Cox–Ingersoll–Ross model

Overview

In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" (short-rate model) as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was introduced in 1985 by John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross as an extension of the Vasicek model, itself an Ornstein–Uhlenbeck process.

The model

The CIR model describes the instantaneous interest rate

r

t

{\displaystyle r_{t}}

with a Feller square-root process, whose stochastic differential equation is

d

r

t

= a ( b −

r

t

)

d t + σ

r

t

d

W

t

,

{\displaystyle dr_{t}=a(b-r_{t})\,dt+\sigma {\sqrt {r_{t}}}\,dW_{t},}

Distribution

Future distribution The distribution of future values of a CIR process can be computed in closed form:

r

t + T

=

Y

2 c

,

{\displaystyle r_{t+T}={\frac {Y}{2c}},}

where

c =

2 a

( 1 −

e

− a T

)

σ

2

{\displaystyle c={\frac {2a}{(1-e^{-aT})\sigma ^{2}}}}

, and Y is a non-central chi-squared distribution with

4 a b

Properties

Mean reversion, Level dependent volatility (

σ

r

t

{\displaystyle \sigma {\sqrt {r_{t}}}}

), For given positive

r

0

{\displaystyle r_{0}}

the process will never touch zero, if

2 a b ≥

σ

2

{\displaystyle 2ab\geq \sigma ^{2}}

; otherwise it can occasionally touch the zero point,

E ⁡ [

r

t

∣

r

Calibration

Ordinary least squares The continuous SDE can be discretized as follows

r

t + Δ t

−

r

t

= a ( b −

r

t

)

Δ t + σ

r

t

Δ t

ε

t

,

{\displaystyle r_{t+\Delta t}-r_{t}=a(b-r_{t})\,\Delta t+\sigma \,{\sqrt {r_{t}\Delta t}}\varepsilon _{t},}

which is equivalent to

r

Simulation

Stochastic simulation of the CIR process can be achieved using two variants:

Discretization Exact

Bond pricing

Under risk-neutral pricing, the drift relevant for bond valuation need not be the same as the statistical drift of the short rate under the real-world probability measure. For example, if

d

r

t

= a ( b −

r

t

)

d

t + σ

r

t

d

W

t

P

{\displaystyle \mathrm {d} r_{t}=a(b-r_{t})\,\mathrm {d} t+\sigma {\sqrt {r_{t}}}\,\mathrm {d} W_{t}^{\mathbb {P} }}

under the real-world measure and the market price of interest-rate risk is specified as

λ (

r

Extensions

The CIR model uses a special case of a basic affine jump diffusion, which still permits a closed-form expression for bond prices. Time varying functions replacing coefficients can be introduced in the model in order to make it consistent with a pre-assigned term structure of interest rates and possibly volatilities. The most general approach is in Maghsoodi (1996). A more tractable approach is in Brigo and Mercurio (2001b) where an external time-dependent shift is added to the model for consistency with an input term structure of rates. A significant extension of the CIR model to the case of stochastic mean and stochastic volatility is given by Lin Chen (1996) and is known as Chen model. A more recent extension for handling cluster volatility, negative interest rates and different distributions is the so-called "CIR #" by Orlando, Mininni and Bufalo (2018, 2019, 2020, 2021, 2023) and a simpler extension focussing on negative interest rates was proposed by Di Francesco and Kamm (2021, 2022), which are referred to as the CIR- and CIR-- models.

See also

Hull–White model Vasicek model Chen model

Further References

Hull, John C. (2003). Options, Futures and Other Derivatives. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-009056-5. Cox, J. C.; Ingersoll, J. E.; Ross, S. A. (1985). "A Theory of the Term Structure of Interest Rates". Econometrica. 53 (2): 385–407. doi:10.2307/1911242. JSTOR 1911242. Maghsoodi, Y. (1996). "Solution of the extended CIR Term Structure and Bond Option Valuation". Mathematical Finance. 6 (6): 89–109. doi:10.1111/j.1467-9965.1996.tb00113.x. Damiano Brigo; Fabio Mercurio (2001). Interest Rate Models — Theory and Practice with Smile, Inflation and Credit (2nd ed. 2006 ed.). Springer Verlag. ISBN 978-3-540-22149-4. Brigo, Damiano; Fabio Mercurio (2001b). "A deterministic-shift extension of analytically tractable and time-homogeneous short rate models". Finance & Stochastics. 5 (3): 369–388. doi:10.1007/PL00013541. S2CID 35316609. Open Source library implementing the CIR process in python Orlando, Giuseppe; Mininni, Rosa Maria; Bufalo, Michele (2020). "Forecasting interest rates through Vasicek and CIR models: A partitioning approach". Journal of Forecasting. 39 (4): 569–579. arXiv:1901.02246. doi:10.1002/for.2642. ISSN 1099-131X. S2CID 126507446.