Filtering Stochastic Volatility Models with Intractable Likelihoods

Transcription

1 Filtering Stochastic Volatility Models with Intractable Likelihoods Katherine B. Ensor Professor of Statistics and Director Center for Computational Finance and Economic Systems Rice University LEAD-Author: Emilian Vankov Collaborator: Michele Guindani SAMSI Games and Reliability, May 20, 2016

2 Outline Intro Stochastic Volatility Model with α stable noise. Filtering and Estimation Contribution - Volatility (SSM) Filtering SMC+ABC Simulation and Application In Summary

3 Financial Markets and Volatility c Source:

4 Typical Financial Data: Dow Jones Industrial Average Prices Returns Squared Returns

5 What is Volatility? Volatility is a measure of variation for price/returns of financial instruments over time. Volatility is not directly observed. Some common features of volatility and prices/returns include: Returns are serially uncorrelated but a dependent process. Mean reversion in volatility to a long-run level. Volatility clustering and persistence results in high autocorrelation. Generally a negative correlation between prices and volatility. Returns are asymmetric as a function of market increases or decreases. Since volatility is not directly observed how do we measure or model volatility?

6 VIX: Implied Volatility Index VIX [ / ] Last 9500 Bollinger Bands (20,2) [Upper/Lower]: / day expected volatility, based on the Black Scholes model, of S&P 500 Index Volume (100,000s): 814,500 Moving Average Convergence Divergence (12,26,9): MACD: Signal: Jun Jun Jun Jun Jun Jul Jul Jul Jul Aug Aug

7 Modeling Volatility from Observed Returns Traditional Time Series Models for Volatility: Conditional Heteroskedastic Volatility Models ARCH (Engle 1982), GARCH (Bolerslev 1986), EGARCH (Nelson 1991), etc. (in R see rugarch and rmgarch, review article Ensor and Koev (2015), realtime volatility and systemic risk estimates NYU Stern Vlab) Stochastic Volatility Model (SVM) - volatility is modeled as a continuous or discrete stochastic process Taylor (1982) - discrete time SVM Hull and White (1998) - continuous time SVM In R see stochvol - relies on "anxilarity-sufficiency interweiving strategy (ASIS)", Kastner and Frühwirth-Schnatter (2014), Yu and Meng (2011).

8 NYU Stern Vlab IBM V-Lab Volatility Institute NYU Stern Documentation» Analysis List» Display: Basic Language: Volatility Analysis GJR-GARCH Type ticker or search (Wildcard=%) Go INTERNATIONAL BUSINESS MACHINES CORP GJR-GARCH VOLATILITY GRAPH Models Assets Volatility Prediction for Friday, May 20, 2016: 22.08% (+0.24) COMPARE SUB PLOT LINE STYLE KEY POSITION COPY GRAPH Date Range: from to Window: 6m 1y 2y 5y 10y all Other International Business Machines Corp Analyses GARCH EGARCH APARCH AGARCH Spline-GARCH Zero Slope Spline-GARCH GAS-GARCH Student T MEM Asy. MEM Asy. Power MEM Documentation Volatility Analysis GJR-GARCH

9 Stochastic Volatility Models (SVM) The focus of our analysis is the Stochastic Volatility Model. y t = exp(x t /2)v t f (y t x t ) x t = µ + φx t 1 + σu t q(x t x t 1 ) x 0 N ( µ/(1 φ), σ 2 /(1 φ 2 ) ) t = 1,...,T y t - returns at time t x t - log-volatility at time t θ = {µ,φ,σ} are parameters v t and u t are noise terms - generally assumed i.i.d. N(0,1) State-Space Model (SSM) where, y t is the observation equation, and x t is the state-equation, i.e. the log volatility is an unobserved latent process.

10 SVM and the Stable Distribution Are u t and v t i.i.d. N(0,1) in practice? The noise of the volatility equation u t, maybe; the noise of the observation equation, v t, probably not. As early as the 1960 s the assumption of normality was questioned as stock returns are often skewed and heavy-tailed.(mandelbrot (1963); Fama (1965)) Let s assume that errors for the return (or observation) equation v t S(α,β,0,1), are distributed as an asymmetric α-stable distribution. y t = exp(x t /2)v t x t = µ + φx t 1 + σu t x 0 N ( µ/(1 φ), σ 2 /(1 φ 2 ) )

11 The α - Stable Distribution: S(α,β,δ,γ) Parametric distribution that captures skewness and heavy tails. Parameters: Stability α (0,2]; Skewness β [ 1,1]; Location δ (, ); Scale γ (0, ) When α = 2 N(δ,2 γ) When α = 1 and β = 0 Cauchy(δ,γ) Characteristic function: { exp { γ α t α [ 1 iβ(tan πα φ(t) = 2 )(sign(t))] + iδt }, if α 1 exp { γ t [ 1 + iβ 2 π (sign(t))ln(t)] + iδt }, if α = 1 The p.d.f is not available in closed form. However, simulating random numbers is possible.

12 Simulation of α-stable SVM Returns from Normal SVM Returns from Stable SVM Returns Returns Time Time

13 Stochastic Volatility Models Filtering and Estimation for SVM y t = exp(x t /2)v t (Observation Eq) x t = µ + φx t 1 + σu t (State Eq) x 0 N ( µ/(1 φ), σ 2 /(1 φ 2 ) ) t = 1,...,T Quantities of interest: Distribution of volatility from time 0 to T, given returns and parameters: p(x 0:T y 1:T,θ) - assumes θ is known Joint distribution of volatility from time 0 to T and parameters, given returns: p(x 0:T,θ y 1:T ) - state filtering and parameter estimation

14 Stochastic Volatility Models - Some History on Estimation Normal SVM Filtering of p(x 0:T y 1:T,θ) Pitt and Shephard (1999) SMC/Particle filter. Estimation of p(x 0:T,θ y 1:T ) Jacquier et al. (1994), Kim et al. (1998) - MCMC methods Harvey et al. (1994) - Kalman filter + quasi-mle Symmetric α-stable SVM; β 0 Filtering of p(x 0:T y 1:T,θ) Jasra et al. (2012) - approximate Bayesian computation (ABC) based SMC Estimation of p(x 0:T,θ y 1:T ) Jasra et al. (2013), Barthelme and Chopin (2014) - ABC based MCMC

15 Our Stochastic Volatility Contributions ASYMMETRIC α-stable SVM Filtering of p(x 0:T y 1:T,θ) ABC based Auxiliary Particle Filter (APF-ABC) The focus of this talk. Estimation of p(x 0:T,θ y 1:T ) Posterior distributions for all parameters, including the asymmetry parameter β. Results - submitted to Bayesian Analysis.

16 State Filtering with Known Parameters Sequential Monte Carlo State filtering with KNOWN parameter via Sequential Monte Carlo (SMC) (density of interest): p(x 0:t y 1:t,θ) p(x 0:t y 1:t,θ) = p(x 0:t 1 y 1:t 1,θ) q(x t x t 1,θ)f (y t x t,θ) p(y t y 1:t 1,θ) p(x 0:t 1 y 1:t 1,θ) is the distribution of interest at time t-1 Assume we have a sample of N particles that we can use to estimate the filter distribution at time t 1; i.e. { ˆX (i) 0:t 1,i = 1,...,N} ˆp(x 0:t 1 y 1:t 1,θ)

17 Sequential Monte Carlo (SMC) How do we go from time t 1 to time t or ˆp(x 0:t 1 y 1:t 1,θ) to ˆp(x 0:t y 1:t,θ)? 1. Sample N particles { X (i) t,i = 1,...,N} to get {( ˆX (i) 0:t 1, X (i) t ),i = 1,...,N} g(x 0:t y 1:t,θ), where g( ) is the importance distribution 2. Weight the N samples based on their importance WEIGHTS are KEY w (i) t p(x(i) 0:t y 1:t) g(x (i) 0:t y 1:t) f (y t x (i) t )q(x (i) t x (i) t 1 ) g(x (i) t x (i) 0:t 1,y 1:t) ˆp(x 0:t y 1:t,θ) = ˆp(y 1:T θ) = T t=1 N i=1 ( 1 N w (i) t δ ˆX (i) 0:t N i=1 w (i) t w (i) t 1 )

18 SMC / APF - Obtaining Filtered Estimates c Consider all random variables involved (X0:T 1:N,A1:N 0:T 1,θ) (X 0:T,A 0:T 1,θ) Source: Modified Figure from Andrieu et al. (2010)

19 Sequential Monte Carlo (SMC) How do we choose the importance distribution g(x (i) t x (i) 0:t 1,y 1:t)? Typically, minimize the variance of the importance weights by setting g(x t x 0:t 1,y 1:t ) = p(x t x t 1,y t ) Problems: Not available in closed form for more complicated models In most applications g(x t x 0:t 1,y 1:t ) q(x t x t 1 ) in which case the ratio simplifies and thus NO DATA IS INVOLVED - i.e. proposal is chosen blindly without looking at the data; dangerous if the data is informative (e.g. outliers)

20 ABC based Sequential Monte Carlo (SMC+ ABC) Can we use the method discussed above directly with the α-stable SVM? Maybe, but the weights depend on the pdf of the α-stable distribution, f (y t x t ), which is not available Solution - Approximate Bayesian Computation Simulate auxiliary data, y s t, from the model Update the weights via a kernel density K ε (y s t,y t ) The SMC-ABC has the marginal target p ε (x 0:T y 1:T,θ) The bias of the SMC-ABC estimate goes to 0 as ε 0

21 ABC based Sequential Monte Carlo (SMC + ABC) Jasra et al. (2012) advocate the use of a Uniform kernel and a proposal g(x t x 0:t 1,y 1:t ) = q(x t x t 1 ) Weights are updated according to: w (i) t = I Qε,y t (ys t )w (i) t 1 Q ε,yt = {y s t : ρ(y s t,y t ) < ε} (use simulated values within a certain radius of the observed values). Drawbacks: Binary weights - either 0 or 1; algorithm can collapse Still sampling from the transition density q(x t x t 1 ) ignoring the data Partial solution by Jasra et al. (2014) At each step resample the particles until N-1 of them have NON-ZERO weight Solves one problem, but induces another, namely the amount of time spent at each iteration is a now a random variable.

22 Auxiliary Particle Filter (APF) and ABC Our Solution Using ideas from Pitt and Shephard (1999) and Carpenter et al. (1999) we propose: Use an auxiliary density to pre-weight the samples before proposing based on the data g(x t x 0:t 1,y 1:t ) = ĝ(y t ξ (x t 1 ))q(x t x t 1 ) Use a Gaussian kernel to evaluate the simulation based weights relative to the observed data (ABC); the variability of the kernel controls the "closeness" required Benefits: Continuous weights the algorithm will not collapse Create better proposals by considering the data No significant increase in computational times

23 Auxiliary Particle Filter (APF) and ABC Example of choices for the auxiliary distribution ĝ(y t ξ (x t 1 )): f (y t E[x t x t 1 ]) problem if there are outliers or heavy tails and not available in our case. Use a t-density with heavy tails centered at E[x t x t 1 ] In general choose densities that are more diffuse than the likelihood and transition densities

24 Auxiliary Particle Filter and ABC Putting our contributions together we get the following algorithm: Initialization t = 0 APF - ABC Sample the initial particles (x (1) 0,..., x(n) 0 ) from the initial distribution. For t = 1,...,T 1. Sample the labels (auxiliary variables) based weights that incorporate information about the data. (Key of APF) 2. Resample the particles using the new labels. 3. Sample a new particle from the transition density of the state equation conditioned on the resampled particles 4. Update weights to account for the discrepancy between the proposal density and the target density based on ABC (instead of likelihood) 5. Normalize weights. 6. Use the new weights and particles to construct empirical density of the latent log volatility.

25 How well does the APF+ABC Method Work Reminder - Stochastic Volatility Models (SVM) parameterization y t = exp(x t /2)v t v t S(α,β,δ,γ) x t = µ + φx t 1 + σu t u t N(0,1) ( ) x 0 N µ/(1 φ), σ 2 /(1 φ 2 )

26 How well does this work? Simulation Study - Volatility Filtering Simulate data for t = 1,..., 500 days from the α-stable SVM Parameter values: α = 1.9, β = 0.1, µ = 0.2, φ = 0.95, σ = 0.7 Estimate the unobserved volatility with N = 5000 particles Auxillary particle distribution ĝ(y t ξ (x (i) t )) is chosen to be a t-distribution with 2 degrees of freedom centered at E[x t x t 1 ] K ε (y s t ) is given by N(y t,ε = 0.25)

27 Simulation APF ABC Filtered Values 0 log volatility 4 8 Volatility Filtered True time Figure: APF-ABC mean log-volatility estimate and the true state volatility. The filtered values are averaged over 100 simulations.

28 Simulation - Our Improvement Error Distributions RMSE ABC APF ABC SMC Figure: Box plots of the root-mean-square error based on 100 simulations.

29 We can now obtain the latent volatility process with known parameters. How do we then obtain the posterior distribution of the parameters? Use SMC/APF+ABC with EM algorithm to obtain MLE s. Use SMC/APF+ABC with MCMC to obtain posterior distribution Single Filter Particle Metropolis within Gibbs (SFPMwG).

30 Modeling Exchange Rate Volatility Series from Literature Exchange rate data for Brazilian Real vs United States Dollar The data spans the period Lombardi and Calzolari (2009) use an indirect estimation approach The parameters for this data are estimated to be: α = , µ = , φ = , σ = Assumes σ y = 1, β = 1, δ = 0, γ = 1 In January 1999, the Central Bank of Brazil was forced to abandon a managed depreciation regime by a speculative attack triggered by a 90-days debt moratorium announced by a provincial governor

31 BRL vs USD plots Series from Literature BRLvsUSD BRLvsUSD Exchange Rate Returns time time Figure: Exchange rate data for the period of the Brazilian Real vs the US Dollar. Real exchange rate(left) and exchange rate returns(right)

32 Filtered Exchange Rate Volatility Series from Literature BRLvsUSD log-vol h.apf_abc x.abc_ukern time Figure: Filtered log-volatility for the exchange rate returns based on ABC-SMC(red dashed line) and ABC-APF(black line)

33 Propane Weekly Spot Prices Propane - clean fuel produced from natural gas or crude oil Uses in the US - heating, agricultural and industrial sectors Some key drivers for price changes Weather, inventory infrastructure and transportation Propane weekly spot prices for Mount Belvieu, TX for 10/01/2007-9/28/2014 Data Source: Energy Information Administration (EIA) Calculate the demeaned weekly spot returns ( y t = 100 log(p t /P t 1 ) 1 T T i=1 log(p i /P i 1 ) )

34 Propane Spot Returns Distribution Density Mean SD Skewness Kurtosis Returns Returns 0 Figure: Histogram (left) and boxplot(right) for the demeaned weekly spot returns for propane 10/01/2007-9/28/2014, Mount Belvieu TX. Jarque-Bera test for Gaussian returns; reject H 0 (p-value < 0.01)

35 Posterior Distribution Summary Table: Summary of the posterior distribution for all parameters of the α-stable stochastic volatility model applied to weekly spot returns for 10/01/2007-9/28/2014. We present the posterior mean, standard deviation and 95% credible interval. α β σ µ φ Mean SDev % CI (1.76, 1.98) ( 0.98, 0.22) (0.25, 0.55) (1.62, 2.64) (0.80, 0.96) The posterior probability that returns are left skewed is given by: P(β < 0) = 0.948

36 Propane Spot Returns Distribution Weekly Absolute Returns Year Weekly Volatility Year Figure: Absolute returns (left) and filtered volatility with 95% CI (right) from α-stable SVM for the demeaned weekly spot returns for propane 10/01/2007-9/28/2014, Mount Belvieu TX.

37 Substituting the conditional mean of the posterior distribution for the parameters, the model for Mean Adjusted Propane Returns is given by: y t = exp(x t /2)v t v t S(1.88, 0.59,0,1) x t = x t u t u t N(0,1) x 0 N ( 2.19/(1 0.89), /( ) )

38 Forecast Distribution of Returns The model is now fully characterized. The forecast distribution of returns conditional on parameters set to their posterior means is given by: VaR 05 =-46; CVaR 05 =-68

39 Defined Stochastic Volatility Model which allows returns to follow heavy-tailed and skewed distributions. Developed APF-ABC to obtain filtered values of the latent state in asymmetric α-stable SVMs (and general SSM). The APF-ABC algorithm is applicable to any state-space model with intractable likelihoods. Compared SMC-ABC and APF-ABC and applied APF-ABC to exchange rate series from the literature. Obtained the posterior distribution for the latent volatility and parameters (methodology not discussed) for asymmetric α-stable SVMs for weekly propane spot prices.

40 THANK YOU FOR YOUR ATTENTION Contact info: For detail and references see Vankov and Ensor arxiv: [stat.co] Vankov 2015 (dissertation) Vankov, Guindani & Ensor 2016 (under review)