GARCH models with jumps

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Garch models with jumps

S.P. Sidorov

Saratov State University, Saratov, Russia

1. Simple GARCH models with Jumps

1.1 Model Description

GARCH-Jump model was proposed and studied in [1]. This paper proposes a model of conditional variance of returns implied by the impact of different type of news.

Let be the log return of a particular stock or the market portfolio from time to time . Let denote the past information set containing the realized values of all relevant variables up to time . Suppose investors know the information in when they make their investment decision at time . Then the relevant expected return to the investors is the conditional expected value of , given , i.e.

The relevant expected volatility to the investors is conditional variance of , given , i.e.

Then

is the unexpected return at time .

In GARCH-Jump model it is supposed that news process have two separate components (normal and unusual news), which cause two types of innovation (smooth and jump-like innovations):

These two news innovations have a different impact on return volatility. It is assumed that the first component reflects the impact of unobservable normal news innovations, while the second one is caused by unusual news events.

The first term in (1) reflects the impact of normal news to volatility:

where be a sequence of i.i.d. random variables such that , is a nonnegative GARCH(1,1) process such that

and . Note that .

The second term in (1) is a jump innovation with . The component is a result of unexpected events and is responsible for jumps in volatility.

The distribution of jumps is assumed to be Poisson distribution. Let be intensity parameter of Poisson distribution. Denote a number of jumps occurring between time and . Then conditional density of is

We suppose that the intensity parameter conditionally varies over time. It is assumed that the conditional jump intensity , i.e. the expected number of jumps occurring between time and conditional on information , has dynamics

The process (3) is called an autoregressive conditional jump intensity and was proposed in the paper [2]. The model based on the assumption that the conditional jump intensity is autoregressive and related both to the last period's conditional jump intensity and to an intensity residual . The intensity residual is defined as

Here is the expected number of jumps occurring from to , and is the conditional expectation of numbers of jumps given the information available at the moment . Thus

i.e. represents the change in the econometrician's conditional forecast of as the information set is updated from to . It is easy to see that , i.e. is a martingale difference sequence with respect to , and therefore , for all .

Denote the size of -th jump that occur from time to , . In the model it is supposed that the jump size is realization of normal distributed random:

Then the cumulative jump size from to is equal to the sum of all jumps occurring from time to :

The jump innovation defined by

garch model jumps

It follows from

that

Therefore we have

1.2 Maximum Likelihood Estimation of GARCH Model with Jumps

The subsection describes quasi-maximum likelihood estimation (QML) of GARCH model with Jumps. The vector of model parameters is

We will assume that belongs to the set

Denote

the vector of the true values of parameters. The aim is to find that maximize a QML function given an observation sequence

of length .

Define the sequence by recursion:

If we assume that the likelihood function is Gaussian, then the log-likelihood function can be written as (see e.g. [2]):

where

and

The sequence of is defined by recursion:

where

and

The maximum likelihood estimator of is defined by

Since the densities (5) has an infinite sum, it is impossible to use them for parameters' estimation. There are two ways of using equation (5):

* taking a finite Taylor expansions of (5);

* truncation of the sum (5), i.e. limitation of the number of terms in the sum.

We useMATLAB software for calibration the GARCH model with jumps. It is should be noted that the calibration problem is non convex and surface of optimized function has a highly complex relief and finding its exact solution is a difficult task. We faced with difficulties when calibrate process via MATLAB function fminsearch. In particular, the calibration process is not robust and extremely sensitive to the choice of a starting point. For this reason, we do not include any empirical results for the GARCH model with jumps (the case of autoregressive jump intensity). However, if we would assume that jump intensity is constant over time then the calibration process converges.

1.3 Empirical Results

Our sample covers a period ranging from July 5, 2005 to July 5, 2008 (i.e. 750 trading days). Our sample is composed of the 92 UK stocks that were part of the FTSE100 index in the beginning of 2005 and which survived through the period of 6 years. We have deleted 8 stocks. In this work we will present empirical results of only 5 company from the FTSE100.

Daily stock closing prices (the last daily transaction price of the security) are obtained from Yahoo Finance database. Results similar to one's presented in the chapter can be verified for all FTSE100 companies. Dataset includes the daily stock closing prices of five companies traded on London Stock Exchange: AVIVA, BP, BT Group, Lloyd Banking Group, HSBC.

Table 1 shows the maximum likelihood estimates of GARCH(1,1) model with Jumps (with constant jump intensity, i.e. it is assumed that ) for log returns of the closing daily prices of the five companies for 3 years (July 5, 2005 - July 5, 2008).

Table 1

Maximum likelihood estimates of GARCH(1,1) model with Jumps for log returns of the closing daily prices

Fig. 1. GARCH model and GARCH model with Jumps performance for BP stock market closing daily prices (January 5, 2005 - December 31, 2010)

2. Individual Stock Volatility Modelling With GARCH--Jumps Model Augmented With News Analytics Data

2.1 Model description

We are going to analyze the impact of news process intensity on stock volatility by extending GARCH-Jump models proposed and studied in [1].

Let be the log return of a particular stock or the market portfolio from time to time . Let denotes the past information set containing the realized values of all relevant variables up to time . Suppose investors know the information in when they make their investment decision at time . Then the relevant expected return to the investors is the conditional expected value of , given , i.e.

The relevant expected volatility to the investors is conditional variance of , given , i.e.

Then

is the unexpected return at time . Following [1] we suppose that news process have two separate components: normal and unusual news,

The first term in (7) reflects the impact of normal news to volatility:

where be a sequence of i.i.d. random variables such that , is a nonnegative process such that

and

The second term in (7) reflects the result of unexpected events and describe jumps in volatility:

where , is a Poisson random variable with conditional jump intensity

where , and is the number of positive and negative news from to respectively. Therefore we directly take into account the qualitative data of news intensity and news sentiment score (source: RavenPack News Scores).

2.2 Empirical results

Table 2 presents maximum likelihood estimates of GARCH(1,1)-Jumps model augmented with news intensity for log returns of the closing daily prices for the five companies (January 5, 2005 - December 31, 2010). It shows that for all companies, i.e. the impact of the number of negative news on the growth of jump intensity much higher than one's of positive news.

Table 2

Maximum likelihood estimates of GARCH(1,1)-Jumps model augmented with news intensity for log returns of the closing daily prices

Note that the GARCH model with jumps (the null model) is a special case of the augmented GARCH-Jumps model (the alternative model). Therefore, to compare the fit of two models it can be used a likelihood ratio test (see e.g. [3]). Results of likelihood ratio test are in Table 3. For tree of five companies the alternative model is preferable with confidence level 5%.

Table 3

Results of the likelihood ratio test for the GARCH model with jumps and the augmented GARCH-Jumps model

Summary

In the paper we have examined two GARCH models with jumps. First we consider the well-known GARCH model with jumps proposed in [1]. Then we introduced the GARCH-Jumps model augmented with news intensity and obtained some empirical results. The main assumption of the model is that jump intensity might change over time and that jump intensity depends linearly on the number of positive and negative news. It is not clear whether news adds any value to a jump-GARCH model. However, the comparison of the values of log likelihood shows that the GARCH-Jumps model augmented with news intensity performs slightly better than "pure" GARCH or the GARCH model with Jumps.

Bibliography

1. J. M. Maheu and T. H. McCurdy. News arrival, jump dynamics, and volatility components for individual stock returns. Journal of Finance, 59(2):755-793, 2004

2. W. H. Chan and J. M. Maheu. Conditional jump dynamics in stock market returns. Journal of Business and Economic Statistics, 20(3):377-389, 2002.

3. D. R. Cox and D. V Hinkley. Theoretical Statistics. Chapman and Hall, 1974.

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