MAXIMUM LIKELIHOOD ESTIMATION OF THE
COX-INGERSOLL-ROSS PROCESS: THE MATLAB
IMPLEMENTATION
Kamil Klad´ıvko
1
Department of Statistics and Probability Calculus, University of Economics, Prague
and Debt Management Department, Ministry of Finance of the Czech Republic
kladivk@vse.cz or kamil.kladivko@mfcr.cz
Abstract
The square root diffusion process is widely used for modeling interest rates
behaviour. It is an underlying process of the well-known Cox-Ingersoll-Ross
term structure model (1985). We investigate maximum likelihood estimation
of the square root process (CIR process) for interest rate time series. The
MATLAB implementation of the estimation routine is provided and tested on
the PRIBOR 3M time series.
1 CIR Process for Interest Rate Modeling
A continuous-time model in finance typically rest on one or more stationary diffusion processes
{X
t
, t ≥ 0}, with dynamics represented by stochastic differential equations:
dX
t
= µ(X
t
)dt + σ(X
t
)dW
t
, (1)
where {W
t
, t ≥ 0} is a standard Brownian motion. The functions µ(·) and σ
2
(·) are, respectively,
the drift and the diffusion functions of the process.
The fundamental process in interest rate modeling is the square root process given by the
following stochastic differential equation:
dr
t
= α(µ − r
t
)dt +
√
r
t
σdW
t
, (2)
where r
t
is the interest rate and θ ≡ (α, µ, σ) are model parameters. The drift function µ(r
t
, θ) =
α(µ − r
t
) is linear and possess a mean reverting property, i.e. interest rate r
t
moves in the
direction of its mean µ at sp ee d α. The diffusion function σ
2
(r
t
, θ) = r
t
σ
2
is proportional to the
interest rate r
t
and ensures that the process stays on a positive domain.
The square ro ot process (2) is the basis for the Cox, Ingersoll, and Ross short-term interest
rate model [1] and therefore often denoted as the CIR process in the financial literature.
1.1 CIR process densities
If α, µ, σ are all positive and 2αµ ≥ σ
2
holds, the CIR process is well-defined and has a steady
state (marginal) distribution. The marginal density is gamma distributed.
For maximum likelihood estimation of the parameter vector θ ≡ (α, µ, σ) transition den-
sities are required. The CIR process is one of few cases, among the diffusion processes, where
the transition density has a closed form expression. We follow the notation given in [1] on page
391. Given r
t
at time t the density of r
t+∆t
at time t + ∆t is
p(r
t+∆t
|r
t
; θ, ∆t) = ce
−u−v
(
v
u
)
q
2
I
q
(2
√
uv), (3)
1
This work was supported by grant of IGA VSE nr. IG410046
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