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Computing option price for Lévy process with fuzzy parameters. (English) Zbl 1177.91132

Summary: In the following paper we propose the method for option pricing based on application of stochastic analysis and theory of fuzzy numbers. The process of underlying asset trajectory belongs to a subclass of Levy processes with jumps. From practical point of view some parameters of such trajectory cannot be precisely described. Therefore, some degree of the market uncertainly has to be reflected in the description of the model itself. For example, the parameters (like interest rate, volatility) of financial market fluctuate from time to time and experts may have different opinions about such parameters. Using theory of fuzzy numbers and stochastic analysis enables us to take into account many sources of uncertainty, not only the probabilistic one. In our paper we apply the theory of Levy characteristics and present some numerical experiments based on Monte Carlo simulations. In detail, we present pricing formula for classical example of European call option.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
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References:

[1] (Bardossy, A.; Duckstein, L., Fuzzy Rule-Based Modeling with Applications to Geophysical, Biological and Engineering Systems (Systems Engineering) (1995), CRC-Press) · Zbl 0857.92001
[2] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-659 (1973) · Zbl 1092.91524
[3] Buckley, J. J., Fuzzy Statistics (2004), Springer · Zbl 1077.62051
[4] Davis, M., Mathematics of financial markets, (Engquist, B.; Schmid, W., Mathematics Unlimited - 2001 & Beyond (2001), Springer: Springer Berlin) · Zbl 1047.91028
[5] Dubois, D.; Prade, H., Fuzzy Sets and Systems - Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049
[6] Duffie, D.; Glynn, P., Efficient Monte Carlo Simulation for Security Prices, Annals of Applied Probability, 5 (1997)
[7] Glasserman, P., Monte Carlo Methods in Financial Engineering (2004), Springer-Verlag · Zbl 1038.91045
[8] Hull, J. C., Options, Futures and Other Derivatives (1997), Prentice-Hall · Zbl 1087.91025
[9] Korn, R.; Korn, E., Option Pricing and Portfolio Optimization (2001), American Mathematical Society · Zbl 0965.91020
[10] Miyahara, Y., (A Note on Esscher Transformed Martingale Measures for Geometric Levy Processes Discussion Papers in Economics, vol. 379 (2004), Nagoya City University), 1-14
[11] P. Nowak, P. Nycz, M. Romaniuk, Dobór optymalnego modelu stochastycznego w wycenie opcji metodami Monte Carlo (in Polish), in: J. Kacprzyk, J. Weglarz, (Eds.) Modelowanie i optymalizacja - Metody i zastosowania. Exit, 2002.; P. Nowak, P. Nycz, M. Romaniuk, Dobór optymalnego modelu stochastycznego w wycenie opcji metodami Monte Carlo (in Polish), in: J. Kacprzyk, J. Weglarz, (Eds.) Modelowanie i optymalizacja - Metody i zastosowania. Exit, 2002.
[12] Shiryaev, A. N.; Kruzhilin, N., Essential of Stochastic Finance (1999/2000), World Scientific Publishing Co. Pvt. Ltd
[13] Wu, H.-Ch., Pricing European options based on the fuzzy pattern of Black-Scholes formula, Computers & Operations Research, 31, 1069-1081 (2004) · Zbl 1062.91041
[14] Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606
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