optimal filtering of jump diffusions extracting latent states from asset prices
Download
Skip this Video
Download Presentation
Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices

Loading in 2 Seconds...

play fullscreen
1 / 31

Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices - PowerPoint PPT Presentation


  • 121 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices' - zareh


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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.
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.
slide10

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
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?
slide17

Simulated Daily Data : SV Model

Returns

Volatility

Discretization Error

slide18

Simulated Monthly Data : SV Model

Returns

Volatility

Discretization Error

slide22

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?
slide27

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