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

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

  • 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, 1987Daily Closing Prices/Returns andOptions Implied Volatilities


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

  • 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

  • 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

  • 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

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

  • 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.


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

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

  • 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

  • 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


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

  • 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?


Simulated Daily Data : SV Model

Returns

Volatility

Discretization Error


Simulated Monthly Data : SV Model

Returns

Volatility

Discretization Error


RMSE: Filtered Mean Volatility

SV model


Filtered density for Spot Volatility

Monthly Data


Simulated Daily Data : SVJ Model


Jump Classification Rate

SVJ model

Percentage of true jumps detected by the filter.


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?


Filtered Volatility: S&P 500 Data


Filtered Volatility: S&P 500 Data


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


Filtered Volatility: October 13-22, 1987

October 13

October 16

Crash 

October 22


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?


SV Model: Filtering with Option Prices


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

October 15

October 16

October 19


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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