Hedging Fixed Income Risk
Overview
Hedging refers to the process of trading instruments in
a portfolio in order to make the portfolio immune to
fluctuations in the interest rate curve.
As a general rule, a portfolio of long only fixed income
instruments is only hedged if all the instruments are
floating rate instruments. This is because any simple bond will
lose value as rates go up, and gain if rates go down, so a portfolio
with these types of bonds will lost value as rates go up. The only
instrument which does not lose value is a floating rate instrument.
Therefore in general, you will only hedge a portfolio that
contains both assets and liabilities, or a portfolio with
optionality.
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{\partial ^2 f}{\partial y ^2} dy %}
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.
