Modelling Shocks from extreme events in finance

Many people model trend in the stock market using either Autoregressive (AR), Moving Average (MA) or Autoregressive Moving Average (ARMA). In these models, shocks to the stock price in prior periods help to determine the price in the current period. Shocks in the more distant past are generally assumed to have less influence on current stock price than that of shocks in the more recent past. However in simple models, the standard deviation of the shocks in each period are assumed to be the same (e.g., σ²_ε,t=σ²_ε ∀ t).

GARCH models estimate volatility of these shocks in each period as a function of both the shocks and their standard deviations in prior periods. Generally, more volatility in the recent past will result in more volatility in the present.

A paper by Bauwens and Storti (2009) examines an interesting phenomenon: “the persistence of the volatility process tend to decrease after extreme events such as those observed in October 1987 and September 2001.” Standard GARCH models equally weight all shocks regardless of whether or not they occur during extreme events. In order to incorporate the observation that volatility decreases after extreme events, the authors propose a Weighted GARCH (WGARCH) model. In their model, shocks are modelled as follows:

• u_t = z_t[s_t-dh_1,t + (1-s_t-d)h_2,t]^1/2
• h_k,t=a_0k + Σ_{i=1 to p} a_iku²_t-i + Σ_{j=1 to q} b_jkh_k,t-j for k=1,2

The term s is in essence a weight which determines the volatility. If h_1,t represents the high volatility period and h_2,t represents the tranquil period, the term s balances the regime in which we are currently situated. For instance, when s approaches 0 we are in the tranquil regime and when s approaches 1, we are in the volatile regime. Model parameters can be estimated using MLE or Bayesian approaches.

This WGARCH model seems to be useful when the persistence of volatility varies into two different types of regimes. It will be interesting to see if the model produces a good fit after analyzing the stock returns after the most recent financial meltdown.

• Luc Bauwens and Giuseppe Storti (2009) “A Component GARCH Model with Time Varying Weights“, Studies in Nonlinear Dynamics & Econometrics: Vol. 13: No. 2, Article 1.