Exact probability distribution function for multifractal random walk models of stocks

We investigate the multifractal random walk (MRW) model, popular in the modelling of stock fluctuations in the financial market. The exact probability distribution function (PDF) is derived by employing methods proposed in the derivation of correlation functions in string theory, including the analytical extension of Selberg integrals. We show that the recent results by Y. V. Fyodorov, P. Le Doussal and A. Rosso obtained with the logarithmic Random Energy Model (REM) model are sufficient to derive exact formulas for the PDF of the log returns in the MRW model. Copyright c © EPLA, 2011 In recent years, the fluctuation dynamics of stocks in financial markets have primarily been described in terms of multiplicative noise (multifractal random walk (MRW)) models [1–8] or coupled random walks [9]. The former models are widely used in financial engineering for risk analysis. The majority of these analyses consider mainly the scaling behaviour of the models. The tails of the distributions [10–17] are fitted to the experimental data, as the probability distribution function (PDF) itself is not known. We aim to reduce the MRW of ref. [3] to the statistical mechanics of a Random Energy Model (REM)-like model [18–20], and calculate the exact PDF. The availability of the exact PDF could dramatically improve the accuracy of the analysis of financial data. In the approach of refs. [1–3], the logarithm of returns ri of financial stock or an exponent such as the S&P 500 is observed with a minimal achievable time resolution τ . The logarithm of the return at the moment of time iτ is described by the following model: ri = xi exp[βyi], 〈ri〉= 0, (1) (a)E-mail: [email protected] where xi are independent random values from the normal distribution and β is the so-called intermittency parameter. If yi are constant, at a value y, the model describes the log-normal distribution of returns. More realistic models are those with stochastic volatility, where yi are random values [21]. For MRW models [3], yi are normally distributed random values, and 〈yiyj〉=Cij , 〈xi 〉= J, i.e. the autocorrelation of x, is described by a constant J , and the correlations between two observations at times i and j are described by a matrix Cij . Logarithmic correlations are generally assumed in y: Cij = 2 ln N dij , dij N ; Cij = 0, dij N, N ≡ T τ , (2) where dijτ is the time interval between two observations at times iτ and jτ , and N 1 is a parameter of the model, corresponding to T , the upper “correlation length” in the observed time series [9]. This model accurately describes experimental data of financial markets as shown, e.g., by Mandelbrot [22]. According to eq. (1) the observable ri (“logarithm of return”) decomposes into a product of two independent multipliers. The dynamical process described by eqs. (1)