi, F. J. (ed.) (2008). Handbook of Finance, Financial Markets and Instruments (vol. 1). John Wiley & Sons.
Forsyth, P. A., Vetzal, K. R. y Zvan, R. (1999). A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Applied Mathematical Finance, 6 (2), 87-106.
Frontczak, R. (2013). Pricing options in jump diffusion models using Mellin transforms. Journal of Mathematical Finance, 3 (03), 366.
Frontczak, R. y Schöbel, R. (2010). On modified Mellin transforms, Gauss–Laguerre quadrature, and the valuation of American call options. Journal of computational and applied mathematics, 234 (5), 1559-1571.
Frontczak, R. y Schöbel, R. (2008). Pricing American options with Mellin transforms (No. 319). Tübinger Diskussionsbeitrag.
Geman, H. y Yor, M. (1993). Bessel Processes, Asian Options, and Perpetuities. Mathematical Finance, 3 (4), 349-375.
Goldman, M. B., Sosin, H. B. y Gatto, M. A. (1979). Path dependent options: “Buy at the low, sell at the high”. The Journal of Finance, 34 (5), 1111-1127.
Goldman, M. B., Sosin, H. B. y Shepp, L. A. (1979). On contingent claims that insure expost optimal stock market timing. The Journal of Finance, 34 (2), 401-413.
Heynen, R. C. y Kat, H. M. (1995). Lookback options with discrete and partial monitoring of the underlying price. Applied Mathematical Finance, 2 (4), 273-284.
Hull, J. C. y White, A. D. (1993). Efficient procedures for valuing European and American path-dependent options. The Journal of Derivatives, 1 (1), 21-31.
Kou, S. G. y Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management science, 50 (9), 1178-1192.
Kou, S. G. (2002). A jump-diffusion model for option pricing. Management Science, 48 (8), 1086-1101.
Krapivsky, P. L. y Ben-Naim, E. (1994). Scaling and multiscaling in models of fragmentation. Physical Review E, 50 (5), 3502.
Linetsky, V. (2004). Lookback options and diffusion hitting times: A spectral expansion approach. Finance and Stochastics, 8 (3), 373-398.
Mellin, H. (1896). Über die fundamentale Wichtigkeit des Satzes von Cauchy für die Theorien der Gamma-und der hypergeometrischen Functionen (vol. 21). ex officina typographica Societatis litterariae fennicae.
Mellin, H. J. (1902). Über den Zusammenhang zwischen den linearen Differential-und Differenzengleichungen. Acta Mathematica, 25 (1), 139-164.
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of financial economics, 3 (1-2), 125-144.
Ohgren, A. (2001). A remark on the pricing of discrete lookback options. Journal of computational finance, 4 (3), 141-147.
Panini, R. y Srivastav, R. P. (2004). Option pricing with Mellin transnforms. Mathematical and Computer Modelling, 40 (1), 43-56.
Panini, R. y Srivastav, R. P. (2005). Pricing perpetual options using Mellin transforms. Applied Mathematics Letters, 18 (4), 471-474.
Petrella, G. y Kou, S. (2004). Numerical pricing of discrete barrier and lookback options via Laplace transforms. Journal of Computational Finance, 8, 1-38.
Riemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse. Ges. Math. Werke und Wissenschaftlicher Nachlaß, 2, 145-155.
Rogers, L. C. G. y Shi, Z. (1995). The value of an Asian option. Journal of Applied Probability, 1077-1088.
SenGupta, I. (2014). Pricing Asian options in financial markets using Mellin transforms. Electronic Journal of Differential Equations, 2014 (234), 1-9.
Spitzer, F. (1956). A combinatorial lemma and its application to probability theory. Transactions of the American Mathematical Society, 82 (2), 323-339.
Tse, W. M., Li, L. K. y Ng, K. W. (2001). Pricing discrete barrier and hindsight options with the tridiagonal probability algorithm. Management Science, 47 (3), 383-393.
Venegas Martínez, F., (2008). Riesgos financieros y económicos / Financial and Economical Risks: productos derivados y decisiones económicas bajo incertidumbre. Cengage Learning Editores.
Wilmott, P., Dewynne, J. y Howison, S. (1993). Option pricing: mathematical models and computation, Oxford Financial Press.
Wilmott, P., Howison, S. y Dewynne, J. (1995). The mathematics of financial derivatives: a student introduction. Cambridge: Cambridge University Press.
Zhang, J. E. (2000). Theory of Continously-sampled Asian Option Pricing (No. 140). City University of Hong Kong, Faculty of Business, Department of Economics and Finance.
Babbs, S. (2000). Binomial valuation of lookback options. Journal of Economic Dynamics and Control, 24 (11), 1499-1525.
Beckers, S. (1980). The constant elasticity of variance model and its implications for option pricing. The Journal of Finance, 35 (3), 661-673.
Black, F. y Scholes, M. (1973). The pricing of options and corporate liabilities. The journal of political economy, 637-654.
Borovkov, K. y Novikov, A. (2002). On a new approach to calculating expectations for option pricing. Journal of Applied Probability, 39 (4), 889-895.
Boyle, P. P. (1999). Pricing lookback and barrier options under the CEV process. Journal of financial and quantitative analysis, 34 (02), 241-264.
Broadie, M., Glasserman, P. y Kou, S. G. (1999). Connecting discrete and continuous path-dependent options. Finance and Stochastics, 3 (1), 55-82.
Brychkov, Y. A., Glaeske, H. J., Prudnikov, A. P. y Tuan, V. K. (1992). Multidimensional Integral Transformations. Gordon and Breach Science Publishers.
Cahen, E. (1894). Sur la fonction ζ(s) de Riemann et sur des functions analogues, In Annales scientifiques de l’École Normale Supérieure (vol. 11, pp. 75-164). Société mathématique de France.
Cai, N. y Kou, S. G. (2011). Option pricing under a mixed-exponential jump diffusion model. Management Science, 57 (11), 2067-2081.
Chandra, S. R. y Mukherjee, D. (2016). Barrier Option Under Lévy Model: A PIDE and Mellin Transform Approach. Mathematics, 4 (1), 2.
Chandra, S. R., Mukherjee, D. y SenGupta, I. (2014). Mellin Transform Approach for Pricing of Lookback Option Under Levy Process with Minimal Martingale Measure. Available at SSRN 2232509.
Conze, A. y R. Vishwanathan (1991). Path-dependent options: The case of Look- back options. Journal Finance, 46, 1893-1907.
Debnath, L. y Bhatta, D. (2014). Integral Transforms and Their Applications (2 ed.). Chapman and Hall/CRC Press.
Dewynne, J. N. y Shaw, W. T. (2008). Differential equations and asymptotic solutions for arithmetic Asian options: ‘Black–Scholes formulae’ for Asian rate calls. European Journal of Applied Mathematics, 19 (04), 353-391.
Elshegmani, Z. A. y Ahmed, R. R. (2011). Analytical solution for an arithmetic Asian option using Mellin transforms. International Journal of Mathematical Analysis, 5 (25-28), 1259-1265.
Elshegmani, Z. A., Ahmad, R. R. y Zakaria, R. H. (2011). New pricing formula for arithmetic Asian options using PDE approach. Appl. Math. Sci., Ruse, 5 (77-80), 3801-3809.
Eltayeb, H. y Kilicman, A. (2007). A note on Mellin transform and partial differential equations. International Journal of Pure and Applied Mathematics, 34 (4), 457.
Fabozzi, F. J. (ed.) (2008). Handbook of Finance, Financial Markets and Instruments (vol. 1). John Wiley & Sons.
Forsyth, P. A., Vetzal, K. R. y Zvan, R. (1999). A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Applied Mathematical Finance, 6 (2), 87-106.
Frontczak, R. (2013). Pricing options in jump diffusion models using Mellin transforms. Journal of Mathematical Finance, 3 (03), 366.
Frontczak, R. y Schöbel, R. (2010). On modified Mellin transforms, Gauss–Laguerre quadrature, and the valuation of American call options. Journal of computational and applied mathematics, 234 (5), 1559-1571.
Frontczak, R. y Schöbel, R. (2008). Pricing American options with Mellin transforms (No. 319). Tübinger Diskussionsbeitrag.
Geman, H. y Yor, M. (1993). Bessel Processes, Asian Options, and Perpetuities. Mathematical Finance, 3 (4), 349-375.
Goldman, M. B., Sosin, H. B. y Gatto, M. A. (1979). Path dependent options: “Buy at the low, sell at the high”. The Journal of Finance, 34 (5), 1111-1127.
Goldman, M. B., Sosin, H. B. y Shepp, L. A. (1979). On contingent claims that insure expost optimal stock market timing. The Journal of Finance, 34 (2), 401-413.
Heynen, R. C. y Kat, H. M. (1995). Lookback options with discrete and partial monitoring of the underlying price. Applied Mathematical Finance, 2 (4), 273-284.
Hull, J. C. y White, A. D. (1993). Efficient procedures for valuing European and American path-dependent options. The Journal of Derivatives, 1 (1), 21-31.
Kou, S. G. y Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management science, 50 (9), 1178-1192.
Kou, S. G. (2002). A jump-diffusion model for option pricing. Management Science, 48 (8), 1086-1101.
Krapivsky, P. L. y Ben-Naim, E. (1994). Scaling and multiscaling in models of fragmentation. Physical Review E, 50 (5), 3502.
Linetsky, V. (2004). Lookback options and diffusion hitting times: A spectral expansion approach. Finance and Stochastics, 8 (3), 373-398.
Mellin, H. (1896). Über die fundamentale Wichtigkeit des Satzes von Cauchy für die Theorien der Gamma-und der hypergeometrischen Functionen (vol. 21). ex officina typographica Societatis litterariae fennicae.
Mellin, H. J. (1902). Über den Zusammenhang zwischen den linearen Differential-und Differenzengleichungen. Acta Mathematica, 25 (1), 139-164.
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of financial economics, 3 (1-2), 125-144.
Ohgren, A. (2001). A remark on the pricing of discrete lookback options. Journal of computational finance, 4 (3), 141-147.
Panini, R. y Srivastav, R. P. (2004). Option pricing with Mellin transnforms. Mathematical and Computer Modelling, 40 (1), 43-56.
Panini, R. y Srivastav, R. P. (2005). Pricing perpetual options using Mellin transforms. Applied Mathematics Letters, 18 (4), 471-474.
Petrella, G. y Kou, S. (2004). Numerical pricing of discrete barrier and lookback options via Laplace transforms. Journal of Computational Finance, 8, 1-38.
Riemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse. Ges. Math. Werke und Wissenschaftlicher Nachlaß, 2, 145-155.
Rogers, L. C. G. y Shi, Z. (1995). The value of an Asian option. Journal of Applied Probability, 1077-1088.
SenGupta, I. (2014). Pricing Asian options in financial markets using Mellin transforms. Electronic Journal of Differential Equations, 2014 (234), 1-9.
Spitzer, F. (1956). A combinatorial lemma and its application to probability theory. Transactions of the American Mathematical Society, 82 (2), 323-339.
Tse, W. M., Li, L. K. y Ng, K. W. (2001). Pricing discrete barrier and hindsight options with the tridiagonal probability algorithm. Management Science, 47 (3), 383-393.
Venegas Martínez, F., (2008). Riesgos financieros y económicos / Financial and Economical Risks: productos derivados y decisiones económicas bajo incertidumbre. Cengage Learning Editores.
Wilmott, P., Dewynne, J. y Howison, S. (1993). Option pricing: mathematical models and computation, Oxford Financial Press.
Wilmott, P., Howison, S. y Dewynne, J. (1995). The mathematics of financial derivatives: a student introduction. Cambridge: Cambridge University Press.
Zhang, J. E. (2000). Theory of Continously-sampled Asian Option Pricing (No. 140). City University of Hong Kong, Faculty of Business, Department of Economics and Finance.
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