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Summary
Magnus Wiktorsson
Summary Magnus Wiktorsson long range dependence? Stylized facts in - - PowerPoint PPT Presentation
Summary Magnus Wiktorsson long range dependence? Stylized facts in - - PowerPoint PPT Presentation
Summary Magnus Wiktorsson long range dependence? Stylized facts in returns No Autocorrelation in returns Unconditional heavy tails Gain/Loss asymmetry. Aggregational Gaussianity Volatility clustering Conditional heavy
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Linear Gaussian models
▶ Model:
Xt+a1Xt−1+. . .+apXt−p = et+c1et−1+. . .+cqet−q
▶ Properties ▶ Estimation (OLS/LS2/MLE) ▶ Identification and model validation
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Non-linear models
▶ Can generate many (too many?) new features ▶ Larger model space! ▶ Difficult to identify - use prior knowledge! ▶ Model selection via absolute tests (e.g. residuals) ▶ and relative tests (LR) or AIC/BIC
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Variance models
▶ Need to transform data ▶ GARCH-family ▶ GAS models ▶ Stochastic volatility ▶ Realized volatility/quadratic variation
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Continuous time I
We restricted most of the course to continuous Semimartingales (no jumps).
▶ It¯
- calculus
▶ Valuation using the RNVF,
πt = p(t, T)EQ[φ(ST)|Ft].
▶ Connections to PDEs.
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Continuous time II
Estimation of parameters for SDEs
▶ Likelihood function generally unknown. ▶ Likelihood approximations (several methods) ▶ GMM
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Partially observed models
In continuous or discrete time
▶ Linear methods - Kalman filters ▶ Approximate non-linear methods - EKFs, UKFs,
IEKFs...
▶ Monte Carlo methods - particle filters
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Ex: 1D Linear Model in MATLAB
% Model % y=c*x + eta % x=a*x +e a=0.8; c=0.9; T=100; q=0.8; r=0.5; x=filter(1, [1 -a],sqrt(q)*randn(T,1)); y=c*x+sqrt(r)*randn(T,1); plot(1:T,x,1:T,y,’o’)
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Ex: Kalman filter in MATLAB
% Kalman filter mf=0; Pf=1; for t=1:T % prediction mpr=a*mf; Ppr=a*Pf*a’+q; % Update KalmanGain=Ppr*c’*inv(c*Ppr*c’+r); mf=mpr+KalmanGain*(y(t)-c*mpr); Pf=(eye(size(Pf))-KalmanGain*c)*Ppr; E_KF(t)=mf; end plot(1:T,E_KF,’kp’)
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Ex: Particle filter in MATLAB
% Particle filter K=10; % initialize xif=randn(K,1); for t=1:T % prediction xipr=a*xif+sqrt(q)*randn(K,1); % update lambdat=normpdf(y(t)*ones(K,1),c*xipr,sqrt(r)); % resampling I=randsample(1:K,K,’true’,lambdat); xif=xipr(I); E_PF(t)=mean(xif); end plot(1:T,E_PF,’g–’)
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