Optimal filtering of jump diffusions extracting latent states from asset prices
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Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices. Jonathan Stroud, Wharton, U. Pennsylvania Stern-Wharton Conference on Statistics in Business April 28 th , 2006 Joint work with Mike Johannes (GSB, Columbia) and Nick Polson (GSB, Chicago). Overview.

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Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices

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Optimal filtering of jump diffusions extracting latent states from asset prices

Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices

Jonathan Stroud, Wharton, U. Pennsylvania

Stern-Wharton Conference on

Statistics in Business

April 28th, 2006

Joint work with Mike Johannes (GSB, Columbia)

and Nick Polson (GSB, Chicago)


Overview

Overview

  • Models in finance

    • Typically specified in continuous-time.

    • Include latent variables such as stochastic volatility and jumps.

  • Two state estimation problems

    • Filtering - sequential estimation of states.

    • Smoothing - off-line estimation of states.

  • Filtering is needed in most financial applications

    • e.g., portfolio choice, derivative pricing, value-at-risk.


S p 500 index october 1987 daily closing prices returns and options implied volatilities

S&P 500 Index, October, 1987Daily Closing Prices/Returns andOptions Implied Volatilities


Outline

Outline

  • Jump diffusion models in finance

  • The filtering problem and the particle filter

  • Application: Double Jump model

    • Simulation study

    • S&P 500 index returns

    • Combining index and options data


Jump diffusion models in finance

Jump Diffusion Models in Finance

  • Yt is observed, Xt is unobserved state variable

  • Nty : latent point processes with intensity ly(Yt-,Xt-).

  • Zny : latent jump sizes with distribution Py(Y(tn-),X(tn-)).

  • Also observe derivative prices (non-analytic)

    .


State space formulation

State-Space Formulation

  • Assuming data at equally-spaced times t, t+1,… the observation and state equation are given by

  • Also have a second observation equation for the derivative prices:

v


The filtering problem

The filtering problem

  • Goal: compute the optimal filtering distribution of all latent variables, given observations up to time t:

  • Existing methods:

    • Kalman filter: linear drifts, constant volatilities.

    • Approximate methods: simple discretization, extended Kalman filter.

    • Quadratic variation estimators: can’t separate jumps and volatility; require high-frequency data; no models.


Our approach

Our approach

We propose an approach which combines two existing ideas:

1) Simulating extra data points

Time-discretize model and simulate additional data points between observations to be consistent with continuous-time specification.

2) Applying particle filtering methods

Sequential importance sampling methods to compute the optimal filtering distribution.


Time discretization

Time-Discretization

  • Simulate M intermediate points using an Euler scheme (other schemes possible)

  • Given the simulated latent variables, we can approximate the (stochastic and deterministic) integrals by summations.


Optimal filtering of jump diffusions extracting latent states from asset prices

Observed Variable, Yt

Yt

time

0

1

2

3

4

5

6

7

8

9

Unobserved Variable, Xt

Xt

time

0

1

2

3

4

5

6

7

8

9


Latent variable augmentation

Latent variable augmentation

Given the augmentation level M, we define the latent variable as Lt = (XtM, JtM, ZtM), where

Then it is easy to simulate from the transition density p(Lt+1|Lt), and to evaluate the likelihood p(Yt+1|Lt+1).


Bayesian filtering

Bayesian filtering

  • Let Lt denote all latent variables. At time t, the filtering (posterior) distribution for the latent variables is given by

  • The prediction and filtering distributions at time t+1 are then given by


The particle filter

The particle filter

  • Gordon, Salmond & Smith (1993) approximate the filtering distribution using a weighted Monte Carlo sample (Lti, pti), i=1…N:

  • The prediction and filtering distributions at time t+1 are then approximated by


Sampling importance resampling particle filter algorithm

Sampling-Importance Resampling Particle Filter Algorithm


Application double jump model

Application: Double-Jump Model

  • Duffie, Pan & Singleton (2000) provide a model with SV and jumps in returns and volatility:

    where Nt ~Poi(lt), Zns ~N(ms,s2s) and Znv ~Exp(lv).

    SV model : Stochastic Volatility

    SVJ model : SV with jumps in returns

    SVCJ model : SV with jumps in returns & volatility


Simulation study

Simulation Study

  • Simulate continuous-time process (M=100) using parameter values from literature.

  • Sample data at daily, weekly & monthly freq’s.

  • Run filter using M=1,2,5,10,25 and N=25,000.

    Questions of interest:

  • How large must M be to recover the “true” filtering distribution?

  • How well can we detect jumps if data are sampled at daily, weekly, monthly frequency?


Optimal filtering of jump diffusions extracting latent states from asset prices

Simulated Daily Data : SV Model

Returns

Volatility

Discretization Error


Optimal filtering of jump diffusions extracting latent states from asset prices

Simulated Monthly Data : SV Model

Returns

Volatility

Discretization Error


Optimal filtering of jump diffusions extracting latent states from asset prices

RMSE: Filtered Mean Volatility

SV model


Optimal filtering of jump diffusions extracting latent states from asset prices

Filtered density for Spot Volatility

Monthly Data


Optimal filtering of jump diffusions extracting latent states from asset prices

Simulated Daily Data : SVJ Model


Optimal filtering of jump diffusions extracting latent states from asset prices

Jump Classification Rate

SVJ model

Percentage of true jumps detected by the filter.


S p 500 example

S&P 500 Example

S&P 500 return data (1985-2002)

  • Daily data (T=4522)

  • SIR particle filter: M=10 and N=25,000.

  • How does volatility differ across models?


Optimal filtering of jump diffusions extracting latent states from asset prices

Filtered Volatility: S&P 500 Data


Optimal filtering of jump diffusions extracting latent states from asset prices

Filtered Volatility: S&P 500 Data


S p 500 index october 1987 filtering results sv svj svcj

S&P 500 Index, October, 1987Filtering Results (SV, SVJ, SVCJ)


Optimal filtering of jump diffusions extracting latent states from asset prices

Filtered Volatility: October 13-22, 1987

October 13

October 16

Crash 

October 22


Filtering with option prices

Filtering with Option Prices

S&P 500 futures options data (1985-1994)

  • At-the-money futures call options

  • Assume 5% pricing error

  • How does option data affect estimated volatility?


Optimal filtering of jump diffusions extracting latent states from asset prices

SV Model: Filtering with Option Prices


Optimal filtering of jump diffusions extracting latent states from asset prices

SV Model: Filtered Densities, Oct. 15-19, 1987

October 15

October 16

October 19


Conclusions

Conclusions

  • Extend particle filtering methods to continuous-time jump-diffusions

  • Incorporate option prices

  • Evaluate accuracy of state estimation

  • Easy to implement

  • Applications


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