Calibration of Stochastic Volatility Models using Particle - KTH

INTRODUCTION PARAMETER ESTIMATION Sequential Monte Carlo Methods Particle MCMC Simulations and Results

Calibration of Stochastic Volatility Modelsusing Particle Markov Chain Monte Carlo

Methods

Jonas Hallgren1

1Department of MathematicsKTH Royal Institute of Technology

Stockholm, Sweden

BFS 2012June 21, 2012

Sydney, Australia

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TABLE OF CONTENTSINTRODUCTION

Financial Times SeriesModel Proposal

PARAMETER ESTIMATION

Bayesian InferenceParameter Simulation

Sequential Monte Carlo MethodsFilters and SmoothersMonte Carlo IntegrationSequential Importance Sampling

Particle MCMCParticle Marginal Metropolis HastingsUnbiased Parallel Metropolis Hastings

Simulations and ResultsEstimationsPrediction

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LOGRETURNS

−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

0.010.020.050.100.25

0.50

0.750.900.950.980.99

0.9970.999

Data

Pro

babili

ty

Normal Probability Plot

Yk = log(

Sk

Sk−1

)3 / 29


MODEL PROPOSAL

Yk = βe12 Xkuk, (1)

Xk = αXk−1 + σwk, (2)(uk,wk) ∼ N (0,Σ), (3)

Σ =[

1 ρρ 1

](4)

4 / 29


BAYESIAN INFERENCE

Bayesian inference, view the parameter as a random variableObservation:

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, x0:n, y0:n) ∝ p(β, α, σ, ρ, x0:n, y0:n) (7)

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POSTERIOR FOR β

p(β|α, σ, ρ, x0:n, y0:n) ∝ p(β, α, σ, ρ, x0:n, y0:n) (8)= p(x0:n, y0:n|β, α, ρ, σ)p(β)p(α, ρ, σ) (9)∝ p(x0:n, y0:n|β, α, ρ, σ)p(β) (10)= p(xn, yn|β, α, ρ, σ, x0:n−1, y0:n−1) (11)× p(x0:n−1, y0:n−1|β, α, ρ, σ)p(β)

= p(β)p(y0, x0|β, α, ρ, σ) (12)

×n∏

k=1

p(xk, yk|β, α, ρ, σ, xk−1)

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POSTERIOR FOR β

p(x, y) =1

|β|σ2π√

1− ρ2(13)

× exp(−

[( y

βe12 x

)2+(x−αxk−1

σ

)2 − 2ρ y(x−αxk−1)

σβe12 x

]2(1− ρ2)

− 12

x)

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PRIOR SELECTION

p(β) =1β2 (14)

p(α) = (α+ 1)δ−1(1− α)γ−1 (15)

p(ρ) =12

(16)

p(σ) =1

σ2σ2(t/2−1)e−

12σ2 S̃0 (17)

Example:

p(β | α, σ, ρ, x0:n, y0:n) ∝ p(x0:n, y0:n | α, β, ρ, σ)p(β) (18)

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IDEA

I Simulate the parameters from the posterior distributions!

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THE GIBBS SAMPLER 1

1. For the first iteration choose ξ(0) = {x(0)0:n, θ

(0)} arbitrarily2. For k = 1, 2, . . . ,N, 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)

...θ(k)D ∼ pX(· | x(k)

0:n, θ(k)1 , θ

(k)2 , . . . , θ

(k−1)D , y0:n)

Now as N tend to infinity, the sequence {ξ(k)}Nk=0 will have pX

as its stationary distribution.New problem: How do we sample θ and x?

1Geman (1984)10 / 29


METROPOLIS-HASTINGS SAMPLER 2

Choose θ0 arbitrarily, then for k = 1, . . . ,N1. 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)

2Metropolis et. al. (1953), Hastings (1970)11 / 29


SEQUENTIAL MONTE CARLO (PARTICLE FILTER)

φk , p(xk | y0:k) (20)

Propose

φk+1(ξ̃) =

∫lk(ξ, ξ̃)φk(ξ) dξ∫

φk(ξ)∫

lk(ξ, ξ̃) dξ̃ dξ(21)

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OUR MODEL

In our setting:

φk+1 = p(xk+1, y0:k+1)/p(y0:k+1) (22)

∝∫

p(yk+1 | xk+1, xk, y0:k) (23)

× p(xk+1 | xk, y0:k)p(xk, y0:k) dxk

=

∫p(yk+1 | xk:k+1)p(xk+1 | xk)p(xk | y0:k)p(y0:k) dxk (24)

=

∫p(yk+1 | xk:k+1)p(xk+1 | xk)φkp(y0:k) dxk (25)

=

∫G(yk+1 | xk:k+1)Q(xk+1 | xk)φkp(y0:k) dxk (26)

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SUMMARIZED

Filter:

φk+1 =

∫G(yk+1 | xk:k+1)Q(xk+1 | xk)φk dxk∫ ∫

G(yk+1 | xk:k+1)Q(xk+1 | xk)φk dxk dxk+1(27)

Smoother:

φ0:k+1|k+1 = p(x0:k+1 | y0:k+1) (28)

=

∫G(yk+1 | xk:k+1)Q(xk+1 | xk)φ0:k|k dx0:k∫ ∫

G(yk+1 | xk:k+1)Q(xk+1 | xk)φ0:k|k dx0:k dx0:k+1

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MONTE CARLO INTEGRATION

We want to evaluate:

µ(f ) =

∫f (x)

dµdν

(x) ν(dx) (29)

We use the estimate:

N−1N∑

j=1

f (ξj)dµdν

(ξj)a.s.−−−−→

N→∞µ(f ) (30)

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SEQUENTIAL IMPORTANCE SAMPLING

1. Sampling: for k = 0, 1, . . .Draw ξ̃1

k+1, . . . , ξ̃Nk+1 | ξ̃

10:k, . . . , ξ̃

N0:k

1.1 Compute the importance weights

ωjk+1 = ω

jkgk+1(ξ̃

jk+1) (31)

2. Resampling: Draw N particles from the N-sizedpopulation where the probability of selecting particle j is

ωjk+1∑N

s ωsk+1

(32)

3. Update the trajectory: Copy the resampled particlestrajectories and replace the ones we discarded.

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EXAMPLE

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 X

Particle trajectories


RECAP

I Object: Model the priceI Need parameters

I Need X trajectories

Which we now have!


Xk = αXk−1 + σwk, (34)(uk,wk) ∼ N (0,Σ), (35)

Σ =[

1 ρρ 1

](36)

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THE GIBBS SAMPLER

1. For the first iteration choose ξ(0) = {x(0)0:n, θ

(0)} arbitrarily2. For k = 1, 2, . . . ,N, 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)

...θ(k)D ∼ pX(· | x(k)

0:n, θ(k)1 , θ

(k)2 , . . . , θ

(k−1)D , y0:n)

Now as N tend to infinity, the sequence {ξ(k)}Nk=0 will have pX

as its stationary distribution

19 / 29


INTRODUCTION

Financial Times SeriesModel Proposal

PARAMETER ESTIMATION

Bayesian InferenceParameter Simulation

Sequential Monte Carlo MethodsFilters and SmoothersMonte Carlo IntegrationSequential Importance Sampling

Particle MCMCParticle Marginal Metropolis HastingsUnbiased Parallel Metropolis Hastings

Simulations and ResultsEstimationsPrediction

20 / 29


METROPOLIS-HASTINGS SAMPLER 2

Choose θ0 arbitrarily, then for k = 1, . . . ,N1. Sample θ∗ ∼ q(· | θ(k))

2. With probability

1 ∧ p(θ∗)q(θ(k) | θ∗)p(θ(k))q(θ∗ | θ(k))

(19)


2Metropolis et. al. (1953), Hastings (1970)21 / 29

PARTICLE MARGINAL METROPOLIS HASTINGS 3

1. Initialization, k = 0

1.1 Set θ0 arbitrarily1.2 Run an 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

2. For iteration k ≥ 12.1 Sample θ∗ ∼ q(· | θk−1)2.2 Run an SMC algorithm targeting pθ∗(x1:T | y1:T), sample the

trajectory of particles as in 1.22.3 With probability

1 ∧ p̂θ∗(y1:T)p(θ∗)q(θk−1 | θ∗)

p̂θk−1(y1:T)p(θk−1)q(θ∗ | θk−1)(37)

put θk = θ∗, ξ(k)1:T = ξ∗1:T, and pθk (y1:T) = pθ∗

3Andrieu et. al. (2010)


UNBIASED PARALLEL METROPOLIS HASTINGS

Choose θm0 arbitrarily, then for k = 1, . . . ,N

1. For each of the C cores (where θ(k) = θmk ):

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)


2. Iterate through the C cores, take the first accepted sampleand put θm

k+γ equal to it. Put θmk:k+γ−1 = θm

k . Throw away allsamples after that. If no sample is accepted, θm

k:C = θmk

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UNBIASED PARALLEL METROPOLIS HASTINGS

I Roughly (Acceptance rate)−1 times faster as numbers ofcores grow large

I Easy to implement

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MODEL PROPOSAL


Xk = αXk−1 + σwk, (2)(uk,wk) ∼ N (0,Σ), (3)

Σ =[

1 ρρ 1

](4)

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ESTIMATES FOR GBP/USD DATA

0.8 0.85 0.9 0.95 10

20

40

60

α0.005 0.01 0.0150

20

40

60

80

β

−0.5 0 0.5 10

20

40

60

80

ρ0.1 0.2 0.3 0.4 0.50

20

40

60

σ

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SIMULATION OF 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

Simulated

Real

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PREDICTION

Dataset Model RMSE (10−3)

GBP/USD SVOL 11.706GBP/USD SVOLρ=0 11.714BIDU SVOL 20.188BIDU SVOLρ=0 20.232S&P500 SVOL 252.94S&P500 SVOLρ=0 252.93XBC/USD SVOL 5.5762XBC/USD SVOLρ=0 5.5920

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Any remark, question or suggestion is welcomed!

[email protected]://www.math.kth.se/∼jhallg/

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