Arguably the most important problem in quantitative finance is to understand the nature of stochastic processes that underlie market dynamics. One aspect of the solution to this problem involves determining characteristics of the distribution of fluctuations in returns. Empirical studies...

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...

We show by explicit closed form calculations that a Hurst exponent H≠1/2 does not necessarily imply long time correlations like those found in fractional Brownian motion. We construct a large set of scaling solutions of Fokker-Planck partial differential equations where H≠1/2. Thus Markov...

There is much confusion in the literature over Hurst exponents. Recently, we took a step in the direction of eliminating some of the confusion. One purpose of this paper is to illustrate the difference between fBm on the one hand and Gaussian Markov processes where H≠1/2 on the other. The...

This paper reports several entirely new results on financial market dynamics and option pricing We observe that empirical distributions of returns are much better approximated by an exponential distribution than by a Gaussian. This exponential distribution of asset prices can be used to develop...

In their path-finding 1973 paper Black and Scholes presented two separate derivations of their famous option pricing partial differential equation (pde). The second derivation was from the standpoint that was Black’s original motivation, namely, the capital asset pricing model (CAPM). We show...

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...

The condition for stationary increments, not scaling, detemines long time pair autocorrelations. An incorrect assumption of stationary increments generates spurious stylized facts, fat tails and a Hurst exponent Hs=1/2, when the increments are nonstationary, as they are in FX markets. The...

The discovery of the dynamics of a time series requires construction of the transition density, 1-point densities and scaling exponents provide no knowledge of the dynamics. Time series require some sort of statistical regularity, otherwise there is no basis for analysis. We state the possible...

The usual derivation of the Fokker-Planck partial differential eqn. (pde) assumes the Chapman-Kolmogorov equation for a Markov process [1,2]. Starting instead with an Ito stochastic differential equation (sde), we argue that finitely many states of memory are allowed in Kolmogorov’s two pdes,...