PROCESS VARIANCE AND VOLATILITY

9.4 PROCESS VARIANCE AND VOLATILITY

Say that a stock price has a time-variant mean and standard deviation given by (µ t ,σ t ). In other words, if we let z t

be a standard random variable then the record of the series can be written as follows : x t =µ t +σ t z t . When the standard deviation is known, the time series can be used to estimate the mean parameter (even if it is time-variant). When the variance is not known, it is necessary to estimate it as well. Such estimation is usually difficult and requires that specific models describing the evolution of the variance be constructed. For example, if we standardize the time series, we obtain a standard normal probability random variable for the error as seen below,

t −µ t

∼ N (0, 1)

We can rewrite this model by setting ε t =σ t z t where the error has a zero mean (usually obtained by de-trending the time series). If the standard deviation is not known, then of course the error is no longer normal and therefore there are statistical problems associated with its estimation. Models of the type ARCH and GARCH seek to estimate this variance by using the residual squared deviations. There are many ways to proceed, however, from both a modelling and a statistical point of view, rendering volatility modelling a challenging task. Empirical finance research has sought to explain volatility in terms of the randomness of incoming information and trading processes. In the first instance, volatility is explained by the effects of external events which were not accounted for initially, while in the latter instance it is based on the behaviour of traders, buyers and sellers that induce greater volatility (such as herd or other systematic and unsystematic behaviours). The number of approaches and statistical techniques one may use for estimating volatility vary as well. For this reason, we shall consider some simple cases, although numerous studies, both methodological and empirical, abound. Many references related to these topics are included as well in the ‘References and additional reading at the end of the chapter.

PROCESS VARIANCE AND VOLATILITY

Example

Let R t +1 , the returns of a firm at time t + 1, be unknown at time t and assume that mean returns forecast at time t are given by the next period expectation µ t =

E t (R t +1 ).This means that the conditional expectation of the one-period returns ‘forecast’ can be calculated. Such a model assumes rational expectations since current returns are strictly an expectation of future ones. At present, hypothesize

a model for the error, given by ε t – also called the innovation. Thus, a one-period ahead return can be written by:

R t +1 =µ t +ε t +1

The volatility (or the return variance) is by definition:

σ 2 2 2 t 2 =E t t +1 t =E t t +1

which is presumed either known or unknown, in which case it is a stochastic volatility model. A simple variance estimate can be based on statistical historical averages. That is to say, using closing daily financial prices P t (spot on stocks for example) and in particular using daily proportional price change: R t = ln P t − ln P t −1 , we obtain (historical) estimates for the mean and the variance:

By the same token we can use the daily range (or a Hi, Lo statistic) for volatil- ity estimation. This is justified by the fact that for identically and independently distributed (iid) large sample statistics, the range and the variance have, approxi- mately, equivalent distributions. Then,

where (H t , L t ) are the high and low prices of the trading day respectively. Historical estimation can be developed further by building weighted estimation schemes, giving greater prominence to recent data compared to past data. In other words, say that a volatility estimate is given by a weighted sum of squares of past returns:

σ ˆ 2 2 t 2 =E t t +1 0 + w i (t)R t +i−1

i =1

where w i (t) denotes the weight at time t associated to past returns. Variance models may be differentiated then by the weighting schemes we use. For the na¨ıve historical model, we have:

1 σ ˆ 2 2 1 t 2 =E t t +1 R

t +i−1 ;w i (t) =

T For an exponential smoothing of volatility forecasts, as done by Riskmetrics, we

i =1

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

have:

t =E t t +1 θ R t +i−1 ;w i (t) = θ , 0≤θ ≤1

σ ˆ 2 2 2 t 2 = θ i −1 R t +i−1 =R t −1 +θ θ i −1 R t −1+i−1

i =1

i =1

and we obtain the recursive scheme:

σ ˆ 2 2 t 2 =R t −1 + θ ˆσ t −1

Extensions were suggested by Engle (1987, 1995) (ARCH models) and Bollerslev (GARCH models). There are other estimation techniques such as nonparametric models that are harder to specify. In these cases, the weighting function w(x t −i ) expresses a memory based on a number of state variables. Such approaches are in general difficult to estimate. The importance of ARCH and GARCH modelling in financial statistics cannot be overestimated, however. Econometric software makes it possible to perform such statistical analyses with great ease, using general models of the variance. For further study we refer to Bollerslev (1986), Nelson and Foster (1994), Taylor (1986), and Engle and Bollerslev (1986).

Example: Stochastic volatility and process discretization

A stochastic volatility model can be obtained by discretization of a plain vanilla continuous-time model. This demonstrates that in handling theoretical models for practical ends and discretizing the model we may also introduce problems associated with stochastic volatility. Say that an asset price is given by the often- used lognormal model:

dS S = µ dt + σ dW

where µ is asset rate of return and σ is its volatility. An application of Ito’s differential rule to Y = ln S, yields:

k Z k ;Z k ∼ N(0, 1); k = 1, 2, . . .

A linear regression provides an estimate of

ity be presumed known and constant for the estimate to be meaningful. If the volatility is not known but it is also estimated by the data at hand, then another regression is needed, supplied potentially by the ARCH–GARCH apparatus and providing a simultaneous estimation of the model’s parameters. Such estimation

281 subsumes, however, a stochastic volatility (since the volatility is error-prone and

IMPLICIT VOLATILITY AND THE VOLATILITY SMILE

estimated using historical values). As a result, discretization, even when it is properly done, can lead to estimation problems that imply stochastic volatility.