Bayesian Inference on a Stochastic Volatility model Using PMCMC methods

Bayesian Inference on a Stochastic Volatilitymodel Using PMCMC methods

Jonas Hallgren

August 1, 2011

OutlineFinancial 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

Financial Time series

2004 2006 2008 20100

2

4

6

8

10

12x 10

9 S&P500 Daily returns

Modeling

We want to model the price of an instrument in order to be ableto:

I Price options

I Evaluate future risks

I Predict future prices


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


LogreturnsSk = log( Sk

Sk−1)

−4 −2 0 2 40

50

100

150

200

250Histogram of 40 years S&P 500 logreturn

1980 2000−4

−2

0

2

4logreturns

year

−3 −2 −1 0 1 2 3

0.0010.003

0.010.020.050.100.250.500.750.900.950.980.99

0.9970.999

Data

Pro

babi

lity

Normal Probability Plot

Model proposal

Yk = βe12Xkuk =

√hkuk

Xk = αXk−1 + σwk = log h2k + b, b , −2 log β

(uk ,wk) ∼ N (0,Σ)

Σ =

[1 ρρ 1

]When ρ = 0, VYk = hk


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


Estimation

Bayesian inference, view the parameter as a random variable:

Observation:Y ∼ p(y |θ), θ ∈ Θ

Parameter posterior distribution:

π(θ|y) =p(y |θ)π(θ)´

Θ p(y |ξ)π(dξ)∝ p(y |θ)π(θ)

p(β|α, σ, ρ, x0:n, y0:n) ∝ p(β, α, σ, ρ, x0:n, y0:n)

= p(x0:n, y0:n|β, α, ρ, σ)p(β)p(. . .)

Prior selection

p(β|α, σ, ρ, x0:n, y0:n) ∝ 1

β2p(x0:n, y0:n| . . .)

p(α|β, σ, ρ, x0:n, y0:n) ∝ (α + 1)δ−1 (1− α)γ−1 p(x0:n, y0:n| . . .)

p(ρ|β, α, σ, x0:n, y0:n) ∝ 1

2p(x0:n, y0:n| . . .)

p(σ|β, α, ρ, x0:n, y0:n) ∝ 1

σ2σ2(t/2−1)e−

12σ2 S0p(x0:n, y0:n| . . .)

p(x , y) =

exp

− 12(1−ρ2)

( y

βe12 x

)2

+(

x−αxk−1σ

)2−2ρ

y(x−αxk−1)

σβe12 x

− 12x

|β|σ2π

√1−ρ2


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


Gibbs sampler

1. For the first iteration we choose ξ0 = {X (0)0:n , θ

(0)}, arbitrarily

2. For k = 1, 2, . . ., draw random samples

2.1 x(k)0:n ∼ pX (·|θ(k−1), y0:n)

2.2 θ(k)1 ∼ pX (·|x (k)

0:n , θ(k−1), y0:n)

...2.3 θ

(k)D ∼ pX (·|x (k)

0:n , θ(k)1 , . . . , θ

(k−1)D , y0:n)

New problem: How do we sample θ and x?

Metropolis-Hastings sampler

Choose θ0 arbitrarily then for k = 0, ...,N

1. Simulate θ∗ ∼ q(·, θk−1)

2. with probability

1 ∧ p(θ∗)q(θ∗, θk)

p(θk)q(θk , θ∗)

set θk+1 = θ∗, otherwise set θk+1 = θk .


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


SMC

φk , p(xk |y0:k)

Propose:

φk(ξ̃) =

´lk−1(ξ, ξ̃)φk−1(ξ)dξ´

φk−1(ξ)´lk−1(ξ, ξ̃)d ξ̃dξ

Our model

In our setting:

φk+1 = p(xk+1, y0:k+1)/p(y0:k+1)

∝ˆ

p(yk+1|xk+1, xk , y0:k)p(xk+1|xk , y0:k)p(xk , y0:k)dxk

=

ˆp(yk+1|xk:k+1)p(xk+1|xk)p(xk |y0:k)p(y0:k)dxk

=

ˆp(yk+1|xk:k+1)p(xk+1|xk)φkp(y0:k)dxk

=

ˆG (yk+1, xk:k+1)Q(xk+1|xk)φkp(y0:k)dxk

Summarized

Filter:

φk+1 =

´G (yk+1, xk:k+1)Q(xk+1, xk)φk|kdxk´ ´

G (yk+1, xk:k+1)Q(xk+1, xk)φk|kdxkdxk+1

Smoother:

φ0:k+1|k+1 =

´G (yk+1, xk:k+1)Q(xk+1, xk)φ0:k|kdx0:k´ ´

G (yk+1, xk:k+1)Q(xk+1, xk)φ0:k|kdx0:kdx0:k+1


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


Monte Carlo Integration

We want to evaluate:

µ(f ) =

ˆf (x)

dµ

dν(x)ν(dx)

We use the estimate:

N−1N∑i=1

f (ξi )dµ

dν(ξi )

a.s.−−−−→N→∞

µ(f )

Sequential Importance Sampling

1. Sampling: for k = 0, 1, . . .

2. Draw ξ̃1k+1, . . . , ξ̃

Nk+1|ξ̃1

0:k , . . . , ξ̃N0:k

2.1 Compute the importance weights

ωik+1 = ωi

kgk+1(ξ̃ik+1)

3. Resampling:

3.1 Draw N particles from the with the probability of success being

the normalized weightsωi

k+1∑Ns ωs

k+1

.

4. Update the trajectory: Copy the resampled particlestrajectories and replace the ones that we did not use.

Example

0 50 100 150 200−1

−0.5

0

0.5

1

1.5

Xk

k

Degeneracy

0 50 100 150 200−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

k

Xk

True XParticle trajectories

Recap

I Object: Model the price

I Need parameters

I Need X trajectories

Which we now have!

Yk = βe12Xkuk =

√hkuk

Xk = αXk−1 + σwk = log h2k + b, b , −2 log β

(uk ,wk) ∼ N (0,Σ)

Σ =

[1 ρρ 1

]

Gibbs sampler

1. For the first iteration we choose ξ0 = {X (0)0:n , θ

(0)}, arbitrarily

2. For k = 1, 2, . . ., draw random samples

2.1 x(k)0:n ∼ pX (·|θ(k−1), y0:n)

2.2 θ(k)1 ∼ pX (·|x (k)

0:n , θ(k−1), y0:n)

...2.3 θ

(k)D ∼ pX (·|x (k)

0:n , θ(k)1 , . . . , θ

(k−1)D , y0:n)


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


Particle MMHStep 1: initialization, i = 0

(a) set θ0 arbitrarily

(b) run a SMC algorithm targeting pθ(0)(x1:T , |y1:T ), sample our

first trajectory of particles ξ̃(0)1:T ∼ p̂θ(0)(·|y1:T ) and denote the

marginal likelihood by p̂θ0(y1:T )

Step 2: for iteration i ≥ 1,

(a) sample θ∗ ∼ q(·|θi−1)

(b) run a SMC algorithm targeting pθ∗(x1:T , |y1:T ), sample ourtrajectory of particles ξ̃∗1:T ∼ p̂θ∗(·|y1:T ) and denote the marginallikelihood by p̂θ∗(y1:T )

(c) with probability

1 ∧ p̂θ∗(y1:T )p(θ∗)

p̂θi−1(y1:T )pθi−1

q(θi−1|θ∗)q(θ∗|θi−1)

put θi = θ∗, ξ(i)1:T = ξ∗1:T and pθi (y1:T ) = pθ∗(y1:T )


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


UPPMMH

1. For t = 0, Choose τ1:C1:N arbitrarily (preferably through an

PMMH-sampler)

2. For t = 1, 2, ...,M

2.1 Simulation step, takes time but does not decrease efficiency asC increases: For γ = 1, 2, . . . ,C

2.1.1 Sample τγNt∼ rγ1:N·t(y , τ

γt·N)

2.2 Merging step, assumed to take zero time to compute: Samplea multidimensional, multinomial variable A1:C

t taking values in1, . . . ,C with equal probability.

2.3 for γ = 1, 2 . . . ,C

2.3.1 put τγ1:N·t = τAγt

1:N·t

3. Sample a multinomial variable Aoutt taking values in 1, . . . ,C

with equal probability and put τout1:K = τAoutt

1:K

PRPMMH

1. For t = 0, Choose τ1:C1:N arbitrarily (preferably through an

PMMH-sampler)

2. For t = 1, 2, ...,M

2.1 For γ = 1, 2, . . . ,C

2.1.1 Sample (ωγ , τγNt) ∼ rγ1:N·t(y , τ

γt·N)

2.2 Normalize weights and resample

2.2.1 For γ = 1, 2, . . . ,C put ω̄(γ) = ω(γ)∑j ω

(j)

2.2.2 Sample a multidimensional, multinomial variable A1:Ct taking

values in 1, . . . ,C with probability (ω̄(1), ω̄(2), . . . , ω̄(M))

2.3 for γ = 1, 2 . . . ,C

2.3.1 put τγ1:N·t = (τAγt

1:N·t , τAγt

Nt)

3. Sample a multinomial variable Aoutt taking values in 1, . . . ,C

with equal probability and put τout1:K = τAoutt

1:K

ImplementationX0 = randn(C,T); theta0 = randn(C,n_theta);

X(:,1) = X0; theta(:,1) = theta0;

for t = 2:M

% Simulationstep

parfor gamma = 2:C % Parallell for-loop

[X(gamma,Nt) theta(gamma,Nt) omega(gamma)] ...

= PMMH_SAMPLER(X(gamma,t), theta(gamma,t), N);

end

% Mergestep

A = randsample(1:C, C, true, omega/sum(omega))

X(:,1:N*t) = X(A,1:N*t);

theta(:,1:t) = theta(A,1:t);

end

A_out = randsample(1:C,1)

tau_out = [X(A_out,:); theta(A_out,:)];


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


Gibbs

0 0.5 10

1000

2000

3000

4000

α0.4 0.6 0.8 10

2000

4000

6000

β

−1 −0.5 0 0.5 10

1000

2000

3000

ρ0 0.1 0.2 0.3 0.4

0

2000

4000

6000

σ

PMMH

0 0.5 10

1000

2000

3000

4000

α0 0.5 1

0

2000

4000

6000

8000

10000

β

−1 −0.5 0 0.5 10

1000

2000

3000

ρ0 0.2 0.4 0.6 0.8

0

2000

4000

6000

σ


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


S&P500

0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

SimulatedReal

Risk measure comparison

VaR and ES answers two questions:

1. VaR: At least how large will a tail event that occurs withsome specific probability occur?

2. Given such a tail event, how large do we expect the loss tobe? 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


Model


Bayesian inference



Sequences

MC-integrals

Particle MCMC

Estimation



PMMH vs. Gibbs

Simulating data


Prediction results

Dataset ModelRMSE(10−3)

MAE(10−3)

Qr PPV

GBP/USD SVOL 11.706 8.417 0.1753 0.55

GBP/USD SVOLρ=0 11.714 8.420 ∼ ∼GBP/USD Long 11.714 8.421 −0.2821 ∼

BIDU SVOL 20.188 15.503 0.1302 0.53

BIDU SVOLρ=0 20.232 15.535 ∼ ∼BIDU Long 20.231 15.531 −0.3031 ∼

S&P500 SVOL 252.94 175.72 0.0825 0.52

S&P500 SVOLρ=0 252.93 175.78 ∼ ∼S&P500 Long 252.93 175.80 −0.4821 ∼S&P500 Longµ 252.91 175.72 ∼ ∼

XBC/USD SVOL 5.5762 3.1417 0.2621 0.35

XBC/USD SVOLρ=0 5.5908 3.1347 ∼ ∼XBC/USD Long 5.5920 3.1323 −0.0477 ∼

Conclusions

I PMMH is nice.

I Correlation is relevant in price behavior.

I Predict risk, perhaps not price.

The End

Questions?

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Bayesian Inference on a Stochastic Volatility model Using PMCMC methods

Technology