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## Multivariate Volatility Models

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**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 discrete points in time, • They are modeled as random variables using techniques from time series analysis**Time Series Concepts - Stationarity**• We observe a univariate time series • 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 current value of and it’s lagged value : • A consistent finite sample estimate is given by:**Time Series Concepts - Models**• Model Y as a linear combination of its’ lagged values (AR) +past errors (MA) + contemporaneous error • Traditionally we assume • Parameter estimation via 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 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**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 ARMA-type process**Univariate Volatility Modeling**• Bollerslev (1987) extended Engle’s model to the now familiar GARCH model: • Parameter estimation via maximum likelihood (Mean equation) (Error term with conditional variance) (Conditional variance equation)**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)**What is Correlation?**• The unconditional correlation between 2 r.v. each with mean 0 is: • 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 varying then we would have**Time Varying Models of Correlation**• Moving averages • Advantage: simplest approach • Problem: equal weight to all the history, need to select window size • 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 • 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 • 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**EWMA of Correlation**• Exponentially weighted moving average (EWMA) is usually written as • 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**EWMA vs. MA50**EWMA reacts more quickly**Generalizing to n-Dimensions**• OK that’s great but most likely our portfolio has more than 2 assets – 1000’s of assets is more realistic • How do we generalize this to n dimensions? • This is most easily expressed in matrix notation**Curse of Dimensionality**• Consider the case of k=2 • In the most general form we need to estimate 21 parameters • For 100 assets we need to estimate 51,010,050 parameters**Dynamic Conditional Correlation**• Estimation procedure: • Estimate univariate GARCH models for all k assets • Standardize the returns by the estimated std. dev. • Estimate Rtfrom the standardized returns, using a simple model**Example: 2 asset case**• Step 1: Construct Dt from the elements of the univariate GARCH models**Example: 2 asset case**• The covariance matrix Ht can be decomposed as:**Example: 2 asset case**• Step 2: construct standardized residuals matrix**Example: 2 asset case**• Recall from the previous discussion that: • Give each ρi,j,ta simple GARCH(1,1) type structure:**Example: 2 asset case**• Step 3: estimate R. • In multivariate form • is the unconditional covariance matrix of the returns/residuals • Variance targeting: • Pre-estimate and then calibrate α, β during estimation of Rt**Example: 2 asset case**• Kevin Sheppard’s UCSD GARCH toolbox, available at http://www.kevinsheppard.com/wiki/UCSD_GARCH**Example: 2 asset case**• Estimated coefficients**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**• Practical problems in finance require forecasts of 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