Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 55

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.