# 9.3 Matching the lognormal distribution

## 9.3 Matching the lognormal distribution

We have assumed the short-term interest rate has a lognormal distribution at any time horizon. This means we require only a mean and standard deviation to fully specify its distribution. However, in the BK model, three factors are required to describe the short-term interest rate process - the target rate, mean reversion speed and local volatility. This means that for a given time horizon, the solution is not unique and the distribution of short- term interest rates may be matched by a family of possible processes.

These processes will differ in their mean reversion and local volatility characteristics. Strong mean reversion means a move away from the target rate is quickly reversed , which is not the case for weaker mean reversion. Hence, a narrow (wide) distribution of the short-term interest rate in the future may result from either strong (weak) mean reversion or low (high) local volatility.

## 9.4 Conclusion

In a simple and concise extension of the BDT model, BK are able to eliminate one of its most frequently cited shortcomings - the direct but artificial link between the current volatility term structure and future values of short-term interest rate volatility. BK introduce a third time-dependent variable, reversion speed, which allows an additional degree of freedom. Now the interest rate and volatility term structures as well as cap prices can be included in the calibration procedure.

Empirical results of calibration exercises for this model are not widely available. However, based on results of other models [1] attempting to include all three term structures (interest rate, volatility and cap prices), one could suspect an over-parameterisation may result, with future volatility term structures taking on unreasonable shapes .

[1] For example the extended-Vasicek Hull White model discussed in Chapter 7.

# Chapter 10: The Ho and Lee Model

Models studied in the previous chapters specify the movement of the short- term interest rate and thereby endogenously determine the form of term structure (including its initial value). Ho and Lee (HL) [ 27 ] developed a model which takes as input, the initial interest rate term structure and derives its subsequent stochastic evolution. Hence the theoretical zero coupon bond prices (that is, those produced by the model) will be exactly consistent with those observed in the market.

HL use all information within the current observed term structure to price contingent claims in such as way as to ensure that profitable arbitrage is precluded.

## 10.1 Assumptions

The assumptions made by HL are the standard assumptions for a perfect capital market in a discrete time framework.

Assumption 1

The market is frictionless, i.e. there are no taxes or transaction costs and securities are perfectly divisible.

Assumption 2

In a discrete time framework each time period is taken to be one unit of time. Hence a zero coupon bond with term to maturity T pays \$1 at the end of the T th time period (taken from valuation time).

Assumption 3

The bond market is complete, with a bond maturing at the end of each time period n , n =0, 1, 2, .

Assumption 4

At each time period n , there are a finite number of possible states of the world. At time n , state i , denote the equilibrium price of a T -maturity zero coupon bond as P ( n ) i ( T ). This function is termed a discount function. At any time n , state i , the interest rate term structure is fully described by a series of discount functions.

By its definition as a discount function, P ( n ) i ( ·) must satisfy certain conditions. That is: