In contrast to mean reversion in prices and returns, for which explanatory mod-els are predominantly microeconomic, mean reversion in volatility is mostly ex-plained in a macroeconomic way. Probably the best example is the large outburst of volatility during the Great Depression in the 1930’s. It is commonly believed that the deflation – loan crisis – contraction spiral described by Fisher (1933) caused the sustained increased volatility on the stock market.
Volatility is seen as the best proxy for the information stream that enters the market. In the efficient market model, the discounted expected value of the dividend stream is the stock price. New information changes the expectations of the dividends, thereby causing price fluctuation.
In this view, uncertainty about the dividend stream is the prime suspect to cause stock price volatility. The dividend stream is itself stochastic and the ne-cessity to estimate it may even exaggerate the stochasticity (Barsky and DeLong 1993). Memory in volatility arises then from memory in the underlying funda-mentals: bad news are followed by bad news, good news by good news.
Mean reversion in volatility is a much better established phenomenon than mean reversion in returns. One of the most frequently reported facts about volatility is that it exhibits long memory, or high persistence (Ding, Engle and Granger 1993, Engle and Patton 2001). That is, it possesses an autoregressive dynamic structure with slowly decaying autocorrelations (such that the sum of the autocorrelation coefficients is possibly not converging).
Figure 4.1 illustrates this for daily observations of the Dow Jones Industrial Average between January 2, 1902, and December 30, 2001. The upper panel shows the sample autocorrelation function of the squared daily returns, the lower panel the sample autocorrelation function for the absolute daily returns, both be-ing a measure for volatility. Both sample autocorrelation functions decay slowly
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exponentially, with significant coefficients up to lag 1,000 or 4 years for the squared returns and up to lag 2,000 or 8 years for the absolute returns. (For the difference in the measures see Ding, Engle and Granger 1993.)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0 0.05 0.1 0.15 0.2
lags sacf(r2 )
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0 0.05 0.1 0.15 0.2 0.25
lags
sacf(abs(r))
0.99 significance level
Figure 4.1: Sample autocorrelation functions for the square of the daily returns (upper panel) and for the absolute daily returns (lower panel) of the Dow Jones Industrial Average between January 2, 1902, and December 30, 2001.
Therefore, volatility is a natural field for the application of mean reversion models. The most commonly used volatility models: GARCH, stochastic volatil-ity models, and long-memory GARCH processes reflect this fact.
The volatility driver of the simplest ARCH specification (Engle 1982) has the structure
ht= ω + αε2t−1,
where ω is some positive real number, ε2t = (rt− µ)2 is the squared excess return, and α is a real number between zero and one. Intuitively, h may be called modelled volatility whereas ε2 is the measured volatility. The model is set up such that the expected value of the measured volatility is equal to the expected value of the modelled volatility. Therefore,
Et−kht= ω + αEt−kht−1.
where k ≥ 2. ARCH is thus a mean reverting model of class I according to Section 2.1 with α being the autoregressive parameter.
The volatility driver of the simplest GARCH specification (Bollerslev 1986) has the structure
ht = ω + αε2t−1+ βht−1,
where β is another parameter between zero and one and α + β < 1. Exactly as in the ARCH specification, the expected value of the measured volatility is equal to the expected value of the modelled volatility. Therefore,
Et−kht = ω + (α + β)Et−kht−1.
GARCH, too, is thus a mean reverting model of class I according to Section 2.1 with the sum of the coefficients being the autoregressive parameter.
In stochastic volatility models, the volatility driver is often specified as an Ornstein-Uhlenbeck process (Fouque et al. 2001):
dVt= λ(EV − Vt) dt + σ dWt,
where Vt is the volatility process at time t and λ is the coefficient that controls the “speed” of the mean reversion. The average time that the process V needs to revert to the mean EV is given by 1/λ. A discretization of this process yields an autoregressive process of order one with negative autoregressive parameter −λ.
Ornstein-Uhlenbeck processes of this type are mean reversion models of class II according to Section 2.1.
GARCH processes have been generalized to capture long memory behavior.
The fractionally integrated GARCH (FIGARCH) approach of Baillie et al. (1996) as well as the multi-component model of Ding and Granger (1996) are essentially ARCH models with an infinite number of lags of the measured volatility ε2 that has to be truncated at some arbitrary lag in estimations.
Baillie et al. arrive at the infinite ARCH order by fractional differencing as described in Section 2.1 in the context of mean reversion models of class III. Ding and Granger obtain it from adding up an infinite number of separate volatility drivers. The FIGARCH model is a mean reverting model of class III whereas the Ding and Granger approach is essentially class I. That the FIGARCH model has a class I representation shows the close relation of the model classes.
All of these models are, again, reduced form models and do not allow for an immediate economic interpretation. Three questions arise naturally:
1. What determines the level of volatility?
2. What determines the long memory of volatility?
3. Is there a connection between prices and volatility?
Except for the obvious fact that macroeconomic conditions influence the level of volatility, all of these questions remain largely elusive until today. Long mem-ory is a “stylized fact” of the data, yet there are but a few explanations around (for example, Kirman and Teyssiere 2000 explain long memory by agent interac-tion and herd behavior).
The third question was addressed by Fischer Black (1976). In his view, a decrease in a company’s equity capital has two effects on the company’s stock price. On the one hand, the lower value induces a decrease in expected future dividends and thereby causes a decline of the stock price. On the other hand, everything else being the same, the debt-equity ratio decreases and this increases the uncertainty about whether the company will be able to meet its liabilities in the future. This increases the volatility of the stock.1 This effect was later labeled the “leverage effect”.
This leverage effect would connect mean reversion in prices and returns with the volatility process. It is conceivable that this is another generator of mean reversion: transmission of mean reversion between volatility and returns.
A reduced form attempt to model this effect is the class of ARCH-in-mean models where the ARCH volatility driver described above also appears in the mean equation of the stock returns (Engle, Lilien, and Robins 1987). This may lead to a better fit to the data but it explains little. As to my knowledge, there is no explanatory model, for example a consumption smoothing approach, that connects mean reversion in returns to mean reversion in volatility.
In this thesis, I will concentrate on the first and second question posed above.
I will employ GARCH models, which are the most commonly used models in practice. I will show that changes in the parameter regime of a GARCH process will cause the sum of the autoregressive parameters to be close to one, when the changepoint is not accounted for in global estimations. This is commonly viewed as evidence of high persistence, or long memory. When accounting for change-points, however, I will show that the average data generating mean reversion of daily stock price volatility is quite fast, of the order of a few days.
This leads us to a partial answer of the second question: Long memory is caused by regime shifts, structural breaks in the time series. These change the level of volatility from regime to regime. We are left with two new questions:
“What causes the changes in regime?” and “What determines the short memory of volatility within regimes?”
1Fischer Black gives the following example: Suppose a company has 10 million worth of assets, 6 million capital and 4 million in bonds outstanding. Now let the value of the firm be cut in half but the own stock is shrinking to 2 million, while bonds drop only to 3 million. The debt-equity ratio has deteriorated significantly, which will cause an increase in volatility of the company’s stock.
One possible cause of regime changes is discussed in Chapter 7 in the context of exchange rates, where interventions by the monetary authorities are a good candidate. For stock prices, good candidates are harder to identify.
Substantially, short memory means that shocks to the stock price die out fast, the time scale involved is of the order of 5 to 10 days. It is conceivable that this is something like a weighted average of the investments horizons of the market participants. This would corroborate to the view expressed in Chapter 3 and in the introduction that investors implement their preferences, views and opinions in the data, and that the aggregate of this implementation over all investors constitutes the data generating process, in not too gross a deviation from certain rationality conditions. The investment horizon is definitely one of the most important parameters an investor has control over, besides his risk preference. At this point, however, there is no economic explanation available that would clearly link the time scale of volatility to the investment horizon of the agents. The issue was shrouded by the long memory ubiquitously measured in the data.
Finally, what are the effects of mean reversion in volatility? It could have been suspected that long memory in volatility carries forward uncertainty. Once a series of high fluctuations arrives, investors are shied away from the stock market, risk averse first. This causes a drop in prices, which, according to the leverage effect, causes another rise in volatility. In this view, long memory in volatility could turn the stock market into a catalyst of depressions. It must be emphasized, though, that this “causal” chain interprets correlations (as opposed to causal relations) found in the data quite liberally. The findings presented in this thesis contradict this picture anyway. When volatility has a short memory, it is much more likely that stock market volatility simply “measures” the state of the economy, without interfering much.