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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

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Multivariate Volatility Models


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    1. Multivariate Volatility Models Scott Nelson July 29, 2008

    2. 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)

    3. 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?

    4. 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

    5. 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

    6. 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

    7. 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

    8. 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:

    9. 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

    10. 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

    11. 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

    12. Stylized Facts About Returns

    13. 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

    14. Univariate Volatility Modeling

    15. 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)

    16. Conditional Correlation

    17. 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)

    18. Data Used in this Section

    19. 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

    20. 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

    21. 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

    22. Moving Average

    23. 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

    24. Impact of Lambda

    25. EWMA vs. MA50 EWMA reacts more quickly

    26. 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

    27. 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

    28. Conditional Variance and Conditional Correlation

    29. Dynamic Conditional Correlation

    30. 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

    31. Example: 2 asset case • Step 1: Construct Dt from the elements of the univariate GARCH models

    32. Example: 2 asset case • The covariance matrix Ht can be decomposed as:

    33. Example: 2 asset case • Step 2: construct standardized residuals matrix

    34. Example: 2 asset case • Recall from the previous discussion that: • Give each ρi,j,ta simple GARCH(1,1) type structure:

    35. 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

    36. Example: 2 asset case • Kevin Sheppard’s UCSD GARCH toolbox, available at http://www.kevinsheppard.com/wiki/UCSD_GARCH

    37. Example: 2 asset case • Estimated coefficients

    38. DCC Results: VCOV Plots

    39. DCC vs. EWMA

    40. 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

    41. 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

    42. THANKS!