Some results on partial differential equations and Asian options. (English) Zbl 1034.35166
Summary: We analyze partial differential equations arising in the evaluation of Asian options. The equations are strongly degenerate partial differential equations in three dimensions. We show that the solution of the no-arbitrage partial differential equation is sufficiently regular and standard numerical methods can be employed to approximate it.
MSC:
35R60 | PDEs with randomness, stochastic partial differential equations |
91B28 | Finance etc. (MSC2000) |
35H10 | Hypoelliptic equations |
Keywords:
ultraparabolic partial differential equations; hypoelliptic equations; finite difference methods; continguent claims valuationReferences:
[1] | DOI: 10.1016/S0378-4266(96)00057-X · doi:10.1016/S0378-4266(96)00057-X |
[2] | Aronson D. G., Colloq. Math. 18 pp 125– (1967) |
[3] | DOI: 10.1142/S0218202595000085 · Zbl 0822.65056 · doi:10.1142/S0218202595000085 |
[4] | Bony J.-M., Grenoble 19 pp 1– (1969) |
[5] | DOI: 10.1016/0378-4266(94)00031-X · doi:10.1016/0378-4266(94)00031-X |
[6] | Caverhill A., RISK 3 pp 25– (1990) |
[7] | DOI: 10.1016/0304-405X(76)90023-4 · doi:10.1016/0304-405X(76)90023-4 |
[8] | Dewynne J., Adv. Futures Options Res. 8 pp 145– (1995) |
[9] | Fichera G., Mem. Cl. Sci. Fis. Mat. Nat. Sez. (8) pp 5– (1956) |
[10] | DOI: 10.1111/j.1467-9965.1993.tb00092.x · Zbl 0884.90029 · doi:10.1111/j.1467-9965.1993.tb00092.x |
[11] | DOI: 10.1016/0304-405X(78)90021-1 · doi:10.1016/0304-405X(78)90021-1 |
[12] | DOI: 10.1007/BF02392081 · Zbl 0156.10701 · doi:10.1007/BF02392081 |
[13] | DOI: 10.1016/0378-4266(90)90039-5 · doi:10.1016/0378-4266(90)90039-5 |
[14] | DOI: 10.1007/BF01452829 · Zbl 0007.02201 · doi:10.1007/BF01452829 |
[15] | DOI: 10.1016/0261-5606(92)90013-N · doi:10.1016/0261-5606(92)90013-N |
[16] | Lanconelli E., Rend. Sem. Mat. Univ. Pol. Tori (52) pp 1– (1994) |
[17] | Manfredini M., Adv. Differential Equations 2 pp 5– (1997) |
[18] | Polidoro S., Le Mat. 49 pp 53– (1994) |
[19] | Polidoro S., Rend. Mat. 15 pp 535– (1995) |
[20] | DOI: 10.1007/BF02575835 · Zbl 0890.65106 · doi:10.1007/BF02575835 |
[21] | DOI: 10.2307/3215221 · Zbl 0839.90013 · doi:10.2307/3215221 |
[22] | Ruttiens A., RISK 3 pp 33– (1990) |
[23] | DOI: 10.2307/2331213 · doi:10.2307/2331213 |
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.