Chan–Karolyi–Longstaff–Sanders process

The CKLS process $X\_t$ is defined by the following stochastic differential equation:

$dX\_t\; =\; (\backslash alpha\; +\; \backslash beta\; X\_t)\; dt\; +\; \backslash sigma\; X\_t^\{\backslash gamma\}dW\_t$

where $W\_t$ denotes the Wiener process. The CKLS process has the following equivalent definition:{{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 }}

$dX\_t\; =\; -k(X\_t\; -\; a)\; dt\; +\; \backslash sigma\; X\_t^\{\backslash gamma\}dW\_t$

• CKLS is an example of a mean-reverting process
• The moment-generating function (MGF) of $X\_t^\{2(1-\backslash gamma)\}$ has a singularity at a critical moment independent of $\backslash gamma$. 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.{{cn|date=June 2022}}
• The CKLS equation has a unique pathwise solution.{{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}}
• 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 $\backslash alpha,\; \backslash beta,\; \backslash sigma,\; \backslash gamma$ are taken to be time-independent.{{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}}
• The CKLS has also been referred to as a one-factor model (also see Factor analysis).{{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}}{{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}}

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, $\backslash sigma$ is assumed to be positive.

The CKLS process is often used to model interest rate dynamics and pricing of bonds, bond options,{{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}} currency exchange rates,{{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}} securities,{{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}} and other options, derivatives, and contingent claims.{{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 }} 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.{{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 }}

One question studied in the literature is how to set the model parameters, in particular the elasticity parameter $\backslash gamma$.{{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}}{{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}} 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 $\backslash gamma\; \backslash ge\; 1$ can model short-term interest rates more accurately than models with .

Later empirical studies by Bliss and Smith have shown the reverse: sometimes lower $\backslash gamma$ values (like 0.5) in the CKLS model can capture volatility dependence more accurately compared to higher $\backslash gamma$ 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 $\backslash gamma\; =\; 1/2$.{{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 }}

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.

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{{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