We can decompose the returns process R_{t} as follows:

While the left hand side of the equation is essentially unforecastable, both of the right-hand-side components of returns display persistent dynamics and hence are forecastable. Both the signs of returns and magnitude of returns are conditional mean dependent and hence forecastable, but their product is conditional mean independent and hence unforecastable. This is an example of a nonlinear “common feature” in the sense of Engle and Kozicki (1993).

Although asset returns are essentially unforecastable, the same is not true for asset return signs (i.e. the direction-of-change). As long as expected returns are nonzero, one should expect sign dependence, given the overwhelming evidence of volatility dependence. Even in assets where expected returns are zero, sign dependence may be induced by skewness in the asset returns process. Hence market timing ability is a very real possibility, depending on the relationship between the mean of the asset returns process and its higher moments. The highly nonlinear nature of the relationship means that conditional sign dependence is not likely to be found by traditional measures such as signs autocorrelations, runs tests or traditional market timing tests. Sign dependence is likely to be strongest at intermediate horizons of 1-3 months, and unlikely to be important at very low or high frequencies. Empirical tests demonstrate that sign dependence is very much present in actual US equity returns, with probabilities of positive returns rising to 65% or higher at various points over the last 20 years. A simple logit regression model captures the essentials of the relationship very successfully.

#### Sign Dynamics Driven By Volatility Dynamics

Let the returns process R_{t} be Normally distributed with mean m and conditional volatility s_{t}.

The probability of a positive return Pr[R_{t+1} >0] is given by the Normal CDF F=1-Prob[0,f]

For a given mean return, m, the probability of a positive return is a function of conditional volatility s_{t}. As the conditional volatility increases, the probability of a positive return falls, as illustrated in Figure 1 below with m = 10% and s_{t} = 5% and 15%.

In the former case, the probability of a positive return is greater because more of the probability mass lies to the right of the origin. Despite having the same, constant expected return of 10%, the process has a greater chance of generating a positive return in the first case than in the second. Thus volatility dynamics drive sign dynamics.

Figure 1

**Email me at jkinlay@investment-analytics.com.com for a copy of the complete article.**

The models are phenomenological and require input of the parameters and so can never be predictive for all scenarios. Sometimes (as in Hull’s book) you find that people use a model such as (if I remember correctly) Black’s model (rather than Black and Scholes) and forgive me if I got the name wrong, to model for example products that depend on interest rates (which according to Hull have mean reversion and so are not behaving like a Brownian motion) and somehow this model “works out” or “works better” than the B-S because although not intended for this it makes the right outputs. The point to remember is that all of these models are phenomenological and depend on estimates of the inputs such as “historical volatility” and volatility of volatility, so that they are both of academic interest and of engineering interest (see how well they might work in practice). As an analogy consider turbulence modelling which makes some assumptions to close the equations – it works for some geometries but needs a lot of adjustment or fails for others. So I guess the thing to do is construct and study these models and then somehow evaluate them for different scenarios and issue recommendations as to how and when to use them? More than one model (and quite differently motivated models) may give the same outputs. In recent years ways of speaking in averages (fuzzy logic) have been as effective as complex control theories in practical engineering.

Daniel HowardDirector at Howard Science Limitedposted 54 minutes ago

The paper evaluates the performance of the model in trading S&P options.

More on the conference here: http://web.incisive-events.com/rma/2009/07/quant-congress-usa/index.html

More details on my Quantitative Investment and Trading blog to come:http://quantinvestment.blogspot.com/

Jonathan Kinlay, PhD (jkinlay@investment-analytics.com)