Multifractal Volatility: Multifractal Volatility: Theory, - PowerPoint PPT Presentation

University of Waterloo University of Waterloo March 27, 2009 March 27, 2009 Multifractal Volatility: Multifractal Volatility: Theory, Forecasting, and Pricing Theory, Forecasting, and Pricing Laurent Calvet Adlai Fisher HEC Paris & Imperial College University of British Columbia

Properties of Financial Data Properties of Financial Data • Foreign Exchange – Thick tails – Volatility Persistence – Volatility comovement across markets • Equity – Skewness – Jumps – Volatility high after down markets (leverage effect/ volatility feedback) • Options – Smile / smirk → (thick tails and volatility asymmetry) – Volatility term-structure and smile decay slowly

Time Scales in Financial Markets Time Scales in Financial Markets • High Frequency – Daily / intraday: macro news, internet bulletin boards, weather (Roll, 1984), analyst reports, liquidity • Medium Term – Monthly, quarterly, business cycle range (Fama and French, 1989) • Long-run – Demographics, technology (Pastor and Veronesi, 2005), natural resource uncertainty, consumption growth (Bansal and Yaron, 2004)

Standard Approaches Standard Approaches • Thick-tailed conditional returns (e.g., Student-t, jumps) – Unpredictable high-frequency shocks • ARCH / GARCH / SV – Good one-step-ahead volatility predictors – Capture medium-run volatility dynamics • The long-run – Fractional Integration (FIGARCH), – Component Models (Engle and Lee, 1989; Heston, 1993) – Markov-switching (Hamilton, 1989) Typically viewed as unrelated modelling choices

Multifractal Approach Multifractal Approach High Low Multifrequency News Shocks Volatility and Returns Arbitrarily many frequencies with 4 parameters Applications: 10 frequencies and over 1,000 states Durations range from minutes to decades Closed form likelihood Improves on standard models in- and out-of-sample Integrates easily into asset pricing applications

OUTLINE OUTLINE 1 – Modelling multifrequency volatility 2 – Volatility comovement 3 – Pricing multifrequency risk

1 – – MULTIFREQUENCY MODEL MULTIFREQUENCY MODEL 1 L. Calvet and A. Fisher Forecasting Multifractal Volatility, Journal of Econometrics , 2001. Multifractality in Asset Returns, Review of Economics and Statistics , 2002. How to Forecast Long-Run Volatility, Journal of Financial Econometrics , 2004. MARKOV-SWITCHING MULTIFRACTAL (MSM) Volatility components with highly heterogeneous durations Parsimonious, tractable, good performance

MSM Definition MSM Definition ( ) 1 / 2 ( ) x ( M ) M M 1 ... M = � � � = � t t t t , t k , t γ k Draw M k,t+1 from M • Independent dynamics: M k,t 1 −γ k M k,t+1 = M k,t • Multipliers: binomial {m0, 2-m0}, equal probability k k � b k k 1 (1 ) k b � • Frequencies: � = � � � � � k k Four parameters Arbitrary number of frequencies

CONSTRUCTION CONSTRUCTION M m or 2 m with equal probability = � 0 0 M 1, t Volatility ( M ) � t M 2, t M 3, t

SIMULATION SIMULATION Dollar-Mark (1973-1996) Multifrequency Model

PROPERTIES PROPERTIES • Multifrequency volatility persistence • Parsimonious • Thick tails • Convenient parameter estimation and forecasting • Out-of-sample volatility forecasts and in-sample measures of fit significantly improve on standard models.

TRACTABILITY OF MSM TRACTABILITY OF MSM A special Markov-switching model Finite State Space State vector M t belongs to finite state space { m ¹ ,...,m d } Transition matrix A Bayesian Updating f ( ; r ) 1 d Conditional distribution ( ,..., ) � = � � = � � t 1 t t 1 t t t + + Closed-Form Likelihood L r ( ,..., r ) 1 T Multistep Forecasting n Given , future states have probability t A � � t

Maximum Likelihood Estimation Maximum Likelihood Estimation of Binomial MSM of Binomial MSM Source: L. Calvet and A. Fisher, How to Forecast Long Run Volatility, Journal of Financial Econometrics , Spring 2004 • Increase in likelihood from k=1 to k=2 is large by any model selection criterion • Constant number of parameters as number of frequencies increases • Models with 7 to 10 frequencies dominate

In-Sample Comparison In-Sample Comparison

OUT-OF-SAMPLE ANALYSIS OUT-OF-SAMPLE ANALYSIS • Estimate MSM(10) • Assess forecasting accuracy on out of sample data n 1 � • Realized volatility = � 2 RV r t n , t i � i 0 = E 2 ( RV ( RV )) � � • Out-of-sample R 2 t n , t n t n , � 1 t = � 2 ( RV RV ) � � t n , t

Volatility Forecasts Volatility Forecasts Source: Calvet and Fisher, How to Forecast Long Run Volatility, Journal of Financial Econometrics , Spring 2004 Results confirmed in CFT (2006), Lux (2008), and Bacry, Kozhemyak, and Muzy (2008).

Forecast Summary – Forecast Summary – p-values against MSM p-values against MSM

2 – – VOLATILITY COMOVEMENT VOLATILITY COMOVEMENT 2 L. Calvet, A. Fisher and S. Thompson (2006), Volatility Comovement: A Multifrequency Approach, Journal of Econometrics . Correlation of Volatility Components DM1 DM2 DM3 DM4 DM5 DM6 DM7 DM8 UK1 0.978 0.979 0.603 0.162 0.040 0.022 0.022 0.009 0.717 0.739 0.620 0.204 0.126 0.050 0.035 0.015 UK2 UK3 0.624 0.618 0.596 0.143 0.113 0.053 0.034 0.013 UK4 0.231 0.240 0.451 0.501 0.330 0.142 0.070 0.029 UK5 0.037 0.046 0.067 0.434 0.589 0.368 0.184 0.082 UK6 0.130 0.138 0.141 0.199 0.402 0.534 0.388 0.200 UK7 0.073 0.077 0.079 0.085 0.177 0.336 0.489 0.401 UK8 0.028 0.030 0.032 0.034 0.076 0.168 0.387 0.503

MULTIVARIATE MSM MULTIVARIATE MSM Two financial series α and β M � � � k t , R 2 M , k {1,..., } k = � � � � k t , + M � � � k t , 1/2 r ( M ... M ) � � � � = � t 1, t t k t , 1/2 ( M ... M ) r � � � � = � t 1, t t k t , � � � = � t IID (0, ) N � � � � � � � t Arrivals in M and M are correlated � � kt kt m 2 m � � � 0 0 p 1/2 p m � � Drawn from bivariate binomial: 0 1/2 p p 2 m � � � 0

VALUE-AT-RISK VALUE-AT-RISK One-day failure rate Bivariate MSM CC GARCH 1% 5% 10% 1% 5% 10% DM and JA Currency � 0.69 4.35 9.10 1.81 5.13 9.01 Currency � 0.95 4.81 9.56 2.30 5.38 9.10 Equal-Weight 0.86 3.92 8.32 1.30 4.66 8.21 Hedge 0.69 5.64 12.21 2.25 6.68 11.81 DM and UK Currency � 0.92 4.92 10.14 1.81 5.13 9.01 Currency � 0.72 5.27 10.68 1.44 4.61 8.29 Equal-Weight 1.07 4.69 10.28 1.87 5.18 8.98 Hedge 0.55 4.72 9.13 0.92 4.00 7.00 JA and UK Currency � 1.01 5.04 9.88 2.30 5.38 9.10 Currency � 0.60 4.41 9.70 1.44 4.61 8.29 Equal-Weight 0.84 4.55 8.78 1.64 4.69 8.03 Hedge 1.15 5.64 11.37 2.25 6.25 10.34 This table displays the frequency of returns that exceed the VaR forecasted by the model. Bivariate MSM uses 5 components. For quantile p % the number reported is the frequency of portfolio returns below quantile p predicted by the model. If the VaR forecast is correct, the observed failure rate should be close to the prediction. Boldface numbers are statistically different from p at the 1% level.

3 – – Pricing Multifrequency Risk Pricing Multifrequency Risk 3 • Idea: When fundamentals (dividends, earnings, consumption) have multifrequency risks, the equilibrium stock price will include endogenous responses to changes in state variables • Volatility feedback: Prices fall when fundamental volatility increases • Overall contribution of endogenous prices responses is 10-40 times larger in multifrequency economy than in single frequency benchmark • Learning about volatility • Generates endogenous skewness

U.S. EQUITY INDEX U.S. EQUITY INDEX Daily excess returns on US aggregate equity 1926-2003: 20,765 observations

Markov-Switching Exchange Economy Markov-Switching Exchange Economy Dividends: ( ) ( ) ( ) d M M / 2 M � = µ � � + � � t d t d t d t d , t M : first order Markov vector, MSM t c g Consumption: � = + � � t c c c , t , ~ N ( 0 , 1 ), correlatio n � � � c , t d , t Preferences: Epstein-Zin, risk-aversion α , EIS ψ Equilibrium: Stock prices, returns, constant r f In equilibrium, P/D ratio driven by the volatility components (feedback) Each component has impact inversely related to its frequency

EMPIRICAL RESULTS EMPIRICAL RESULTS  Calibration Consumption parameters (g , ó , ñ ) • c c cd 0.5 ≤ψ≤ 1.5 ø and ä appear only in r Set r 1% • = f f δ ≈ 0.98 Dividend growth: ì - r = 1.2%, = 11% per year • � d f d Long-run P/D = 25 Implies a unique α •  Maximum Likelihood Estimation 3 free parameters (m ,ã ,b) • 0 k Daily US excess stock returns •

ML ESTIMATION ML ESTIMATION Postwar (1952 - 2003) Var( ) r Annual equity premium = 4.2% Feedback = t 1 � Var( d ) � t

VOLATILITY FEEDBACK VOLATILITY FEEDBACK  Estimation on 1926-2003 sample Feedback = 40%  Campbell and Hentschel (JFE, 1992) Based on skewed unifrequency process (QGARCH). Multifrequency economy outperforms Campbell and Hentschel in sample. CH : 1% - 2% Feedback Multifrequency economy: 30% - 40%