×

Probabilistic aspects of finance. (English) Zbl 1279.91053

The paper is devoted to the analysis of the modern mathematical aspects of financial markets. A brief historical review is presented. It is emphasized that there is a broad interdisciplinary consensus across mathematics, finance, and economics that the discounted price fluctuation of a liquid financial asset should be viewed as a stochastic process even if the central limit theorem does not work. So, the standard setting in mathematical finance is probabilistic, and it involves two types of probability measures. In view of this, market efficiency is related to the existence of a martingale measure. Derivatives together with the paradigm of perfect hedging are discussed. A great attention is paid to the notion of Knightian uncertainty that distinguishes between “risk” and “uncertainty” in the context of economic decision theory.
Price formation, market microstructure, and the emergence of algorithmic trading are studied. The reason to consider algorithmic trading is an increasing need to complement the classical microeconomic picture of noise traders and information traders by taking into account a variety of trading algorithms which are actually used on the financial market. This may make the analysis of the resulting price dynamics more tractable, since the structure of trading algorithms is more transparent and easier to model than the behavioral characteristics of individual agents. The key to understanding algorithmic trading and its potential benefits and risks is the phenomenon of price impact which is studied in the concluding section.

MSC:

91B02 Fundamental topics (basic mathematics, methodology; applicable to economics in general)
91B24 Microeconomic theory (price theory and economic markets)
91G10 Portfolio theory
91B06 Decision theory
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Alfonsi, A., Fruth, A. and Schied, A. (2010). Optimal execution strategies in limit order books with general shape functions. Quant. Finance 10 143-157. · Zbl 1185.91199
[2] Alfonsi, A., Schied, A. and Slynko, A. (2012). Order book resilience, price manipulation, and the positive portfolio problem. SIAM J. Financial Math. 3 511-533. · Zbl 1255.91412
[3] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203-228. · Zbl 0980.91042
[4] Avellaneda, M. and Stoikov, S. (2008). High-frequency trading in a limit order book. Quant. Finance 8 217-224. · Zbl 1152.91024
[5] Bachelier, L. (1995). Théorie de la Spéculation : Théorie Mathématique du Jeu. Les Grands Classiques Gauthier-Villars . [ Gauthier-Villars Great Classics ]. Sceaux: Éditions Jacques Gabay. Reprint of the 1900 original.
[6] Bernoulli, D. (1738). Specimen theoriae novae de mensura sortis. Commentarii Academiae Scientiarum Imperialis Petropolitanae 5 175-1926. Translated by L. Sommer: Econometrica 22 (1954) 23-36. · Zbl 0055.12004
[7] Biagini, F., Föllmer, H. and Nedelcu, S. (2011). Shifting martingale measures and the birth of a bubble as a submartingale. Unpublished manuscript. · Zbl 1336.91089
[8] Bick, A. and Willinger, W. (1994). Dynamic spanning without probabilities. Stochastic Process. Appl. 50 349-374. · Zbl 0801.90010
[9] Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy 637-654. · Zbl 1092.91524
[10] Borkar, V.S., Konda, V.R. and Mitter, S.K. (2004). On De Finetti coherence and Kolmogorov probability. Statist. Probab. Lett. 66 417-421. · Zbl 1081.28004
[11] Bovier, A., Černý, J. and Hryniv, O. (2006). The opinion game: Stock price evolution from microscopic market modeling. Int. J. Theor. Appl. Finance 9 91-111. · Zbl 1131.91021
[12] Brown, H., Hobson, D. and Rogers, L.C.G. (2001). Robust hedging of barrier options. Math. Finance 11 285-314. · Zbl 1047.91024
[13] Brunnermeier, M.K. and Pedersen, L.H. (2005). Predatory trading. J. Finance 60 1825-1863.
[14] Bühler, H. (2006). Consistent variance curve models. Finance Stoch. 10 178-203. · Zbl 1101.91031
[15] Bühler, H. (2006). Volatility markets: Consistent modeling, hedging and practical implementation. Ph.D. thesis, TU Berlin.
[16] Carlin, B.I., Lobo, M.S. and Viswanathan, S. (2007). Episodic liquidity crises: Cooperative and predatory trading. J. Finance 65 2235-2274.
[17] Cassidy, J. (2009). How Markets Fail : The Logic of Economic Calamities . New York: Farrar, Straus & Giroux.
[18] CFTC-SEC. (2010). Findings regarding the market events of May 6, 2010. Technical report.
[19] Cont, R. andde Larrard, A. (2010). Linking volatility with order flow: Heavy traffic approximations and diffusion limits of order book dynamics. Unpublished manuscript.
[20] Cont, R. andde Larrard, A. (2013). Price dynamics in a Markovian limit order market. SIAM J. Financial Math. · Zbl 1288.91092
[21] Cont, R. and Fournie, D.A. (2013). Functional Ito calculus and stochastic integral representation of martingales. Ann. Probab. 41 109-133. · Zbl 1272.60031
[22] Cont, R., Kukanov, A. and Stoikov, S. (2010). The price impact of order book events. Unpublished manuscript. Available at . 1011.6402
[23] Cox, A.M.G. and Obłój, J. (2011). Robust hedging of double touch barrier options. SIAM J. Financial Math. 2 141-182. · Zbl 1228.91067
[24] Davis, M., Obłó, J. and Raval, V. (2013). Arbitrage bounds for weighted variance swap prices. Math. Finance . · Zbl 1314.91209
[25] de Finetti, B. (1990). Theory of Probability : A Critical Introductory Treatment. Vol. 1. Wiley Classics Library . Chichester: Wiley. · Zbl 0694.60001
[26] de Finetti, B. (1990). Theory of Probability : A Critical Introductory Treatment. Vol. 2. Wiley Classics Library . Chichester: Wiley. · Zbl 0694.60001
[27] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463-520. · Zbl 0865.90014
[28] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215-250. · Zbl 0917.60048
[29] Delbaen, F. and Schachermayer, W. (2006). The Mathematics of Arbitrage. Springer Finance . Berlin: Springer. · Zbl 1106.91031
[30] Dellacherie, C. (1971). Quelques commentaires sur les prolongements de capacités. In Séminaire de Probabilités , V ( Univ. Strasbourg , Année Universitaire 1969 - 1970). Lecture Notes in Math. 191 77-81. Berlin: Springer.
[31] Deprez, O. and Gerber, H.U. (1985). On convex principles of premium calculation. Insurance Math. Econom. 4 179-189. · Zbl 0579.62090
[32] Doeblin, W. (2000). Sur L’équation de Kolmogoroff , Par W. Doeblin . Paris: Éditions Elsevier. C. R. Acad. Sci. Paris Sér. I Math. 3 31 (2000), Special Issue. · Zbl 0973.00016
[33] Dupire, B. (1993). Model art. Risk 6 118-124.
[34] Dupire, B. (2009). Functional Itô Calculus. Bloomberg Portfolio Research paper.
[35] Dynkin, E.B. (1992). Superdiffusions and parabolic nonlinear differential equations. Ann. Probab. 20 942-962. · Zbl 0756.60074
[36] El Karoui, N., Jeanblanc-Picqué, M. and Shreve, S.E. (1998). Robustness of the Black and Scholes formula. Math. Finance 8 93-126. · Zbl 0910.90008
[37] El Karoui, N. and Quenez, M.C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29-66. · Zbl 0831.90010
[38] Föllmer, H. (1981). Calcul d’Itô sans probabilités. In Seminar on Probability , XV ( Univ. Strasbourg , Strasbourg , 1979 / 1980) ( French ). Lecture Notes in Math. 850 143-150. Berlin: Springer. · Zbl 0461.60074
[39] Föllmer, H. (2001). Probabilistic aspects of financial risk. In European Congress of Mathematics , Vol. I ( Barcelona , 2000). Progr. Math. 201 21-36. Basel: Birkhäuser. · Zbl 1047.91041
[40] Föllmer, H. and Gundel, A. (2006). Robust projections in the class of martingale measures. Illinois J. Math. 50 439-472 (electronic). · Zbl 1099.94016
[41] Föllmer, H., Horst, U. and Kirman, A. (2005). Equilibria in financial markets with heterogeneous agents: A probabilistic perspective. J. Math. Econom. 41 123-155. · Zbl 1118.91044
[42] Föllmer, H. and Kabanov, Y.M. (1998). Optional decomposition and Lagrange multipliers. Finance Stoch. 2 69-81. · Zbl 0894.90016
[43] Föllmer, H. and Leukert, P. (2000). Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4 117-146. · Zbl 0956.60074
[44] Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints. Finance Stoch. 6 429-447. · Zbl 1041.91039
[45] Föllmer, H. and Schied, A. (2011). Stochastic Finance : An Introduction in Discrete Time , third revised and extended ed. Berlin: de Gruyter. · Zbl 1126.91028
[46] Föllmer, H., Schied, A. and Weber, S. (2009). Robust preferences and robust portfolio choice. In Mathematical Modelling and Numerical Methods in Finance (P. Ciarlet, A. Bensoussan and Q. Zhang, eds.) 15 29-88. Amsterdam: Elsevier/North-Holland. · Zbl 1180.91274
[47] Föllmer, H. and Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis ( London , 1989). Stochastics Monogr. 5 389-414. New York: Gordon and Breach. · Zbl 0738.90007
[48] Föllmer, H. and Schweizer, M. (1993). A microeconomic approach to diffusion models for stock prices. Math. Finance 3 1-23. · Zbl 0884.90027
[49] Föllmer, H. and Schweizer, M. (2010). The minimal martingale measure. In Encyclopedia of Quantitative Finance (R. Cont, ed.) 1200-1204. Hoboken, NJ: Wiley.
[50] Föllmer, H. and Sondermann, D. (1986). Hedging of nonredundant contingent claims. In Contributions to Mathematical Economics 205-223. Amsterdam: North-Holland. · Zbl 0663.90006
[51] Frittelli, M. and Rosazza Gianin, E. (2002). Putting order in risk measures. Journal of Banking & Finance 26 1473-1486.
[52] Friz, P.K. and Victoir, N.B. (2010). Multidimensional Stochastic Processes as Rough Paths : Theory and Applications. Cambridge Studies in Advanced Mathematics 120 . Cambridge: Cambridge Univ. Press. · Zbl 1193.60053
[53] Gatheral, J. (2006). The Volatility Surface. A Practitioner’s Guide . Hoboken, NJ: Wiley Finance.
[54] Gatheral, J. (2010). No-dynamic-arbitrage and market impact. Quant. Finance 10 749-759. · Zbl 1194.91208
[55] Gatheral, J. and Schied, A. (2013). Dynamical models of market impact and algorithms for order execution. In Handbook on Systemic Risk (J.-P. Fouque and J. Langsam, eds.). Cambridge: Cambridge University Press.
[56] Gilboa, I. and Schmeidler, D. (1989). Maxmin expected utility with nonunique prior. J. Math. Econom. 18 141-153. · Zbl 0675.90012
[57] Goovaerts, M.J., De Vylder, F. and Haezendonck, J. (1984). Insurance Premiums : Theory and Applications . Amsterdam: North-Holland. · Zbl 0532.62082
[58] Harrison, J.M. and Kreps, D.M. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20 381-408. · Zbl 0431.90019
[59] Heath, D. (2000). Back to the Future. Plenary Lecture, First World Congress of the Bachelier Finance Society, Paris.
[60] Hellwig, M. (2009). Systemic risk in the financial sector: An analysis of the subprime-mortgage financial crisis. De Economist 157 129-207.
[61] Hernández-Hernández, D. and Schied, A. (2007). A control approach to robust utility maximization with logarithmic utility and time-consistent penalties. Stochastic Process. Appl. 117 980-1000. · Zbl 1121.91047
[62] Hobson, D.G. (1998). Robust hedging of the lookback option. Finance Stoch. 2 329-347. · Zbl 0907.90023
[63] Huber, P.J. (1981). Robust Statistics . New York: Wiley. · Zbl 0536.62025
[64] Huber, P.J. and Strassen, V. (1973). Minimax tests and the Neyman-Pearson lemma for capacities. Ann. Statist. 1 251-263. · Zbl 0259.62008
[65] Huberman, G. and Stanzl, W. (2004). Price manipulation and quasi-arbitrage. Econometrica 72 1247-1275. · Zbl 1141.91450
[66] Jarrow, R.A., Protter, P. and Shimbo, K. (2007). Asset price bubbles in complete markets. In Advances in Mathematical Finance. Appl. Numer. Harmon. Anal. 97-121. Boston, MA: Birkhäuser. · Zbl 1154.91452
[67] Jarrow, R.A., Protter, P. and Shimbo, K. (2010). Asset price bubbles in incomplete markets. Math. Finance 20 145-185. · Zbl 1205.91069
[68] Kabanov, Y.M. (1997). On the FTAP of Kreps-Delbaen-Schachermayer. In Statistics and Control of Stochastic Processes ( Moscow , 1995 / 1996) 191-203. River Edge, NJ: World Scientific. · Zbl 0926.91017
[69] Karatzas, I., Lehoczky, J.P. and Shreve, S.E. (1987). Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optim. 25 1557-1586. · Zbl 0644.93066
[70] Karatzas, I. and Shreve, S.E. (1998). Methods of Mathematical Finance. Applications of Mathematics ( New York ) 39 . New York: Springer. · Zbl 0941.91032
[71] Kirman, A. (2010). The economic crisis is a crisis for economic theory. CESifo Economic Studies 56 498-535.
[72] Klöck, F., Schied, A. and Sun, Y. (2011). Price manipulation in a market impact model with dark pool. Unpublished manuscript. · Zbl 1398.91529
[73] Knight, F. (1921). Risk , Uncertainty , and Profit . Boston: Houghton Mifflin.
[74] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904-950. · Zbl 0967.91017
[75] Kramkov, D. and Schachermayer, W. (2003). Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13 1504-1516. · Zbl 1091.91036
[76] Kramkov, D.O. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Related Fields 105 459-479. · Zbl 0853.60041
[77] Kratz, P. and Schöneborn, T. (2010). Optimal liquidation in dark pools. Unpublished manuscript. · Zbl 1402.91709
[78] Kreps, D.M. (1979). Three Essays on Capital Markets . Stanford University: Institute for Mathematical Studies in the Social Sciences. Reprinted in Revista Española de Economica 4 (1987), 111-146.
[79] Lehalle, C.A. (2013). Market microstructure knowledge needed to control an intra-day trading process. In Handbook on Systemic Risk (J.-P. Fouque and J. Langsam, eds.). Cambridge: Cambridge University Press.
[80] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths. Oxford Mathematical Monographs . Oxford: Oxford Univ. Press. · Zbl 1029.93001
[81] Maccheroni, F., Marinacci, M. and Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74 1447-1498. · Zbl 1187.91066
[82] Merton, R.C. (1973). Theory of rational option pricing. Bell J. Econom. and Management Sci. 4 141-183. · Zbl 1257.91043
[83] Mittal, H. (2008). Are you playing in a toxic dark pool? A guide to preventing information leakage. Journal of Trading 3 20-33.
[84] Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 1293-1311. · Zbl 0267.60050
[85] Monroe, I. (1978). Processes that can be embedded in Brownian motion. Ann. Probab. 6 42-56. · Zbl 0392.60057
[86] Musiela, M. and Zariphopoulou, T. (2009). Portfolio choice under dynamic investment performance criteria. Quant. Finance 9 161-170. · Zbl 1158.91387
[87] Musiela, M. and Zariphopoulou, T. (2010). Stochastic partial differential equations and portfolio choice. In Contemporary Quantitative Finance 195-216. Berlin: Springer. · Zbl 1217.91173
[88] Neuberger, A. (1994). The log contract. The Journal of Portfolio Management 20 74-80.
[89] Obizhaeva, A. and Wang, J. (2013). Optimal trading strategy and supply/demand dynamics. J. Financial Markets 16 1-32.
[90] Poincaré, H. (1908). Science et méthode. Revue scient. (5) 10 417-423. · JFM 39.0095.03
[91] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Berlin: Springer. · Zbl 0917.60006
[92] Samuelson, P.A. (1965). Proof that properly anticipated prices fluctuate randomly. Industrial Management Review 6 .
[93] Savage, L.J. (1972). The Foundations of Statistics , revised ed. New York: Dover Publications Inc. · Zbl 0276.62006
[94] Schervish, M.J., Seidenfeld, T. and Kadane, J.B. (2008). The fundamental theorems of prevision and asset pricing. Internat. J. Approx. Reason. 49 148-158. · Zbl 1185.91087
[95] Schied, A. (2005). Optimal investments for robust utility functionals in complete market models. Math. Oper. Res. 30 750-764. · Zbl 1082.91052
[96] Schied, A. (2007). Optimal investments for risk- and ambiguity-averse preferences: A duality approach. Finance Stoch. 11 107-129. · Zbl 1143.91021
[97] Schied, A. (2013). A control problem with fuel constraint and Dawson-Watanabe superprocesses. Ann. Appl. Probab. · Zbl 1288.60100
[98] Schied, A. and Stadje, M. (2007). Robustness of delta hedging for path-dependent options in local volatility models. J. Appl. Probab. 44 865-879. · Zbl 1210.91136
[99] Schmeidler, D. (1986). Integral representation without additivity. Proc. Amer. Math. Soc. 97 255-261. · Zbl 0687.28008
[100] Schöneborn, T. (2008). Trade execution in illiquid markets. Optimal stochastic control and multi-agent equilibria. Ph.D. thesis, TU Berlin.
[101] Schöneborn, T. and Schied, A. (2009). Liquidation in the face of adversity: Stealth vs. sunshine trading. Unpublished manuscript.
[102] Schweizer, M. (2010). Mean-variance hedging. In Encyclopedia of Quantitative Finance (R. Cont, ed.) 1177-1181. Wiley.
[103] Sondermann, D. (2006). Introduction to Stochastic Calculus for Finance : A New Didactic Approach. Lecture Notes in Economics and Mathematical Systems 579 . Berlin: Springer. · Zbl 1136.91014
[104] Stoikov, S.F. and Zariphopoulou, T. (2005). Dynamic asset allocation and consumption choice in incomplete markets. Australian Economic Papers 44 414-454.
[105] Turner, A. (2009). The Turner Review: A regulatory response to the global banking crisis. FSA, March.
[106] von Neumann, J. and Morgenstern, O. (1980). Theory of Games and Economic Behavior , 3rd ed. Princeton, NJ: Princeton Univ. Press. · Zbl 0452.90092
[107] Weber, P. and Rosenow, B. (2005). Order book approach to price impact. Quant. Finance 5 357-364. · Zbl 1134.91379
[108] Yan, J.A. (2002). A numeraire-free and original probability based framework for financial markets. In Proceedings of the International Congress of Mathematicians , Vol. III ( Beijing , 2002) 861-871. Beijing: Higher Ed. Press. · Zbl 1005.60058
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.