Joseph L. McCauley

Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with application to finance

We show by explicit closed form calculations that a Hurst exponent H≠12 does not necessarily imply long time correlations like those found in fractional Brownian motion (fBm). We construct a large set of scaling solutions of Fokker–Planck partial differential equations (pdes) where H≠12. Thus Markov processes, which by construction have no long time correlations, can have H≠12. If a Markov process scales with Hurst exponent H≠12 then it simply means that the process has nonstationary increments. For the scaling solutions, we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient D(x,t) is specified. As an example, we generate a class of student-t-like densities from the class of quadratic diffusion coefficients. Notably, the Tsallis density is one member of that large class. The Tsallis density is usually thought to result from a nonlinear diffusion equation, but instead we explicitly show that it follows from a Markov process generated by a linear Fokker–Planck equation, and therefore from a corresponding Langevin equation. Having a Tsallis density with H≠12 therefore does not imply dynamics with correlated signals, e.g., like those of fBm. A short review of the requirements for fBm is given for clarity, and we explain why the usual simple argument that H≠12 implies correlations fails for Markov processes with scaling solutions. Finally, we discuss the question of scaling of the full Green function g(x,t;x′,t′) of the Fokker–Planck pdes.

Response to “Worrying Trends in Econophysics”

This article is a response to the recent Worrying Trends in Econophysics critique written by four respected theoretical economists. Two of the four have written books and papers that provide very useful critical analyses of the shortcomings of the standard textbook economic model, neo-classical economic theory and have even endorsed my book. Largely, their new paper reflects criticism that I have long made and that our group as a whole has more recently made. But I differ with the authors on some of their criticism, and partly with their proposed remedy.

Introduction to multifractals in dynamical systems theory and fully developed fluid turbulence

Abstract We review the theory of multifractals within the context of iterated maps of the interval and illustrate both the f (α) and s (λ) formulations within the context of simple maps. Statistical mechanics and thermodynamics of strange sets are discussed in the s (λ) formalism and the ideas of inverse temperature and disorder are interpreted in terms of lattice configurations that are generated by the symbolic dynamics of the map. The transfer matrix is introduced, and phase transitions are discussed within a mean-field approximation. It is emphasized that nonuniform fractals (multifractals) are generated by variations in the Liapunov exponent over different classes of symbol sequences, and the inverse temperature is the control parameter for classifying these sequences. We review also the Kolmogorov 1941 theory of turbulence, the uniform β-model, and “random” β-model and discuss all three within the context of the multifractal formalism.

Nonstationary Increments, Scaling Distributions, and Variable Diffusion Processes in Financial Markets

Fat-tailed distributions have been reported in fluctuations of financial markets for more than a decade. Sliding interval techniques used in these studies implicitly assume that the underlying stochastic process has stationary increments. Through an analysis of intraday increments, we explicitly show that this assumption is invalid for the Euro–Dollar exchange rate. We find several time intervals during the day where the standard deviation of increments exhibits power law behavior in time. Stochastic dynamics during these intervals is shown to be given by diffusion processes with a diffusion coefficient that depends on time and the exchange rate. We introduce methods to evaluate the dynamical scaling index and the scaling function empirically. In general, the scaling index is significantly smaller than previously reported values close to 0.5. We show how the latter as well as apparent fat-tailed distributions can occur only as artifacts of the sliding interval analysis.

Thermodynamic analogies in economics and finance: instability of markets

Interest in thermodynamic analogies in economics is older than the idea of von Neumann to look for market entropy in liquidity, advice that was not taken in any thermodynamic analogy presented so far in the literature. In this paper, we go further and use a standard strategy from trading theory to pinpoint why thermodynamic analogies necessarily fail to describe financial markets, in spite of the presence of liquidity as the underlying basis for market entropy. Market liquidity of frequently traded assets does play the role of the ‘heat bath‘, as anticipated by von Neumann, but we are able to identify the no-arbitrage condition geometrically as an assumption of translational and rotational invariance rather than (as finance theorists would claim) an equilibrium condition. We then use the empirical market distribution to introduce an assets entropy and discuss the underlying reason why real financial markets cannot behave thermodynamically: financial markets are unstable, they do not approach statistical equilibrium, nor are there any available topological invariants on which to base a purely formal statistical mechanics. After discussing financial markets, we finally generalize our result by proposing that the idea of Adam Smiths Invisible Hand is a falsifiable proposition: we suggest how to test nonfinancial markets empirically for the stabilizing action of The Invisible Hand.

Shadowing by computable chaotic orbits

Abstract We report new results on the shadowing of computable nonperiodic pseudo-orbits by computable chaotic orbits. While ordinary machine truncation/roundoff decisions can produce only periodic pseudo-orbits, we show how nonperiodic pseudo-orbits of chaotic maps can be generated by using for the truncation decision an algorithm for a computable irrational number. The resulting nonperiodic pseudo-orbits are shadowed by unique chaotic orbits of the dynamical system. We illustrate this by constructing examples of nonperiodic pseudo-orbits along with their unique chaotic shadowing orbits for a hyperbolic system. We conclude that the β-shadowing lemma without additional hypotheses provides no information for inferring chaotic attractor statistics from pseudo-orbit statistics in computation.

Martingale Option Pricing

We show that our earlier generalization of the Black–Scholes partial differential equation (pde) for variable diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, the equivalence of Black–Scholes to a Martingale was proven for the case of the Gaussian returns model by Harrison and Kreps, but we prove it for a much larger class of returns models where the returns diffusion coefficient depends irreducibly on both returns x and time t. That option prices blow up if fat tails in logarithmic returns x are included in market return is also proven.

Martingales, detrending data, and the efficient market hypothesis

We discuss martingales, detrending data, and the efficient market hypothesis for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t). Beginning with x-independent drift coefficients R(t) we show that Martingale stochastic processes generate uncorrelated, generally nonstationary increments. Generally, a test for a martingale is therefore a test for uncorrelated increments. A detrended process with an x- dependent drift coefficient is generally not a martingale, and so we extend our analysis to include the class of (x,t)-dependent drift coefficients of interest in finance. We explain why martingales look Markovian at the level of both simple averages and 2-point correlations. And while a Markovian market has no memory to exploit and presumably cannot be beaten systematically, it has never been shown that martingale memory cannot be exploited in 3-point or higher correlations to beat the market. We generalize our Markov scaling solutions presented earlier, and also generalize the martingale formulation of the efficient market hypothesis (EMH) to include (x,t)- dependent drift in log returns. We also use the analysis of this paper to correct a misstatement of the ‘fair game’ condition in terms of serial correlations in Fama’s paper on the EMH. We end with a discussion of Levy’scharacterization of Brownian motion and prove that an arbitrary martingale is topologically inequivalent to a Wiener process.

A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations” by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker–Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker–Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker–Planck equation from a Chapman–Kolmogorov equation, but no proof was offered that a Chapman–Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker–Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker–Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the we present the theory of Fokker–Planck pdes and Chapman–Kolmogorov equations for stochastic processes with finite memory.

MULTIFRACTAL DESCRIPTION OF THE STATISTICAL EQUILIBRIUM OF CHAOTIC DYNAMICAL SYSTEMS

In this review, we apply the method of backward iteration to chaotic maps of the interval to illustrate how both the f(α) spectrum and its underlying statistical mechanics follow directly from the dynamics in statistical equilibrium. The sizes of intervals in the coarse-grained phase space are expressed directly in terms of finite-time average Liapunov exponents, representing the reverse of the information flow that is the underlying cause of deterministic chaos. The transfer matrix formulation follows directly from the method of backward iteration when addresses generated from symbolic dynamics are assigned to tree-branches. The inverse temperature is interpreted in terms of classes of initial data of the dynamical system. Finally, the usefulness of the thermodynamic formalism is illustrated by showing how the pore distribution of sandstone can be modeled by a certain two-scale Cantor set on an octal tree.