Stochastic Portfolio Theory Weights with Discrete Rebalancing

# 85579


# Under Fernholz's Stochastic Portfolio Theory, Which is the formula for optimal porfolio weights? (specific question details in main text)












Under Fernholz's Stochastic Portfolio Theory, Which is the formula for optimal porfolio weights in a process with weight's sizes' actualization done in discrete times when the individual assets behave as Geometric Brownian Motions? Original question, shorten because of text limits.

Intro

Imagine I have $N$ assets with prices that behave as Geometric Brownian Motions as defined in Wikipedia: $dS_{it}=\mu_i S_{it}+\sigma_iS_{it}dW_{it}dt$ for a constant drifts $\mu_i$, constant volatilities $\sigma_i$, and correlations $\rho_{ij}$ among assets not necessarily zero, so each asset is described between weights actualizations as:

$$S_{it}=S_{i0}e^{\left(\mu_i-\frac{\sigma_i^2}{2}\right)t+\sigma_iW_{it}}\tag{Eq. 1}\label{Eq. 1}$$ and $W_{it}$ is a Wiener process.

The portfolio $Z_t$ is built as a weighted combination $Z_t = \sum\limits_{i=1}^N w_iS_{it}$ where the weights $0\leq w_i \leq 1$ with $\sum\limits_{i=1}^N w_i = 1$ are kept constant between each actualization process, which will happen as follows: after some time intervals, not necessarily periodic, let call them $T_k$, that will been triggered when the maximum absolute deviation of an asset from the average become higher that some specific pre-define threshold, i.e. if the average of the assets prices at any time is $\bar{S}_t = \frac{1}{N} \sum\limits_{i=1}^N S_{it}$, and the predefined maximum deviation is $\Delta$, then every time it happens that if

$$t=t^*\ /\ \exists i\ /\ \max_i\{|S_{it}-\bar{S}_t|\}\geq \Delta \Rightarrow T_k:= t^* \text{ weight actualization}\tag{Eq. 2}\label{Eq. 2}$$

then a new weights actualization is realized, where each constant $\mu_i$, $\sigma_i$, $\rho_{ij}$, and $S_{0i}$ are actualized through some parameter estimation from historical values, and then the optimal weights formula is calculated for all the assets, and some sells/buys are done accordingly to equalize the portfolio for the next period such current proportions match the optimal weights for this period.

I know beforehand that \eqref{Eq. 2} must be ill-described, was my best attempt to represent in a rigorous math language the actualization process I am trying to represent. In this sense, as example, the weights $w_i$ should be instead $w_{iT_k}$, but it becomes cumbersome in the notation.

Question

Now with this kind of process, I wonder which is the formula for the optimal weights as it was developed by Fernholtz / Karatzas / Kardaras in Stochastic Portfolio Theory.

Motivation

I have tried to read the authors papers, but I got lost into the mathematical formalism of SDEs/Martingales/Arbitrage theory: please first just give the formula, then please comment the caveats like what is being optimized, assumptions that are different from the specific process I defined if needed, and with the formula you present, if it is possible that some weights get attributed zero importance $\exists i\,\, /\ w_i=0$: this generate issues since $N$ will be different from the pool of assets I study from the total amount of assets I pick for taking the average - be explicit if those quantities diverge.

Please try to keep it as simple as possible (take into account I already didn't understood the authors formal presentation).