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Option pricing under incompleteness and stochastic volatility. (English) Zbl 0900.90095

Summary: We consider a very general diffusion model for asset prices which allows the description of stochastic and past-dependent volatilities. Since this model typically yields an incomplete market, we show that for the purpose of pricing options, a small investor should use the minimal equivalent martingale measure associated to the underlying stock price process. Then we present stochastic numerical methods permitting the explicit computation of option prices and hedging strategies, and we illustrate our approach by specific examples.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
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[1] Ansel J.-P., Lois de Martingale, Densites et Decomposition de Follmer-Schweizer (1991)
[2] Bensoussan A., Acta Appl. Math 2 pp 139– (1984)
[3] DOI: 10.1086/260062 · Zbl 1092.91524
[4] Clark J. M. C., Lecture Notes in Control and Information Sciences 42 pp 69– (1982)
[5] DOI: 10.1016/0022-0531(89)90067-7 · Zbl 0678.90011
[6] Karoui N., Programmation Dynamique et Evaluation des Actifs Contingents en Marche Incomplet (1991) · Zbl 0736.90009
[7] Follmer H., Applied Stochastic Analysis pp 389– (1991)
[8] DOI: 10.1016/0022-0531(79)90043-7 · Zbl 0431.90019
[9] He H., Consumption and Portfolio Policies with Incomplete Markets and Short-Sale Constraints: the Infinite Dimensional Case (1990)
[10] DOI: 10.2307/2328253
[11] Hull J., Adv. Futures Options Res 3 pp 29– (1988)
[12] DOI: 10.2307/2330709
[13] DOI: 10.1137/0327063 · Zbl 0701.90008
[14] Karatzas I., Math. Oper. Res 11 pp 261– (1986)
[15] DOI: 10.1137/0325086 · Zbl 0644.93066
[16] Karatzas I., Math. Oper. Res 15 pp 80– (1990)
[17] DOI: 10.1137/0329039 · Zbl 0733.93085
[18] Karatzas I., Brownian Motion and Stochastic Calculus (1988) · Zbl 0638.60065
[19] Kind P., Ann. Appl. Probab 1 pp 379– (1991)
[20] Kloeden P. E., The Numerical Solution of Stochastic Differential Equations (1991)
[21] P. E. Kloeden, E. Platen, and I. Wright (1991 ): ”The Approximation of Multiple Stochastic Integrals,” forthcoming. · Zbl 0761.60048
[22] DOI: 10.2307/1926560
[23] DOI: 10.1016/0022-0531(71)90038-X · Zbl 1011.91502
[24] DOI: 10.2307/1913811 · Zbl 0283.90003
[25] Merton R. C., Continuous-Time Finance (1990) · Zbl 1019.91502
[26] Mikulevicius R., Math. Nachr 138 pp 93– (1988)
[27] Milstein G. N., Theory Probab. Appl 19 pp 557– (1974)
[28] DOI: 10.1137/1123045 · Zbl 0422.60048
[29] Milstein G. N., The Numerical Integration of Stochastic Differential Equations (1988) · Zbl 0810.65144
[30] Newton N., Stochastics 19 pp 175– (1986) · Zbl 0618.60053
[31] Pages H., Three Essays in Optimal Consumption (1989)
[32] DOI: 10.1007/BF01438265 · Zbl 0554.60062
[33] Platen E., Lietuvos Mat. Rinkinys 21 pp 121– (1981)
[34] Platen E., Sankhya 44 pp 163– (1982)
[35] Platen E., Zur zeitdiskreten Approximation von Itoprozessen (1984)
[36] DOI: 10.1137/0719041 · Zbl 0496.65038
[37] Schweizer M., Mean-Variance Hedging for General Claims (1990) · Zbl 0742.60042
[38] DOI: 10.1016/0304-4149(91)90053-F · Zbl 0735.90028
[39] Schweizer M., Martingale Densities for General Asset Prices (1991) · Zbl 0762.90014
[40] DOI: 10.2307/2330793
[41] Talay D., Stochastics 9 pp 275– (1983) · Zbl 0512.60041
[42] Talay D., Lecture Notes in Control and Information Sciences 61 pp 294– (1984)
[43] Wagner W., Approximation of Ito Integral Equations (1978) · Zbl 0413.60056
[44] DOI: 10.1016/0304-405X(87)90009-2
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