Asset Prices as Ito Processes
Overview
Modeling assets as
Ito Processes
utililizes the methods of
Stochastic Calculus
to calculate the relevant risk numbers.
Bond
The bond follows a simple equation. The dollar amount of interest in a given period is the interest rate times the
current value.
{% dB(t) = r(t) B(t) dt %}
We assume that r(t) is a constant, just called r. Then, the equation for the risk free rate becomes:
{% B(t) = B_0 e^{rt} %}
Without loss of generality, we set {% B_0 = 1 %}.
Stock
The stock price is modeled as a
Geometric Brownian Motion.
{% dS(t) = \alpha(S,t) S(t) dt + \sigma(S, t) S(t) dW(t) %}
When there are multiple stocks, then each stock price is modeled as an individual brownian motion, where each brownian motion
is correlated to the others through a correlation matrix.
Within a factor model framework, asset prices are modeled as the sum of a set of brownian motions. When using a factor modeling
approach, one can assume that the brownian motions are uncorrelated, becuase asset correlations can then be assumed to
arise from a dependence on common risk factors.
Portfolio Mechanics
Portfolios are key to derivative pricing. The fundamental insight is that the cost of trading a self financing portfolio
which replicates the derivative payoff is the price of the derivative (according to the law of one price).
Within the context of stocahstic differential equations, it is therefore necessary to be able to state the equations of
how a portfolio evolves, given the evolution of the underlying assets. (For a complete treatment, please see:
portfolio dynamics)
First, we assume we have a
vector of assets {% \vec{S} %}, each of which can be modeled
by the standard geometric brownian motion (given above). Now we assume that we have a portfolio of these assets, where we
specify {% \vec{h} %} is a vector of the units of each asset held, and {% \vec{\pi} %} is a vector representing the
portfolio weights invested in each asset. (here, the sum of the component of {% \vec{\pi} %} should equal 1.)
We define the portfolio value to be V, and hence we have
{% V(t) = \vec{h}(t)^T \times \vec{S}(t) %}
A self financing portfolio is one where
{% dV(t) = h(t) dS(t) %}
Here we have suppressed the vector notation.
Modeling Techniques
Additional Topics
