Continuous-time trading and the emergence of probability

This paper establishes a non-stochastic analog of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. We consider an idealized financial security with continuous price paths, without making any stochastic assumptions. It is shown that typical price paths possess quadratic variation, where “typical” is understood in the following game-theoretic sense: there exists a trading strategy that earns infinite capital without risking more than one monetary unit if the process of quadratic variation does not exist. Replacing time by the quadratic variation process, we show that the price path becomes Brownian motion. This is essentially the same conclusion as in the Dubins–Schwarz result, except that the probabilities (constituting the Wiener measure) emerge instead of being postulated. We also give an elegant statement, inspired by Peter McCullagh’s unpublished work, of this result in terms of game-theoretic probability theory.

[1]  J. Lindeberg.  Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung , 1922 .

[2]  G. Burton.  Sobolev Spaces , 2013 .

[3]  P. Mykland.  Conservative delta hedging , 2000 .

[4]  S. Peng.  Nonlinear Expectations and Stochastic Calculus under Uncertainty , 2010, Probability Theory and Stochastic Modelling.

[5]  Jerzy Neyman.  Contributions to probability theory , 1956 .

[6]  Walter Schachermayer,et al.  A direct proof of the Bichteler-Dellacherie Theorem and connections to arbitrage , 2010, 1004.5559.

[7]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[8]  Richard M. Dudley,et al.  Sample Functions of the Gaussian Process , 1973 .

[9]  T. Björk.  Arbitrage Theory in Continuous Time , 2019 .

[10]  G. Parisi.  Brownian motion , 2005, Nature.

[11]  Ondřej Klobušník,et al.  ArXiv.org e-print archive , 2004 .

[12]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[13]  H. L. Le Roy,et al.  Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability; Vol. IV , 1969 .

[14]  C. Stricker,et al.  Sur la p-variation des surmartingales , 1979 .

[15]  Peter,et al.  Continuous-time trading and emergence of randomness , 2007 .

[16]  Nizar Touzi,et al.  Paris-Princeton Lectures on Mathematical Finance 2002 , 2003 .

[17]  Rimas Norvaisa.  Modelling of stock price changes: A real analysis approach , 2000, Finance Stochastics.

[18]  Krzysztof Burdzy,et al.  On Nonincrease of Brownian Motion , 1990 .

[19]  Terry Lyons,et al.  Uncertain volatility and the risk-free synthesis of derivatives , 1995 .

[20]  Akimichi Takemura,et al.  Implications of contrarian and one-sided strategies for the fair-coin game , 2007 .

[21]  L. Denis,et al.  A THEORETICAL FRAMEWORK FOR THE PRICING OF CONTINGENT CLAIMS IN THE PRESENCE OF MODEL UNCERTAINTY , 2006, math/0607111.

[22]  Kei Takeuchi,et al.  A new formulation of asset trading games in continuous time with essential forcing of variation exponent , 2007, 0708.0275.

[23]  W. Gasarch,et al.  The Book Review Column 1 Coverage Untyped Systems Simple Types Recursive Types Higher-order Systems General Impression 3 Organization, and Contents of the Book , 2022 .

[24]  A. P. Dawid,et al.  Present position and potential developments: some personal views , 1984 .

[25]  L. Young.  DIFFERENTIAL EQUATIONS DRIVEN BY ROUGH SIGNALS ( I ) : AN EXTENSION OF AN INEQUALITY OF , 2004 .

[26]  G. Schwarz,et al.  On time-free functions , 1972 .

[27]  R. M. Dudley,et al.  Concrete Functional Calculus , 2010 .

[28]  D. Lépingle,et al.  La variation d'ordre p des semi-martingales , 1976 .

[29]  P. M. Neumann,et al.  Groups and Geometry , 1994 .

[30]  Rajeeva L. Karandikar,et al.  On the quadratic variation process of a continuous Martingale , 1983 .

[31]  Vladimir Vovk,et al.  Continuous-time trading and the emergence of randomness , 2007, 0712.1275.

[32]  David Hobson,et al.  Robust hedging of the lookback option , 1998, Finance Stochastics.

[33]  Kei Takeuchi,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS Sequential Optimizing Strategy in Multi-dimensional Bounded Forecasting Games , 2009, 0911.3933.

[34]  W. Rudin.  Real and complex analysis , 1968 .

[35]  A. Dawid,et al.  Insuring against loss of evidence in game-theoretic probability , 2010, 1005.1811.

[36]  A. Dawid,et al.  Prequential probability: principles and properties , 1999 .

[37]  W. Vega,et al.  On Almost Sure Convergence of Quadratic Brownian Variation , 1974 .

[38]  Rimas Norvaisa,et al.  Quadratic variation, p-variation and integration with applications to stock price modelling , 2001, math/0108090.

[39]  A. Dvoretzky,et al.  Nonincrease Everywhere of the Brownian Motion Process , 1961 .

[40]  Mark H. A. Davis,et al.  THE RANGE OF TRADED OPTION PRICES , 2007 .

[41]  Vladimir Vovk,et al.  A Game-Theoretic Explanation of the √(dt) Effect , 2003 .

[42]  Akimichi Takemura,et al.  Lévy’s Zero–One Law in Game-Theoretic Probability , 2009, Journal of Theoretical Probability.

[43]  Kei Takeuchi,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS , 2006 .

[44]  F. Delbaen.  Probability and Finance: It's Only a Game! , 2002 .

[45]  D. Hobson.  The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices , 2011 .

[46]  S. Peng.  G -Expectation, G -Brownian Motion and Related Stochastic Calculus of Itô Type , 2006, math/0601035.

[47]  Peter Jagers,et al.  SÉMINAIRE DE PROBABILITÉS(STRASBOURG) , 1983 .

[48]  Kei Takeuchi,et al.  Multistep Bayesian Strategy in Coin-Tossing Games and Its Application to Asset Trading Games in Continuous Time , 2008, 0802.4311.

[49]  Hrvoje Kraljević,et al.  Functional Analysis II , 1987 .

[50]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[51]  Rimas Norvai.  Modelling of stock price changes: A real analysis approach , 2000 .

[52]  Vladimir Vovk,et al.  Forecasting point and continuous processes: Prequential analysis , 1993 .

[53]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[54]  Avi Bick,et al.  Quadratic-Variation-Based Dynamic Strategies , 1995 .

[55]  Jan Oblój,et al.  Robust pricing and hedging of double no-touch options , 2009, Finance Stochastics.

[56]  A. Shiryaev.  Essentials of stochastic finance , 1999 .

[57]  Gianluca Cassese,et al.  ASSET PRICING WITH NO EXOGENOUS PROBABILITY MEASURE , 2007 .

[58]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[59]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[60]  J. Hoffmann-jorgensen,et al.  The general marginal problem , 1987 .

[61]  Akimichi Takemura,et al.  The generality of the zero-one laws , 2008, 0803.3679.

[62]  Kei Takeuchi,et al.  Capital Process and Optimality Properties of a Bayesian Skeptic in Coin-Tossing Games , 2005, math/0510662.

[63]  G. Shafer,et al.  Probability and Finance: It's Only a Game! , 2001 .

[64]  A. M. Davie,et al.  Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation , 2007, 0710.0772.

[65]  Michel Bruneau.  Sur la p-variation d'une surmartingale continue , 1979 .

[66]  M. Lévy.  Le Mouvement Brownien Plan , 1940 .

[67]  J. Michael Harrison,et al.  Arbitrage Pricing of Russian Options and Perpetual Lookback Options , 1993 .

[68]  K. E. Dambis,et al.  On the Decomposition of Continuous Submartingales , 1965 .

[69]  G Schwarz,et al.  Time-free continuous processes. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[70]  Vladimir Vovk,et al.  Continuous-time trading and the emergence of volatility , 2008 .

[71]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[72]  L. Dubins,et al.  ON CONTINUOUS MARTINGALES. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[73]  Stephen Taylor,et al.  Exact asymptotic estimates of Brownian path variation , 1972 .

[74]  Walter Willinger,et al.  Dynamic spanning without probabilities , 1994 .

[75]  Hans Föllmer,et al.  Calcul d'ito sans probabilites , 1981 .

[76]  Vladimir Vovk,et al.  Game-theoretic Brownian motion , 2008, 0801.1309.

[77]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[78]  Vladimir Vovk,et al.  Rough paths in idealized financial markets , 2010, 1005.0279.

[79]  Akimichi Takemura,et al.  On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game , 2005, math/0508190.