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Wellposedness of the boundary value formulation of a fixed strike Asian option. (English) Zbl 1181.91314

Summary: This work is the follow up to [J. Hugger, in: Numerical mathematics and advanced applications. Proceedings of ENUMATH 2001, the 4th European conference, Ischia, July 2001, Berlin: Springer. 409–418 (2003; Zbl 1069.91052)] where a partial differential equation equivalent to the stochastic formulation for a fixed strike Asian option was derived.
In the present work the differential equation is complemented with boundary value conditions that are derived from financial conditions.
With the complete boundary value formulation thus recovered, well-posedness of the problem is addressed. It turns out that the problem takes the form of a degenerated parabolic boundary value problem with a second-order, linear, time-dependent PDE with non-negative characteristic form. Apart from the degeneracy in the PDE, also the boundary conditions (derived from the financial understanding) are “the wrong ones” or at least are non-standard. There are conditions on boundaries where none are expected to be needed because of the degeneracy and there are boundaries where conditions are expected to be needed but none can be found.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
35K20 Initial-boundary value problems for second-order parabolic equations
91G80 Financial applications of other theories

Citations:

Zbl 1069.91052
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Full Text: DOI

References:

[1] Alziary, B.; Décamps, J.-P.; Koehl, P.-F., A P.D.E. approach to Asian options: analytical and numerical evidence, J. Banking & Finance, 21, 613-640 (1997)
[2] Barucci, E.; Polidoro, S.; Vespri, V., Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci., 11, 3, 475-497 (2001) · Zbl 1034.35166
[3] Boyle, P. P., New life forms on the option landscape, J. Financial Eng., 2, 3, 217-252 (1993)
[4] Fichera, G., Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8), 5, 1-30 (1956) · Zbl 0075.28102
[5] Hörmander, L., Hypoelliptic second order differential equations, Acta Math., 119, 147-171 (1967) · Zbl 0156.10701
[10] Večeř, J., A new PDE approach for pricing arithmetic average Asian options, J. Comput. Finance, 4, 105-113 (2001)
[11] Zhang, P. G., Flexible Asian options, J. Financial Eng., 3, 1, 65-83 (1994)
[12] Zvan, R.; Forsyth, P. A.; Vetzal, K. R., Robust numerical methods for PDE models of Asian options, J. Comput. Finance, 1, 39-78 (1998) · Zbl 0945.65005
[13] Zvan, R.; Forsyth, P. A.; Vetzal, K. R., A finite volume approach for contingent claims valuation, IMA J. Numer. Anal., 21, 703-731 (2001) · Zbl 1004.91032
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