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# Adapt Signal Proc EECS 659

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This 53 page Class Notes was uploaded by Ophelia Ritchie on Thursday October 29, 2015. The Class Notes belongs to EECS 659 at University of Michigan taught by Staff in Fall. Since its upload, it has received 44 views. For similar materials see /class/231560/eecs-659-university-of-michigan in Engineering Computer Science at University of Michigan.

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Date Created: 10/29/15

Adaptive Algorithms for Tracking Volatility Dinesh Krithivasan EECS 659 Adaptive Signal Processing Presentation Outline 0 Motivation for SV Models 9 The Discrete SV Model a Gibbs Sampling A Filtering Application 6 Results Conclusions 0 Motivation for SV Models Derivatives Pricing o Derivatives Financial instruments whose values derive from underlying assets 0 Examples include options forwards swaps etc 0 Asset prices modeled as a stochastic process 0 Problem Price derivatives in a fair and arbitrage free manner Black Scholes Model 0 Asset price 5 modeled as Brownian motion KITS utdtatth 0 pt drift of the stock price at volatility variance of the stock price 0 at crucial in pricing any derivative of St Larger at means more risk involved and hence investors expect greater returns 0 Black Scholes model Gives optimal price CStt of an European callput option 2 lazsz r56C 6t 2 652 Ezrc Black Scholes Model d5 5 Asset price 5 modeled as Brownian motion utdtatth pt drift of the stock price at volatility variance of the stock price at crucial in pricing any derivative of St Larger at means more risk involved and hence investors expect greater returns Black Scholes model Gives optimal price CStt of an European callput option ac 1 02C 2 2 E a 5 6 52r5 6C ErC Critical simplifying assumption at 0 is constant over time i Stochastic Volatility Models 0 In real financial markets volatility changes over time 0 Stochastic Volatility SV models view at as a stochastic process 0 Generic SV model dSt pStdt a t5tth d0 ashtdf shtdBt o asht sht specific to the SV model used 0 Objective Tune the parameters based on the observations e The Discrete SV Model Discrete SV Model 0 Model studied here Discrete SV Model 0 Model studied here Vt elk2Q etwo1iid a yr observations 2 ht underlying volatilities a at observation noise Discrete SV Model a Model studied here yteh3926t etW0liid hm bht 1 PM aunt ltlgt s 1 m JV01 iid a yr observations 3 p Instantaneous volatility a ht underlying volatilities o b Persistence of volatility 90392 0 5t observation noise r volatility of ht Discrete SV Model a Model studied here yteh3926t etW0liid ht ht l pannt s 11 JV01 iid 02 hl N Wl vl sz a yr observations 3 p Instantaneous volatility a ht underlying volatilities o b Persistence of volatility 90392 0 5t observation noise r volatility of ht Discrete SV Model a Model studied here yteh3926t etW0liid ht1 Pht l39 1 u0nnt 0 1 72 s 17t Jl0liid h1 W01 a yr observations 3 p Instantaneous volatility a ht underlying volatilities o b Persistence of volatility 2 0 5t observation noise 9 Ur volatility 0f ht 0 Objective Estimate 6 494105 and hl t from yl t i Discrete SV Model contd yteh3926t etW0liid hnl mll uamhstlmtwqdl id 05 1 42 h1 N jlHv State space model with ht as the hidden state 0 State update is linear and Markovian 0 Observation yr is a non linear function of the state ht i 9 Gibbs sampling lClC Sampling 0 Obtain samples from a pdf by repeatedly sampling from a Markov chain c The Markov chains invariant distribution is the target density of interest a In our case we would sample from the posterior 7rh1 t lyl t a With enough samples can compute features of 7rh1 t In t Gibbs Sampling 0 Special case of a Monte Carlo Markov Chain sampling Gibbs Sampling 0 Special case of a Monte Carlo Markov Chain sampling 0 Suppose we need to sample from pXy Gibbs Sampling 0 Special case of a Monte Carlo Markov Chain sampling 0 Suppose we need to sample from pXy a Marginalizing the joint density is hard Gibbs Sampling 0 Special case of a Monte Carlo Markov Chain sampling 0 Suppose we need to sample from pXy a Marginalizing the joint density is hard a Easy to sample from pXiy and pin Gibbs Sampling 0 Special case of a Monte Carlo Markov Chain sampling 0 Suppose we need to sample from pXy o Marginalizing the joint density is hard 9 Easy to sample from pXiy and pin o Gibbs sampler proceeds as below Gibbs Sampling 0 Special case of a Monte Carlo Markov Chain sampling 0 Suppose we need to sample from pXy o Marginalizing the joint density is hard 9 Easy to sample from pXiy and pin o Gibbs sampler proceeds as below a Pick yo from some distribution that has same support as pyy Gibbs Sampling 0 Special case of a Monte Carlo Markov Chain sampling 0 Suppose we need to sample from pXy o Marginalizing the joint density is hard 9 Easy to sample from pXiy and pin o Gibbs sampler proceeds as below a Pick yo from some distribution that has same support as pyy a Pick X0 from the distribution PX yle0 Gibbs Sampling 0 Special case of a Monte Carlo Markov Chain sampling 0 Suppose we need to sample from pXy o Marginalizing the joint density is hard 0 Easy to sample from pXiy and pin o Gibbs sampler proceeds as below a Pick yo from some distribution that has same support as pyy a Pick X0 from the distribution PX yle0 a For i1N pick y from Py Xylx1 and x from PX yxly Factorizing the Posterior Distribution 0 Use Gibbs sampler to sample from 71h11t ly17t Factorizing the Posterior Distribution 0 Use Gibbs sampler to sample from 71h11t ly17t a Initialize I71 t and 9 parzl l Factorizing the Posterior Distribution 0 Use Gibbs sampler to sample from 71h17t ly17t a Initialize I71 t and 9 parzl a Sample ht from 71htl hwyl t9 t1N Factorizing the Posterior Distribution 0 Use Gibbs sampler to sample from 71h17t ly17t a Initialize I71 t and 9 parzl a Sample ht from 71htl hwyl t9 t1N a Sample of from 7105 lyl hhl why Factorizing the Posterior Distribution 0 Use Gibbs sampler to sample from 71h17t ly17t a Initialize I71 t and 9 parzl a Sample ht from 71htl hwyl t9 t1N a Sample of from 7105 lyl hhl why a Sample p from 71pl I71 tp039T2 Factorizing the Posterior Distribution 0 Use Gibbs sampler to sample from 7rh1 t lyl t a Initialize I71 t and 9 parzl a Sample ht from 71htl hty1 t9 t1N a Sample of from 7105 lyl hhl why a Sample b from 71pl I71 tp039T2 a Sample p from 71pl I71 ham Factorizing the Posterior Distribution 0 Use Gibbs sampler to sample from 7rh1 t lyl t a Initialize I71 t and 9 parzl a Sample ht from 71htl hty1 t9 t1N a Sample of from 7105 lyl hhl why a Sample b from 71pl I71 tp039T2 a Sample p from 71pl I71 ham a Iterate the sampling steps as needed Factorizing the Posterior Distribution 0 Use Gibbs sampler to sample from 7rh1 t lyl t a Initialize I71 t and 9 parzl a Sample ht from 71htl hty1 t9 t1N a Sample of from 7105 lyl hhl why a Sample b from 71pl I71 tp039T2 a Sample p from 71pl I71 ham a Iterate the sampling steps as needed 0 Easy to sample from each of these conditional distributions Conjugate Priors 0 Setup is Bayesian need priors for h and H Conjugate Priors 0 Setup is Bayesian need priors for h and H 0 Use Conjugate priors when possible Conjugate Priors 0 Setup is Bayesian need priors for h and H 0 Use Conjugate priors when possible 9 I71 t multivariate Gaussian Conjugate Priors 0 Setup is Bayesian need priors for h and H 0 Use Conjugate priors when possible 9 I71 t multivariate Gaussian a 0 Inverse Gamma distribution Conjugate Priors 0 Setup is Bayesian need priors for h and H 0 Use Conjugate priors when possible 9 I71 t multivariate Gaussian a 0 Inverse Gamma distribution a p Scaled and shifted Beta distribution Conjugate Priors 0 Setup is Bayesian need priors for hl t and H 0 Use Conjugate priors when possible 9 71 t multivariate Gaussian o of Inverse Gamma distribution a b Scaled and shifted Beta distribution a p Diffuse prior 71p olt c Sampling from Conditional Distributions 0 Use different sampling methods to sample from conditional distributions Sampling from Conditional Distributions 0 Use different sampling methods to sample from conditional distributions 0 Critical to sample 71hfl hty17t9 efficiently Sampling from Conditional Distributions 0 Use different sampling methods to sample from conditional distributions 0 Critical to sample 71htl hty1 t efficiently a ht sampled from a Gaussian distribution with acceptance rejection sampling Sampling from Conditional Distributions 0 Use different sampling methods to sample from conditional distributions 0 Critical to sample 71htl hty1 t efficiently a ht sampled from a Gaussian distribution with acceptance rejection sampling 2 9039 sampled from the Inverse Gamma distribution Sampling from Conditional Distributions 0 Use different sampling methods to sample from conditional distributions 0 Critical to sample 71htl hty1 t efficiently a ht sampled from a Gaussian distribution with acceptance rejection sampling a 0 sampled from the Inverse Gamma distribution a p sampled from a Gaussian with acceptance rejection sampling Sampling from Conditional Distributions 0 Use different sampling methods to sample from conditional distributions 0 Critical to sample 71htl hty1 t efficiently ht sampled from a Gaussian distribution with acceptance rejection 3 sampling a 0 sampled from the Inverse Gamma distribution a p sampled from a Gaussian with acceptance rejection sampling a p sampled from a Gaussian a A Filtering Application An Application Filtering 0 Assume H is known For ex 6 set as empirical mean of the Gibbs sampler output 0 Can use a particle filter approach to filter ht given data yl t 0 Generate samples from 7rht y1 t given a set of M particles h f1h j 1 from the distribution 71ht1 y1 tcl 0 These samples generated using importance sampling to draw from a Gaussian mixtu re 3 Results Sfmm e m and H gmgmm Hsmvmns am mum a am am nnnn nnnx am am am am am Figure Evolution of the Iterates Figure Histogram of the Iterates I w 294m Simulated vs Actual Volatility Actual vs Simulated Volatilities i Autucunelatiun ulthevulatilities Crass cunelatiun between slmul Correlation mun lZEIEI Mun mun mun 2n m Shi ed Lags mm mm EDD Figure Simulated vs Actual volatility I l Predicted vs Actual Volatility Actual vs Predicted Volatilities ActuaWulatilW Pvedictequlatilitv Volatility ht lm Sn i u ztin z u Time Steps Figure Actual vs Predicted volatility 7 I l Outline ConclusiOns Conclusions and Future Work 0 Conclusions 0 Future Work Conclusions and Future Work 0 Conclusions a Presented a MCMC based sampling method for a SV model a Derived a filtering application based on particle filtering 6 Compared MCMC with a Kalman filter based likelihood maximization method 0 Future Work

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