4. (The Cox–Ingersoll–Ross (CIR) Model)

We examine the CIR interest-rate model (also known as square-root diffusion) defined as the following SDE with constant coefficients (Glasserman, 2004):

dr = a(b − r)dt + ????

√

rdW.

This is a model of the short rate and it is referred to as the CIR model. In this case the short rate r(t) is pulled towards b at a speed controlled by

a. Furthermore, if r(0) > 0 then r(t) will never be negative and r(t) remains strictly positive for all t if the following Feller condition is satisfied:

ab >

1 2????

2

.

The objective of this exercise is to introduce this model into the framework in Figure 14.6 and gain an insight into its analytical and numerical properties.

Answer the following questions:

a) Create a class for the CIR SDE and integrate it into the framework.

b) The CIR SDE does not have an explicit solution. We then resort to numerical methods.

Apply the explicit Euler and Heun methods for various time-step sizes and expirations.

c) Investigate the finite difference schemes when the Feller condition is not satisfied.

d) Experiment with the finite difference schemes and determine positivity for small and large values of drift, volatility and expiration parameters.

We remark that the transition density for the CIR process is known and it can be represented in terms of a non-central chi-squared distribution (Glasserman, 2004, p. 122).