Actuarial Applications of Multifractal Modeling Part II Time Series Applications Yakov Lantsman, Ph.D. NetRisk, Inc.

Financial Time Series: Existing Solutions Modeling financial time series are based on assumptions of Markov chain stochastic processes (rejection of long-term correlation). Efficient Market Hypothesis (EMH) and Capital Assets Pricing Model (CAPM). Lognormal distribution framework is prevailing to model uncertainty. Existing models possess large set of parameters (ARIMA, GARCH) which contribute to high degree of instability and uncertainty of conclusions.

Financial Time Series: Proposed Approach Multifractal modeling framework to model financial time series: interest rate, CPI, exchange rate, etc. Multiplicative Levy cascade as a mechanism to simulate multifractal fields. Application of Extreme Value Theory (EVT) to model probabilities of extreme events.

Some References on Multifractal Modeling Multifractal Analysis of Foreign Exchange Data, Schmitt, Schertzer, Lovejoy. Multifractality of Deutschemark / US Dollar Exchange Rates, Fisher, Calvet, Mandelbrot. Multifractal Model of Asset Returns, Mandelbrot, Fisher, Calvet. Volatilities of Different Time Resolutions, Muller, et al. Chaotic Analysis on US Treasury Interest Rates, Craighead Temperature Fluctuations, Schmitt, et al.

Financial Time Series: Modeling Hierarchy Continuous time diffusion models: one-factor (Cox, Ingersoll and Ross) multi-factor (Andersen and Lund) Discrete time series analysis: ARIMA GARCH ARFIMA, HARCH (Heterogeneous) MMAR (Multifractal Model of Asset Return).

Financial Time Series: MMAR Information contained in the data at different time scales can identify a model. Reliance upon a single scale leads to inefficiency and forecasts that vary with the time-scale of the chosen data. Multifractal processes will be defined by a restrictions on the behavior in their moments as the time-scale of observation changes.

Three Pillars of MMAR MMAR incorporates long (hyperbolic) tail, but not necessarily imply an infinite variance (additive Levy models); Long-dependence, the characteristic feature of fractional Brownian motion (FBM); Concept of trading time that is the cumulative distribution function of multifractal measure.

MMAR Definition {P(t); 0  t  T } price of asset and X(t)=Ln(P(t)/P(0)) Assumption: X(t) is a compound process: X(t)  BH [ (t)], BH (t) is FBM with index H, and  (t) stochastic trading time;  (t) is a multifractal process with continuous, non-decreasing paths and stationary increments satisfies: {BH (t)} and { (t)} are independent. Theorem: X(t) is multifractal with scaling function X (q)   (Hq) and stationary increments.

MMAR: Statistical Properties (Structure Function) Self-Similarity: Universality: Link to Power Spectrum:

Q-Q Plots for Error Term Distributions Treasury Yields (Normal) Industrial B1 Bond Yields (Normal) Treasury Yields (t-distribution) Industrial B1 Bond Yields (t-distribution)

Interest Rate Modeling 3-month Treasury Bill Rate (weekly observations)

Interest Rate Modeling K(q) function log-log plot of power spectrum function

Exchange Rate Modeling $/DM spot rate (weekly observations)

Exchange Rate Modeling K(q) function log-log plot of power spectrum function

Actuarial Applications of Multifractal Modeling Part II Time Series Applications Yakov Lantsman, Ph.D. NetRisk, Inc.