On CAPM and Black-Scholes, differing risk-return strategies

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 here, in contrast, that the option valuation is not uniquely determined; in particular, strategies based on the delta-hedge and CAPM provide different valuations of an option although both hedges are instantaneouly riskfree. Second, we show explicitly that CAPM is not, as economists claim, an equilibrium theory.Capital asset pricing model (CAPM); nonequilibrium; financial markets; Black-Scholes; option pricing strategies;

An empirical model of volatility of returns and option pricing

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 a new pricing model for options (in closed algebraic form) that is shown to provide valuations that agree very well with those used by traders. We show how the Fokker-Planck formulation of fluctuations can be used with a local volatility (diffusion coeffficient) to generate an exponential distribution for asset returns, and also how fat tails for extreme returns are generated dynamically by a simple generalization of our new volatility model. Nonuniqueness in deducing dynamics from empirical data is discussed and is shown to have no practical effect over time scales much less than one hundred years. We derive an option pricing pde and explain why it‘s superfluous, because all information required to price options in agreement with the delta-hedge is already included in the Green function of the Fokker-Planck equation for a special choice of parameters. Finally, we also show how to calculate put and call prices for a stretched exponential returns density.Market instability; market dynamics; finance; option pricing

Hurst exponents, Markov processes, and nonlinear diffusion equations

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 processes, which by construction have no long time correlations, can have H≠1/2. If a Markov process scales with Hurst exponent H≠ 1/2 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≠1/2 therefore does not imply dynamics with correlated signals, e.g., like those of fractional Brownian motion. A short review of the requirements for fractional Brownian motion is given for clarity, and we explain why the usual simple argument that H≠1/2 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 pde.Hurst exponent; Markov process; scaling; stochastic calculus; autocorrelations; fractional Brownian motion; Tsallis model; nonlinear diffusion

Martingales, the efficient market hypothesis, and spurious stylized facts

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 nonstationarity arises from systematic uneveness in noise traders’ behavior. Spurious results arise mathematically from using a log increment with a ‘sliding window’. We explain why a hard to beat market demands martingale dynamics , and martingales with nonlinear variance generate nonstationary increments. The nonstationarity is exhibited directly for Euro/Dollar FX data. We observe that the Hurst exponent Hs generated by the using the sliding window technique on a time series plays the same role as does Mandelbrot’s Joseph exponent. Finally, Mandelbrot originally assumed that the ‘badly behaved second moment of cotton returns is due to fat tails, but that nonconvergent behavior is instead direct evidence for nonstationary increments. Summarizing, the evidence for scaling and fat tails as the basis for econophysics and financial economics is provided neither by FX markets nor by cotton price data

Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets

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 conducted over the last decade have reported that they are non-Gaussian, scale in time, and have power-law (or fat) tails [1–5]. However, because they use sliding interval methods of analysis, these studies implicitly assume that the underlying process has stationary increments. We explicitly show that this assumption is not valid for the Euro-Dollar exchange rate between 1999-2004. In addition, we find that fluctuations in returns of the exchange rate are uncorrelated and scale as power laws for certain time intervals during each day. This behavior is consistent with a diffusive process with a diffusion coefficient that depends both on the time and the price change. Within scaling regions, we find that sliding interval methods can generate fat-tailed distributions as an artifact, and that the type of scaling reported in many previous studies does not exist.Nonstationary increments; autocorrelations; scaling; Hurst exponents; Markov process

Martingales, the efficient market hypothesis, and spurious stylized facts

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 nonstationarity arises from systematic uneveness in noise traders’ behavior. Spurious results arise mathematically from using a log increment with a ‘sliding window’. We explain why a hard to beat market demands martingale dynamics , and martingales with nonlinear variance generate nonstationary increments. The nonstationarity is exhibited directly for Euro/Dollar FX data. We observe that the Hurst exponent Hs generated by the using the sliding window technique on a time series plays the same role as does Mandelbrot’s Joseph exponent. Finally, Mandelbrot originally assumed that the ‘badly behaved second moment of cotton returns is due to fat tails, but that nonconvergent behavior is instead direct evidence for nonstationary increments. Summarizing, the evidence for scaling and fat tails as the basis for econophysics and financial economics is provided neither by FX markets nor by cotton price data.Nonstationary increments; martingales; fat tails; Hurst exponent scaling

Empirically Based Modeling in the Social Sciences and Spurious Stylized Facts

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 tests for statistical regularity in terms of increments. 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 nonstationarity arises from systematic uneveness in noise traders’ behavior. Spurious results arise mathematically from using a log increment with a ‘sliding window’. The Hurst exponent Hs generated by the using the sliding window technique on a time series plays the same role as Mandelbrot’s Joseph exponent. Mandelbrot originally assumed that the ‘badly behaved second moment of cotton returns is due to fat tails, but that nonconvergent behavior providess instead direct evidence for nonstationary increments.Stylized facts, nonstationary time series analysis,regression, martingales, uncorrelated increments, fat tails, efficient market hypothesis,sliding windows

Hurst exponents, Markov processes, and fractional Brownian motion

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 difference lies in the increments, which are stationary and correlated in one case and nonstationary and uncorrelated in the other. The two- and one-point densities of fBm are constructed explicitly. The two-point density doesn’t scale. The one-point density for a semi-infinite time interval is identical to that for a scaling Gaussian Markov process with H≠1/2 over a finite time interval. We conclude that both Hurst exponents and one point densities are inadequate for deducing the underlying dynamics from empirical data. We apply these conclusions in the end to make a focused statement about ‘nonlinear diffusion’.Markov processes; fractional Brownian motion; scaling; Hurst exponents; stationary and nonstationary increments; autocorrelations