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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.<ref name=":5">{{Cite journal |last1=Chan |first1=K. C. |last2=Karolyi |first2=G. Andrew |last3=Longstaff |first3=Francis A. |last4=Sanders |first4=Anthony B. |date=July 1992 |title=An Empirical Comparison of Alternative Models of the Short-Term Interest Rate |journal=The Journal of Finance |language=en |volume=47 |issue=3 |pages=1209–1227 |doi=10.1111/j.1540-6261.1992.tb04011.x|doi-access=free }}</ref>{{sfn|Chan|Karolyi|Longstaff|Sanders|1992}}

== Definition ==
The CKLS process <math>X_t</math> is defined by the following [[stochastic differential equation]]:
<blockquote><math>dX_t = (\alpha + \beta X_t) dt + \sigma X_t^{\gamma}dW_t </math></blockquote>
where <math>W_t </math> denotes the [[Wiener process]]. The CKLS process has the following equivalent definition:<ref name=":0">{{Cite journal |last1=Kokabisaghi |first1=Somayeh |last2=Pauwels |first2=Eric J. |last3=Van Meulder |first3=Katrien |last4=Dorsman |first4=André B. |date=2018-09-02 |title=Are These Shocks for Real? Sensitivity Analysis of the Significance of the Wavelet Response to Some CKLS Processes |journal=International Journal of Financial Studies |language=en |volume=6 |issue=3 |pages=76 |doi=10.3390/ijfs6030076 |issn=2227-7072|doi-access=free }}</ref>
<blockquote><math>dX_t = -k(X_t - a) dt + \sigma X_t^{\gamma}dW_t </math></blockquote>

== Properties ==

* CKLS is an example of a [[mean-reverting process]]<ref name=":0" />
* The [[moment-generating function]] (MGF) of <math>X_t^{2(1-\gamma)}</math> has a [[Singularity (mathematics)|singularity]] at a critical moment independent of <math>\gamma</math>. 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]].{{cn|date=June 2022}}
* The CKLS equation has a unique pathwise solution.<ref name=":1">{{Cite journal |last1=Cai |first1=Yujie |last2=Wang |first2=Shaochen |date=2015-03-01 |title=Central limit theorem and moderate deviation principle for CKLS model with small random perturbation |url=https://www.sciencedirect.com/science/article/pii/S0167715214003939 |journal=Statistics & Probability Letters |language=en |volume=98 |pages=6–11 |doi=10.1016/j.spl.2014.11.017 |issn=0167-7152}}</ref>
* Cai and Wang (2015) have derived a [[central limit theorem]] and deviation principle for the CKLS model while studying its asymptotic behavior.<ref name=":1" />
* CKLS has been referred to as a time-homogeneous model as usually the parameters <math>\alpha, \beta, \sigma, \gamma</math> are taken to be time-independent.<ref name=":2">{{Cite journal |last1=Fan |first1=Jianqing |last2=Jiang |first2=Jiancheng |last3=Zhang |first3=Chunming|author3-link=Chunming Zhang |last4=Zhou |first4=Zhenwei |title=Time-Dependent Diffusion Models for Term Structure Dynamics |date=2003 |url=https://www.jstor.org/stable/24307157 |journal=Statistica Sinica |volume=13 |issue=4 |pages=965–992 |jstor=24307157 |issn=1017-0405}}</ref>
* The CKLS has also been referred to as a [[Multiple factor models|one-factor model]] (also see [[Factor analysis]]).<ref name=":3">{{Cite journal |last1=Dell'Aquila |first1=Rosario |last2=Ronchetti |first2=Elvezio |last3=Trojani |first3=Fabio |date=2003-05-01 |title=Robust GMM analysis of models for the short rate process |url=https://www.sciencedirect.com/science/article/pii/S0927539802000506 |journal=Journal of Empirical Finance |language=en |volume=10 |issue=3 |pages=373–397 |doi=10.1016/S0927-5398(02)00050-6 |issn=0927-5398}}</ref><ref name=":4">{{Cite journal |last=Nowman |first=K. B. |date=September 1997 |title=Gaussian Estimation of Single-Factor Continuous Time Models of The Term Structure of Interest Rates |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1540-6261.1997.tb01127.x |journal=The Journal of Finance |language=en |volume=52 |issue=4 |pages=1695–1706 |doi=10.1111/j.1540-6261.1997.tb01127.x}}</ref>

== Special cases ==
Many [[interest rate]] models and [[short-rate model]]s are special cases of the CKLS process which can be obtained by setting the CKLS model parameters to specific values.<ref name=":5" /><ref name=":4" /> In all cases, <math>\sigma</math> is assumed to be positive.

{| class="wikitable"
|+ Family of CKLS process under different parametric specifications.
|-
! Model/Process !! <math>\alpha</math> !! <math>\beta</math> !! <math>\gamma</math>
|-
| [[Merton model|Merton]] || Any || 0 || 0
|-
| [[Vasicek model|Vasicek]] || Any || Any || 0
|-
| [[Cox–Ingersoll–Ross model|CIR or square root process]] || Any || Any || 1/2
|-
| Dothan || 0 || 0 || 1
|-
| [[Geometric Brownian motion]] or [[Black–Scholes model|Black–Scholes–Merton model]] || 0 || Any || 1
|-
| Brennan and Schwartz || Any || Any || 1
|-
| CIR VR || 0 || 0 || 3/2
|-
| [[Constant elasticity of variance model|CEV]] || 0 || Any || Any
|}

== Financial applications ==
The CKLS process is often used to model interest rate dynamics and pricing of [[Bond (finance)|bonds]], [[bond option]]s,<ref>{{Cite journal |last1=Tangman |first1=D. Y. |last2=Thakoor |first2=N. |last3=Dookhitram |first3=K. |last4=Bhuruth |first4=M. |date=2011-12-01 |title=Fast approximations of bond option prices under CKLS models |url=https://www.sciencedirect.com/science/article/pii/S1544612311000158 |journal=Finance Research Letters |language=en |volume=8 |issue=4 |pages=206–212 |doi=10.1016/j.frl.2011.03.002 |issn=1544-6123}}</ref> [[currency exchange rate]]s,<ref>{{Cite journal |last1=Sikora |first1=Grzegorz |last2=Michalak |first2=Anna |last3=Bielak |first3=Łukasz |last4=Miśta |first4=Paweł |last5=Wyłomańska |first5=Agnieszka |date=2019-06-01 |title=Stochastic modeling of currency exchange rates with novel validation techniques |url=https://www.sciencedirect.com/science/article/pii/S0378437119304674 |journal=Physica A: Statistical Mechanics and Its Applications |language=en |volume=523 |pages=1202–1215 |doi=10.1016/j.physa.2019.04.098 |bibcode=2019PhyA..523.1202S |s2cid=149884892 |issn=0378-4371}}</ref> [[Security (finance)|securities]],<ref>{{Cite journal |last1=Nowman |first1=K. Ben |last2=Sorwar |first2=Ghulam |date=1999-03-01 |title=Pricing UK and US securities within the CKLS model Further results |url=https://www.sciencedirect.com/science/article/pii/S1057521999000198 |journal=International Review of Financial Analysis |language=en |volume=8 |issue=3 |pages=235–245 |doi=10.1016/S1057-5219(99)00019-8 |issn=1057-5219}}</ref> and other [[Option (finance)|options]], [[Derivative (finance)|derivatives]], and [[contingent claim]]s.<ref>{{Citation |last1=Dinenis |first1=E. |title=Valuation of Derivatives Based on CKLS Interest Rate Models |last2=Allegretto |first2=W. |last3=Sorwar |first3=G. |last4=N |first4=Quaderno |last5=Barone-adesi |first5=Giovanni |last6=Dinenis |first6=Elias |last7=Sorwar |first7=Ghulam|citeseerx=10.1.1.24.6963 }}</ref><ref name=":2" /> 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.<ref>{{Cite journal |last1=Koedijk |first1=Kees G. |last2=Nissen |first2=François G. J. A. |last3=Schotman |first3=Peter C. |last4=Wolff |first4=Christian C. P. |date=1997-04-01 |title=The Dynamics of Short-Term Interest Rate Volatility Reconsidered |journal=Review of Finance |language=en |volume=1 |issue=1 |pages=105–130 |doi=10.1023/A:1009714314989 |issn=1572-3097|doi-access=free }}</ref>

One question studied in the literature is how to set the model parameters, in particular the elasticity parameter <math>\gamma</math>.<ref>{{Cite journal |last1=Mishura |first1=Yuliya |last2=Ralchenko |first2=Kostiantyn |last3=Dehtiar |first3=Olena |date=2022-05-01 |title=Parameter estimation in CKLS model by continuous observations |url=https://www.sciencedirect.com/science/article/pii/S0167715222000153 |journal=Statistics & Probability Letters |language=en |volume=184 |pages=109391 |doi=10.1016/j.spl.2022.109391 |arxiv=2105.13724 |s2cid=235248362 |issn=0167-7152}}</ref><ref>{{Cite journal |last1=Nowman |first1=K. Ben |last2=Sorwar |first2=Ghulam |date=1999-09-01 |title=An Evaluation of Contingent Claims Using the CKLS Interest Rate Model: An Analysis of Australia, Japan, and the United Kingdom |url=https://doi.org/10.1023/A:1010013604561 |journal=Asia-Pacific Financial Markets |language=en |volume=6 |issue=3 |pages=205–219 |doi=10.1023/A:1010013604561 |s2cid=150454155 |issn=1573-6946}}</ref> [[Robust statistics]] and [[Nonparametric statistics|nonparametric estimation techniques]] have been used to measure CKLS model parameters.<ref name=":3" /><ref name=":2" />

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 <math>\gamma \ge 1 </math> can model [[short-term interest rates (disambiguation)|short-term interest rates]] more accurately than models with <math>\gamma < 1 </math>.<ref name=":5" />

Later empirical studies by Bliss and Smith have shown the reverse: sometimes lower <math>\gamma</math> values (like 0.5) in the CKLS model can capture volatility dependence more accurately compared to higher <math>\gamma</math> 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 <math>\gamma = 1/2</math>.<ref>{{Cite SSRN |last1=Bliss |first1=Robert R. |last2=Smith |first2=David C. |date=1998-03-01 |title=The Elasticity of Interest Rate Volatility: Chan, Karolyi, Longstaff, and Sanders Revisited|ssrn=99894 }}</ref>

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.<ref name=":3" />

== References ==
{{Reflist}}
{{Stochastic processes}}

{{DEFAULTSORT:Chan-Karolyi-Longstaff-Sanders process}}
[[Category:Stochastic differential equations]]
[[Category:Markov processes]]
[[Category:Variants of random walks]]
[[Category:Financial models]]
[[Category:Financial risk modeling]]
[[Category:Credit risk]]
[[Category:Options (finance)]]
[[Category:Derivatives (finance)]]
[[Category:Equity securities]]
[[Category:Interest rates]]
[[Category:Fixed income analysis]]
[[Category:Stochastic models]]
[[Category:Short-rate models]]