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

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.

T.1,...,t,Y|)(Z ,N,Xp t:1tτ1tntt n

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.

y

M

1t

y

M

1t

y

M

1t

tty-1

tty

M

1t

ZJ)εX,(Yσ)MX,(YμY

x

M

1t

x

M

1t

x

M

1t

tx-1

tx

M

1t

ZJ)ε(Xσ)M(XμX

Xt

Yt

time

time

Observed Variable, Yt

Unobserved Variable, Xt

1 2 3 4 5 6 7 8 90

1 2 3 4 5 6 7 8 90

Latent variable augmentation

Given the augmentation level M, we define the latent variable as Lt = (Xt

M, JtM, Zt

M), 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).

M

1-t

M

1-M-t

M

M-t

Mt

M

1-t

M

1-M-t

M

M-t

Mt

M

1-t

M

1-M-t

M

M-t

Mt

Z...,,Z,ZZ

J...,,J,JJ

X...,,X,XX

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

t:1t Y|Lp

1t1tt:11t1t:11t L|YpY|LpY|Lp :Filtering

tt:1tt1tt:11t dL Y|Lp L|LpY|Lp:Prediction

The particle filter• Gordon, Salmond & Smith (1993) approximate the

filtering distribution using a weighted Monte Carlo sample (Lt

i, ti), i=1…N:

• The prediction and filtering distributions at time t+1 are then approximated by

N

1i

it

ittt:1t

N )πLδ(L)Y|(Lp

it

N

1i

it1tt:11t

N π L|LpY|Lp:Prediction

L|Yp πL|LpY|Lp:Filtering 1t1tN

1i

ii1t1t:11t

Ntt

Sampling-Importance Resampling Particle Filter Algorithm

.Mset πand }{πweightsingcorrespond

with}{LondistributidiscretethefromResample3.

π)L|p(Y π

: weightslikelihood theCompute 2.

).L|p(L~L

:n transitiostate the viaforward particles Propagate 1.

1...N.i)},π,{(Lset particle initialan Start with0.

1i1t

i1t

i1t

it

i1t1t

i1t

it1t

i1t

it

it

Application: Double-Jump Model

• Duffie, Pan & Singleton (2000) provide a model with SV and jumps in returns and volatility:

where Nt ~Poi(t), Zns ~N(s,2

s) and Znv ~Exp(v).

SV model : Stochastic Volatility SVJ model : SV with jumps in returns SVCJ model : SV with jumps in returns & volatility

t

t

N

1n

vn

vt-tvtvvt

N

1n

sn

st-tttvtt

ZddWVσdtVθκdV

ZddWVdt2/V-Vηr)dlog(S

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?

Simulated Daily Data : SV Model

Returns

Volatility

Discretization Error

Simulated Monthly Data : SV Model

Returns

Volatility

Discretization Error

RMSE: Filtered Mean VolatilitySV model

M Daily Weekly Monthly

1 0.81 2.17 6.44

2 0.32 0.52 3.02

5 0.28 0.26 0.80

10 0.28 0.24 0.30

25 0.28 0.24 0.22

Filtered density for Spot Volatility Monthly Data

Simulated Daily Data : SVJ Model

Jump Classification RateSVJ model

Observation Frequency

Daily Weekly Monthly

.60 .30 .03

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)

Date Return Volatility (annual) P(Jump)

Oct 14 -3.0 20.9 20.8 18.7 .04 .10

Oct 15 -2.4 21.7 21.9 20.2 .01 .02

Oct 16 -5.3 25.8 23.6 27.1 .59 .59

Oct 19 -22.9 35.7 25.3 44.4 1.00 .98

Oct 20 5.2 35.6 27.7 43.0 .06 .00

Oct 21 8.7 36.6 29.0 42.9 .69 .00

Oct 22 -4.0 36.4 31.4 42.8 .03 .01

Filtered Volatility: October 13-22, 1987

October 13

October 16

October 22

Crash

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