×

zbMATH — the first resource for mathematics

A semigroup approach to generalized Black-Scholes type equations in incomplete markets. (English) Zbl 1422.91683
Summary: In this paper we will study an option pricing problem in incomplete markets by an analytic point of view. The incompleteness is generated by the presence of a non-traded asset. The aim of this paper is to use the semigroup theory in order to prove existence and uniqueness of solutions to generalized Black-Scholes type equations that are non-linearly associated with the price of European claims written exclusively on non-traded assets. Then, we derive analytic expressions of the solutions. An approximate representation in terms of a generalized Feynman-Kac type formula is derived in cases where an explicit closed form solution is not available. Numerical examples are also given (see Appendix E) where theoretical approximations and numerical tests reveal a remarkable agreement.
MSC:
91G20 Derivative securities (option pricing, hedging, etc.)
47D60 \(C\)-semigroups, regularized semigroups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Achdou, Y.; Pironneau, O., Computational Methods for Option Pricing, SIAM Frontiers in Applied Mathematics, vol. 30, (2005), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 1078.91008
[2] Black, F.; Scholes, M., The pricing of options on corporate liabilities, J. Polit. Econ., 81, 637-659, (1973) · Zbl 1092.91524
[3] Bordag, L. A.; Frey, R., Nonlinear option pricing models for illiquid markets: scaling properties and explicit solutions, (Ehrhardt, M., Nonlinear Models in Mathematical Finance. New Research Trends in Option Pricing, (2009), Nova Science Publishers: Nova Science Publishers New York), 103-130
[4] Brennan, M. J.; Schwartz, E. S., Savings bonds, retractable bonds and callable bonds, J. Financ. Econ., 5, 67-88, (1977)
[5] Canale, A.; Mininni, R. M.; Rhandi, A., Analytic approach to solve a degenerate PDE for the Heston model, Math. Methods Appl. Sci., 40, 4982-4992, (2017) · Zbl 1370.35184
[6] Cheng, W.; Costanzino, N.; Liechty, J.; Mazzucato, A.; Nistor, V., Closed-form asymptotics and numerical approximations of 1D parabolic equations with applications to option pricing, SIAM J. Financial Math., 2, 901-934, (2011) · Zbl 1242.35131
[7] Courtadon, G., The pricing of options on default-free bonds, J. Financ. Quant. Anal., 17, 75-100, (1982)
[8] Duffie, D., Dynamic Asset Pricing Theory, (2001), Princeton University Press: Princeton University Press Princeton · Zbl 1140.91041
[9] Duffie, D.; Filipović, D.; Schachermayer, W., Affine processes and applications in finance, Ann. Appl. Probab., 13, 984-1053, (2003) · Zbl 1048.60059
[10] Engel, K. J.; Nagel, R., One-Parameter Semigroup for Linear Evolution Equations, (2000), Springer-Verlag: Springer-Verlag New York
[11] Farad, O. S., Linearization and nonlinear stochastic differential equations with locally Lipschitz condition, Appl. Math. Sci., 1, 2553-2563, (2007) · Zbl 1140.65013
[12] Frittelli, M., Introduction to a theory of value coherent with the no-arbitrage principle, Finance Stoch., 4, 275-297, (2000) · Zbl 0965.60046
[13] Goldstein, J. A., Semigroup of Linear Operators & Applications, (2017), Dover Publications, Inc.: Dover Publications, Inc. Mineola, New York
[14] Goldstein, G. R.; Goldstein, J. A.; Mininni, R. M.; Romanelli, S., The semigroup governing the generalized Cox-Ingersoll-Ross equation, Adv. Differential Equations, 21, 235-264, (2016) · Zbl 1341.47051
[15] Goldstein, J. A.; Mininni, R. M.; Romanelli, S., Markov semigroup and groups of operators, Commun. Stoch. Anal., 1, 247-267, (2007)
[16] Goldstein, J. A.; Mininni, R. M.; Romanelli, S., A new explicit formula for the solution of the Black-Merton-Scholes equation, (Sengupta, A. N.; Sundar, P., Infinite Dimensional Stochastic Analysis. In Honor of Hui-Hsiung Kuo. Infinite Dimensional Stochastic Analysis. In Honor of Hui-Hsiung Kuo, QP-PQ: Quantum Probability and White Noise Analysis, vol. 22, (2008), World Scientific), 226-235 · Zbl 1145.91346
[17] Lunardi, A.; Miranda, M.; Pallara, D., Lecture Notes of the 19th Internet Seminar 2015/2016: Infinite DimensionaI Analysis
[18] Musiela, M.; Zariphopoulou, T., An example of indifference price under exponential preferences, Finance Stoch., 8, 229-239, (2004) · Zbl 1062.93048
[19] Musiela, M.; Zariphopoulou, T., Indifference Prices and Related Measures, (2001), The University of Texas at Austin, Technical Report
[20] Oksendal, B., Stochastic Differential Equations and Applications: An Introduction with Applications, (2003), Springer: Springer Berlin · Zbl 1025.60026
[21] Papageorgiou, A.; Traub, J. F., Faster evaluation of multidimensional integrals, Comput. Phys., 11, 574-578, (1997)
[22] Rouge, R.; El Karoui, N., Pricing via utility maximization and entropy, Math. Finance, 10, 259-276, (2000) · Zbl 1052.91512
[23] Shiryaev, A. N., Essential of Stochastic Finance: Facts, Models, Theory, (1999), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Singapore
[24] Svensson, L. E.O., Nontraded assets in incomplete markets: pricing and portfolio choice, Econ. Rev., 37, 1149-1168, (1993)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.