A fuzzy approach to option pricing in a Levy process setting

• Autorzy: Piotr Nowak , Maciej Romaniuk
• Języki publikacji: EN

Abstrakty

EN

In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets. We assume that some parameters of the financial instrument cannot be precisely described and therefore they are introduced to the model as fuzzy numbers. Application of fuzzy arithmetic enables us to consider various sources of uncertainty, not only the stochastic one. To obtain the European call option pricing formula we use the minimal entropy martingale measure and Levy characteristics.

Słowa kluczowe

EN

option pricing; Levy processes; minimal entropy martingale measure; fuzzy sets; Monte Carlo simulation

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2013
• Tom: 23
• Numer: 3
• Strony: 613-622

Daty

wydano

2013

otrzymano

2012-07-17

poprawiono

2013-01-24

Twórcy

autor

Piotr Nowak

• Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland

autor

Maciej Romaniuk

• Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland
• The John Paul II Catholic University of Lublin, ul. Konstantynów 1H, 20-708 Lublin, Poland

Bibliografia

• Bakshi, G., Cao, C. and Chen, Z. (1997). Empirical performance of alternative option pricing models, The Journal of Finance LII(5): 2003-2049.
• Bardossy, A. and Duckstein, L. (1995). Fuzzy Rule-Based Modeling with Applications to Geophysical, Biological and Engineering Systems (Systems Engineering), CRC Press, Boca Raton, FL.
• Barndorff-Nielsen, O.E. (1998). Processes of normal inverse Gaussian type, Finance and Stochastics 2(1): 41-68.
• Bates, D. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutschemark options, The Review of Financial Studies 9(1): 69-107.
• Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities, Journal of Political Economy 81(3): 637-659.
• Brigo, D., Pallavicini, A. and Torresetti, R. (2007). Credit derivatives: Calibration of CDO tranches with the dynamical GPL model, Risk Magazine 20(5): 70-75.
• Davis, M. (2001). Mathematics of financial markets, in B. Engquist and W. Schmid (Eds.), Mathematics Unlimited-2001 & Beyond, Springer, Berlin, pp. 361-380.
• Dubois, D. and Prade, H. (1980). Fuzzy Sets and Systems - Theory and Applications, Academic Press, New York, NY.
• El Karoui, N. and Rouge, R. (2000). Pricing via utility maximization and entropy, Mathematical Finance 10(2): 259-276.
• Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets, Mathematical Finance 10(1): 39-52.
• Fujiwara, T. and Miyahara, Y. (2003). The minimal entropy martingale measures for geometric Levy processes, Finance and Stochastics 7(1): 509-531.
• Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering, Springer-Verlag, New York, NY.
• Hull, J.C. (1997). Options, Futures and Other Derivatives, Prentice Hall, Upper Saddle River, NJ.
• Jacod, J. and Shiryaev, A. (1987). Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin/Heidelberg/New York, NY.
• Kou, S.G. (2002). A jump-diffusion model for option pricing, Management Science 48(8): 1086-1101.
• Kou, S.G. and Wang, H. (2004). Option pricing under a double exponential jump diffusion model, Management Science 50(9): 1178-1192.
• Li, C. and Chiang, T.-W. (2012). Intelligent financial time series forecasting: A complex neuro-fuzzy approach with multi-swarm intelligence, International Journal of Applied Mathematics and Computer Science 22(4): 787-800, DOI: 10.2478/v10006-012-0058-x.
• Madan, D.B. and Seneta, E. (1990). The variance gamma (v.g.) model for share market returns, The Journal of Business 63(4): 511-524.
• Merton, R. (1976). Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics 3(1): 125-144.
• Miyahara, Y. (2004). A note on Esscher transformed martingale measures for geometric Levy processes, Discussion Papers in Economics, No. 379, Nagoya City University, Nagoya, pp. 1-14.
• Nowak, P. (2011). Option pricing with Levy process in a fuzzy framework, in K.T. Atanassov, W. Homenda, O. Hryniewicz, J. Kacprzyk, M. Krawczak, Z. Nahorski, E. Szmidt and S. Zadrożny (Eds.), Recent Advances in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics, Polish Academy of Sciences, Warsaw, pp. 155-167.
• Nowak, P., Nycz, P. and Romaniuk, M. (2002). On selection of the optimal stochastic model in the option pricing via Monte Carlo methods, in J. Kacprzyk and J. Węglarz (Eds.), Modelling and Optimization-Methods and Applications, Exit, Warsaw, pp. 59-70, (in Polish).
• Nowak, P. and Romaniuk, M. (2010). Computing option price for Levy process with fuzzy parameters, European Journal of Operational Research 201(1): 206-210.
• Shiryaev, A.N. (1999). Essential of Stochastic Finance, World Scientific Publishing, Singapore.
• Ssebugenyi, C.S. (2011). Using the minimal entropy martingale measure to valuate real options in multinomial lattices, Applied Mathematical Sciences 67(5): 3319-3334.
• Wu, H.-C. (2004). Pricing European options based on the fuzzy pattern of Black-Scholes formula, Computers & Operations Research 31(7): 1069-1081.
• Wu, H.-C. (2007). Using fuzzy sets theory and Black-Scholes formula to generate pricing boundaries of European options, Applied Mathematics and Computation 185(1): 136-146.
• Xu, W.D., Wu, C.F. and Li, H.Y. (2011). Foreign equity option pricing under stochastic volatility model with double jumps, Economic Modeling 28(4): 1857-1863.
• Yoshida, Y. (2003). The valuation of European options in uncertain environment, European Journal of Operational Research 145(1): 221-229.
• Zadeh, L.A. (1965). Fuzzy sets, Information and Control 8(47): 338-353.
• Zhang, L.-H., Zhang, W.-G., Xu, W.-J. and Xiao, W.-L. (2012). The double exponential jump diffusion model for pricing European options under fuzzy environments, Economic Modelling 29(3): 780-786.
• Zhou, C. (2002). Fuzzy-arithmetic-based Lyapunov synthesis in the design of stable fuzzy controllers: A computing-with-words approach, International Journal of Applied Mathematics and Computer Science 12(3): 411-421.

Identyfikatory

DOI

10.2478/amcs-2013-0046