Stochastic But Trendy

Market prices are deeply stochastic, but trendy nevertheless. The Analytix perspective integrates this seeming contradiction in a consistent statistical framework. Micro-returns are indeed random, but the nature of the randomness is always changing.

Given this evolving randomness, returns are addressed as a non-stationary random variable. Like an ever-changing probability distribution precipitating a series of random draws. The distribution itself is hidden, materializing only in the direction, range and magnitude of successive random draws.

What's in a price?

In this context, a price history is a cumulative path of these successive draws. Uptrends accumulate from distributions with positive mean; downtrends from distributions with negative mean. [**Rain Simulation**]
Distributions of negative mean generally have greater variance than distributions of positive mean, which is to say, downtrends are more volatile.

Up or down, the variance is large relative to the mean. So distinguishing a state of positive-mean or negative-mean distributions is not easy. To see how difficult this is, compare these distributions from the recent bull and bear markets...

The two distributions "look" almost identical, both in general shape, and in the respective means (+11/100 of 1% and -9/100 of 1%) seeming almost indistinguishable from zero. Yet the cumulative outcomes (=+55% and -30%) are vastly different. The difference between means is statistically significant beyond any common threshold. The probability these two samples were drawn from some one enduring probability set (as implicitly assumed in "long term" studies) is vanishingly small.

As individual “draws” are essentially random, actual prediction is not reasonable. What is reasonable is to monitor the returns stream (and other data) for indications of the probability state governing current behavior. Market states have persistence. Absent evidence of change, today’s market state is usually not far different from yesterday’s. Tomorrow’s state will be hard enough to estimate tomorrow, and almost impossible to estimate today.

**Non-Determinism**

It's not just the market future that is stochastically indeterminate. In the stochastic context, even the past is not deterministic. Seen as an accumulation of random draws, an actualized price history is but one of many possible paths that might have emerged from the given probability states.

So even if we could “re-wind” history to run it again, we would be replaying not a specific sequence of prices, but a new sequence of draws from the evolving probability sets. New trials of the same probability sets would generate new, alternative price paths. Such paths would show similar general direction and volatility, but individually distinct fluctuations. Open
**Rewind** for simulations on this theme.

**Knowability**

In this view, the future is unknowable… not for lack of data or understanding, nor because the market is purely efficient. The future is unknowable because the future is existentially indeterminate. The future -- like the past -- is a probability set from which some one outcome will emerge.

Accepting what we cannot know, we can deal with the limited and probabilistic things we can know. The critical question is; What is the current state of the probabilities distribution? When we can answer that question (which is not always), we can assess probable and improbable outcomes. Avoiding predictions and projections, we live in a universe of expected values, confidence levels, probabilities.

This probabilistic context underlies various models, metrics, simulations and strategies offered at this site.

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