Geometric Brownian Motion

Overview


Using the Stochastic Calculus, portfolios consiting of Assets Modeled as Geometric Brownian Motions can be modeled and optimized.

Asset Definition


The basic model of a GBM asset is:
{% \frac{dp}{p} = \mu dt + \sigma dW %}
Here

  • {% \mu %} is the instaneious average return
  • {% \sigma %} is the volatility

Setup


A portfolio is a collection of assets. The exact composition is given by a set of portfolio weights, {% w_1,...,w_n %}. The instanteous arithmetic portfolio return is then given by
{% \frac{d \pi}{\pi} = \sum w_i \frac{d p_i}{p_i} %}
where

  • {% \pi %} is the portfolio value
  • {% w_i %} are portfolio weights for each asset

Portfolio Growth


The variance of a portfolio of geometric brownian motions is then given by
{% Var(\pi) = \mathbb{E}[\sum_i w_i \sigma_i dW_i]^2 %}
{% Var(\pi) = \mathbb{E}[(\sum w_i \sigma_i dW_i)(\sum w_j \sigma_j dW_j)] = \sum_{i,j} w_i \sigma_{ij}w_j dt %}
Comparing with the GBM formulas
{% \mathbb{E}[ln \frac{\pi(t)}{\pi(0)}] = \sum_i w_i \mu_i t - \frac{1}{2} \sum_{i,j} w_i \sigma_{i,j}w_j t %}
Hence, the portfolio growth rate is maximized by maximizing
{% \vec{w}^T \vec{\mu} - \vec{w}^T \Sigma \vec{w} %}
(see Luenberger)

For a derivation of the weights, see Optimal Growth Derivation

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