Overview
The binomial tree is a short rate model (term structure models) that models the short rate of the interest rate curve and constructs an implied curve from the short rate.
Specification
Short Rate Binomial Tree The binomial short rate tree is a model that assumes that there is a current short rate, and at each subsequent period, the rate can move to one of two pre-specified rates.
- {% \sigma %} - volatility of the 1-period forward rate
- {% r_{1,L} %} - the lower value of the 1-period forward rate, one period from now
- {% r_{1,H} %} - the higher value of the 1-period forward rate, one period from now
Given the specified volatility, we must have
{% r_{1,H} = r_{1,L} \times e^{2 \sigma} %}
This assumes that the probability of moving to either the up rate or the down rate is 0.5.
Constructing the Rate Tree
- Choose the Period
The binomial tree is a discrete representation of the evolution of interests over a specified period.
Each layer of the tree represents the movement from one time period to the next. The analyst must construct the
tree to match the problem at hand. There are two time periods to be chosen:
- The overall size of the tree, the time period over which rates are modelled. This is typically constructed to match the length of a bond that is being priced. That is, if the bond is a 10 year bond, the tree will be constructed to model the short rate over 10 years
- The length of the period between layers of the tree. This length will determine the size of the tree. That is, if the overall tree period is 10 years, and each layer represents a single year, there will be 10 layers to the tree. Typically, the layers are designed so that each cash flow of the bond being analyzed will fall on a period representing a layer in the tree. As an example, for a 10 year bond that pays twice a year, the tree will be constructed to have 2 layers per year, or 20 layers.
- Obtain the Current Short Rate and Volatility - The short rate that is chosen should match the period between layers of the tree. That is, if each layer represents a year, the current rate should be chosen to be the current 1-year rate. If the current rate is given as {% r %}, then the standard deviation of the rate is {% \sigma \times r %}. That is, some standard deviation is measured or assumed, and the volatility is chosen to match it.
- Calibrate the tree - the difficult process is to decide what values to use for the upper and lower rates, {% r_H %} and {% r_L %} at each layer of the tree. Given the volatility assumption, once we know one value, the other value is determined.
Iteratively Discounting
Once the tree has been constructed, it can be used to value a bond with an embedded option. The process runs by defining the value of the bond for each node of the tree at the bonds maturity. Typically, the bond at maturity is just the value of the principal plus the last interest payment. This is a fixed amount that is the same at each node of the last level of the tree.
The next step is to discount each value back one level of the tree, using the interest rate and probability specified at each node of the tree. Note, this will be the value of the bond at that node, that is, if the state of the short rate evolves to that node.
For a callable bond, the value of the bond cannot exceed the threshold (strike price) of the bond, otherwise, it will be called. Therefore, for any node where the discounted value exceeds the strike, the bond is called and the value is equal to the strike. That is, any node where the value has exceeded the strike needs to be edited so that the value is equal to the strike at that node.
Next, if any cash flows are to occur at the given level, these need to be added to the value.
Lastly, this discounting process is iterated until you arrive at the beginning of the tree.
Valuing the Embedded Option
The value of the option is the difference value of the bond without the option, and then bond with the option.
{% OptionValue = BondValue - BondWithOption %}