Calibration of Stochastic Volatility Models using Particle Markov Chain Monte Carlo Methods

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1 Calibration of Stochastic Volatility Models using Particle Markov Chain Monte Carlo Methods Jonas Hallgren 1 1 Department of Mathematics KTH Royal Institute of Technology Stockholm, Sweden BFS 2012 June 21, 2012 Sydney, Australia 1 / 29

2 TABLE OF CONTENTS INTRODUCTION Financial Times Series Model Proposal PARAMETER ESTIMATION Bayesian Inference Parameter Simulation Sequential Monte Carlo Methods Filters and Smoothers Monte Carlo Integration Sequential Importance Sampling Particle MCMC Particle Marginal Metropolis Hastings Unbiased Parallel Metropolis Hastings Simulations and Results Estimations Prediction 2 / 29

3 LOGRETURNS 250 Histogram of 40 years S&P 500 logreturn 4 logreturns year Normal Probability Plot Probability Data ( ) Sk Y k = log S k 1 3 / 29

4 MODEL PROPOSAL Y k = βe 1 2 X k u k, (1) X k = αx k 1 + σw k, (2) (u k, w k ) N (0, Σ), ] (3) Σ = (4) [ 1 ρ ρ 1 4 / 29

5 BAYESIAN INFERENCE Bayesian inference, view the parameter as a random variable Observation: Y θ = θ p(y θ = θ), θ p(θ) (5) Parameter posterior distribution: p(θ y) = p(y θ)p(θ) Θ p(y θ )p(dθ p(y θ)p(θ) (6) ) Example: p(β α, σ, ρ, x 0:n, y 0:n ) p(β, α, σ, ρ, x 0:n, y 0:n ) (7) 5 / 29

6 POSTERIOR FOR β p(β α, σ, ρ, x 0:n, y 0:n ) p(β, α, σ, ρ, x 0:n, y 0:n ) (8) = p(x 0:n, y 0:n β, α, ρ, σ)p(β)p(α, ρ, σ) (9) p(x 0:n, y 0:n β, α, ρ, σ)p(β) (10) = p(x n, y n β, α, ρ, σ, x 0:n 1, y 0:n 1 ) (11) p(x 0:n 1, y 0:n 1 β, α, ρ, σ)p(β) = p(β)p(y 0, x 0 β, α, ρ, σ) (12) n p(x k, y k β, α, ρ, σ, x k 1 ) k=1 6 / 29

7 POSTERIOR FOR β 1 p(x, y) = β σ2π (13) 1 ρ 2 [( ( y ) 2 ( x αxk 1 ) 2 y(x αx + βe 1 2 exp x σ 2ρ k 1 )] σβe 1 2 x 2(1 ρ 2 1 ) ) 2 x 7 / 29

8 PRIOR SELECTION p(β) = 1 β 2 (14) p(α) = (α + 1) δ 1 (1 α) γ 1 (15) Example: p(ρ) = 1 (16) 2 1 p(σ) = σ 2 σ 2(t/2 1) e 1 2σ 2 S 0 (17) p(β α, σ, ρ, x 0:n, y 0:n ) p(x 0:n, y 0:n α, β, ρ, σ)p(β) (18) 8 / 29

9 IDEA Simulate the parameters from the posterior distributions! 9 / 29

10 THE GIBBS SAMPLER 1 1. For the first iteration choose ξ (0) = {x (0) 0:n, θ(0) } arbitrarily 2. For k = 1, 2,..., N, draw random samples 2.1 x (k) 0:n p X( θ (k 1), y 0:n ) 2.2 θ (k) 1 p X ( x (k) 0:n, θ(k 1), y 0:n ) θ (k) D. p X( x (k) 0:n, θ(k) 1, θ(k) 2,..., θ(k 1) D, y 0:n ) Now as N tend to infinity, the sequence {ξ (k) } N k=0 will have p X as its stationary distribution. New problem: How do we sample θ and x? 1 Geman (1984) 10 / 29

11 METROPOLIS-HASTINGS SAMPLER 2 Choose θ 0 arbitrarily, then for k = 1,..., N 1. Sample θ q( θ (k) ) 2. With probability 1 p(θ )q(θ (k) θ ) p(θ (k) )q(θ θ (k) ) (19) set θ (k+1) = θ, otherwise set θ (k+1) = θ (k) 2 Metropolis et. al. (1953), Hastings (1970) 11 / 29

12 SEQUENTIAL MONTE CARLO (PARTICLE FILTER) Propose φ k p(x k y 0:k ) (20) φ k+1 ( ξ) = lk (ξ, ξ)φ k (ξ) dξ φk (ξ) l k (ξ, ξ) d ξ dξ (21) 12 / 29

13 OUR MODEL In our setting: φ k+1 = p(x k+1, y 0:k+1 )/p(y 0:k+1 ) (22) p(y k+1 x k+1, x k, y 0:k ) (23) p(x k+1 x k, y 0:k )p(x k, y 0:k ) dx k = p(y k+1 x k:k+1 )p(x k+1 x k )p(x k y 0:k )p(y 0:k ) dx k (24) = p(y k+1 x k:k+1 )p(x k+1 x k )φ k p(y 0:k ) dx k (25) = G(y k+1 x k:k+1 )Q(x k+1 x k )φ k p(y 0:k ) dx k (26) 13 / 29

14 SUMMARIZED Filter: φ k+1 = G(yk+1 x k:k+1 )Q(x k+1 x k )φ k dx k G(yk+1 x k:k+1 )Q(x k+1 x k )φ k dx k dx k+1 (27) Smoother: φ 0:k+1 k+1 = p(x 0:k+1 y 0:k+1 ) (28) = G(yk+1 x k:k+1 )Q(x k+1 x k )φ 0:k k dx 0:k G(yk+1 x k:k+1 )Q(x k+1 x k )φ 0:k k dx 0:k dx 0:k+1 14 / 29

15 MONTE CARLO INTEGRATION We want to evaluate: µ(f ) = f (x) dµ (x) ν(dx) (29) dν We use the estimate: N 1 N j=1 f (ξ j ) dµ dν (ξj ) a.s. N µ(f ) (30) 15 / 29

16 SEQUENTIAL IMPORTANCE SAMPLING 1. Sampling: for k = 0, 1,... Draw ξ 1 k+1,..., ξ N k+1 ξ 1 0:k,..., ξ N 0:k 1.1 Compute the importance weights ω j k+1 = ωj k g k+1( ξ j k+1 ) (31) 2. Resampling: Draw N particles from the N-sized population where the probability of selecting particle j is ω j k+1 N s ωs k+1 (32) 3. Update the trajectory: Copy the resampled particles trajectories and replace the ones we discarded. 16 / 29

17 EXAMPLE True X Particle trajectories X k k

18 RECAP Object: Model the price Need parameters Need X trajectories Which we now have! Y k = βe 1 2 X k u k, (33) X k = αx k 1 + σw k, (34) (u k, w k ) N (0, Σ), ] (35) Σ = (36) [ 1 ρ ρ 1 18 / 29

19 THE GIBBS SAMPLER 1. For the first iteration choose ξ (0) = {x (0) 0:n, θ(0) } arbitrarily 2. For k = 1, 2,..., N, draw random samples 2.1 x (k) 0:n p X( θ (k 1), y 0:n ) 2.2 θ (k) 1 p X ( x (k) 0:n, θ(k 1), y 0:n ) θ (k) D. p X( x (k) 0:n, θ(k) 1, θ(k) 2,..., θ(k 1) D, y 0:n ) Now as N tend to infinity, the sequence {ξ (k) } N k=0 will have p X as its stationary distribution 19 / 29

20 INTRODUCTION Financial Times Series Model Proposal PARAMETER ESTIMATION Bayesian Inference Parameter Simulation Sequential Monte Carlo Methods Filters and Smoothers Monte Carlo Integration Sequential Importance Sampling Particle MCMC Particle Marginal Metropolis Hastings Unbiased Parallel Metropolis Hastings Simulations and Results Estimations Prediction 20 / 29

21 METROPOLIS-HASTINGS SAMPLER 2 Choose θ 0 arbitrarily, then for k = 1,..., N 1. Sample θ q( θ (k) ) 2. With probability 1 p(θ )q(θ (k) θ ) p(θ (k) )q(θ θ (k) ) (19) set θ (k+1) = θ, otherwise set θ (k+1) = θ (k) 2 Metropolis et. al. (1953), Hastings (1970) 21 / 29

22 PARTICLE MARGINAL METROPOLIS HASTINGS 3 1. Initialization, k = Set θ 0 arbitrarily 1.2 Run an SMC algorithm targeting p θ (0)(x 1:T y 1:T ), sample our (0) first trajectory of particles ξ 1:T ˆp θ (0)( y 1:T) and denote the marginal likelihood by ˆp θ0 2. For iteration k Sample θ q( θ k 1 ) 2.2 Run an SMC algorithm targeting p θ (x 1:T y 1:T ), sample the trajectory of particles as in With probability 1 ˆp θ (y 1:T)p(θ )q(θ k 1 θ ) ˆp θk 1 (y 1:T )p(θ k 1 )q(θ θ k 1 ) (37) put θ k = θ, ξ (k) 1:T = ξ 1:T, and p θ k (y 1:T ) = p θ 3 Andrieu et. al. (2010)

23 UNBIASED PARALLEL METROPOLIS HASTINGS Choose θ0 m arbitrarily, then for k = 1,..., N 1. For each of the C cores (where θ (k) = θk m): 1.1 Sample θ q( θ (k) ) 1.2 With probability 1 p(θ )q(θ (k) θ ) p(θ (k) )q(θ θ (k) ) (38) set θ (k+1) = θ, otherwise set θ (k+1) = θ (k) 2. Iterate through the C cores, take the first accepted sample and put θk+γ m equal to it. Put θm k:k+γ 1 = θm k. Throw away all samples after that. If no sample is accepted, θk:c m = θm k 23 / 29

24 UNBIASED PARALLEL METROPOLIS HASTINGS Roughly (Acceptance rate) 1 times faster as numbers of cores grow large Easy to implement 24 / 29

25 MODEL PROPOSAL Y k = βe 1 2 X k u k, (1) X k = αx k 1 + σw k, (2) (u k, w k ) N (0, Σ), ] (3) Σ = (4) [ 1 ρ ρ 1 25 / 29

26 ESTIMATES FOR GBP/USD DATA α ρ β σ 26 / 29

27 SIMULATION OF S&P Simulated Real / 29

28 PREDICTION Dataset Model RMSE (10 3 ) GBP/USD SVOL GBP/USD SVOL ρ= BIDU SVOL BIDU SVOL ρ= S&P500 SVOL S&P500 SVOL ρ= XBC/USD SVOL XBC/USD SVOL ρ= / 29

29 Any remark, question or suggestion is welcomed! jhallg/ 29 / 29

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Exercises Tutorial at ICASSP 216 Learning Nonlinear Dynamical Models Using Particle Filters Andreas Svensson, Johan Dahlin and Thomas B. Schön March 18, 216 Good luck! 1 [Bootstrap particle filter for