MEMORY AND VOLATILITY

9.2 MEMORY AND VOLATILITY

‘Memory’ represents quantitatively the effects of past states on the current one and how we use it to construct forecasts of future states. A temporal ‘indepen- dence’ is equivalent to a ‘timeless’ situation in which the events reached at one point in time are independent of past and future states. In this circum- stance, there is no ‘memory’ and volatility tends to be smaller. A temporal de- pendence induces time correlations, however, and thereby a process variance (volatility) growth. Time and memory, in both psychological and quantitative senses, can also form the basis for distinguishing among past, present and future. Objectively, the present is now; subjectively, however, the present also consists of past and future. This idea has been stated clearly by St Augustine (Confessions, Book XI, xx):

yet perchance it might be properly said, ‘there be three times; a present of things past, a present of things present, and a present of things future.’ For these three do exist in some sort, in the soul, but otherwise I do not see them; present of things past, memory; present of things present, sight; present of things future, expectation.

We are thus always in the present. But the present has three dimensions: (1) The present of the past.

(2) The present of the present. (3) The present of the future.

Technically, we construct the past with experiences and empirical observa- tions of the (price) process as it unfolds over time; our construction of the future (prices), on the other hand, must be in terms of indeterminate and uncertain events which are our best assessment of the future (price) at a given (filtered) present time. To a large extent, ‘technical analysis’ in finance uses such an approach. We

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

have different mechanisms for establishing things past and establishing things future. Our ability to relate the past and the future to each other – i.e. to make sense of temporal change – by means of a temporal ‘sequentiality’ is the prime reason for studying memory processes. For example, ‘remembering that stock markets behave cyclically’ might induce a cyclical behaviour of prices (which need not, of course, be the case). ‘Remembering’, i.e. recording the claims history of an insured over the last years, may be used to determine a premium payments schedule. The ‘health’ history of a patient might provide important clues to deter- mining the probabilities of his survival over time as he approaches ages where a population has a tendency to be depleted. In finance, these issues are particularly relevant. Rational expectations and its risk-neutral pricing framework squarely states that ‘there is no memory of the past’ since all current price values are ‘an estimate of future prices’. In this sense, in a rational expectations framework, the ‘present is the anticipation of the future at the known risk-free rate’. By the same token, the SDF (stochastic discount factor) claims essentially the same but with- out specifying a deterministic kernel for discounting future states. By contrast, charting approaches in finance state that there is a memory of the past which is used through modelling based on past data to determine current prices. The financial dilemma regarding rational expectations and charting is thus reduced to a memory issue and how it affects the process of price formation. Potential approaches can be summarized by:

(1) No memory in which case the past and the future have no effect on current prices. (2) Anticipative (rational expectations or SDF) memory in which current prices are defined in terms of a predictable ‘expectation’ of future prices. (3) Long-run memory, expressing the inter-relationship of past events and current prices and therefore the omnipresent effects of the past in any present.

For example, if speculative prices exhibit dependency, then the existence of such dependency would be inconsistent with rational expectations and would thus make a strong case for technical forecasting on stock prices (contrary to the con- ventional assumption that prices fluctuate randomly and are thus unpredictable). In addition, the notion of market efficiency is dependent on ‘market memory’. Fama (1970) defines explicitly an efficient market as one in which information is instantly reflected in the market price. This means that, provided all the past infor- mation F(t) at time t is used, a market is efficient if its expected price conditioned by this information equals the current price. Thus for a given time t + T and price p(t), we have, as seen previously: p(t) = E[ p(t + T ) |F(t)] . As time goes by, additional information is obtained and F(t) grows to include more informa- tion (a new filtration) F(t + 1) and thus p(t + 1) = E[ p(t + 1 + T ) |F(t + 1)] . This property of markets efficiency (assuming that it exists) underlies the martin- gale approach to finance, as we saw earlier. Without it, markets have a mea- sure of ‘predictability’ and can thus lead to some investor making arbitrage profits.

VOLATILITY , EQUILIBRIUM AND INCOMPLETE MARKETS