×

A general version of the fundamental theorem of asset pricing. (English) Zbl 0865.90014

The fundamental theorem of asset pricing is that for a stochastic process \(S=(S_t)_{t\in\mathbb{R}}\), the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. The present paper focuses on the term essentially in the above statement. The general idea underlying the no arbitrage condition is that there should be no trading strategy \(H\) for the process \(S\) such that the final payoff described by the stochastic integral \((H.S)_\infty\) is a nonnegative function strictly positive with positive probability which has been termed as the no free lunch with vanishing risk (NFLVR). The major result of this paper is contained in the following: Let \(S\) be a bounded real valued semi-martingale. There is an equivalent martingale measure for \(S\) if and only if \(S\) satisfies NFLVR. The fact that NFLVR guarantees the existence of an equivalent martingale measure for \(S\) allows wide applicability of martingale theory. Several versions of the no free lunch condition are also introduced and their relationship studied.

MSC:

91G99 Actuarial science and mathematical finance
91B24 Microeconomic theory (price theory and economic markets)
60G44 Martingales with continuous parameter
60H30 Applications of stochastic analysis (to PDEs, etc.)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Ansel, J.P., Stricker, C. (1992): Couverture des actifs contingents. (Preprint) · Zbl 0796.60056
[2] Ansel, J.P., Stricker, C. (1993): Lois de martingale, densit?s et d?composition de F?llmer-Schweizer. Ann. Inst. Henri Poincar?28, 375-392 · Zbl 0772.60033
[3] Artzner, Ph., Heath, D. (1990): Completeness and non unique pricing. Strasborg: Universit? Louis Pasteur (Preprint)
[4] Back, K., Pliska, S. (1991): On the fundamental theorem of asset pricing with an infinite state space. J. Math. Econ.20, 1-18 · Zbl 0721.90016 · doi:10.1016/0304-4068(91)90014-K
[5] Banach, S. (1932): Th?orie des op?rations lin?aires. Monogr. Mat., Warsawa 1. Reprint by Chelsea Scientific Books (1963) · Zbl 0005.20901
[6] Black, F., Scholes, M. (1973): The pricing of options and corporate liabilities. J. Pol. Econ.81, 637-654 · Zbl 1092.91524 · doi:10.1086/260062
[7] Chou, C.S., Meyer, P.A., Stricker, S. (1980): Sur les int?grales stochastiques de processus pr?visibles non born?s. In: Az?ma, J., Yor, M. (eds.) S?minaire de probabilit?s XIV. (Lect. Notes Math., vol. 784, pp. 128-139) Berlin, Heidelberg New York: Springer
[8] Dalang, R.C., Morton, A., Willinger, W. (1989): Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stochastics and Stochastics Rep.29, 185-202 · Zbl 0694.90037
[9] Delbaen, F. (1992): Representing martingale measures when asset prices are continuous and bounded. Math. Finance2, 107-130 · Zbl 0900.90101 · doi:10.1111/j.1467-9965.1992.tb00041.x
[10] Delbaen, F., Schachermayer, W. (1993): Arbitrage and free lunch with bounded risk for unbounded continuous processes. (to appear) · Zbl 0884.90024
[11] Delbaen, F., Schachermayer, W. (1993b): Forthcoming paper on Bes3 (1) process
[12] Delbaen, F., Schachermayer, W. (1994): An inequality for the predictable projection of an adapted process. (in preparation) · Zbl 0831.60031
[13] Dellacherie, C., Meyer, P. (1980): Probabilit?s et potentiel, chap. V ? VIII, th?orie des martingales. Paris: Hermann · Zbl 0464.60001
[14] Diestel, J. (1975): Geometry of Banach spaces-selected topics. (Lect. Notes Math., vol. 485) Berlin Heidelberg New York: Springer · Zbl 0307.46009
[15] Dothan, M. (1990): Prices in financial markets. New York: Oxford University Press · Zbl 0744.90010
[16] Duffie, D. (1992): Dynamic asset pricing theory. Princeton: Princeton University Press · Zbl 1140.91041
[17] Duffie, D., Huang, C.F. (1986): Multiperiod security markets with differential information. J. Math. Econ.15, 283-303 · Zbl 0608.90006 · doi:10.1016/0304-4068(86)90017-0
[18] Dybvig, P., Ross, S. (1987): Arbitrage. In: Eatwell, J., Milgate, M., Newman, P. (eds.) The new Palgrave dictionary of economics, vol. 1, pp. 100-106, London: Macmillan
[19] ?mery, M. (1979): Une topologie sur l’espace des semimartingales. In: Dellacherie, C. et al. (eds.) S?minaire de probabilit?s XIII. (Lect. Notes Math. vol. 721, pp. 260-280) Berlin Heidelberg New York: Springer
[20] ?mery, M. (1980): Compensation de processus non localement int?grables. In: Az?ma, J., Yor, M. (eds.) S?minaire de probabilit?s XIV (Lect. Notes Math., vol. 784, pp. 152-160) Berlin Heidelberg New York: Springer
[21] F?llmer, H., Schweizer, M. (1991): Hedging of contingent claims under incomplete information. In: Davis M.H.A., Elliott, R.J., (eds.) Applied stochastic analysis. (Stochastic Monogr., vol. 5, pp. 389-414) London New York: Gordon and Breach
[22] Grothendieck, A. (1954): Espaces vectoriels topologiques. S?o Paulo: Sociedade de Matematica de S?o Paulo · Zbl 0058.33401
[23] Harrison, M., Kreps, D. (1979): Martingales and arbitrage in multiperiod security markets. J. Econ. Theory20, 381-408 · Zbl 0431.90019 · doi:10.1016/0022-0531(79)90043-7
[24] Harrison, M., Pliska, S. (1981): Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes Appl.11, 215-260 · Zbl 0482.60097 · doi:10.1016/0304-4149(81)90026-0
[25] Huang, C.F., Litzenberger, R. (1988): Foundations for financial economics. Amsterdam: Noord Holland · Zbl 0677.90001
[26] Jacka, S.D. (1992): A martingale representation result and an application to incomplete financial markets. Math. Finance2, 239-250 · Zbl 0900.90044 · doi:10.1111/j.1467-9965.1992.tb00031.x
[27] Kabanov, Yu.M., Kramkov, D.O. (1993): No arbitrage and equivalent martingale measures: An elementary proof of the Harrison-Pliska theorem. Moscow: Central Economics and Mathematics Institute. (Preprint) · Zbl 0834.60045
[28] Karatzas, I., Lehoczky, J.P., Shreve, S.E., Xu, G.L. (1991): Martingale and duality methods for utility maximisation in an incomplete Market. SIAM J. Control Optimization29, 702-730 · Zbl 0733.93085 · doi:10.1137/0329039
[29] Karatzas, I., Shreve, S.E. (1988): Brownian motion and stochastic calculus. Berlin Heidelberg New York: Springer · Zbl 0638.60065
[30] Kreps, D. (1981): Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ.8, 15-35 · Zbl 0454.90010 · doi:10.1016/0304-4068(81)90010-0
[31] Kusuoka, S. (1993): A remark on arbitrage and martingale and martingale measures. Publ. Res. Inst. Math. Sci. (to appear) · Zbl 0807.90009
[32] Lakner, P. (1992): Martingale measures for right continuous processes which are bounded below. (Preprint)
[33] L?pingle, D. (1978): Une in?galit? de martingales. In: Dellacherie, C. et al. (eds.) S?min. de Probab. XII. (Lect. Notes Math., vol. 649, pp. 134-137) Berlin Heidelberg New York: Springer
[34] Lo?ve, M. (1978): Probability theory, 4th ed. Berlin Heidelberg New York: Springer
[35] Mc Beth, D.W. (1991): On the existence of equivalent martingale measures. Thesis Cornell University
[36] Merton, R. (1973): The theory of rational option pricing. Bell J. Econ. Manag. Sci.4, 141-183 · Zbl 1257.91043 · doi:10.2307/3003143
[37] Memin, J. (1980): Espaces de semi martingales et changement de probabilit?. Z. W. Verw. Geb.52, 9-39 · Zbl 0416.60046 · doi:10.1007/BF00534184
[38] Meyer, P.A. (1976): Un cours sur les int?grales stochastiques. In: Meyer, P.A. (ed.) S?minaire de Probabilit? X. (Lect. Notes Math., vol. 511, pp. 245-400) Berlin Heidelberg New York: Springer
[39] Protter, Ph (1990): Stochastic integration and differential equations, a new approach. Berlin Heidelberg New York: Springer · Zbl 0694.60047
[40] Revuz, D., Yor, M. (1991): Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer · Zbl 0731.60002
[41] Rogers, C. (1993): Equivalent martingale measures and no-arbitrage. Queen Mary and Westfield College (Preprint) · Zbl 0851.60042
[42] Schachermayer, W. (1992): A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insur. Math. Econ.11, 1-9 · Zbl 0781.90010 · doi:10.1016/0167-6687(92)90013-2
[43] Schachermayer, W. (1993): Martingale measures for discrete time processes with infinite horizon. Math. Finance, Vol 4, 1994, 25-56 · Zbl 0893.90017 · doi:10.1111/j.1467-9965.1994.tb00048.x
[44] Schachermayer, W. (1993b). A counterexample to several problems in the theory of asset pricing. Math. Finance, Vol 3, 1993, 217-230 · Zbl 0884.90050 · doi:10.1111/j.1467-9965.1993.tb00089.x
[45] Stein, E. (1970): Topics in harmonic analysis. (Ann. Math. Stud., vol. 63) Princeton: Princeton University Press · Zbl 0193.10502
[46] Stricker, C. (1990): Arbitrage et lois de martingale. Ann. Inst. Henri Poincar?26, 451-460 · Zbl 0704.60045
[47] Yan, J.A. (1980): Caract?risation d’une classe d’ensembles convexes deL 1 ouH 1. In: Az?ma, J., Yor, M. (eds.) S?minaire de Probabilit? XIV. (Lect. Notes Math., vol. 784, pp. 220-222) Berlin Heidelberg New York: Springer
[48] Yor, M. (1978): Sous-espaces denses dansL 1 ouH 1 et repr?sentation des martingales. In: Dellacherie, C. et al. (es.) S?minaire de Probabilit? XII. (Lect. Notes Math., vol. 649, pp. 265-309) Berlin Heidelberg New York: Springer
[49] Yor, M. (1978b): Inegalit?s entre processus minces et applications. C.R. Acad. Sci., Paris, Ser. A286, 799-801 · Zbl 0389.60039
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.