Chan–Karolyi–Longstaff–Sanders process explained

In mathematics, the Chan–Karolyi–Longstaff–Sanders process (abbreviated as CKLS process) is a stochastic process with applications to finance. In particular it has been used to model the term structure of interest rates. The CKLS process can also be viewed as a generalization of the Ornstein–Uhlenbeck process. It is named after K. C. Chan, G. Andrew Karolyi, Francis A. Longstaff, and Anthony B. Sanders, with their paper published in 1992.^[1]

Definition

The CKLS process

X _{t} is defined by the following stochastic differential equation:

dX _{t} = (\alpha + \beta X _{t)} dt + \sigma X ^{\gamma} _{t} dW _{t}

where W _{t} denotes the Wiener process. The CKLS process has the following equivalent definition:^[2]

dX _{t} = - k(X _{t} - a) dt + \sigma X ^{\gamma} _{t} dW _{t}

Properties

• CKLS is an example of a mean-reverting process
• The moment-generating function (MGF) of

X ^{2 ( 1- \gamma)} _{t} has a singularity at a critical moment independent of \gamma . Moreover, the MGF can be written as the MGF of the CIR model plus a term that is a solution to a Nonlinear partial differential equation.

• The CKLS equation has a unique pathwise solution.^[3]
• Cai and Wang (2015) have derived a central limit theorem and deviation principle for the CKLS model while studying its asymptotic behavior.
• CKLS has been referred to as a time-homogeneous model as usually the parameters

\alpha, \beta, \sigma, \gamma are taken to be time-independent.^[4]

• The CKLS has also been referred to as a one-factor model (also see Factor analysis).^{[5] [6]}

Special cases

Many interest rate models and short-rate models are special cases of the CKLS process which can be obtained by setting the CKLS model parameters to specific values. In all cases,

\sigma is assumed to be positive.

Family of CKLS process under different parametric specifications.

Model/Process	\alpha	\beta	\gamma
	Any	0	0
	Any	Any	0
	Any	Any	1/2
Dothan	0	0	1
	0	Any	1
Brennan and Schwartz	Any	Any	1
CIR VR	0	0	3/2
	0	Any	Any

Financial applications

The CKLS process is often used to model interest rate dynamics and pricing of bonds, bond options,^[7] currency exchange rates,^[8] securities,^[9] and other options, derivatives, and contingent claims. It has also been used in the pricing of fixed income and credit risk and has been combined with other time series methods such as GARCH-class models.^[10]

One question studied in the literature is how to set the model parameters, in particular the elasticity parameter

\gamma .^{[11] [12]} Robust statistics and nonparametric estimation techniques have been used to measure CKLS model parameters.

In their original paper, CKLS argued that the elasticity of interest rate volatility is 1.5 based on historical data, a result that has been widely cited. Also, they showed that models with

\gamma \ge 1 can model short-term interest rates more accurately than models with \gamma < 1 .

Later empirical studies by Bliss and Smith have shown the reverse: sometimes lower

\gamma values (like 0.5) in the CKLS model can capture volatility dependence more accurately compared to higher \gamma values. Moreover, by redefining the regime period, Bliss and Smith have shown that there is evidence for regime shift in the Federal Reserve between 1979 and 1982. They have found evidence supporting the square root Cox-Ingersoll-Ross model (CIR SR), a special case of the CKLS model with \gamma = 1 / 2 .^[13]

The period of 1979-1982 marked a change in monetary policy of the Federal Reserve, and this regime change has often been studied in the context of CKLS models.

Notes and References

1. Chan . K. C. . Karolyi . G. Andrew . Longstaff . Francis A. . Sanders . Anthony B. . July 1992 . An Empirical Comparison of Alternative Models of the Short-Term Interest Rate . The Journal of Finance . en . 47 . 3 . 1209–1227 . 10.1111/j.1540-6261.1992.tb04011.x. free .
2. Kokabisaghi . Somayeh . Pauwels . Eric J. . Van Meulder . Katrien . Dorsman . André B. . 2018-09-02 . Are These Shocks for Real? Sensitivity Analysis of the Significance of the Wavelet Response to Some CKLS Processes . International Journal of Financial Studies . en . 6 . 3 . 76 . 10.3390/ijfs6030076 . 2227-7072. free .
3. Cai . Yujie . Wang . Shaochen . 2015-03-01 . Central limit theorem and moderate deviation principle for CKLS model with small random perturbation . Statistics & Probability Letters . en . 98 . 6–11 . 10.1016/j.spl.2014.11.017 . 0167-7152.
4. Fan . Jianqing . Jiang . Jiancheng . Zhang . Chunming. Chunming Zhang . Zhou . Zhenwei . Time-Dependent Diffusion Models for Term Structure Dynamics . 2003 . Statistica Sinica . 13 . 4 . 965–992 . 24307157 . 1017-0405.
5. Dell'Aquila . Rosario . Ronchetti . Elvezio . Trojani . Fabio . 2003-05-01 . Robust GMM analysis of models for the short rate process . Journal of Empirical Finance . en . 10 . 3 . 373–397 . 10.1016/S0927-5398(02)00050-6 . 0927-5398.
6. Nowman . K. B. . September 1997 . Gaussian Estimation of Single-Factor Continuous Time Models of The Term Structure of Interest Rates . The Journal of Finance . en . 52 . 4 . 1695–1706 . 10.1111/j.1540-6261.1997.tb01127.x.
7. Tangman . D. Y. . Thakoor . N. . Dookhitram . K. . Bhuruth . M. . 2011-12-01 . Fast approximations of bond option prices under CKLS models . Finance Research Letters . en . 8 . 4 . 206–212 . 10.1016/j.frl.2011.03.002 . 1544-6123.
8. Sikora . Grzegorz . Michalak . Anna . Bielak . Łukasz . Miśta . Paweł . Wyłomańska . Agnieszka . 2019-06-01 . Stochastic modeling of currency exchange rates with novel validation techniques . Physica A: Statistical Mechanics and Its Applications . en . 523 . 1202–1215 . 10.1016/j.physa.2019.04.098 . 2019PhyA..523.1202S . 149884892 . 0378-4371.
9. Nowman . K. Ben . Sorwar . Ghulam . 1999-03-01 . Pricing UK and US securities within the CKLS model Further results . International Review of Financial Analysis . en . 8 . 3 . 235–245 . 10.1016/S1057-5219(99)00019-8 . 1057-5219.
10. Koedijk . Kees G. . Nissen . François G. J. A. . Schotman . Peter C. . Wolff . Christian C. P. . 1997-04-01 . The Dynamics of Short-Term Interest Rate Volatility Reconsidered . Review of Finance . en . 1 . 1 . 105–130 . 10.1023/A:1009714314989 . 1572-3097. free .
11. Mishura . Yuliya . Ralchenko . Kostiantyn . Dehtiar . Olena . 2022-05-01 . Parameter estimation in CKLS model by continuous observations . Statistics & Probability Letters . en . 184 . 109391 . 10.1016/j.spl.2022.109391 . 2105.13724 . 235248362 . 0167-7152.
12. Nowman . K. Ben . Sorwar . Ghulam . 1999-09-01 . An Evaluation of Contingent Claims Using the CKLS Interest Rate Model: An Analysis of Australia, Japan, and the United Kingdom . Asia-Pacific Financial Markets . en . 6 . 3 . 205–219 . 10.1023/A:1010013604561 . 150454155 . 1573-6946.
13. Bliss . Robert R. . Smith . David C. . 1998-03-01 . The Elasticity of Interest Rate Volatility: Chan, Karolyi, Longstaff, and Sanders Revisited. 99894 .