MCMC-Based Sampling Method for Stochastic Volatility Models, Study notes of Electrical and Electronics Engineering

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Adaptive Algorithms for Tracking Volatility

Dinesh Krithivasan

EECS 659: Adaptive Signal Processing

Presentation Outline

1 Motivation for SV Models

2 The Discrete SV Model

3 Gibbs Sampling

4 A Filtering Application

5 Results

6 Conclusions

Derivatives Pricing

Derivatives - Financial instruments whose values derive from underlying

assets.

Examples include options, forwards, swaps etc.

Asset prices modeled as a stochastic process.

Problem: Price derivatives in a fair and arbitrage-free manner.

Black Scholes Model

Asset price St modeled as Brownian motion:

dSt St

= μ t dt + σ t dWt.

μ t - drift of the stock price, σ t - volatility, variance of the stock price.

σ t crucial in pricing any derivative of St. Larger σ t means more risk

involved and hence investors expect greater returns.

Black Scholes model: Gives optimal price C (St , t) of an European

call/put option.

∂ C

∂ t

σ

2

S

2

C

∂ S^2

+ rS

∂ C

∂ S

= rC

Critical simplifying assumption: σ t = σ is constant over time.

Stochastic Volatility Models

In real financial markets, volatility changes over time.

Stochastic Volatility (SV) models view σ t as a stochastic process.

Generic SV model

dSt = μ St dt +

p

σ t St dWt

d σ t = α St ,t dt + β St ,t dBt

α St ,t , β St ,t - specific to the SV model used.

Objective: Tune the parameters based on the observations.

Outline

1 Motivation for SV Models

2 The Discrete SV Model

3 Gibbs Sampling

4 A Filtering Application

5 Results

6 Conclusions

Discrete SV Model

Model studied here:

yt = e

ht / 2

² t ² t ∼ N ( 0 , 1 )i.i.d

ht+ 1 = φ ht + ( 1 − φ ) μ + σηη t , | φ | ≤ 1 , η t ∼ N ( 0 , 1 )i.i.d

h 1 ∼ N ( μ ,

σ

2 η

1 − φ^2

yt - observations

ht - underlying volatilities

² t - observation noise

μ - Instantaneous volatility

φ - Persistence of volatility

σ

2 η - volatility of^ ht

Objective: Estimate θ = ( φ , μ , σ

2

η )^ and^ h^1 :t^ from^ y^1 :t^.

Discrete SV Model

Model studied here:

yt = e

ht / 2

² t ² t ∼ N ( 0 , 1 )i.i.d

ht+ 1 = φ ht + ( 1 − φ ) μ + σηη t , | φ | ≤ 1 , η t ∼ N ( 0 , 1 )i.i.d

h 1 ∼ N ( μ ,

σ

2 η

1 − φ^2

yt - observations

ht - underlying volatilities

² t - observation noise

μ - Instantaneous volatility

φ - Persistence of volatility

σ

2 η - volatility of^ ht

Objective: Estimate θ = ( φ , μ , σ

2

η )^ and^ h^1 :t^ from^ y^1 :t^.

Discrete SV Model

Model studied here:

yt = e

ht / 2

² t ² t ∼ N ( 0 , 1 )i.i.d

ht+ 1 = φ ht + ( 1 − φ ) μ + σηη t , | φ | ≤ 1 , η t ∼ N ( 0 , 1 )i.i.d

h 1 ∼ N ( μ ,

σ

2 η

1 − φ^2

yt - observations

ht - underlying volatilities

² t - observation noise

μ - Instantaneous volatility

φ - Persistence of volatility

σ

2 η - volatility of^ ht

Objective: Estimate θ = ( φ , μ , σ

2

η )^ and^ h^1 :t^ from^ y^1 :t^.

Discrete SV Model contd.

yt = e

ht / 2

² t ² t ∼ N ( 0 , 1 )i.i.d

ht+ 1 = φ ht + ( 1 − φ ) μ + σηη t , | φ | ≤ 1 , η t ∼ N ( 0 , 1 )i.i.d

h 1 ∼ N ( μ ,

σ

2 η

1 − φ^2

State space model with ht as the hidden state.

State update is linear and Markovian.

Observation yt is a non-linear function of the state ht.

MCMC Sampling

Obtain samples from a pdf by repeatedly sampling from a Markov chain.

The Markov chain’s invariant distribution is the target density of

interest.

In our case, we would sample from the posterior π (h 1 :t , θ | y 1 :t ).

With enough samples, can compute features of π (h 1 :t , θ | y 1 :t ).

Gibbs Sampling

Special case of a Monte Carlo Markov Chain sampling.

Suppose we need to sample from p(x, y )

Marginalizing the joint density is hard.

Easy to sample from pX |Y and pY |X.

Gibbs sampler proceeds as below

Pick y 0 from some distribution that has same support as pY (y ).

Pick x 0 from the distribution pX |Y (x | y 0 ).

For i = 1 ,... , N, pick yi from PY |X (y | xi− 1 ) and xi from PX |Y (x | yi ).

Gibbs Sampling

Special case of a Monte Carlo Markov Chain sampling.

Suppose we need to sample from p(x, y )

Marginalizing the joint density is hard.

Easy to sample from pX |Y and pY |X.

Gibbs sampler proceeds as below

Pick y 0 from some distribution that has same support as pY (y ).

Pick x 0 from the distribution pX |Y (x | y 0 ).

For i = 1 ,... , N, pick yi from PY |X (y | xi− 1 ) and xi from PX |Y (x | yi ).

Gibbs Sampling

Special case of a Monte Carlo Markov Chain sampling.

Suppose we need to sample from p(x, y )

Marginalizing the joint density is hard.

Easy to sample from pX |Y and pY |X.

Gibbs sampler proceeds as below

Pick y 0 from some distribution that has same support as pY (y ).

Pick x 0 from the distribution pX |Y (x | y 0 ).

For i = 1 ,... , N, pick yi from PY |X (y | xi− 1 ) and xi from PX |Y (x | yi ).