×

Some mathematical results in the pricing of American options. (English) Zbl 0797.60051

The authors elaborate on the problem of pricing an American call option, i.e. the right to buy an asset for a prescribed price at any time up to the expiry date. It is known that this problem requires the solution of a free boundary problem for an associated PDE if the asset price is modelled as a suitable stochastic differential equation. Apart from deriving the equivalence of the pricing and free boundary problems, the authors examine the large- and small-time behaviour, reformulate the problem as a variational inequality, address the question of numerical solution and emphasize the relationship with certain physically motivated free boundary problems.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
91G20 Derivative securities (option pricing, hedging, etc.)
35K20 Initial-boundary value problems for second-order parabolic equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barbu, Optimal Control of Variational Inequalities (1983)
[2] DOI: 10.1016/0304-405X(76)90024-6 · doi:10.1016/0304-405X(76)90024-6
[3] Kinderlehrer, An Introduction to Variational Inequalities and Their Applications (1980) · Zbl 0457.35001
[4] DOI: 10.1111/j.1467-9965.1991.tb00007.x · Zbl 0900.90109 · doi:10.1111/j.1467-9965.1991.tb00007.x
[5] DOI: 10.1007/BF00047211 · Zbl 0714.90004 · doi:10.1007/BF00047211
[6] DOI: 10.1093/rfs/3.4.573 · Zbl 1386.91152 · doi:10.1093/rfs/3.4.573
[7] Hull, Options, Futures and Other Derivative Securities (1989)
[8] DOI: 10.2307/2328161 · doi:10.2307/2328161
[9] Grinberg, Sov. Phys. Tech. Phys. 15 pp 1579– (1971)
[10] Friedman, Stochastic Differential Equations I, II (1976)
[11] Elliott, Weak and Variational Methods for Moving Boundary Problems (1982) · Zbl 0476.35080
[12] Dewynne, J. Austral. Math. Soc. 31 pp 81– (1989)
[13] DOI: 10.1137/0309028 · doi:10.1137/0309028
[14] DOI: 10.1093/imamat/10.1.19 · Zbl 0247.65064 · doi:10.1093/imamat/10.1.19
[15] Crank, Free and Moving Boundary Problems (1984)
[16] Cox, Options Markets (1985)
[17] DOI: 10.1086/260062 · Zbl 1092.91524 · doi:10.1086/260062
[18] ?ksendal, Stochastic Differential Equations (1992) · doi:10.1007/978-3-662-02847-6
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.