Overview
Stochastic Portfolio Theory applies the tools of stochastic calculus to the dynamics of a portfolio of assets. The discussion here follows that of Fernholz.
Asset Definitions
For the purposes of stochastic portfolio theory, we define the price of a stock to follow the following Ito process dynamics:
{% d log S(t) = \alpha(t) dt + \sum_{j=1}^n \sigma_j dW_j(t) %}
Note, this definition defines a stock as being driven by mulitple
Brownian Motions.
The brownian motions here are independent. (no
correlations)
This departs a bit from the standard defintion, where each stock is driven by a single Brownian motion, and the Brownian motions are correlated. The definitions are equivalent though, as a sum of Brownian motions is a Brownian motion, and correlations will arise when the Brownian motions that driven individual stocks are shared.
Applying Itos lemma givens
{% dS(t) = S(t) (\alpha(t) + \frac{1}{2} \sum \sigma_j^2(t)) dt +S(t)\sum\sigma_j(t)dW_j(t) %}
Which can be re-written as
{% \frac{d S(t)}{S(t)} = \alpha'(t) dt + \sum \sigma_j(t) dW_j(t) %}
Market
A market is defined to be a set of assets that are driven by the same set of Brownian motions, here each asset is indexed by {% i %}
{% d log S_i(t) = \alpha_i(t) dt + \sum_{j=1}^n \sigma_{i,j} dW_j(t) %}
In this definition, there is a common set of Brownian motions driving all the assets. Nothing in the definition precludes
any of the {% \sigma_{i,j} %} from being equal to {% 0 %}. As noted above, even though we assume the Brownian motions to be uncorrated,
correlations will exists among the assets through the sharing of the Brownian motions.
(see multiple brownian motions for information regarding multiple processes and correlations)
Portfolio
Given a set of stocks, a portfolio is an adapted vector process, {% \pi(t) %} such that {% \pi_i(t) %} represents the percentage of the portfolio invested in asset {% i %}. The following represents the portfolio constraint.
{% \sum_{i=1}^n \pi_i(t) = 1 %}
The value of the portfolio is labeled {% Z_{\pi}(t) %}. The amount of money invested in the {% i^{th} %}
asset is {% \pi_i(t)Z_{\pi}(t) %}.
When the price of the {% i^{th} %} asset {% S_i(t) %} changes by {% dS_i(t) %}, the change in the value of the portfolio is
{% \pi_i(t)Z_{\pi}(t) \frac{dS_i(t)}{S_i(t)} %}
which then implies
{% \frac{dZ_{\pi}(t)}{Z_{\pi}(t)} = \sum \pi_i(t) \frac{dS_i(t)}{S_i(t)} %}
The market portfolio is defined to be the one with weights
{% \mu_i(t) = \frac{X_i(t)}{X_1(t) + ... + X_n(t)} %}
Portfolio Growth
The Portfolio Growth is then given by
{% d log Z_{\pi}(t) = \alpha_{\pi}(t)dt + \sum_{i,j}^n \pi_i(t) \sigma_{i,j}(t) dWj %}
where we define {% \alpha_{\pi}(t) %} as
{% \alpha_{\pi}(t) = \sum_i \pi_i(t) \alpha_i(t) + \frac{1}{2} (\sum_i \pi_i(t)\sigma_{ii}(t) - \sum_{i,j} \pi_i(y) \pi_j(t) \sigma_{ij}(t) ) %}
Optimizing portfolio is akin to maximizing {% \alpha_{\pi}(t) %} (because the last term in the above is a
martingale)