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Proposal for Funding of the Project: The Multi-Fractal Model of Asset Returns: Multivariate Extensions, Estimation, and Applications for Risk Management Antrag auf Gewährung einer Sachbeihilfe eingereicht

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Proposal for Funding of the Project: The Multi-Fractal Model of Asset Returns: Multivariate Extensions, Estimation, and Applications for Risk Management Antrag auf Gewährung einer Sachbeihilfe eingereicht von: Prof. Dr. Thomas Lux Lehrstuhl für Geld, Währung und Internationale Finanzmärkte Institute für Volkswirtschaftlehre und Institut für Weltwirtschaft University of Kiel Olshausenstr. 40, Kiel, Germany 2 1 Proposal for Funding of the Project: The Multi-Fractal Model of Asset Returns: Multivariate Extensions, Estimation, and Applications for Risk Management. This a new project that has not received financial support by the DFG or any other research funding agency before. 1.1 Applicant Professor Dr. Thomas Lux Born: , German citizen Universität Kiel Institut für Volkswirtschaftslehre und Institut für Weltwirtschaft Lehrstuhl für Geld, Währung und Internationale Finanzmärkte Olshausenstrasse 40, D Kiel Tel. (office): (+49) Fax: +(49) Private Address: Bölskamp 20, D Neuwittenbek Tel. (private): (+49) Topics Fractality, long memory, Markov-switching, realized volatility and Student-t innovations in asset returns. GMM estimation, Maximum Likelihood, Simulated Maximum Likelihood and non-parametric estimation techiques for multifractal models of asset returns. Applications to forecasting and risk management for different asset types and cross-sections. 1.3 Research Areas Financial Econometrics, Risk Management, Econophysics 3 1.4 Anticipated Duration A total of 3 years. The funding by the DFG would be requested for the total anticipated duration of the project. 1.5 Proposal Period 36 Months Desired commencement of the project: Funded persons: Leonardo Morales-Arias, M.Sc. for the first period of about 12 months, who will be followed by Payam Norouzzadeh, M.Sc. in months 13 through Summary of the Project Multifractal processes have been originally developed in physics for modeling turbulent fluids and related phenomena. They have recently also attracted attention in empirical finance because of their ability to replicate the major stylized facts of asset returns: varying degrees of long-term correlation of different measures of volatility, and fat tails in the unconditional distribution of price changes. While the multifractal apparatus developed in statistical physics is mainly concerned with combinatorial operations on measures, analogous causal multifractal models have been designed for financial applications. Although this literature is still at a very early stage, it has already developed a range of statistical techniques for proper estimation of multifractal models and has demonstrated successful applications in forecasting of volatility. The present project builds upon the earlier work on inference methods for multifractal models and their practical applications by our group. One major task will be the development of appropriate statistical methodology for multivariate multifractal models. We will investigate the behavior of various extensions and explore the use of multifractal models for risk management and portfolio management. Further research includes the analysis of the role of innovations vis-à-vis the intrinsic volatility dynamics as well as the adaptation of the multifractal model to measures of realized volatility. Given the evidence on multi-scaling of many physical time series, we also expect that the methodological innovations in financial applications will generate spill over effects to the application of multifractal models in the natural science (for example, improved forecasts of precipitation). 4 2 State of the Literature and Own Work on the Subject 2.1 State of the Literature While so-called uni-fractal of self-similar processes (like fractional Brownian motion) are widely used in financial economics, more general multi-fractal processes have only been considered very recently as candidate data-generating mechanisms for financial prices. However, the typical signature of multi-fractality, namely variations in the scaling behavior of different moments of the data, is a well-established feature of price changes in both stock and foreign exchange markets. In the economics literature (which did not use the concept of scaling or fractality until recently) the equivalent feature of variations in temporal dependence of various powers of the raw data has been reported first by Ding, Granger and Engle (1993) and has been confirmed by a number of later studies (e.g. Lux, 1996; Mills, 1997; Lobato and Savin, 1998). Almost at the same time, a number of papers authored by physicists (Schmitt, Schertzer and Lovejoy, 1999; Vandewalle and Ausloos, 1998a,b; Vassilicos, Demos and Tata, 1993) using a different set of analytical tools arrived at essentially the same conclusions concerning the multi-fractal nature of various financial records. These findings have also stimulated comparisons between the statistical behavior of data from both financial markets and experimental data of turbulent flows (Vassilicos, 1995; Mantegna and Stanley, 1996; Voit, 2001). The basic principle for construction of multifractal models in statistical physics is a cascading process of iterative splitting of initially uniform probability mass into more and more heterogeneous subsets. Starting with a uniform distribution over a certain interval, one splits this interval into two subintervals that receive fractions, say p 1 and 1 p 1, of the overall mass. In the next step, the same procedure is repeated for the newly created subsets so that one ends up with four intervals with probability mass p 2 1, p 1 (1 p 1 ) and (1 p 1 ) 2, respectively. 1 In principle, this process can be repeated ad infinitum. One thus obtains a hierarchical structure of components, where smaller ones (small whirls is the application to turbulent flows) emanate from the higher levels of the hierarchy via this probabilistic split of energy. By its very construction, a combinatorial multifractal along the above lines exhibits different degrees of scaling or long-term dependence for different powers of the resulting measure. This makes adaptation of the multifractal apparatus a promising avenue of reseach in empirical finance. Interestingly, empirical research in finance also provides us with more direct evidence in favor of the hierarchical structure implied by these cascade models: namely, 1 The two intervals in the center both have measure p 1 (1 p 1 ). 5 it has been found that volatility on fine scales (tick-by-tick data) can be explained to a larger extent by coarse-grained volatility than vice versa (Müller, 1997). This feature would be expected in a multifractal generating process, but not for traditional models of asset returns. Although the analogy between both turbulence and financial prices does not hold in all details, some authors have proposed to formalise the financial multi-fractality using cascade models developed for velocity differences in turbulent flows (e.g. Ghasghaie et al., 1996). Recently, Calvet, Fisher and Mandelbrot (1997) proceeded one step further by proposing a compound stochastic process as a data generating mechanism for financial prices in which a multi-fractal cascade plays the role of a time transformation or time-varying scale function of the variance of the incremental process. 2 In their model 3 an incremental Brownian motion is subordinate to the cumulative distribution function of a multifractal measure. However, this multifractal component is of combinatorial rather than causal nature (it is actually identical to the model proposed for turbulent flows by Mandelbrot, 1974). Unfortunately, the original MMAR suffers from non-stationarity since the combinatorial construction of the multifractal measure is restricted to a (predefined) bounded interval (in space or time). While Calvet and Fisher (2002) develop estimators and diagnostic tests for this particular model, its non-causal nature makes it intrinsically difficult to apply to financial data. As a consequence, although the potential of this new approach for generating multi-scaling in returns is not shared by traditional models in finance, rigorous comparison of its performance to, for example, more traditional GARCH processes (Bollerslev, Chou and Kroner, 1992), is hampered by the lack of statistical theory for the parameter estimates and statistical tools for comparison of alternative models. These severe limitations have been overcome by the development of an iterative, causal analogue of the combinatorial MMAR, cf. Breymann et al. (2000) and Calvet and Fisher (2001). Calvet and Fisher define a continuous-time multifractal model with random times for the changes of its volatility components and demonstrate weak convergence of the discretized version of this process to its continuous-time limit. This approach preserves the hierarchy of volatility components of the original MMAR but dispenses with its restriction to a bounded interval. In the discrete-time version of this model, the volatility dynamics is driven by a Markov-switching process with a large number of states. One important advantage of the multifractal model is that this large state space does not come along with a similarly large number of parameters. Quite in contrast to standard Markov-switching models, the hierarchical structure leads to 2 Cf. also Mandelbrot (1999) and Mandelbrot and Hudson (2004) for a non-technical introduction to the subject and the relationship between the MMAR and multifractal models in turbulence. 3 denoted the Multifractal Model of Asset Returns (MMAR) 6 a transition matrix whose probabilities are determined by only a few parameters, or might even be defined in a way to be parameter-free. To concretize ideas, we will reproduce here the structure of the Markov-Switching Multifractal Model (MSM). Returns are modeled as r t = σ t u t (1) with innovations u t taken from a stationary distribution (e.g. the standard Normal distribution). Instantaneous volatility σ t is determined by the product of k volatility components or multipliers M (1) t, M (2) t... and a scale factor σ: σ 2 t = σ 2 k i=1 M (i) t. (2) Following the basic hierarchical principle of the multifractal approach, each volatility component will be renewed at time t with a probability γ i depending on its rank within the hierarchy of multipliers and remains unchanged with probability 1 γ i. The convergence property demonstrated by Calvet and Fisher (2001) requires to formalize transition probabilities according to: γ i = 1 (1 γ k ) (bi k ) (3) with γ k and b parameters to be estimated. However, a large part of the literature uses simpler specification of hierarchical probabilities (used in, for example, Liu and Lux, 2006 and Lux, 2007) such as: γ i = 2 (k i) (4) which also allows for components of very different life-times. While the parameterfree version of eq. (4) greatly facilitates estimation, it is also germaine to the structure of renewal of multipliers in the physical applications of multifractals and the previous MMAR. The MSM model is fully specified once we have determined the number k of volatility components and their distribution. In the small body of available literature, the multipliers have been assumed to follow either a Binomial or a Lognormal distribution. Since one would normalize the distribution so that E[M (i) t ] = 1, only one parameter has to be estimated for the distribution of volatility components in these cases. Taking into account the scale parameter, σ, we end up with a very parsimonious family of stochastic processes that is parameterized by only two parameters although the number of states could be arbitrary large (for large k). 7 Calvet and Fisher (2004) develop a maximum likelihood algorithm (MLE) for estimation of the parameters of this MSM model. Since MLE requires an evaluation of the transition matrix, it works only for discrete distributions of the multipliers and is not applicable for, e.g., the alternative proposal of a Lognormal distribution. It also encounters bounds of computational feasibility for specifications with more than about 10 volatility components. Despite these limitations, the estimated Binomial MSM specifications in Calvet and Fisher (2004) did provide gains in forecasting accuracy of future volatility over standard GARCH and fractional GARCH models (FIGARCH) over medium-term and long horizons. Calvet, Fisher and Thompson (2006) add two additional innovations to this literature: first, they propose a bi-variate generalization of the Markov-Switching Multifractal Model (which is different from the one by Liu and Lux which had been developed independently). Second, they develop a simulated ML or particle filter approach for estimation of the parameters that is somewhat more broadly applicable than the MLE algorithm of their previous paper. 2.2 Own Work on the Subject The applicant and his group have been working on various aspects of this new class of stochastic volatility models over the last couple of years. Lux (2001) provides parameter estimates for two specifications of the early combinatorial MMAR and demonstrates that simulations of the estimated models yield an unconditional distribution that is much closer to the empirical distribution than returns from standard GARCH-type models. Another paper (Lux, 2004) demonstrates the unreliability of the so-called scaling estimator adopted from statistical physics by the early literature. At least for the subordinate multifractal processes of Mandelbrot et al. (1997) and Calvet and Fisher (2002), this popular estimator is shown to give extremely volatile parameter estimates and tests based on this estimator are unable to reject the null hypothesis of multi-fractal behavior for uni-fractal processes. Lux (2007) takes up the question of estimating the second generation Markov-Switching Multifractal model. This paper introduces a Generalized Method of Moments (GMM) estimator that is universally applicable to all possible specifications of MSM processes. In particular, it can be applied in all those cases where ML is not applicable or computationally unfeasible. In all cases, it is advantageous computationally as it requires CPU time of (at least) one or two orders of magnitude less than that of ML or SML estimation algorithms. In order to forecast volatility, these estimates are combined with best linear forecasts (Brockwell and Dahlhaus, 2004) on the base of analytical autocovariances of the process. Numerical experiments show that despite the loss of efficiency implied by using GMM rather than ML, the accuracy of forecasting is almost the same. The empirical application shows the advantage of new specifications of MSM models that can be handled via GMM but not by ML estimation. In particular, allowing for a larger state space (k higher than 10) often allows for further improvements in forecasting power against the specifications explored by Calvet and Fisher (2004). Lux and Kaizoji (2007) perform a more comprehensive forecasting exercise with the MF asset pricing model for the Tokyo Stock Market using GMM estimation together with best linear forecasts. It is found that the GMM-MF model can outperform both short and long memory models in terms of forecastability. More precisely, it is shown that long memory models such as FIGARCH, ARFIMA and the GMM-MF outperform short memory models such as GARCH and ARMA models. Morever, when comparing the long memory models against each other, it is found that the GMM-MF model often improves upon the forecasts of both FIGARCH and ARFIMA models. Lux and Kaizoji also show that an appropriately modified version of the MSM model can also be used as a model for trading volume and again performs better in forecasting this quantity than more standard time series models like ARMA and ARFIMA. Some other aspects of multifractal models and their applications have been successfully dealt with in recent Ph.D. theses under the supervision of the applicant. Lee (2007) considers multinomial extensions of the Binomial version of MSM which improves their in-sample and out-of-sample fit in comparison to the baseline Binomial specification. Liu, di Matteo and Lux (2007) have explored the behavior of estimated MF models under various statistical tests. Liu and Lux (2006) extend the MF model to the bivariate case and estimate their bivariate multifractal model (BMF) via GMM and Maximum Likelihood. The BMF asset pricing model is assesed empirically by computing the Value-at-Risk of portfolios employing stock, bond and foreign exchange rate indices. In subsequent work, Liu (2007), the multi-variate approach of Liu and Lux (2006) is contrasted with the alternative multi-variate model published by Calvet, Fisher and Thompson (2006). Both approaches are characterized by somewhat different mechanisms for the correlation of volatility between assets: Calvet et al. assume that the distributions of volatility components might have different parameters for both assets, but that they are correlated over all hierarchical levels. Liu and Lux (2006), in contrast, suppose that the parameters are the same, but that volatility correlation is restricted to a subset of components k 1 k. In Liu s comparative analysis of both specifications, the Liu and Lux (2006) version appeared to be somewhat better able to capture the empirical distribution of Value-at-Risk in various bi-variate test portfolios. 8 9 3 Proposed Research Programme for the Project 3.1 Objectives As previously indicated one of the main aims of this project is to extend the bivariate multifractal model proposed by Liu and Lux (2006) for forecasting and risk management. For this purpose we will start by resorting to the best linear forecasts computed via the Durbin-Levinson algorithm given the highly non-linear structure of the BMF model along the lines of Lux (2007) for the uni-variate case. A first important step will be the generalization of the so far relatively preliminary state of development of these forecasts. Best linear forecasts are computed separately for two assets of a simple portfolio in Liu and Lux (2006). In order to fully exploit the rich structure of dependencies offered by MF models, this approach needs to be generalized by the development of truely multi-variate forecasts taking into account the structure of autocorrelations between assets. These more accurate forecasts of portfolio risk would then have to be generalized for large portfolios with an arbitrary number of assets. For these MMF portfolio forecasts, we will design a comprehensive framework for forecasting VaR which can be summarized in four main steps. First, we will apply rolling windows and recursive procedures to investigate the optimal scheme when forecasting with the BMF model. Studies by Granger (1989) and Aiolfi and Timmermann (2006) show that it is often preferable to combine alternative forecasts of different forecasting schemes or nested models in a linear fashion and thereby obtain a new predictor. Thus, we will also consider whether optimal forecast combinations (e.g. MMF models with normal and t-distributed innovations) can improve forecastability in comparison singular schemes. Second, a rigorous set of forecast evaluation techniques will be employed. In particular, we will employ (i) a CUSUM type test for parameter stability, (i) a test on average predictive accuracy such as the one proposed by Diebold and Mariano (1995), (ii) a test on forecast encompassing as proposed by Harvey (1998), (iii) a test of outof-sample distributional features of returns as proposed by Hendriksson and Merton (1981) and (iv) a test for serial correlation on forecasting errors. Steps 1 to 2 will be carried out for different frequencies of the data (tic by tic, daily, monthly, quaterly). Third, we will employ state-of-the-art tests for Value-at-Risk that consider both the unconditional coverage hypothesis (the probability of an ex-post loss exceeding
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