Portfolio Duration
Overview
Continuous time risk
The duration measure also falls out of continuous time price modeling. If we start with the yield to maturity as a function
of time, y(t), then
the price equation can be restated as
{% P(t) = f(t, y(t)) = \sum_j e^{y(t)(u_i - t)} C_j %}
If we now we model y(t) as an Ito process., then we get from Itos formula
{% dP = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial y} dy + \frac{1}{2} \frac{\partial ^2 f}{\partial y ^2} dt %}
where
{% \frac{\partial f}{\partial y} %} is equal to -Duration {% \times P %}
(
Back pg.244)
This shows how the stochastic calculus methodologies can be used to also derive how to hedge a fixed income portfolio using
duration.
