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)
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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 and
Options Implied Volatilities
Date Price($) Return ImpVol SpotVol Jump
Oct 14 305.2 -3.0 21.5
Oct 15 298.1 -2.4 22.7
Oct 16 282.7 -5.3 24.1
Oct 19 224.8 -22.9 62.3 ? ?
Oct 20 236.8 5.2 86.1
Oct 21 258.4 8.7 88.5
Oct 22 248.3 -4.0 66.9
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 y(Yt-,Xt-).
• Zny : latent jump sizes with distribution y(Y(n-),X(n-)).
• Also observe derivative prices (non-analytic) .
xt
yt
N
1n
xt-t
xt
xt
N
1n
yt-t-t
ytt
yt
Zd)dW(Xσ)dt(XμdX
Zd)dWX,(Yσ)dtX,(YμdY
xn
yn
ttT
dsrQtttt X,Y|YfeEX,YgY
~T
ts
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
y
1t
yt
N
Nn
yn
1t
t
ys-s-s
y1t
tss
yt1t Z)dWX,(Yσ)dsX,(YμYY
x
1t
xt
N
Nn
xn
1t
t
xs-s
x1t
ts
xt1t Z)dW(Xσ)ds(XμXX
tttt εX,YglogY~
log
The filtering problem• Goal: compute the optimal filtering distribution of all latent