Actuarial Applications of Multifractal Modeling

1. Actuarial Applications of Multifractal Modeling Part IITime Series ApplicationsYakov Lantsman, Ph.D.NetRisk, Inc.

2. 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.

3. 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.

4. 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.

5. 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).

6. 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.

7. 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.

8. 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 FBMwith 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.

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

10. 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)

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

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

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

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

15. Actuarial Applications of Multifractal Modeling Part IITime Series ApplicationsYakov Lantsman, Ph.D.NetRisk, Inc.