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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)

- 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

- 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

- 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)
.

- 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

- 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.

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.

- 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

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).

- 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

- 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

- 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

- 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 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

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

- Extend particle filtering methods to continuous-time jump-diffusions
- Incorporate option prices
- Evaluate accuracy of state estimation
- Easy to implement
- Applications