Daniel J. Duffy - Numerical Methods in Computational Finance

(3.8)

In the cases and and when P and Q are constants, this equation is linear, and an exact solution can then be found. We reduce Equation (3.8)to a linear one by the substitution to produce the following linear ODE:

(3.9)

The Riccati ODE is a non-linear ODE of the form:

(3.10)

This ODE has many applications, for example to interest-rate models (Duffie and Kan (1996)). In some cases a closed-form solution to Equation (3.10)is possible, but in this book our focus is on approximating it using the finite difference method.

We now discuss the relationship between the Riccati equation and the pricing of a zero-coupon bond P ( t , T ), which is a contract that offers one dollar at maturity T . By definition, an affine term structure model assumes that P ( t , T ) has the form:

Let us assume that the short-term interest rate is described by the following stochastic differential equation (SDE):

where is a standard Brownian motion under the risk-neutral equivalent measure and and are given functions.

Duffie and Kan proved that P ( t , T ) is exponential-affine if and only if the drift and volatility have the form:

where and are given functions of t .

The coefficients A ( t , T ) and B ( t , T ) in this case are determined by the following ordinary differential equations:

(3.11)

and:

(3.12)

The first Equation (3.11)for B ( t , T ) is the Riccati equation and the second one (3.12)is solved easily from the first one by integration.

ODEs can be used as simple models of population growth, for example, by assuming that the rate of reproduction of a population of size P is proportional to the existing population and to the amount of available resources. The ODE is:

where r is the growth rate and K is the carrying capacity . The initial population is . It is easy to check the following identities:

Transformation of this equation leads to the logistic ODE:

(3.13)

where n is the population in units of carrying capacity and measures time in units of 1/ r .

For systems, we can consider the predator-prey model in an environment consisting of foxes and rabbits:

(3.14)

where:

	=	number of rabbits at time t
	=	number of foxes at time t
	=	birth rate of rabbits
	=	death rate of rabbits
	=	unit birth rate of rabbits
	=	death rate of foxes
	=	birth rate of foxes
	=	unit birth rate of foxes.

The ODE system (3.14)is a model of a closed ecological environment in which foxes and rabbits are the only kinds of animals. Rabbits eat grass (of which there is a constant supply), procreate and are eaten by foxes. All foxes eat rabbits, procreate and die of geriatric diseases.