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Actuarial Applications of Multifractal ModelingActuarial Applications

of Multifractal Modeling

Part IITime Series Applications

Yakov Lantsman, Ph.D.NetRisk, Inc.

Part IITime Series Applications

Yakov Lantsman, Ph.D.NetRisk, Inc.

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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.

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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.

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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.

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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).

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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.

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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.

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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.

1)()())(( qqtqctE

1)()())(( qqtqctE

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MMAR: Statistical Properties (Structure Function)

• Self-Similarity:

• Universality:

• Link to Power Spectrum:

1)()())(( qqtqctE

)()()( tZtZtZ

)(1

1)( qq

CqHqK

0,20 q

)2(1 K

)(

)())((qK

qT

q

TZEtZE

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Q-Q Plots for Error Term Distributions

Treasury Yields (Normal) Industrial B1 Bond Yields (Normal)

Industrial B1 Bond Yields (t-distribution)Treasury Yields (t-distribution)

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Interest Rate Modeling

0 180 360 540 720 900 1080 1260 1440 1620 18000

2

4

6

8

10

12

14

16

18

20

3-month Treasury Bill Rate (weekly observations)

0 200 400 600 800 1000 1200 1400 1600 180030

20

10

0

10

20

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Interest Rate Modeling

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

K(q) function

7 6.3 5.6 4.9 4.2 3.5 2.8 2.1 1.4 0.7 010

8

6

4

2

0

2

4

6

8

10

log-log plot of power spectrum function

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Exchange Rate Modeling

$/DM spot rate (weekly observations)

0 100 200 300 400 500 600 700 80010

5

0

5

10

0 85 170 255 340 425 510 595 680 765 8501

1.25

1.5

1.75

2

2.25

2.5

2.75

3

3.25

3.5

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Exchange Rate Modeling

K(q) function

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.515

13

11

9

7

5

3

1

1

3

5

log-log plot of power spectrum function

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Actuarial Applications of Multifractal ModelingActuarial Applications

of Multifractal Modeling

Part IITime Series Applications

Yakov Lantsman, Ph.D.NetRisk, Inc.

Part IITime Series Applications

Yakov Lantsman, Ph.D.NetRisk, Inc.