BDT make use of a binomial tree to specify their lognormal model. Within the binomial tree they are able to match two inputs: the interest rate and volatility term structures. This is done using the location and spacing of the nodes at each time point.
To match three input values, one could use a trinomial tree. However, to avoid the additional complexity of a trinomial tree, BK approach this problem with a binomial tree, but vary the time spacing during its life. This introduces another degree of freedom, allowing all three inputs to be matched. The computational simplicity of a binomial tree is maintained and the risk-neutral probabilities are ½.
If the input functions defining the model, i.e. ¼ ( t ), ( t ) and ƒ ( t ) are known, the binomial tree of short-term interest rates is constructed so as to match these values at each time step. The tree has the following specifications:
At each time, the (vertical) spacing of the nodes must match the local volatility (volatility of the short-term interest rate). Since volatility is a function of time only, spacings for a given time are equal.
The drift of the nodes from one time to the next is determined by the target rate.
The time (horizontal) spacing differs over the life of the binomial tree. This time spacing is calibrated to the mean reversion speed.
Define the following variables :
_{ n } = t _{ n +1 } ˆ’ t ^{ n } | - | time period between two consecutive time nodes, |
_{ n } = ( t _{ n } ) | - | mean reversion speed at time t = n , |
ƒ _{ n } = ƒ ( t _{ n } ) | - | local volatility at time t = n . |
Mean reversion is defined as the speed with which the short-term interest rate tends towards the target rate. As the short-term interest rate gets closer to this target rate, the local volatility decreases. Hence, the mean reversion may be equated to the rate of change of local volatility, which is represented by:
For positive mean reversion ƒ _{ n } ˆ _{ n } < ƒ _{ n ˆ’ 1 } ˆ _{ n ˆ’ 1 } and gives the percentage decrease in volatility from time period _{ n ˆ’ 1 } to _{ n } . From (9.1) we may find _{ n } as a function of _{ n ˆ’ 1 } and the speed of mean reversion. Hence, at each time node, we may determine the size of the next time step, dependent on the speed of mean reversion. Using equation (9.1) we have:
which is a quadratic polynomial in with roots:
Since by definition only one of the roots is an admissible solution to (9.2) and:
Hence the time spacing is dynamically constructed, with the next time step size determined at each node after all its associated variables ƒ _{ n } , _{ n } etc. have been determined. The initial time step _{ } is chosen according to the required accuracy. Small _{ } equates to very fine time spacing, which produces more accurate results. For positive mean reversion speed, the time spacing decreases through time, and the higher the speed of mean reversion the more pronounced this decrease.
If we already have the model outputs, that is the interest rate, volatility and cap term structures, we need to find the corresponding values of the model inputs, ¼ ( t ), ( t ) and ƒ ( t ). Here, time is divided into segments which are subdivided into time steps. The values of ¼ , and ƒ , applicable for the first time segment, are chosen so as to match the outputs at the end of this segment. Similarly ¼ , and ƒ applicable during the second time segment are chosen such that the outputs are matched at the end of this segment. Using this methodology, we find the implied target rate, mean reversion speed and short-term interest rate volatility for each time segment. These implied values do not specify the real-world evolution of the short-term interest rate, but they do specify a short-term interest rate process in a one-factor world which produces the required security prices.
When the model outputs change, that is the interest rate and volatility term structures observed in the market shift, the tree needs to be recalibrated to determine new parameters of the implied process.
Ideally, we would like to determine a general interest rate process as a function of several parameters. Re-estimation of the process should yield the same parameters. This type of model would be a true description of the interest rate process and could be used to give valuations at any time.