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1 Fin285a:Computer Simulations and Risk Assessment Section 7.1 Modeling Volatility: basic models Daníelson, ,

2 Overview Moving average model Exponentially weighted moving average (EWMA) GARCH models Estimating GARCH models Other volatility models Fall 2016: LeBaron Fin285a: / 34

3 Moving average model Exponentially weighted moving average (EWMA) GARCH models Estimating GARCH models Other volatility models Moving average model Fall 2016: LeBaron Fin285a: / 34

4 Mean adjust For most of these models we will assume E(R t ) = 0. This makes things easier, and at short horizons it is not an unreasonable assumption. You can also simply use R t mean(r t ) too Fall 2016: LeBaron Fin285a: / 34

5 Moving average model ˆσ 2 t = 1 m m 1 j=0 R 2 t j (7.1.1) Fall 2016: LeBaron Fin285a: / 34

6 Plot volatility Dow retusvolplt.m Moving average of returns and scrambled returns Big difference Note: ma function Fall 2016: LeBaron Fin285a: / 34

7 help ma >> help ma usage [ma, range] = ma(x,nma) x = price series nma = moving average length ma = moving average of price series with zeros in periods where t<nma range = range of valid ma s note: ma(range) gives valid ma s (p(range)>ma(range) could be used as a signal >> Fall 2016: LeBaron Fin285a: / 34

8 Moving average model One parameter, m Trade-off Estimate accuracy: big m Adjust to changing volatility: small m Evaluation difficult (don t know σ 2 t ) Volatility is Latent Fall 2016: LeBaron Fin285a: / 34

9 Moving average model Exponentially weighted moving average (EWMA) GARCH models Estimating GARCH models Other volatility models Exponentially weighted moving average (EWMA) Fall 2016: LeBaron Fin285a: / 34

10 Exponentially weighted moving average ˆσ 2 t = λˆσ2 t 1 +(1 λ)r2 t 1 (7.1.2) Fall 2016: LeBaron Fin285a: / 34

11 Exponentially weighted moving average ˆσ 2 t = λˆσ2 t 1 +(1 λ)r2 t 1 (7.1.2) ˆσ 2 t 1 = λˆσ2 t 2 +(1 λ)r2 t 2 (7.1.3) ˆσ 2 t = λ(λˆσ 2 t 2 +(1 λ)r 2 t 2)+(1 λ)r 2 t 1 (7.1.4) ˆσ 2 t = λ(λ 2ˆσ t 3 +λ(1 λ)r 2 t 3)+λ(1 λ)r 2 t 2+(1 λ)r 2 t 1 (7.1.5) Eventually, Fall 2016: LeBaron Fin285a: / 34

12 Exponentially weighted moving average ˆσ 2 t = λˆσ2 t 1 +(1 λ)r2 t 1 (7.1.2) ˆσ 2 t 1 = λˆσ2 t 2 +(1 λ)r2 t 2 (7.1.3) ˆσ 2 t = λ(λˆσ 2 t 2 +(1 λ)r 2 t 2)+(1 λ)r 2 t 1 (7.1.4) ˆσ t 2 = λ(λ 2ˆσ t 3 +λ(1 λ)rt 3)+λ(1 λ)r 2 t 2+(1 λ)r 2 t 1 2 (7.1.5) Eventually, ˆσ t 2 = (1 λ)(rt 1 2 +λrt 2 2 +λ 2 Rt ) (7.1.6) Smooth weighting into the past! Fall 2016: LeBaron Fin285a: / 34

13 Riskmetrics VaR ˆσ 2 t = (1 λ)(r 2 t 1 +λr 2 t 2 +λ 2 R 2 t ) Riskmetrics original VaR system (back to J.P. Morgan) Set λ = 0.94 Pretty arbitrary, but works well in many situations Matlab: use ema function exponential moving average Fall 2016: LeBaron Fin285a: / 34

14 Quick note on sums S = (x+λx+λ 2 x+...) λs = (λx+λ 2 x+λ 3 x+...) S λs = x S = x 1 λ (1 λ)s = x (7.1.7) For x = 1, (1 λ)(1+λ1+λ ) = 1 (7.1.8) Weights in formulas sum to 1. ˆσ 2 t = (1 λ)(r2 t 1 +λr2 t 2 +λ2 R 2 t ) Fall 2016: LeBaron Fin285a: / 34

15 help ema >> help ema usage [ma, range] = ema(x,lambda,nback) Exponential moving average x = time series to filter lambda = exponetial weight parameter nback = max lags to use >> Fall 2016: LeBaron Fin285a: / 34

16 Moving average model Exponentially weighted moving average (EWMA) GARCH models Estimating GARCH models Other volatility models GARCH models Fall 2016: LeBaron Fin285a: / 34

17 GARCH:Generalized Autoregressive Conditional Heteroskedasticity Complete model for changing variances Engle (1982) and Bollerslev(1986) GARCH(1,1): R t = σ t u t u t N(0,1) (7.1.9) σt 2 = α 0 +α 1 Rt 1 2 +βσt 1 2 (7.1.10) Other options: Nonzero mean, R t = σ t u t +µ u t t ν (student-t) ARCH(1): β = 0 Fall 2016: LeBaron Fin285a: / 34

18 Nonzero mean GARCH R t = µ+σ t u t u t N(0,1) (7.1.11) σt 2 = α 0 +α 1 (R t 1 µ) 2 +βσt 1 2 (7.1.12) For short horizons this doesn t matter Subtracting a constant mean is not very difficult Changing means, µ t, is tricky Fall 2016: LeBaron Fin285a: / 34

19 Quick notes on ARCH (not GARCH) Autoregressive Conditional Heteroskedasticity Developed for inflation series For persistence in financial data requires many lags ARCH(q) q σt 2 = α 0 + α j Rt j 2 (7.1.13) j=1 GARCH works better GARCH(1,1) is kind of a benchmark Fall 2016: LeBaron Fin285a: / 34

20 GARCH(p,q) q p σt 2 = α 0 + α j Rt j 2 + j=1 i=1 ARCH(q) corresponds to all p = 0. β i σ 2 t i (7.1.14) Back to GARCH(1,1): σ 2 t = α 0 +α 1 R 2 t 1 +βσ 2 t 1 (7.1.15) Fall 2016: LeBaron Fin285a: / 34

21 GARCH(1,1) σ 2 = E(σ 2 t) = α 0 +α 1 E(R 2 t 1)+βE(σ 2 t 1) (7.1.16) σ 2 = E(σ 2 t) = α 0 +α 1 σ 2 +βσ 2 (7.1.17) σ 2 = Parameter restrictions, α 0 (1 α 1 β) (7.1.18) α 0,α 1,β > 0 α 1 +β < 1 Fall 2016: LeBaron Fin285a: / 34

22 Properties of GARCH(1,1) Persistent volatility Leptokurtosis (fat tails) Example model: β = 0.89,α 1 = 0.1,α 0 = 2x10 6 T = 10,000 Code: simgarch.m See Skewness, Kurtosis, JB test Not normal, zero skew, kurtosis > 3 Next figures Fall 2016: LeBaron Fin285a: / 34

23 Simulated GARCH time series Fall 2016: LeBaron Fin285a: / 34

24 Simulated GARCH histogram Fall 2016: LeBaron Fin285a: / 34

25 GARCH(1,1) variance forecast (1 period) Assume you know all information up to t 1. Notation: E t 1 (R t+h ) Similar to our riskmetrics equations σ 2 t = α 0 +α 1 R 2 t 1 +βσ2 t 1 (7.1.19) σt 2 = α 0 +α 1 Rt 1 2 +β(α 0 +α 1 Rt 2 2 +βσt 2) 2 (7.1.20) σt 2 = α m 0(1+β +β )+α 1 β i 1 Rt i 2 +βm σt m 2 (7.1.21) i=1 This looks like the EWMA forecast. For longer horizons it is different. Fall 2016: LeBaron Fin285a: / 34

26 GARCH(1,1) forecast more than 1-step σ 2 t = α 0 +(α+β)σ 2 +α 1 (R 2 t 1 σ 2 )+β(σ 2 t 1 σ 2 ) (7.1.22) Since, then, σ 2 = α 0 1 α β (7.1.23) σ 2 t = σ2 +α 1 (R 2 t 1 σ2 )+β(σ 2 t 1 σ2 ) (7.1.24) σ 2 t+1 = σ 2 +α 1 (E t 1 (R 2 t) σ 2 )+β(σ 2 t σ 2 ) (7.1.25) σ 2 t+1 = σ 2 +α 1 (σ 2 t σ 2 )+β(σ 2 t σ 2 ) (7.1.26) σ 2 t+1 = σ 2 +(α 1 +β)(σ 2 t σ 2 ) (7.1.27) Fall 2016: LeBaron Fin285a: / 34

27 GARCH(1,1) variance forecast σ 2 t+1 = σ 2 +(α 1 +β)(σ 2 t σ 2 ) (7.1.28) σ 2 t+2 = σ 2 +(α 1 +β)(σ 2 t+1 σ 2 ) (7.1.29) σ 2 t+2 = σ 2 +(α 1 +β) 2 (σ 2 t σ 2 ) (7.1.30) σ 2 t+m = σ 2 +(α 1 +β) m (σ 2 t σ 2 ) (7.1.31) As m gets big this converges to σ 2, the unconditional variance. Fall 2016: LeBaron Fin285a: / 34

28 Compare to: Exponentially weighted moving average ˆσ 2 t = λˆσ 2 t 1 +(1 λ)r 2 t 1 (7.1.32) ˆσ 2 t+1 = λˆσ 2 t +(1 λ)e t 1 (R 2 t) (7.1.33) ˆσ t+1 2 = λˆσ t 2 +(1 λ)ˆσ t 2 = ˆσ t 2 (7.1.34) Similar logic gives, ˆσ t+m 2 = ˆσ t 2 m > 0 (7.1.35) It stays the same out into the future. No long run variance to converge back to. Fall 2016: LeBaron Fin285a: / 34

29 Moving average model Exponentially weighted moving average (EWMA) GARCH models Estimating GARCH models Other volatility models Estimating GARCH models Fall 2016: LeBaron Fin285a: / 34

30 Estimating GARCH(1,1) Maximimum likelihood See page 43 Many matlab routes: doc garch Matlab example: garchdemo.m Fall 2016: LeBaron Fin285a: / 34

31 Matlab code % demo of GARCH estimation in matlab (Danielason 50) % This has been modified to handle updated econometrics TB load../data/retus.mat ret = ret-mean(ret); spec = garch(1,1); % set up type of GARCH model fit = estimate(spec, ret); % estimate model v = infer(fit,ret); % generate conditional variances ht = sqrt(v); % ht Conditional standard deviation alpha0 = fit.constant; % constant in variance equation alpha1 = fit.arch; % squared return beta = fit.garch; % lagged variance sret = ret./ht; % standardized residuals Fall 2016: LeBaron Fin285a: / 34

32 Residual diagnostic Key test Divide returns by estimated std., R t ˆσ t Should be constant variance For normal GARCH, should be N(0,1) Diagnostics Plot time series, and histfit Kurtosis, skewness Autocorrelations of squares Fall 2016: LeBaron Fin285a: / 34

33 Moving average model Exponentially weighted moving average (EWMA) GARCH models Estimating GARCH models Other volatility models Other volatility models Fall 2016: LeBaron Fin285a: / 34

34 GARCH in Mean R t = µ t +σ t Z t (7.1.36) µ t = δσ 2 t (7.1.37) Mean/variance dynamic relationship. Doesn t work all that well. Fall 2016: LeBaron Fin285a: / 34

35 The leverage effect Volatility rises when prices fall Strong in equity markets, but not FX markets One possible model: σt 2 = α 0 +βσt 1 2 +α 1 Rt 1 2 +φrt 1I 2 t 1 (7.1.38) I t 1 = 0 if (R t 1 0) (7.1.39) Many other models for this. I t 1 = 1 if (R t 1 < 0) (7.1.40) Fall 2016: LeBaron Fin285a: / 34

36 Overview Moving average model Exponentially weighted moving average (EWMA) GARCH models Estimating GARCH models Other volatility models Fall 2016: LeBaron Fin285a: / 34

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