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Multifractal Volatility:Multifractal Volatility:Theory, Forecasting, and PricingTheory, Forecasting, and Pricing

Laurent CalvetHEC Paris & Imperial College

Adlai FisherUniversity of British Columbia

University of WaterlooUniversity of WaterlooMarch 27, 2009March 27, 2009

Properties of Financial DataProperties 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 MarketsTime 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 ApproachesStandard 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 modellingchoices

Multifractal ApproachMultifractal Approach

Volatility and ReturnsArbitrarily many frequencies with 4 parameters

Applications: 10 frequencies and over 1,000 statesDurations range from minutes to decades

Closed form likelihoodImproves on standard models in- and out-of-sample

Integrates easily into asset pricing applications

Multifrequency News ShocksHigh Low

OUTLINEOUTLINE

3 – Pricing multifrequency risk

1 – Modelling multifrequency volatility

2 – Volatility comovement

1 1 –– MULTIFREQUENCY MODEL MULTIFREQUENCY MODEL

MARKOV-SWITCHING MULTIFRACTAL (MSM)

Volatility components with highly heterogeneous durationsParsimonious, tractable, good performance

L. Calvet and A. FisherForecasting 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.

MSM DefinitionMSM Definition

Mk,t+1 = Mk,t

γk

1−γk

Draw Mk,t+1 from MMk,t• Independent dynamics:

• Multipliers: binomial {m0, 2-m0}, equal probability

1 (1 )k kb k k

k k kb! ! !

" "= " " #• Frequencies:

( ) ( ) 2/1,,1 ... tktt

MMM !! =ttt

Mx !" )(=

Four parametersArbitrary number of frequencies

CONSTRUCTIONCONSTRUCTION

0 0 or 2 with equal probabilityM m m= !

Volatility ( )t

M!

1,tM

2,tM

3,tM

Multifrequency Model

Dollar-Mark (1973-1996)

SIMULATIONSIMULATION

PROPERTIESPROPERTIES

• Multifrequency volatility persistence

• Parsimonious

• Convenient parameter estimation and forecasting

• Out-of-sample volatility forecasts and in-sample measures of fit significantly improve on standard models.

• Thick tails

TRACTABILITY OF MSMTRACTABILITY OF MSMA special Markov-switching model

Finite State SpaceState vector Mt belongs to finite state space {m¹,...,md}

Transition matrix A

! = ! !1

Conditional distribution ( , ..., )d

t t t 1 1( ; )

t t tf r

+ +! = !

Bayesian Updating

Multistep Forecasting

Given , future states have probability n

t tA! !

Closed-Form Likelihood

1( , ..., )

TL r r

Maximum Likelihood Estimation Maximum Likelihood Estimation of Binomial MSMof Binomial MSM

• 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

Source: L. Calvet and A. Fisher, How to Forecast Long Run Volatility, Journal of Financial Econometrics, Spring 2004

In-Sample ComparisonIn-Sample Comparison

OUT-OF-SAMPLE ANALYSISOUT-OF-SAMPLE ANALYSIS

2

, ,

2

,

( ( ))1

( )

t n t n t nt

t nt

RV RV

RV RV

!!= !

!

""

E

• Estimate MSM(10)

12

,

0

n

t n t i

i

RV r

!

!=

="• Realized volatility

• Out-of-sample R2

• Assess forecasting accuracy on out of sample data

Volatility ForecastsVolatility 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 2 –– VOLATILITY COMOVEMENT VOLATILITY COMOVEMENT

Correlation of Volatility Components

L. Calvet, A. Fisher and S. Thompson (2006), Volatility Comovement: A Multifrequency Approach, Journal of Econometrics.

0.5030.3870.1680.0760.0340.0320.0300.028UK8

0.4010.4890.3360.1770.0850.0790.0770.073UK7

0.2000.3880.5340.4020.1990.1410.1380.130UK6

0.0820.1840.3680.5890.4340.0670.0460.037UK5

0.0290.0700.1420.3300.5010.4510.2400.231UK4

0.0130.0340.0530.1130.1430.5960.6180.624UK3

0.0150.0350.0500.1260.2040.6200.7390.717UK2

0.0090.0220.0220.0400.1620.6030.9790.978UK1

DM8DM7DM6DM5DM4DM3DM2DM1

MULTIVARIATE MSMMULTIVARIATE MSM

Two financial series α and β

, 2

,

,

, {1,..., }k t

k t

k t

MM k k

M

!

" +

# $= % %& '( )

R

1/2

1, ,

1/2

1, ,

( ... )

( ... )

t t tk t

t t tk tr

r M M

M M

! ! ! !

" " " "

#

#

=

=

IID (0, )t

t

!

"

#

#

$ %= &' (

) *N

correlated are and in Arrivals!"

ktktMM

Drawn from bivariate binomial:

0 0

0

0

2

1/ 2

1/ 22

m m

p pm

p pm

! !

"

"

#

#

##

VALUE-AT-RISKVALUE-AT-RISKOne-day failure rate

This table displays the frequency of returns that exceed the VaR forecasted by the model. Bivariate MSMuses 5 components. For quantile p% the number reported is the frequency of portfolio returns below quantile ppredicted by the model. If the VaR forecast is correct, the observed failure rate should be close to theprediction. Boldface numbers are statistically different from p at the 1% level.

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

3 3 –– Pricing Multifrequency Risk Pricing Multifrequency Risk

• Idea: When fundamentals (dividends, earnings, consumption)have multifrequency risks, the equilibrium stock price willinclude endogenous responses to changes in state variables

• Volatility feedback: Prices fall when fundamental volatilityincreases• 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 INDEXU.S. EQUITY INDEX

Daily excess returns on US aggregate equity1926-2003: 20,765 observations

Markov-Switching Exchange EconomyMarkov-Switching Exchange Economy

( ) ( ) ( )

MSM vector,Markovorder first :

2/ ,

t

tdtdtdtdt

M

MMMd !""µ +#=$

Consumption:

Dividends:

!""

"#

n correlatio ),1,0(~, ,,

,

N

gc

tdtc

tccct +=$

Preferences: Epstein-Zin, risk-aversion α, EIS ψ

Equilibrium: Stock prices, returns, constant rf

In equilibrium, P/D ratio driven by the volatility components (feedback)Each component has impact inversely related to its frequency

Calibration

0.5≤ψ≤1.5δ ≈ 0.98

Implies a unique α

EMPIRICAL RESULTSEMPIRICAL RESULTS

d f d Dividend growth: ì - r = 1.2%, = 11% per year!•

Long-run P/D = 25 •

f f ø and ä appear only in r Set r 1%• =

c c cd Consumption parameters (g , ó , ñ )•

Maximum Likelihood Estimation

0 3 free parameters (m , ã ,b)

k•

Daily US excess stock returns•

ML ESTIMATIONML ESTIMATIONPostwar (1952 - 2003)

Var( ) Feedback = 1

Var( )

t

t

r

d!

"Annual equity premium = 4.2%

VOLATILITY FEEDBACKVOLATILITY FEEDBACK

Estimation on 1926-2003 sample Feedback = 40%

Multifrequency economy: 30% - 40% Feedback

CH : 1% - 2%

Campbell and Hentschel (JFE, 1992) Based on skewed unifrequency process (QGARCH).

Multifrequency economy outperforms Campbell and Hentschel in sample.

Learning EquilibriumLearning Equilibrium

• Noisy signals: I)N(0,~z ,z 1t1t1 +++ + !"tM

• P:D linear in investor beliefs: ( ) )(d

1j

!=

"=" jj

tt mQQ

LEARNING ABOUT VOLATILITY IS ASYMMETRICLEARNING ABOUT VOLATILITY IS ASYMMETRIC

Investors learn abruptly about volatility increases,gradually about decreases.

Learning

EquilibriumFewer large positive returns than under full informationEndogenous skewnessTradeoff between skewness and kurtosis

Calibrated ResultsCalibrated Results

Skewness / Kurtosis TradeoffSkewness / Kurtosis Tradeoff

LONG-RUNLONG-RUNCONSUMPTION RISKCONSUMPTION RISK

IID consumption Consumption parameters (gc,σc, ρc,d ) of Bansal and Yaron(2004) Implied risk-aversion α ≈ 35, comparable to Lettau Ludvigson and Wachter (2006)

Long-run / multifrequency consumption risk Add switches in dividend drift, consumption drift, and consumption volatility Generate reasonable equity premium with α = 10 Dividend volatility feedback > 20%

4 4 –– Continuous-time MSM and Endogenous Continuous-time MSM and EndogenousJump-diffusionsJump-diffusions

Use equilibrium valuation to generate a parsimonious model of multifrequency price jumps

“Multifrequency Jump-Diffusions: An Equilibrium Approach”Journal of Mathematical Economics, January 2008.

JUMP-DIFFUSIONS IN FINANCEJUMP-DIFFUSIONS IN FINANCE

Option pricing

• Stock price follows exogenous jump-diffusion (Merton, 1976)

• Statistical refinements of price process: - Stochastic volatility (Bakshi, Cao and Chen, 1997; Bates, 2000) - Infinite number of jumps in a finite time interval

(Carr, Géman, Madan and Yor, 2002) - Exogenous correlation between price jumps and volatility

(Duffie, Pan and Singleton, 2000; Carr and Wu, 2003)

Equilibrium

• Exogenous jumps in endowment process - Equity premium (Liu, Pan and Wang, 2005)

Equilibrium SpecificationEquilibrium Specification

( ) ( ) ( )t

C t C t C

t

dCg M dt M dZ t

C!= +Endowment

,

,

1( ): Brownian with zero drift and covariance matrix

1( )

C DC

C DD

Z t

Z t

!

!

" #$ %& '( )

* + , -

Dividend ( ) ( ) ( )t

D t D t D

t

dDg M dt M dZ t

D!= +

The drift and volatility of each process are deterministic functions ofMt

Representative Agent

00

Expected isoelastic utility ( )

'( )

t

te u c dt

u c c

!

"

+#$

$=

%E

Observes state Mt and receives consumption flow

EQUILIBRIUM STOCK PRICEEQUILIBRIUM STOCK PRICE

,0

( ) ( ) ( ) ( )

0( ) ln

s

f t h D t h C t h D t h C Dr M g M M M

t tq M e ds

! ! "+ + + +# $+% & & +'( )* +,

= - ./ 0,E

Endogenous volatility feedback

The log price follows the jump-diffusion

pt = dt + q(Mt)

where dt is the log dividend, and q(Mt) is the log of the P/D ratio.

STOCK DYNAMICSSTOCK DYNAMICS

Many small jumps, some moderate jumps, a few large jumpsVolatility and price jumps endogenously correlated

CONCLUSIONCONCLUSION

• Tractable Multifrequency Equilibrium Feedback increases with likelihood and number of frequencies Information quality generates an endogenous trade-off between skewness and kurtosis

• Jump-Diffusions Price jumps endogenously driven by volatility changes Endogenous jump size: small jumps common, rare large jumps

• MSM Parsimoniously specifies shocks of heterogeneous durations Captures persistence and high variability of financial volatility Performs well in- and out-of-sample

ADDITIONAL SLIDESADDITIONAL SLIDES

INFINITY OF FREQUENCIESINFINITY OF FREQUENCIES

1/2

1,, ,Volatility ( ) ( ) degenerate when

t D tD k k tM M M k! !" # $K

2

,0Time deformation ( ) ( )

t

sk D kt M ds! "= #

{ ( )} is a positive martingale with bounded expectation

Sequence { ( )} converges to a random variable

k k

k k

t

t

!

!

"

0 1Fixed parameters ( , , , )

Dm b! "

When , fundamentals include components of increasing frequencyk !"

LIMITING DIVIDEND PROCESSLIMITING DIVIDEND PROCESS

2 If ( ) , the sequence of time-deformations

converges to a limit ,

which has continuous sample paths.

M b

!"

<E

( )Local Hölder exponent: | ( ) ( ) | ( )

t

tX t t X t C t

!+ " # $ "

β(t)

Continuous Itô processes ½

Traditional Jump diffusion 0 or ½

Multifractal θ∞ Continuum

LIMITING STOCK PRICELIMITING STOCK PRICE

(1 )( ) ( )

2

0

If , 1 and - (1 - ) 0,

the log-price weakly converges to

( ),

where ( ) ln .

t t D

t

s t s t

t

C D g

d q t

q t e ds M

! !" # #

! " $ !

% %

%

&+% & & + &' () *

%

= + = >

+

, -. .= / 0

. .1 23E

The limiting price process is a multifractal jump-diffusionwith countably many frequencies and infinite activity.

1

1

k

kb! ! "

=

FREQUENCIES

Distribution: M ≥ 0, Ε M = 1

COMPONENTS

γk dt

1 − γkdt

Draw Mk,t+dt from distribution M

Mk,t+dt = Mk,t

Mk,t

Continuous-time MSMContinuous-time MSM

0 1

Four parameters

( , , , )m b! "

1/ 2

1, ,( ) ( )

D t D t k tM M M! !" …

CONSTRUCTIONCONSTRUCTION

0 0 or 2 with equal probabilityM m m= !

Volatility ( )t

M!

1,tM

2,tM

3,tM