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Stochastic Equations


At MVS, we consider the market to be semi-predictable: that is, on an instantaneous timeframe, the market behaves like a Brownian motion and its randomness can be modeled by a stochastic function, where each future outcome does not depend on historical outcomes. A variation of the Black-Scholes equation is utilized to model the "random walk" phenomenon. Below is an image of a stochastic process that depicts six randomly generated lines using the standard equation.

This graphic demonstrates that the variance or "spread" of the graph increases with time. This spread can be represented as a probability distribution, where different financial indicators contribute to the rate at which the spread increases with time. As an example, a lower value for market capitalization would increase the rate at which the future value of market price diverges from the starting point. Therefore, market capitalization is inversely related to the future variance of values generated by this function. Since variance is essentially synonymous with risk in this situation, a lower market cap company will tend to exhibit greater risk (and potential return) than that of a company with a higher market capitalization. The number of terms that can be used as future predictors of variance is essentially infinite and their magnitudes of contribution to the Brownian function differ greatly.

Random Verses Predictable Outcomes

A well-known shortcoming of many functions that describe Brownian type processes is that they are too insular and do not possess components that incorporate dramatic and atypical outcomes. For example, a stochastic equation that models the motion of an ion as it moves around in a vessel will not properly decribe how this movement will be altered if the temperature is increased dramatically or if the vessel is subjected to a high voltage differential. This is why a hybrid model that combines a stochastic function with linear, sinusoidal, and exponential functions is necessary to model the market over longer timeframes. See The Business Cycle.

The graph below is depicting Brownian scaling, where the time interval (a variable in this case) is decreasing at an exponential rate for the same equation. In a real world situation involving the stock market, the equation itself would change depending on the time interval and would begin to include sinusoidal, exponential, and polynomial functions as well, which would give rise to a directional and sometimes cyclical shape.

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