by

Adjoa Numatsi and Erick Rengifo Economics Department, Fordham University, New York

Exploratory analysis of GARCH Models versus Stochastic Volatility Models with

Jumps in Returns and Volatility (Work in Progress)

Conference on Quantitative Social Science Research Using R

Fordham University – 18-19 June, 2009

OUTLINE

Introduction Models and Data Preliminary Results Conclusion Further research

Introduction

Multiple attempts to improve equity pricing models to reconcile theory with empirical features

Important because misspecified models mistakes in forecasting

Example in option pricing: Black Scholes formula has shown biases (Rubinstein, 1985) due to 2 major assumptions on the underlying stock pricing process Stock prices: continuous path through time, their distribution is

lognormal Variance of stock returns is constant (Black and Scholes, 1973)

But asset returns are leptokurtic, and display volatility clustering (Chernov et al.(1999))

Both assumptions have been relaxed allowing for: discontinuities in the form of jump diffusion models (Merton (1976), Cox and Ross

(1976)) stochastic volatility (Hull and White (1987), Scott (1987), Heston (1993))

Bates (1996) and Scott (1997) combined stochastic volatility models with jumps in returns

However, the volatility process is still misspecified (Bakshi, Cao, and Chen (1997), Bates (2000), and Pan (2002)): Jumps in returns can generate large movement, but the impact is temporary Diffusive stochastic volatility is persistent but because its dynamics are driven by a

Brownian motion small normally distributed increments.

Need for conditional volatility to move rapidly and also be persistent

Duffie, Pan, and Singleton (2000) : models with jumps in both returns and volatility

Introduction, cont…

Estimated by Eraker, Johannes, and Polson (2003). Results showed almost no misspecification

But with all the features that these new models have, they are complex and it is time consuming to work with them.

Our study will address this issue by comparing models with jumps and simple GARCH models

GARCH models: Introduced by Bollerslev (1986) and Taylor (1986) Have time varying variance Discrete time models empirically favored compared to continuous

time models There are attempts to model GARCH with jumps (Duan, Ritchken, Sun

(2006)) but we are interested in simple GARCH

Introduction, cont…

Objectives

Given the complexity of stochastic volatility models with jumps in both returns and volatility (SV2J thereafter), we want to provide a model that will allow to choose SV2J models only when they are relevant

More specifically: We want to compare the performance of SV2J models and

simple GARCH models in order to identify the market situation in which their respective performances are significantly different.

Clusters vs. Jumps

Models should be able to capture specific behavior in the data

Index returns display clusters and jumps Jumps do not always imply existence of autoregressive

conditional heteroskedasticity (ARCH) process ARCH type Models cannot capture dynamics

However we do not always have jumps in mean and variance, but a smooth diffusion process where clusters can be found ARCH type Models can do a good job then.

Clusters vs. Jumps, cont…

1st case: a jump but no clusters.

Ho = Homoscedasticity

ARCH LM test (p_value) = 0.321236.

We do not reject the null there is no ARCH effect. GARCH model is not good here

2nd case: clusters.

ARCH LM test (p_value) = 0.011259.

We reject the null There is ARCH process, which means GARCH is appropriate here.

Clusters vs. Jumps, cont…

> GARCH_withARCH Title: GARCH Modelling Call: garchFit(formula = ~arma(0, 0) + garch(1, 1), data = Y, trace = FALSE) Mean and Variance Equation: data ~ arma(0, 0) + garch(1, 1) [data = Y] Conditional Distribution: norm Coefficient(s): mu omega alpha1 beta1 0.035196 0.079091 0.083386 0.814847 Std. Errors: based on Hessian Error Analysis: Estimate Std. Error t value Pr(>|t|) mu 0.03520 0.02540 1.385 0.165915 omega 0.07909 0.03072 2.575 0.010037 * alpha1 0.08339 0.02444 3.412 0.000644 *** beta1 0.81485 0.05389 15.121 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Log Likelihood: -1392.634 normalized: -1.263733

•Alpha1 and beta1 are the GARCH and ARCH coefficients. They are significant

•R package used: fGarch

Clusters vs. Jumps, cont…

We can also have both jumps and clusters (which is actually the case in our last example).

Clusters are smooth movements increasing and decreasing. Jumps are not well defined in the literature, but they are characterized by one big move up or down. In our study we consider that differences in returns above 3 standard deviations from the mean are jumps

What we are doing is to find situations in which SV models give results significantly different from GARCH, and therefore are worth the effort of estimating them.

Data

FTSE100 daily returns from July 3, 1984 to Dec 29, 2006 (5686 observations)

The volatility was generated by a rolling window approach

Differences in returns and volatility above 3 standard deviations from the mean are considered as jumps

Data, cont…

Y

Mean 0.031348

Standard Error 0.013494

Median 0.062642

Mode 0

Standard Deviation 1.017553

Sample Variance 1.035415

Kurtosis 8.268878

Skewness -0.5567

Range 20.62556

Minimum -13.0286

Maximum 7.596966

Sum 178.2447

Count 5686

V

Mean 1.042229

Standard Error 0.015588

Median 0.685022

Mode #N/A

Standard Deviation 1.17543

Sample Variance 1.381636

Kurtosis 18.20166

Skewness 3.866057

Range 9.187442

Minimum 0.182197

Maximum 9.36964

Sum 5926.115

Count 5686

Jumps in returns and volatility

Models: GARCH

Models: SV2J

Models: SV2J, cont…

Methodology

Estimate a simple GARCH model and a SV2J model on a period where the market is stable, and on a period where there is instability in the market, and compare the performances. Market stability will be measured in standard deviations from long-term mean returns.

Compare the results in terms of out-of-sampling forecast errors and the use of resources (basically time)

Estimation method

There are packages in R dealing with GARCH (tseries, fGarch packages), and with Stochastic Differential Equations (sde package).

But there are no packages yet for sde equation with double jumps we have written a function in R to estimate our SV models, using MCMC method.

We first derived the posterior distribution from the prior information, the distribution of the state variables and the likelihood.

Then we derived the full conditional distributions from the posterior and programmed them in R.

R packages used: Rlab, MCMCpack, msm

Estimation method: R code

## R program library(Rlab) library(MCMCpack) library(msm) ## 1- Read in the data FTSE= read.table("c:/Users/Adjoa/FTSE_02june09.txt", header=T) attach(FTSE) summary(FTSE) ## 2- MCMC function ## Markov chain Monte Carlo algorithm for SV model with jumps mhsampler= function(NUMIT,data) {………… ## sample from full conditional distribution of xi_y (Gibbs update) xi_y = rnorm(1, mean=((Y_t-mu)/V_past + (mu_y-rho_J*xi_v)/sigmasq_y)/(1/V_past +1/sigmasq_y) , sd=1/(1/V_past +1/sigmasq_y) ) ……………………….} ## 3- Run MCMC and get results of estimations NUMIT=100 mchain <- mhsampler(NUMIT,dat=FTSE) # call the function with appropriate arguments warmup= round(0.1*NUMIT) ## number of warmup draws warmup parameter_1=mchain$parameter_1 parameter=parameter_1[,(warmup+1):NUMIT] ## Discard 10% of iterations as warmup or burn-in period ……………………………………

R packages used: Rlab, MCMCpack, msm

Steps:

1. Read in the data

2. MCMC function to estimate the parameters and generate the state variables

3. Run the function and analyze the results

Preliminary Results

We estimate:

GARCH (1,1)Stochastic Volatility Model with jumps

Preliminary Results, cont…

  parameters sd median

mu 0.928826 4.49E-01 0.928147

mu_y -0.9443744 4.60E-01 -0.97721

mu_v 141.31438 2.97E+02 67.77618

theta 0.1780038 2.79E-02 0.171543

sigmasq_y 0.68103638 4.68E-02 0.694145

sigmasq_v 0.0097323 1.36E-02 0.007103

k 0.97460276 5.32E-02 0.995949

rho 0.23444102 1.63E-01 0.204819

rho_J 2.95602216 2.84E+00 2.562103

lamda_y 0.00130032 1.76E-03 0.000904

lamda_v 0.00130032 1.76E-03 0.000904

sigma_y 0.82524929 NaN NaN

sigma_v 0.09865241 NaN NaN

Estimation challenges and future steps

Convergence Choice of starting values, mostly for the state

variables (especially jumps in returns and volatility)

Explore R interface with WinBUGS (Bayesian Inference Using Gibbs Sampling)

Thank you!!!