Multivariate Volatility Models Scott Nelson July 29, 2008 Outline of Presentation Introduction to Quantitative Finance Time Series Concepts Stationarity, Autocorrelation, Time Series Models Univariate Volatility Models Stylized facts about return series GARCH Multivariate Volatility Models Moving averages EWMA

Dynamic Conditional Correlation (DCC) Motivation from Quant Finance Most of the stuff in this talk is motivated by problems from quantitative finance Financial econometrics is one part of a larger field which goes under various names (quantitative finance, mathematical finance, computational finance, etc) The field applies quantitative models and theories to solve problems in the financial markets Some questions we can answer better than others What will be the closing price of IBM tomorrow? What is the fair price today of a call option on IBM, expiring in 3 months with a strike price of $57?

Motivation From Finance Other examples (Alexander, 2000) What is the volatility forecast for asset XYZ? Need this to price options written on the asset (option pricing) How can we optimally structure our positions to minimize our risk? (portfolio optimization) What is the overall risk exposure of our firm, so we can set aside adequate capital reserves? (value at risk) All of these questions depend on modeling and forecasting of volatility and correlations of asset prices Efficient Market Hypothesis Standard economic theory states that stock price

movements are unpredictable Efficient market hypothesis: prices completely reflect all available information If the future price of the stock is expected to increase, the current stock price will fully adjust to account for this Since future news is unpredictable (by definition), future price movements are also unpredictable (follow a random walk) According to the weakest form of this theory, it is impossible to make consistent above-average returns by studying only the historical price The Statistical Approach to QF

We observe a sequence of asset prices at { pt },int 1,...., T discrete points time, They are modeled as random variables using techniques from time series analysis Time Series Concepts - Stationarity We observe a univariate time Yseries { yt }, t 1,...., T Most time series models assume Y is stationary A time series is covariance stationary if it has a constant mean, variance and autocovariances

In other words the distribution is invariant to time shift If Y is nonstationary, we can difference it to make it stationary Time Series Concepts - Autocorrelation We can define the correlation between the currentytvalue of and itsylagged value : t i Cov( yt , yt i ) Cov( yt , yt i ) i if Var ( yt ) Var ( yt i )

Var ( yt ) Var ( yt )Var ( yt i ) A consistent finite sample estimate is given by: T (r r )(r t i t i T t i r)

2 ( r r ) t t 1 , 0 i T - 1 Time Series Concepts - Models Model Y as a linear combination of its lagged values (AR) +past errors (MA) + contemporaneous error AR(p) MA(q)

ARMA(p,q) p yt i yt i t i 1 Q yt i t i t i 1 P yt t i yt 1 t i t 1 t i 1 ARIMA(p,1, q)

Q j 1 P Q zt yt yt 1 , zt t i zt 1 t i t 1 t i 1 j 1 Traditionally we assume 2 ~ N ( 0 ,

Parameter estimation viat maximum )likelihood Model selection can be done based on goodness of fit stats The Statistical Approach to QF What to model: prices or returns? Prices are nonstationary Define the return, log rt log( pt / pt 1 ) Log returns are stationary and approximately normally distributed with a mean of 0 and a possibly time varying variance Stylized Facts About Returns Returns difficult to predict Volatility is time-varying with persistent

autocorrelation Positive skewness in the distribution of returns (long left tail) Extreme crashes Fat tails in the distribution of returns Fatter than a normal distribution would suggest Stylized Facts About Returns What is Volatility? Volatility = variance Volatility is a measure of the variability of the returns Need to distinguish between unconditional

volatility and conditional volatility. Volatility cannot be directly observed As a proxy we take Squared Returns Engle (1981) noticed that volatility of time series clusters, and could be modeled using an ARMAtype process Univariate Volatility Modeling Univariate Volatility Modeling Bollerslev (1987) extended Engles model to the now familiar GARCH model: Yt e t (Mean equation)

e t ~ N(0, h t ) (Error term with conditional variance) q p i 1 j1 h t 0 i e 2t i jh t j (Conditional variance equation)

Parameter estimation via maximum likelihood Conditional Correlation Multivariate Models Why are multivariate models better than just building a bunch of univariate models? Multivariate models allow the analyst to model the important variables in the system together These models allow for dynamic relationships between the variables (more realistic) Data Used in this Section

What is Correlation? The unconditional correlation between 2 r.v. Cov (r1, r2 ) 12 each with mean 0 is: 12 2 2 Var(r1 )Var(r2 ) 1 2 This is the covariance standardized to lie in [- 1,1] Here we are assuming there exists a true correlation, and the observed correlation at any time is just random variation around this

If instead we believe the correlation is time Cov(r1,t have , r2,t ) 12,t varying then we would 12,t Var (r12,t )Var (r22,t ) 1,t 2,t Time Varying Models of Correlation 1. Moving averages Advantage: simplest approach Problem: equal weight to all the history, need to

select window size 2. Exponentially weighted moving averages Advantage: uses all the history, recent history given more weight than older history Disadvantage: need to select smoothing parameter, the model yields restrictive dynamics 3. Multivariate GARCH Advantage: realistic dynamics informed by the data Disadvantage: can be difficult to ensure covariance matrix is positive definite Moving Average of Correlation Instead of averaging over the entire sample, we can use a rolling window estimate of correlation 12,t

t 1 t 1 r r s t n 1 1, s 2 , s 2 r s t n 1 1, s t 1

2 r s t n 1 2 , s This depends on an appropriate window size (n) Small values of n will result in a choppy correlation Large value of n will smooth out the correlation Old observations have the same weight as recent values When an old observation drops out of the window, we will see a large change in the correlation, even though nothing has happened recently Moving Average

EWMA of Correlation Exponentially weighted moving average (EWMA) is usually written as 12,t 122 ,t 12,t 22,t where 122 ,t (1 )r1,t 1r2,t 1 122 ,t 1 12,t (1 )r12,t 1 12,t 1 22,t (1 )r22,t 1 22,t 1 Nice thing about this is it uses the entire history, and attaches exponentially decreasing weights to the

observations In other words recent history counts more than old history Larger lambda -> smoother estimate Impact of Lambda EWMA vs. MA50 EWMA reacts more quickly Generalizing to n-Dimensions OK thats great but most likely our portfolio has more than 2 assets 1000s of assets is more realistic How do we generalize this to n dimensions? This is most easily expressed in matrix

rt | t 1 ~ N (0, H t ) notation H t is k k time - varying covariance matrix 11 1k H t k1 kk H t must be positive definite Curse of Dimensionality Consider the case of k=2 h11,t H t

h21,t h12,t , where h12,t h21,t h22,t In the most general form we need to estimate 21 parameters h11,t 1 11 12,t 1 12 22,t 1 13 1,t 1 2,t 1 11h11,t 1 12 h12,t 1 13h12,t 1 h12,t 2 21 12,t 1 22 22,t 1 23 1,t 1 2,t 1 21h11,t 1 22h12,t 1 23h12,t 1 h 22,t 3 31 12,t 1 32 22,t 1 33 1,t 1 2,t 1 31h11,t 1 32h12,t 1 33h12,t 1 For 100 assets we need to estimate 51,010,050

parameters Conditional Variance and Conditional Correlation Following Engle (2002), the conditional correlation between two r.v. each with mean 0 is : Et 1 (r1,t , r2,t ) 12,t Et 1 (r12,t ) Et 1 (r22,t ) Recall the conditional variance is defined as hi ,t Et 1 (ri 2,t ) We can define the return as a N(0,1) scaled by it' s conditional variance ri,t hi ,t i ,t , i ,t ~ N (0,1), i 1,2 Therefore we can write the conditional correlation as Et 1 ( 1,t , 2,t ) 12,t Et 1 ( 1,t , 2,t ) 2 2

Et 1 ( 1,t ) Et 1 ( 2,t ) which is the conditional covariance of the disturbances Dynamic Conditional Correlation The model is : rt | t 1 ~ N (0, H t ) H t Dt Rt Dt Rt diag (Qt ) 1 Qt diag (Qt ) 1 Qt (1 )Q ( t -1 t -1 ) Qt 1 t Dt 1rt where H t is the time varying covariance matrix D t is a diagonal matrix of time - varying standard deviations from univariate GARCH models Rt is the time - varying correlation matrix Qt is the time - varying covariance matrix of the standardized residuals

t are the standardized returns Dynamic Conditional Correlation Estimation procedure: 1. Estimate univariate GARCH models for all k assets 2. Standardize the returns by the estimated std. dev. 3. Estimate Rt from the standardized returns, using a simple model Example: 2 asset case Step 1: Construct Dt from the elements of the univariate GARCH models h 1,t Dt

0 0 h2,t Example: 2 asset case The covariance matrix Ht can be decomposed as: h 0 1 12 h1,t 0 1,t

Dt Rt Dt 0 h2,t 21 1 0 h2,t h 0 h1,t 12 h2,t 1,t 0 h2,t 21 h1,t h2,t h12,t h1,t 12 h2,t

2 h2,t 21 h1,t h2,t 12,t 2 1,t 1,t 2 ,t 12,t

1 , t 2 , t 21,t 21,t 2 1,t 2 ,t 2 ,t

2,t 1,t 12,t 22,t Example: 2 asset case Step 2: construct standardized residuals matrix 1 / 1,t t D r 0 1 t t

r1,t 0 r1,t 1,t r 1 / 2,t r2,t 2,t 2 ,t Example: 2 asset case Recall from the previous discussion that: 1 Et 1 ( t t ) Rt 21,t 1 12,t

1 q12,t q1,t q2,t q12,t q1,t q2,t 1 Give each i,j,t a simple GARCH(1,1) type structure: qi , j ,t i , j 1 i ,t 1 j ,t 1 qi , j ,t 1 .

Example: 2 asset case Step 3: estimate R. In multivariate form Qt (1 )Q ( t 1 t 1 ) Qt 1 Q is the unconditional covariance matrix of the returns/residuals Variance targeting: Pre-estimate Q estimation of Rt and then calibrate , during

Example: 2 asset case Kevin Sheppards UCSD GARCH toolbox, available at http://www.kevinsheppard.com/wiki/UCSD_G ARCH Example: 2 asset case Estimated coefficients DCC Results: VCOV Plots -4 8 -4 Nasdaq Variance

x 10 6 6 Nasdaq-DJ Covariance x 10 4 4 2 2 0

0 2000 4000 6000 8000 10000 0 0 2000

-3 1 4000 6000 8000 10000 8000 10000 Dow Jones Variance

x 10 0.8 0.6 0.4 0.2 0 0 2000 4000 6000 DCC vs. EWMA

Advantages & Disadvantages of DCC Advantages Relatively easy to estimate Should work for large dimensional covariance matrices More flexible dynamics than exponential smoothing Disadvantages Imposes the same dynamics on all the assets Conclusions 1. Practical problems in finance require forecasts of 2. 3. 4.

5. conditional variances and conditional covariances/correlations Univariate GARCH models can provide forecasts of conditional variances Conditional correlation forecasts are plagued by the curse of dimensionality Simple methods are widely used (rolling window, EWMA) but they lack a firm statistical basis The DCC estimator offers a practical multivariate GARCH framework that overcomes some of these problems THANKS!