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Scott Nelson July 29, 2008

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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)

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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?

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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

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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

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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

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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

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Time Series Concepts - Autocorrelation We can define the correlation between the current value of and its lagged value : A consistent finite sample estimate is given by:

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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 AR(p) MA(q) ARMA(p,q) ARIMA(p,1,q)

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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

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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

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Stylized Facts About Returns

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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

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Univariate Volatility Modeling

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Bollerslev (1987) extended Engles model to the now familiar GARCH model: Parameter estimation via maximum likelihood (Mean equation) (Error term with conditional variance) (Conditional variance equation)

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Conditional Correlation

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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)

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Data Used in this Section

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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

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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

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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

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Moving Average

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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

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Impact of Lambda

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EWMA vs. MA50 EWMA reacts more quickly

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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 notation

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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

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Conditional Variance and Conditional Correlation

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Dynamic Conditional Correlation

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Estimation procedure: 1. Estimate univariate GARCH models for all k assets 2. Standardize the returns by the estimated std. dev. 3. Estimate R t from the standardized returns, using a simple model

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Example: 2 asset case Step 1: Construct D t from the elements of the univariate GARCH models

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Example: 2 asset case The covariance matrix H t can be decomposed as:

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Example: 2 asset case Step 2: construct standardized residuals matrix

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Example: 2 asset case Recall from the previous discussion that: Give each ρ i,j,t a simple GARCH(1,1) type structure:

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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 R t

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Example: 2 asset case Kevin Sheppards UCSD GARCH toolbox, available at http://www.kevinsheppard.com/wiki/UCSD_GARCH http://www.kevinsheppard.com/wiki/UCSD_GARCH

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Example: 2 asset case Estimated coefficients

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DCC Results: VCOV Plots

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DCC vs. EWMA

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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

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Conclusions 1. Practical problems in finance require forecasts of conditional variances and conditional covariances/correlations 2. Univariate GARCH models can provide forecasts of conditional variances 3. Conditional correlation forecasts are plagued by the curse of dimensionality 4. Simple methods are widely used (rolling window, EWMA) but they lack a firm statistical basis 5. The DCC estimator offers a practical multivariate GARCH framework that overcomes some of these problems

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THANKS!

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