The solution of fuzzy linear systems by nonlinear programming: a financial application. (English) Zbl 1111.90105
Summary: Fuzzy linear systems of equations play a major role in various financial applications. In this paper we analyse a particular fuzzy linear system: the derivation of the risk neutral probabilities in a fuzzy binary tree. This system has previously been investigated and different solutions to different forms of the same system have been proposed. The aim of this paper is twofold. First, we highlight that the different solutions proposed, arise from different forms of the same system. Second, in order to find a unique vector solution for the system, we propose a practical algorithm that boils down to the solution of a nonlinear optimization problem.
References:
| [1] | Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654 (1973) · Zbl 1092.91524 |
| [2] | Boyle, P.; Vorst, T., Option replication in discrete time with transaction costs, The Journal of Finance, 47, 1, 271-293 (1992) |
| [3] | Buckley, J. J.; Qu, Y., Solving systems of linear fuzzy equations, Fuzzy Sets and Systems, 43, 33-43 (1991) · Zbl 0741.65023 |
| [4] | Cox, J.; Ross, S.; Rubinstein, S., Option pricing, a simplified approach, Journal of Financial Economics, 7, 229-263 (1979) · Zbl 1131.91333 |
| [5] | Cox, J.; Rubinstein, M., Option Markets (1985), Prentice-Hall |
| [6] | Derman, E.; Kani, I., Riding on a smile, Risk, 7, 2, 32-39 (1994) |
| [7] | Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press · Zbl 0444.94049 |
| [8] | Dupire, B., Pricing with a smile, Risk, 7, 1, 18-20 (1994) |
| [9] | Hull, J.; White, A., The pricing of options on assets with stochastic volatilities, The Journal of Finance, 42, 2, 281-300 (1987) |
| [10] | Karatzas, I.; Kou, S., On the pricing of contingent claims under constraints, The Annals of Applied Probability, 6, 2, 321-369 (1996) · Zbl 0856.90012 |
| [11] | Leland, H., Option Pricing and Replication with Transaction Costs, The Journal of Finance, 40, 5, 1283-1301 (1985) |
| [12] | Moore, R. E., Methods and applications of interval analysis, SIAM Studies in Applied Mathematics (1966) · Zbl 0176.13301 |
| [13] | Muzzioli, S., 2003. A note on fuzzy linear systems, Discussion Paper no. 447, Economics Department, University of Modena and Reggio Emilia.; Muzzioli, S., 2003. A note on fuzzy linear systems, Discussion Paper no. 447, Economics Department, University of Modena and Reggio Emilia. |
| [14] | Muzzioli, S., Reynaerts, H., in press. Fuzzy linear systems of the form \(A_1xb_1A_2xb_2\); Muzzioli, S., Reynaerts, H., in press. Fuzzy linear systems of the form \(A_1xb_1A_2xb_2\) · Zbl 1095.15004 |
| [15] | Muzzioli, S., Torricelli, C., 2001. A multiperiod binomial model for pricing options in a vague world. In: Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications, pp. 255-264.; Muzzioli, S., Torricelli, C., 2001. A multiperiod binomial model for pricing options in a vague world. In: Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications, pp. 255-264. · Zbl 1179.91245 |
| [16] | Muzzioli, S.; Torricelli, C., A multiperiod binomial model for pricing options in a vague world, Journal of Economic Dynamics and Control, 28, 861-887 (2004) · Zbl 1179.91245 |
| [17] | Neumaier, A., Interval Methods for Systems of Equations (1990), Cambridge University Press · Zbl 0706.15009 |
| [18] | Reynaerts, H., Vanmaele, M., 2003. A sensitivity analysis for the pricing of European call options in a binary tree model. In: Proceedings of the third International Symposium on Imprecise Probabilities and their Applications, pp. 467-481.; Reynaerts, H., Vanmaele, M., 2003. A sensitivity analysis for the pricing of European call options in a binary tree model. In: Proceedings of the third International Symposium on Imprecise Probabilities and their Applications, pp. 467-481. |
| [19] | Rubinstein, M., Implied binomial trees, Journal of Finance, 49, 3, 771-818 (1994) |
| [20] | Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, International Journal of Fuzzy Sets and Systems, 1, 1, 3-28 (1978) · Zbl 0377.04002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
