# 4.4 Conclusion

## 4.4 Conclusion

In developing a two factor model, LS overcome one of the most frequently cited criticisms of one factor models: the perfect correlation of instantaneous returns on bonds of all maturities. The model produces closed-form option prices for the case of stochastic volatility; this is a highly desirable feature few other models can produce. Additionally, the very flexible functional form of the model allows for very complicated shapes of the yield curve to be obtained with relative ease. However, this flexible functional form makes calibration rather difficult. The flexible functionality allows almost any market- observed term structure to be fitted, but this does not necessarily ensure meaningful term structure dynamics. One of the inevitable side effects of increasing the numbers of factors is the increased complexity; here, pricing of a simple European option requires evaluation of the bivariate non-central chi-square distribution.

# Chapter 5: Langetieg's Multi-Factor Equilibrium Framework

The term structure of interest rates is embedded in the macro-economic system and is related to various economic factors. For this reason, Langetieg [ 36 ] proposes a model that can accommodate an arbitrary number of economic variables . The model is essentially an extension of Vasicek's term structure model [ 50 ], studied in Chapter 1, with multiple sources of uncertainty.

## 5.1 Underlying assumptions

Langetieg makes certain assumptions which allow for a mathematically tractable, intuitively sound model:

Assumption 1

The set of stochastic economic factors which are related to the interest rate term structure follow a joint elastic random walk.

Assumption 2

The instantaneous risk-free rate of interest may be expressed as a linear combination of these factors.

Assumption 3

The market prices of risk of the factors are deterministic, that is, they are either constants or function of time only.

The assumption of an elastic random walk means that the Vasicek model, which incorporates a univariate elastic random walk, is extended to a multivariate elastic random walk. Vasicek does not assume the functional form of the bond price, but derives it from the following assumptions (which apply to Langetieg's model as well):

• Bond prices are functionally related to certain stochastic factors.

• These underlying factors follow a specific stochastic process.

• The markets are sufficiently perfect to allow for a no arbitrage equilibrium to be reached.

## 5.2 Choice of generating process

There exists empirical evidence to support both the random walk and the elastic random walk as generating processes for stochastic factors within a macro-economic system. Therefore we may conclude that the generating process for the short- term interest rate is adequately described by:

where a ( t ), b ( t ) and ƒ ( t ) are either constants or functions of time. a ( t )+ b ( t ) r ( t ) is the stochastic [1] instantaneous drift and ƒ ( t ) the deterministic instantaneous variance of r ( t ). The behaviour of r ( t ) is determined by the value of b ( t ) since, for:

• b < 0, r tends to ˆ’

• b = 0, the generating process for r simplifies to a random walk,

• b > 0, r explodes in finite time since it is repelled by the level ˆ’ .

Under the random walk generating process, short-term interest rates drift to positive and negative infinity with probability one. The elastic random walk with b < 0 eliminates this problem. It does, however, allow transient occurrences of negative interest rates and hence is not an appropriate model when short-term interest rates are close to zero. Negative interest rates are completely eliminated by setting the variance coefficient proportional to r ± , ± > 0. In the case of the Cox, Ingersoll and Ross model [ 18 ], ± = ½ (see Chapter 2). This creates a natural reflecting barrier at r = 0, but introduces mathematical complexity which is difficult to implement in the multivariate case where the underlying factors are stochastic. Langetieg makes use of an elastic random walk process, with the assumption that the short-term interest rate is sufficiently above zero to make the probability of negative interest rates, in finite time, negligible.

[1] It is stochastic due to the functional dependence on r ( t ).