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