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Bayesian Inference on a Stochastic Volatility model Using PMCMC methods
1. Bayesian Inference on a Stochastic Volatility
model Using PMCMC methods
Jonas Hallgren
August 1, 2011
2. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
3. Financial Time series
9 S&P500 Daily returns
x 10
12
10
8
6
4
2
0
2004 2006 2008 2010
4. Modeling
We want to model the price of an instrument in order to be able
to:
Price options
Evaluate future risks
Predict future prices
5. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
6. Logreturns
Sk = log( SSk )
k−1
Histogram of 40 years S&P 500 logreturn logreturns
250 4
2
200 0
−2
150
−4
1980 2000
year
Normal Probability Plot
100
0.999
0.997
0.99
0.98
0.95
0.90
Probability
0.75
50 0.50
0.25
0.10
0.05
0.02
0.01
0.003
0.001
0
−4 −2 0 2 4 −3 −2 −1 0 1 2 3
Data
7. Model proposal
1
Yk = βe 2 Xk uk = hk uk
2
Xk = αXk−1 + σwk = log hk + b, b −2 log β
(uk , wk ) ∼ N (0, Σ)
1 ρ
Σ =
ρ 1
When ρ = 0, VYk = hk
8. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
11. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
12. Gibbs sampler
(0)
1. For the first iteration we choose ξ0 = {X0:n , θ(0) }, arbitrarily
2. For k = 1, 2, . . ., draw random samples
(k)
2.1 x0:n ∼ pX (·|θ(k−1) , y0:n )
(k) (k)
2.2 θ1 ∼ pX (·|x0:n , θ(k−1) , y0:n )
.
.
.
(k) (k) (k) (k−1)
2.3 θD ∼ pX (·|x0:n , θ1 , . . . , θD , y0:n )
New problem: How do we sample θ and x?
13. Metropolis-Hastings sampler
Choose θ0 arbitrarily then for k = 0, ..., N
1. Simulate θ∗ ∼ q(·, θk−1 )
2. with probability
p(θ∗ )q(θ∗ , θk )
1∧
p(θk )q(θk , θ∗ )
set θk+1 = θ∗ , otherwise set θk+1 = θk .
14. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
18. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
19. Monte Carlo Integration
We want to evaluate:
ˆ
dµ
µ(f ) = f (x) (x)ν(dx)
dν
We use the estimate:
N
dµ i a.s.
N −1 f (ξ i ) (ξ ) − − → µ(f )
−−
dν N→∞
i=1
20. Sequential Importance Sampling
1. Sampling: for k = 0, 1, . . .
˜1 ˜N ˜1 ˜N
2. Draw ξk+1 , . . . , ξk+1 |ξ0:k , . . . , ξ0:k
2.1 Compute the importance weights
i i ˜i
ωk+1 = ωk gk+1 (ξk+1 )
3. Resampling:
3.1 Draw N particles from the with the probability of success being
i
ωk+1
the normalized weights N s .
s ωk+1
4. Update the trajectory: Copy the resampled particles
trajectories and replace the ones that we did not use.
21. Example
1.5
1
0.5
k
X
0
−0.5
−1
0 50 100 150 200
k
23. Recap
Object: Model the price
Need parameters
Need X trajectories
Which we now have!
1
Yk = βe 2 Xk uk = hk uk
2
Xk = αXk−1 + σwk = log hk + b, b −2 log β
(uk , wk ) ∼ N (0, Σ)
1 ρ
Σ =
ρ 1
25. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
26. Particle MMH
Step 1: initialization, i = 0
(a) set θ0 arbitrarily
(b) run a SMC algorithm targeting pθ(0) (x1:T , |y1:T ), sample our
˜(0)
first trajectory of particles ξ1:T ∼ pθ(0) (·|y1:T ) and denote the
ˆ
marginal likelihood by pθ0 (y1:T )
ˆ
Step 2: for iteration i ≥ 1,
(a) sample θ∗ ∼ q(·|θi−1 )
(b) run a SMC algorithm targeting pθ∗ (x1:T , |y1:T ), sample our
˜∗
trajectory of particles ξ1:T ∼ pθ∗ (·|y1:T ) and denote the marginal
ˆ
likelihood by pθ∗ (y1:T )
ˆ
(c) with probability
pθ∗ (y1:T )p(θ∗ ) q(θi−1 |θ∗ )
ˆ
1∧
pθi−1 (y1:T )pθi−1 q(θ∗ |θi−1 )
ˆ
(i)
put θi = θ∗ , ξ1:T = ξ1:T and pθi (y1:T ) = pθ∗ (y1:T )
∗
27. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
28. UPPMMH
1:C
1. For t = 0, Choose τ1:N arbitrarily (preferably through an
PMMH-sampler)
2. For t = 1, 2, ..., M
2.1 Simulation step, takes time but does not decrease efficiency as
C increases: For γ = 1, 2, . . . , C
γ γ γ
2.1.1 Sample τNt ∼ r1:N·t (y , τt·N )
2.2 Merging step, assumed to take zero time to compute: Sample
a multidimensional, multinomial variable A1:C taking values in
t
1, . . . , C with equal probability.
2.3 for γ = 1, 2 . . . , C
γ
γ A
t
2.3.1 put τ1:N·t = τ1:N·t
3. Sample a multinomial variable Aout taking values in 1, . . . , C
t
out
out = τ At
with equal probability and put τ1:K 1:K
29. PRPMMH
1:C
1. For t = 0, Choose τ1:N arbitrarily (preferably through an
PMMH-sampler)
2. For t = 1, 2, ..., M
2.1 For γ = 1, 2, . . . , C
γ γ γ
2.1.1 Sample (ω γ , τNt ) ∼ r1:N·t (y , τt·N )
2.2 Normalize weights and resample
ω (γ)
2.2.1 For γ = 1, 2, . . . , C put ω (γ) =
¯ (j)
j ω
2.2.2 Sample a multidimensional, multinomial variable A1:C taking t
values in 1, . . . , C with probability (¯ (1) , ω (2) , . . . , ω (M) )
ω ¯ ¯
2.3 for γ = 1, 2 . . . , C
γ γ
γ A A
2.3.1 put τ1:N·t = (τ1:N·t , τNtt )
t
3. Sample a multinomial variable Aout taking values in 1, . . . , C
t
out Aout
t
with equal probability and put τ1:K = τ1:K
31. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
34. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison
36. Risk measure comparison
VaR and ES answers two questions:
1. VaR: At least how large will a tail event that occurs with
some specific probability occur?
2. Given such a tail event, how large do we expect the loss to
be? Expressed in mathematical terms: ES EY · IY <VaR
Model VaR ES
Empirical −0.2581 −0.3766
SVOL −0.2781 − 0.2772 −0.3561 − 0.3550
SVOLρ=0 −0.2735 − 0.2728 −0.3484 − 0.3474
37. Outline
Financial Time Series
Model
Parameter Estimation
Bayesian inference
Parameter simulation
Sequential Monte Carlo methods
Sequences
MC-integrals
Particle MCMC
Estimation
Parallel computation
Simulations and results
PMMH vs. Gibbs
Simulating data
Prediction comparison