Term Structure - Discrete Single Factor Models
Overview
Discrete factor short rate models are typically built using a
tree model,
which is familiar from standard
derivative pricing.
Binomial Tree
The binomial tree short rate model, models the current spot rate for a given time frame. For example, it
models the evolution of the 1 year rate over time. Each node of the tree represents a given 1 year rate.
The tree starts with a given rate (the current short rate) and a volatility.
{% \sigma %} = the volatility of the rate.
{% r_{up} = e^{2\sigma} \times r_{down} %}
This relationship assumes that the probability of up and down are both 0.5.
(
Fabozzi pg 384)
Pricing
Once a tree has been constructed, it can be used to price fixed income payments. The fixed income payments must occur
at the period of one of the levels of the tree. That is, if the perdiod specified for the tree is 1 year,
(each layer of the tree represents another year out), then only payments that occur at round numbers of years can be priced.
The pricing works through
risk neutral valuation.
Consider a path trhough the tree. Starting at {% r_0 %}, the rate environment moves up to {% r_{1,H} %}. Next, the rate moves down
to {% r_{2,HL} %} (Here we specify a node by the level in the tree, and then the path to get there, here up and then down)
Because the up path and down path each have a risk neutral probability of 0.5, each path has a probability of {% \frac{1}{2}^n %}.
The discount rate is the multiplication of each rate along the path. In this case,
{% r = r_0 \times r_{1,H} \times r_{2,HL} %}
{% Price = \mathbb{E}(r) = \sum_i r_i %}
Here we are pricing a single dollar paid at the given date.
Calibration
The next step is to choose the rates. That is, we need to choose {% r_{down} %} and then {% r_{up} %} would
follow from the relationship above. The value of {% r_{down} %} chosen needs to be the one that replicates
current bond prices correctly.
Examples
