Applications of the likelihood theory in finance: modelling and pricing. (English. French summary) Zbl 1416.62586
Summary: This paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of Le Cam’s theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into statistical experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness, and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black-Scholes price of a European option is related to Neyman-Pearson tests and it has an interpretation as Bayes risk. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Itô type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
statistical experiment; filtered likelihood ratio process; martingale measure; pricing formula; Bayes risk; Neyman-Pearson test; completeness; contiguity; local asymptotic normalityReferences:
| [1] | Andersen, P.K., Borgan, O., Gill, R.D. & Keiding, N. (1997). Statistical Models Based on Counting Processes. New York : Springer. |
| [2] | Delbaen, F. & Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. , 300, 463-520. · Zbl 0865.90014 |
| [3] | Föllmer, H. & Leukert, P. (1999). Quantile hedging. Finan. Stochast. , 3, 251-273. · Zbl 0977.91019 |
| [4] | Föllmer, H. & Schied, A. (2004). Stochastic Finance. An Introduction in Discrete Time. Berlin: Wolter de Gruyter. · Zbl 1126.91028 |
| [5] | Gerber, H.U. & Shiu, E.S. (1994). Option pricing by Esscher transforms. Trans. Soc. Actuar. , XLVI, 98-140. |
| [6] | Gushchin, A.A. & Mordecki, E. (2002). Bounds on option prices for semimartingale market models. Proc. Steklov Inst. Math. , 273, 73-113. |
| [7] | Hájek, J., Šidák, Z. & Sen, P.K. (1999). Theory of Rank Tests. San Diego : Acad. Press. · Zbl 0944.62045 |
| [8] | Hubalek, F. & Schachermayer, W. (1998). When does convergence of asset prices imply convergence of option prices? Math. Finan. , 8(4), 385-403. · Zbl 1020.91026 |
| [9] | Jacod, J. (1989). Convergence of filtered statistical models and Hellinger processes. Stochastic Process. Appl. , 32, 47-68. · Zbl 0684.60030 |
| [10] | Jacod, J. & Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes. Berlin : Springer‐Verlag. · Zbl 1018.60002 |
| [11] | Janssen, A. & Milbrodt, H. (1993). Rényi type goodness of fit tests with adjusted principal direction of alternatives. Scand. J. Stat. , 20, 177-194. · Zbl 0798.62060 |
| [12] | Kabanov, Yu. M. & Kramkov, D.O. (1994). Large financial markets: Asymptotic arbitrage and contiguity. Theory Probab. Appl. , 39, 182-187. · Zbl 0834.90018 |
| [13] | Karatzas, I. & Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus. New York : Springer‐Verlag. · Zbl 0734.60060 |
| [14] | Kelley, J.L. (1975). General Topology. New York : Springer‐Verlag. · Zbl 0306.54002 |
| [15] | Korn, R. & Korn, E. (2001). Option Pricing and Portfolio Optimization. American Math. Soc. · Zbl 0965.91020 |
| [16] | LeCam, L. (1986). Asymptotic Methods in Statistical Decision Theory. New York : Springer‐Verlag. · Zbl 0605.62002 |
| [17] | Lehmann, E.L. & Romano, J.P. (2005). Testing Statistical Hypotheses. New York : Springer‐Verlag. · Zbl 1076.62018 |
| [18] | Norberg, E. (2004). On filtered experiments and the information contained in likelihood ratios. Math. Methods Statist. , 13, 50-68. · Zbl 1129.62002 |
| [19] | Rudin, W. (1991). Functional analysis. New York : McGraw‐Hill. · Zbl 0867.46001 |
| [20] | Rudloff, B. & Karatzas, I. (2010). Testing composite hypotheses via convex du ality. Bernoulli , 16, 1224-1239. · Zbl 1207.62101 |
| [21] | Rüschendorf, L. (1988). Asymptotische Statistik. Stuttgart : B.G. Teubner. · Zbl 0661.62001 |
| [22] | Schied, A. (2004). On the Neyman‐Pearson Problem for law‐invariant risk measures and robust utility functionals. Ann. Appl. Probab. , 14, 1398-1423. · Zbl 1121.91054 |
| [23] | Schied, A. (2005). Optimal investments for robust utility functionals in complete market models. Math. Oper. Res. , 30, 750-764. · Zbl 1082.91052 |
| [24] | Shiryaev, A.N. (1999). Essentials of Stochastic Finance. Facts, Models, Theory. Singapore : World Scientific. · Zbl 0926.62100 |
| [25] | Shiryaev, A.N. & Spokoiny, V.G. (2000). Statistical Experiments and Decisions: Asymptotic Theory. World Scientific. · Zbl 0967.62002 |
| [26] | Strasser, H. (1985a). Mathematical Theory of Statistics. Berlin: Wolter de Gruyter. · Zbl 0594.62017 |
| [27] | Strasser, H. (1985b). Einführung in die lokale asymptotische Theorie der Statistik. Bayreuther Mathematische Schriften, Vol. 19, 1-58. · Zbl 0575.62024 |
| [28] | Strasser, H. (1987). Stability of Filtered Experiments. Contributions to Stochastics, In honour of the 75th birthday of W. Eberl, pp. 202-213. Heidelberg : Physica‐Verlag. · Zbl 0659.62010 |
| [29] | Torgersen, E.N. (1991). Comparison of Statistical Experiments. Cambridge : Cambridge Univ. Press. · Zbl 0732.62009 |
| [30] | Witting, H. (1985). Mathematische Statistik I. Parametrische Verfahren bei festem Stichprobenumfang. B.G. Teubner. · Zbl 0581.62001 |
| [31] | Witting, H. & Müller‐Funk, U. (1995). Mathematische Statistik II. B.G. Teubner. · Zbl 0838.62001 |
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.
