Narrow Results

It has become popular recently to apply the multifractal formalism of statistical physics (scaling analysis of structure functions and f(α) singularity spectrum analysis) to financial data. The outcome of such studies is a nonlinear shape of the structure function and a nontrivial behavior of the spectrum. Eventually, this literature has moved from basic data analysis to estimation of particular variants of multifractal models for asset returns via fitting of the empirical τ(q) and f(α) functions. Here, we reinvestigate earlier claims of multifractality using four long time series of important financial markets. Taking the recently proposed multifractal models of asset returns as our starting point, we show that the typical "scaling estimators" used in the physics literature are unable to distinguish between spurious and "true" multiscaling of financial data. Designing explicit tests for multiscaling, we can in no case reject the null hypothesis that the apparent curvature of both the scaling function and the Hölder spectrum are spuriously generated by the particular fat-tailed distribution of financial data. Given the well-known overwhelming evidence in favor of different degrees of long-term dependence in the powers of returns, we interpret this inability to reject the null hypothesis of multiscaling as a lack of discriminatory power of the standard approach rather than as a true rejection of multiscaling. However, the complete "failure" of the multifractal apparatus in this setting also raises the question whether results in other areas (like geophysics) suffer from similar shortcomings of the traditional methodology.

The issue in this paper is to measure the memory parameter in several transformations of historical data of asset returns in the UK (x_t) by means of fractionally integrated techniques. We use both parametric and semiparametric methods, and the results show that the power transformations of the absolute values of the returns (i.e., |x_t|^θ, with θ < 2 to avoid the fourth moment problem) are long memory, with the memory parameter fluctuating around 0.25.

We derive a discrete Log-Normal Asset Pricing Model (LAPM) based on log-normal distributed risky asset returns. Providing an analytical description of the efficient frontier in E(Log(R))-STD(Log(R)) space, we than show that under the log-normality of returns' assumption a segmented market equilibrium is created. The LAPM overcomes some of the drawbacks of the CAPM, hence better conforms with empirical observation; it shows how different portfolios of risky assets may be optimal for different investors; it shows why optimal portfolios may contain only a small number of risky assets, as well as why even with homogeneous expectations optimal portfolios for some investors may include risky assets held in short positions.

This paper develops a class of recursive, but not necessarily expected utility, preferences over intertemporal consumption lotteries. An important feature of these general preferences is that they permit risk attitudes to be disentangled from the degree of intertemporal substitutability. Moreover, in an infinite horizon, representative agent context these preference specifications lead to a model of asset returns in which appropriate versions of both the atemporal CAPM and the intertemporal consumption-CAPM are nested as special cases. In our general model, systematic risk of an asset is determined by covariance with both the return to the market portfolio and consumption growth, while in each of the existing models only one of these factors plays a role. This result is achieved despite the homotheticity of preferences and the separability of consumption and portfolio decisions. Two other auxiliary analytical contributions which are of independent interest are the proofs of (i) the existence of recursive intertemporal utility functions, and (ii) the existence of optima to corresponding optimization problems. In proving (i), it is necessary to define a suitable domain for utility functions. This is achieved by extending the formulation of the space of temporal lotteries in Kreps and Porteus (1978) to an infinite horizon framework.

A final contribution is the integration into a temporal setting of a broad class of atemporal non-expected utility theories. For homogeneous members of the class due to Chew (1985) and Dekel (1986), the corresponding intertemporal asset pricing model is derived.