CIR介绍及OLS+ML估计程序

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MAXIMUM LIKELIHOOD ESTIMATION OF THE

COX-INGERSOLL-ROSS PROCESS: THE MATLAB

IMPLEMENTATION

Kamil Klad´ıvko

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

, t ≥ 0}, with dynamics represented by stochastic diﬀerential equations:

= µ(X

)dt + σ(X

)dW

, (1)

where {W

, t ≥ 0} is a standard Brownian motion. The functions µ(·) and σ

(·) 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:

= α(µ − r

)dt +

√

σdW

, (2)

where r

is the interest rate and θ ≡ (α, µ, σ) are model parameters. The drift function µ(r

, θ) =

α(µ − r

) is linear and possess a mean reverting property, i.e. interest rate r

moves in the

direction of its mean µ at sp ee d α. The diﬀusion function σ

, θ) = r

is proportional to the

interest rate r

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

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

at time t the density of r

t+∆t

at time t + ∆t is

p(r

t+∆t

; θ, ∆t) = ce

−u−v

(

)

√

uv), (3)

This work was supported by grant of IGA VSE nr. IG410046

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