Calibration of the Model
One of the most important issues for the practical applications is the estimation of the three unknown parameters of MRW model (ct, A, L) with the real data.
where At is the scale oflog-returns (e. g, 1-, 5-, 10-, 20-min etc.). The parameter a2 can be then estimated with the linear regression of the Var [1AtXAt [k]] on At as it is shown in Fig. 8.
Estimation of intermittency coefficient A and integral scale L is much more complicated, because they define the unobserved log-volatility process! At [k]. In Bacry et al. (2001) it is shown that the magnitude correlation function
Cp(x,/) = E [|1xX[k C l] |p, |1xX[k] n,
where the pre-factor is defined as
K2p = Lpa2p (2p – 1)!! dui… dUp |u – Uj |
C(r,/) ~-A2 ln^(23)
In other words, magnitude correlation function, for small enough r, has similar behavior to the correlation function of underlying log-volatility process [k]. Thus, regressing C (r, l) on log l one can estimate the parameter A2. Finally, the integral scale L can be obtained as the scale l after which autocorrelation function (23) is indistingushable from noise.
Figure 9 illustrates fits of the A2 and L using relation (23). Measures of the slope and intercept of Cr (l) ~ ln (l) provide good estimate of respectively A2 and L, though the estimation of the integral scale L is typically worse in comparison with estimation of A2. The algorithm of determining L could be summarized as follows:
1. Set the size of small rolling window;
2. Scan values of magnitude function within rolling window;
3. Stop scanning if all elements within rolling window belong to the interval of insignificance;
4. Set L for the index of the middle point of rolling window at its last position.
Fig. 9 Magnitude correlation function C (r, l) increments of MRW process sample of length 221 for A2 = 0.06, a = 7.5 • 10~5, L = 2048 and r = 15. Black solid line represents linear regression (23). Horizontal dashed lines represent insignificance interval and vertical dashed line denotes estimated value of L. The estimated A2 equals to 0.0623 and estimated L is 1905
However, this algorithm strongly depends on the choice of the rolling window size x and requires additional validation of the results.
Alternative way of estimation of intermittency coefficient A2 involves estimation of the multifractal spectrum £q = f (q) of the process. Given the analytical expression (11) one can then estimate A2 with the least squares estimator. Straightforward estimation of £q requires calculation of moments of increments Mq(l) as a function of scale l using the definition (4) and then regressing log Mq(l) on log l for different values of q, implying relation (5). Results of estimation of the multifractal spectrum for MRW process are presented in Fig. 10. One can see good agreement of the empirical spectrum with theoretical prediction up to orders of q = 6. The divergence of analytical and theoretical spectrum for higher values of q results from the insufficient sample size. Alternative methods of estimation of multifractal spectrum are based on the wavelet transform—so – called Wavelet Transform Modulus Maxima (WTMM) (Arneodo et al. 1998a) and detrended fluctuation analysis: Multifractal Detrended Fluctuation Analysis (MF-DFA) (Kantelhardt et al. 2002) and Multifractal Detrended Moving Average (MF-DMA) (Gu and Zhou 2010). However it should be noted that all these methods are subjected to the bias for large values of q and in real cases due to short observed realizations are not efficient with respect to estimation of A2.
Arneodo, A., Bacry, E., & Muzy, J.-F. (1998a). Random cascades on wavelet dyadic trees. Journal of Mathematical Physics, 39(8), 4142-4164.
Arneodo, A., Muzy, J.-F., & Sornette, D. (1998b). “Direct” causal cascade in the stock market. The European Physical Journal B – Condensed Matter and Complex Systems, 2(2), 277-282.
Bachelier, L. (1900). Theorie de la speculation. Annales scientifiques de I’Ecole normale superieure, 3(17), 21-86.
Bacry, E., Delour, J., & Muzy, J.-F. (2001). Multifractal random walk. Physical Review E, 64(2), 456, 026103C.
Bacry, E. & Muzy, J.-F. (2003). Log-infinitely divisible multifractal processes. Communications in Mathematical Physics, 236(3), 449-475.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327.
Bollerslev, T. (2010). Glossary to ARCH (GARCH). Volatility and Time Series Econometrics, 28, 137-164.
Bouchaud, J.-P., Matacz, A., & Potters, M. (2001). Leverage effect in financial markets: the retarded volatility model. Physical Review Letters, 87(22), 228701 + .
Bouchaud, J.-P. & Potters, M. (2000). Theory of financial risks: from statistical physics to risk management. Cambridge: Cambridge University Press.
Breymann, W., Ghashghaie, S., & Talkner, P. (2000). A stochastic cascade model for FX dynamics. International Journal of Theoretical and Applied Finance, (3), 357-360.
Calvet, L. E. & Fisher, A. J. (2002). Multifractality in asset returns: theory and evidence. Review of Economics and Statistics, 84(3), 381-406.
Calvet, L. E. & Fisher, A. J. (2004). How to forecast long-run volatility: regime switching and the estimation of multifractal processes. Journal of Financial Econometrics, 2(1), 49-83.
Calvet, L. E. & Fisher, Adlai J. (2008). Multifractal volatility theory, forecasting, and pricing. Burlington, MA: Academic Press. ISBN 9780080559964.
Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1, 223-236.
Davis, M. (1987). Production of conditional simulations via the LU triangular decomposition of the covariance matrix. Mathematical Geology, 19(2), 91-98.
Davis, R. A., & Mikosch, T. (2009). Extreme value theory for GARCH processes. In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. Mikosch (Eds.), Handbook of financial time series (pp. 187-200). Springer: New York.
Dembo, A., Mallows, C. L., & Shepp, L. A. (1989). Embedding nonnegative definite Toeplitz matrices in nonnegative definite circulant matrices, with application to covariance estimation. IEEE Transactions on Information Theory, 35(6), 1206-1212.
Dietrich, C. R. & Newsam, G. N. (1997). Fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix. SIAM Journal on Scientific and Statistical Computing, 18(4), 1088-1107.
Engle, R. (2001). GARCH 101: The use of ARCH/GARCH models in applied econometrics. The Journal of Economic Perspectives, 15(4), 157-168.
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 50(4), 9871007.
Fama, E. F. (1970). Efficient capital markets: a review of theory and empirical work. The Journal of Finance, 25(2), 383-417.
Fama, E. F. (1991). Efficient capital markets: II. Journal of Finance, 46(5), 1575-1617.
Filimonov, V. & Sornette, D. (2011). Self-excited multifractal dynamics. Europhysics Letters, 94(4), 46003.
Gu, G.-F. & Zhou, W.-X. (2010). Detrending moving average algorithm for multifractals. Physical Review E, 82(1), 011136+.
Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A., & Stanley, H. E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A: Statistical Mechanics and its Applications, 316(1-4), 87-114.
Kolmogorov, A. N. (1941). The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokladi Akademii Nauk SSSR, XXXI, 299-303.
Kolmogorov, A. N. (1962). A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. Journal of Fluid Mechanics, 13(1), 82-85.
Liu, R., Matteo, T., & Lux, T. (2008). Multifractality and Long-Range Dependence of Asset Returns: The Scaling Behaviour of the Markov-Switching Multifractal Model with Lognormal Volatility Components. Kiel Working Papers, (pp. 1-15).
Lux, T. (2009). Stochastic Behavioral Asset-Pricing Models and the Stylized Facts. In Handbook ofFinancial Markets: Dynamics and Evolution (pp. 161-215). Amsterdam: North Holland.
Mandelbrot, B. B. (1975). Les objets fractals: forme, hasard et dimension. Paris: Flammarion.
Mandelbrot, B. B. (1982). The fractal geometry of nature. San Francisco: W. H. Freeman and Company.
Mandelbrot, B. B. (1985). Self-affine fractals and fractal dimension. Physica Scripta, 32(4), 257-260.
Mandelbrot, B. B. & Van Ness, J. W. (1968). Fractional Brownian Motions, fractional noises and applications. SIAM Review, 10(4), 422-437.
Muzy, J.-F., Delour, J., & Bacry, E. (2000). Modelling fluctuations of financial time series: from cascade process to stochastic volatility model. The European Physical Journal B – Condensed Matter and Complex Systems, 17(3), 537-548.
Pochart, B. & Bouchaud, J.-P. (2002). The skewed multifractal random walk with applications to option smiles. Quantitative Finance, 2(4), 303-314.
Richardson, L. F. (1961). The problem of contiguity: an appendix of statistics of deadly quarrels. General Systems Yearbook, 6, 139-187.
Saichev, A. & Filimonov, V. (2007). On the spectrum of multifractal diffusion process. Journal of Experimental and Theoretical Physics, 105(5), 1085-1093.
Saichev, A. & Sornette, D. (2006). Generic multifractality in exponentials of long memory processes. Physical Review E, 74(1), 011111 + .
Sornette, D. (2003). Why stock markets crash: critical events in complex financial systems. Princeton: Princeton University Press.