The Bond Pricing Equation
Overview
The bond pricing equation is a differential equation that is derived using logic similar to that of
derivative pricing to compute the price of a bond, given the volatilities of the relevant rates.
The basis of the equation is that there are a lot of bonds in the market, many more than there
are rates (unless one views the curve as an infinite set of rates....)
Given these facts, on may conclude that one could hedge the risk in a bond with other bonds.
The derivation then proceeds to calculate how to hedge the bond, and then computes a price from
the resulting hedge portfolio.
Derivation
The derivation follows the derivation of the Black Sholes equation in spirit. The derivation constructs a portfolio
for which all source of randomness has been hedged out, and then asserts that in an arbitrage free world, the
drift of this portfolio must equal the risk free rate.
In order to construct the hedge, we must know how many sources of randomness are present in our model. The most
basic fixed income model are the short rate models, that is, models for which the short rate is assumed to
follow a pre-specified stochastic equation, and that the rest of the term structure is derivable from
properties of the short rate. (Indeed, if we can price bonds based on the short rate, then we can find all
the other rates in the curve.)
We label the short rate, {% r %}. Then because we have only one source of randomness, we should be able to hedge out
that source of randomness from one bond, with only one other bond. That is, we will seek to construct a portfolio
that has zero risk from two different bonds. We will label the portfolio value as {% \Pi %}
We will assume that we hold one unit of bond 1, and {% \Delta %} units of bond 2. Then, our portfolio value
is given by.
{% \Pi = B_1 - \Delta B_2 %}
using
Itos lemma, we have
{% d\Pi = \partial{B_1} / \partial{t} \; dt + \partial{B_1} / \partial{r} \; dr + \frac{1}{2} \sigma_1^2 \partial^2{B_1} / \partial{r}^2 \; dt %}
{% - \Delta ( \partial{B_2} / \partial{t} \; dt + \partial{B_2} / \partial{r} \; dr + \frac{1}{2} \sigma_2^2 \partial^2{B_2} / \partial{r}^2 \; dt)%}
The choice of delta given below eliminates the dr terms in the above equation.
{% \Delta = \frac{\partial{B_1}}{r} / \frac{\partial{B_2}}{r} %}
{% d\Pi = r\Pi dt = r(B_1 - ( \frac{\partial{B_1}}{r} / \frac{\partial{B_2}}{r}) B_2) dt %}
{% (\partial{B_1} / \partial{t} \; dt + \partial{B_1} / \partial{r} \; dr + \frac{1}{2} \sigma_1^2 \partial^2{B_1} / \partial{r}^2 \; dt) / \partial{B_1} / \partial{r} = %}
{% (\partial{B_2} / \partial{t} \; dt + \partial{B_2} / \partial{r} \; dr + \frac{1}{2} \sigma_1^2 \partial^2{B_2} / \partial{r}^2 \; dt) / \partial{B_2} / \partial{r} %}
{% (\partial{B_1} / \partial{t} \; dt + \partial{B_1} / \partial{r} \; dr + \frac{1}{2} \sigma_1^2 \partial^2{B_1} / \partial{r}^2 \; dt) / \partial{B_1} / \partial{r} = a(r,t) %}
