1. 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)

2. 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.

3. S&P 500 Index, October, 1987 Daily Closing Prices/Returns and Options Implied Volatilities DatePrice($)ReturnImpVolSpotVolJump Oct 14305.2 -3.021.5 Oct 15298.1 -2.422.7 Oct 16282.7 -5.324.1 Oct 19224.8-22.962.3?? Oct 20236.8 5.286.1 Oct 21258.4 8.788.5 Oct 22248.3 -4.066.9

4. 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

5. Jump Diffusion Models in Finance Y t is observed, X t is unobserved state variable N t y : latent point processes with intensity y (Y t-,X t- ). Z n y : latent jump sizes with distribution  y (Y(  n- ),X(  n- )). Also observe derivative prices (non-analytic).

6. 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

7. 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.

8. 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.

9. 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.

10. XtXt YtYt time Observed Variable, Y t Unobserved Variable, X t 1234567890 1234567890

11. Latent variable augmentation Given the augmentation level M, we define the latent variable as L t = (X t M, J t M, Z t M ), where Then it is easy to simulate from the transition density p(L t+1 |L t ), and to evaluate the likelihood p(Y t+1 |L t+1 ).

12. Bayesian filtering Let L t 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

13. The particle filter Gordon, Salmond & Smith (1993) approximate the filtering distribution using a weighted Monte Carlo sample (L t i,  t i ), i=1…N: The prediction and filtering distributions at time t+1 are then approximated by

14. Sampling-Importance Resampling Particle Filter Algorithm

15. Application: Double-Jump Model Duffie, Pan & Singleton (2000) provide a model with SV and jumps in returns and volatility: where N t ~Poi( t), Z n s ~N(  s,  2 s ) and Z n v ~Exp( v ). SV model : Stochastic Volatility SVJ model : SV with jumps in returns SVCJ model : SV with jumps in returns & volatility

16. 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: 1)How large must M be to recover the “true” filtering distribution? 2)How well can we detect jumps if data are sampled at daily, weekly, monthly frequency?

17. Simulated Daily Data : SV Model Returns Volatility Discretization Error

18. Simulated Monthly Data : SV Model Returns Volatility Discretization Error

19. RMSE: Filtered Mean Volatility SV model MDailyWeeklyMonthly 10.812.176.44 20.320.523.02 50.280.260.80 100.280.240.30 250.280.240.22

20. Filtered density for Spot Volatility Monthly Data

21. Simulated Daily Data : SVJ Model

22. Jump Classification Rate SVJ model Observation Frequency DailyWeeklyMonthly.60.30.03 Percentage of true jumps detected by the filter.

23. 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?

24. Filtered Volatility: S&P 500 Data

26. S&P 500 Index, October, 1987 Filtering Results (SV, SVJ, SVCJ) DateReturnVolatility (annual)P(Jump) Oct 14 -3.020.920.818.7.04.10 Oct 15 -2.421.721.920.2.01.02 Oct 16 -5.325.823.627.1.59 Oct 19-22.935.725.344.41.00.98 Oct 20 5.235.627.743.0.06.00 Oct 21 8.736.629.042.9.69.00 Oct 22 -4.036.431.442.8.03.01

27. Filtered Volatility: October 13-22, 1987 October 13 October 16 October 22 Crash 

28. 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?

29. SV Model: Filtering with Option Prices

30. SV Model: Filtered Densities, Oct. 15-19, 1987 October 15 October 16 October 19

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