exploring time-varying jump intensities: evidence from s&p500 returns and options

Exploring Time-Varying Jump Intensities: Evidence from

S&P500 Returns and Options

Peter Christoffersen, Kris Jacobs and Chayawat Ornthanalai

McGill University

FDIC Risk Management ConferenceApril 12th 2008

Objectives

To understand how should jump dynamics be specified. Should jump intensity be time-varying? How should the jump and normal components be

specified jointly in the returns of S&P500 index?

To investigate the implication of different jump specifications on option pricing

The implication of our results on the existing estimates of the continuous-time models that are based on the jump-diffusion framework

Objectives

The Compound Poisson (CP) process is distributed as

For the Merton jump:

The number of jumps that arrive over a finite intervalis:

The structure of Merton jumps

jtt xy

21 ,Normal~ j

yt hn Poisson~

Intuitions behind the time-varying jump intensity? Jumps clustering: Maheu and Mccurdy (2004) The likelihood of jumps (Fear of crash): Bates (1991)

Evidence for time-varying (TV) jump intensities in the returns dynamics are mixed.

Andersen, Benzoni, and Lund (ABL, 2002) - Find no time series evidence for the TV jump intensity.

Bates (2006) – Uses his AML method and finds evidence for the TV jump intensity.

In the mix of opinions regarding the TV jump intensity Recent studies have become more supportive of the model with jumps in

the volatility: SVCJ, Eraker, Johannes, and Polson (EJP 2003) Li, Wells and Yu (2006), Broadie, Chernov, and Johannes (BCJ 2007,2008),

and Johannes, Polson, and Stroud (2008)

Background/Motivations (1)

Summary of selected jump-diffusion models in the literature

Jumps Correl Return State Dependent Stochastic Returns Options Joint Returnsin Volatility and Vol. Jumps Jump Intensity Jump Intensity only only and Options

Cont. Time Models

BCC (1997)

Bates (2000)

Pan (2002)

ABL(2003)

CGGT (2003)

EJP (2003)

Eraker (2004)

Huang, Wu (2004)

Bates (2006)

Li, Wells, Yu (2007)

BCJ (2007)

Discrete Time Models

Maheu, McCurdy (2004)

DRS (2006)

J-GARCH

Notes on the abbreviations: Bakshi,Cao and Chen (BCC,1997), Andersen, Bezoni, and Lund (ABL, 2002), Chernov, Gallant, Ghysels, and Tauchen (CGGT, 2003), Eraker, Johannes, and Polson (EJP, 2002), Broadie, Chernov, and Johannes (BCJ,2007), and Duan, Ritchken, and Sun (DRS,2006).

Summary of selected jump-diffusion models in the literature

Jumps Correl Return State Dependent Stochastic Returns Options Joint Returnsin Volatility and Vol. Jumps Jump Intensity Jump Intensity only only and Options

Cont. Time Models

BCC (1997)

Bates (2000)

Pan (2002)

ABL(2003)

CGGT (2003)

EJP (2003)

Eraker (2004)

Huang, Wu (2004)

Bates (2006)

Li, Wells, Yu (2007)

BCJ (2007)

Discrete Time Models

Maheu, McCurdy (2004)

DRS (2006)

J-GARCH

Notes on the abbreviations: Bakshi,Cao and Chen (BCC,1997), Andersen, Bezoni, and Lund (ABL, 2002), Chernov, Gallant, Ghysels, and Tauchen (CGGT, 2003), Eraker, Johannes, and Polson (EJP, 2002), Broadie, Chernov, and Johannes (BCJ,2007), and Duan, Ritchken, and Sun (DRS,2006).

It is far from simple to estimate continuous-time modelsthat are based on the jump-diffusion framework.

Challenge: the presence of latent factors.

Estimation of complex jump models are often based on option prices. This is dangerous when the latent factors are treated as part of the structural parameters in the estimation.

Time-series estimates of the SVCJ from returns rely on discrete-time approximation. EJP, Eraker (2004), and Li, Wells, and Yu (2007) approximate CP using the Bernoulli jump process.

Why do we use a discrete-time framework?

Main Findings1. Yes! It is important to have time-varying jump intensities in

modeling the returns dynamic.

2. Option pricing results favour the model with jump intensity that is linear with the variance of the normally distributed return component.

3. The estimates from our model support the presence of multiple jumps per day. Hence, parameters that are estimated from the studies that use Bernoulli distribution to approximate the CP process are likely to be biased.

4. In order to produce significant improvement in option valuation, jump models must allow for sizable magnitude of jump risk premia.

The general setup: physical measure

The conditional equity premium is

Various nested specifications8/18

MLE results: overview

Parameters Normal Jump Normal Jump Normal Jump Normal Jump

w -1.210E-06 8.053E-03 2.612E-05 5.472E-06 -1.073E-06 -1.005E-06 7.549E-09

(1.458E-07) (2.002E-03) (1.180E-06) (2.457E-06) (1.647E-07) (1.252E-07) (4.905E-09)

θ -1.254E-02 -4.211E-04 -2.628E-03 -1.662E-03

(4.511E-03) (1.501E-08) (1.411E-03) (8.355E-04)

δ 2.861E-02 1.012E-02 1.924E-02 1.613E-02

(3.260E-03) (3.318E-04) (1.583E-03) (8.355E-04)

k 5.209E+02(1.21E+02)

PropertiesPersistence 0.98345 0.99999 0.98168 0.95826 0.99524

% Annual Var 90.24 9.76 31.75 68.25 83.57 16.43 81.28 18.72

Llhood# Pars 9 9 9 12

37554 36581 37573 37576

J-GARCH(4) J-GARCH(1) J-GARCH(2) J-GARCH(3)

GARCHLlhood 3734235573 36580

BSM Merton

J-GARCH models

w -1.210E-06 8.053E-03 2.612E-05 5.472E-06 -1.073E-06 -1.005E-06 7.549E-09

(1.458E-07) (2.002E-03) (1.180E-06) (2.457E-06) (1.647E-07) (1.252E-07) (4.905E-09)

θ -1.254E-02 -4.211E-04 -2.628E-03 -1.662E-03

(4.511E-03) (1.501E-08) (1.411E-03) (8.355E-04)

δ 2.861E-02 1.012E-02 1.924E-02 1.613E-02

(3.260E-03) (3.318E-04) (1.583E-03) (8.355E-04)

k 5.209E+02(1.21E+02)

% Annual Var 90.24 9.76 31.75 68.25 83.57 16.43 81.28 18.72

Llhood# Pars 9 9 9 12

37554 36581 37573 37576

GARCHLlhood 3734235573 36580

BSM Merton

J-GARCH models

w -1.210E-06 8.053E-03 2.612E-05 5.472E-06 -1.073E-06 -1.005E-06 7.549E-09

(1.458E-07) (2.002E-03) (1.180E-06) (2.457E-06) (1.647E-07) (1.252E-07) (4.905E-09)

θ -1.254E-02 -4.211E-04 -2.628E-03 -1.662E-03

(4.511E-03) (1.501E-08) (1.411E-03) (8.355E-04)

δ 2.861E-02 1.012E-02 1.924E-02 1.613E-02

(3.260E-03) (3.318E-04) (1.583E-03) (8.355E-04)

k 5.209E+02(1.21E+02)

% Annual Var 90.24 9.76 31.75 68.25 83.57 16.43 81.28 18.72

Llhood# Pars

37554 36581 37573 375769 9 9 12

GARCHLlhood 3734235573 36580

BSM Merton

J-GARCH models

MLE results: conditional jump intensity

Option Pricing Results: At a fixed level of the equity premium We price call options on each Wednesday from 1996-2005

We set the long-run equity premium to be 6% across all models.

IVRMSE ratio IVRMSE ratio

J-GARCH(1) 1.03 0.91

J-GARCH(2) 1.33 1.32

J-GARCH(3) 0.98 0.82

J-GARCH(4) 0.99 0.92

Option Pricing Performance for Jump Models relative to GARCH

Model Specification All Normal Risk All Jump Risk

Option Pricing Results: At a fixed level of the equity premium We price call options on each Wednesday from 1996-2005

We set the long-run equity premium to be 6% across all models.

IVRMSE ratio IVRMSE ratio

J-GARCH(1) 1.03 0.91J-GARCH(2) 1.33 1.32J-GARCH(3) 0.98 0.82J-GARCH(4) 0.99 0.92

Option Pricing Performance for Jump Models relative to GARCH

Model Specification All Normal Risk All Jump Risk

Option Pricing Results: Changing the level of the equity premium

J-GARCH(3) IVRMSE ratio

0.0 2.0 4.0 6.0 8.0 10.00.0 0.97 0.92 0.86 0.82 0.81 0.88

2.0 0.97 0.92 0.86 0.82 0.834.0 0.98 0.92 0.87 0.836.0 0.98 0.92 0.878.0 0.98 0.93

10.0 0.98

Normal Risk

Premium

Total Equity Premium (%)

Option Pricing Results: Changing the level of the equity premium

J-GARCH(3) IVRMSE ratio

0.0 2.0 4.0 6.0 8.0 10.00.0 0.97 0.92 0.86 0.82 0.81 0.88

2.0 0.97 0.92 0.86 0.82 0.834.0 0.98 0.92 0.87 0.836.0 0.98 0.92 0.878.0 0.98 0.93

10.0 0.98

Normal Risk

Premium

Total Equity Premium (%)

Option Pricing Results: The impact of changes in the equity premium level on the IV smirk

Further Analysis : Decomposition of daily returns by particle filter (1/2)

Further Analysis : Decomposition of daily returns by particle filter (2/2)

Further Analysis : The impact of using Bernoulli approximation

Parameters Jump Jump

w 9.689E-04(2.920E-04)

θ -3.91E-02 -4.87E-02(2.12E-02) (2.10E-02)

δ 4.84E-02 6.16E-02(6.70E-03) (6.66E-03)

k 1.66E+01(4.35E+00)

LogLikelihood 37530 37528

J-GARCH(1) J-GARCH(3)

Result: no time-series evidence for time-varying jump intensities

Further Analysis : What is the bias from applying Bernoulli approximation to J-GARCH(3)?

With Bernoulli approximation, jumps are perceived as extremely large and rare events.

Conclusions

1. Models that are based on the finite-activity Merton Jump should allow for time-varying jump intensity.

2. Jumps arrive in clusters, and their arrival rate is dependent on the level of the market variance.

3. Option pricing models require the presence of jump risk premium for explaining the smirk patterns in the implied volatilities at the reasonable equity premium levels.

4. It is not acceptable to approximate a Compound Poisson process using a Bernoulli distribution. Studies that assume this will produce biases in their estimates.

exploring time-varying jump intensities: evidence from s&p500 returns and options

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