CKLS Model Research Articles

A New CKLS Model with GARCH-type Volatility and SSAEPD Error Terms

A New CKLS Model with GARCH-type Volatility and SSAEPD Error Terms

Parameter estimation in CKLS model by continuous observations

We consider a stochastic differential equation of the form drt=(a−brt)dt+σrtβdWt, where a, b and σ are positive constants, β∈(12,1). We study the estimation of an unknown drift parameter (a,b) by continuous observations of a sample path {rt,t∈[0,T]}. We prove the strong consistency and asymptotic normality of the maximum likelihood estimator. We propose another strongly consistent estimator, which generalizes an estimator proposed in Dehtiar et al. (2021) for β=12. The identification of the diffusion parameters σ and β is discussed as well.

Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient

We study a class of fractional stochastic differential equations (FSDEs) with coefficients that may not satisfy the linear growth condition and non-Lipschitz diffusion coefficient. Using the Lamperti transform, we obtain conditions for positivity of solutions of such equations. We show that the trajectories of the fractional CKLS model with β>1 are not necessarily positive. We obtain the almost sure convergence rate of the backward Euler approximation scheme for solutions of the considered SDEs. We also obtain a strongly consistent and asymptotically normal estimator of the Hurst index H>1/2 for positive solutions of FSDEs.

PMCMC for Term Structure of Interest Rates under Markov Regime Switching and Jumps

AbstractA parameter estimation method, called PMCMC in this paper, is proposed to estimate a continuous-time model of the term structure of interests under Markov regime switching and jumps. There is a closed form solution to term structure of interest rates under Markov regime. However, the model is extended to be a CKLS model with non-closed form solutions which is a typical nonlinear and non-Gaussian state-space model(SSM) in the case of adding jumps. Although the difficulty of parameter estimation greatly prevents from researching models, we prove that the nonlinear and non-Gaussian state-space model has better performances in studying volatility. The method proposed in this paper will be implemented in simulation and empirical study for SHIBOR. Empirical results illustrate that the PMCMC algorithm has powerful advantages in tackling the models.

Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application

A method based on higher-order partial differential equation (PDE) numerical scheme are proposed to obtain the transition cumulative distribution function (CDF) of the diffusion process (numerical differentiation of the transition CDF follows the transition probability density function (PDF)), where a transformation is applied to the Kolmogorov PDEs first, then a new type of PDEs with step function initial conditions and 0, 1 boundary conditions can be obtained. The new PDEs are solved by a fourth-order compact difference scheme and a compact difference scheme with extrapolation algorithm. After extrapolation, the compact difference scheme is extended to a scheme with sixth-order accuracy in space, where the convergence is proved. The results of the numerical tests show that the CDF approach based on the compact difference scheme to be more accurate than the other estimation methods considered; however, the CDF approach is not time-consuming. Moreover, the CDF approach is used to fit monthly data of the Federal funds rate between 1983 and 2000 by CKLS model.

The Explicit Solution and Precise Distribution of CKLS Model under Girsanov Transform

The relation between CKLS model and CIR model will be investigatedin this paper. It will be shown that under a suitabletransformation, any CKLS model of order $\frac{1}{2} 1$ corresponds to a CIR model under a new probabilityspace. Moreover, the explicit solution and the precise distributionof the CKLS model at any time $t$ are obtained under the newprobability measure. The moment estimation of CKLS model will be given finally.

Gaussian Estimation and Forecasting of the UK Yield Curve with Multi-Factor Continuous Time Models

In this paper we estimate the term structure of daily UK interest rates using more flexible continuous time models. The multivariate CKLS framework is employed for dynamic estimation and forecasting of four classical models over the eventful period of 2000-2013. The extensions are applied in two stages to four and five factor formulations, allowing us to assess the potential benefit of gradually increasing the model-flexibility. The Gaussian estimation methods for dynamic continuous time models yield insightful comparative results concerning the two different segments of the yield curve, short-term and long-term, respectively. In terms of in-sample performance the multi-factor general CKLS model is superior to all the other restricted models. When compared to benchmark discrete time models, the out-of-sample performance of the extended continuous time models seem to be consistently superior with regards only to the short-term segment of the yield curve.

Fast approximations of bond option prices under CKLS models

A new computational method for approximating prices of zero-coupon bonds and bond option prices under general Chan–Karolyi–Longstaff–Schwartz models is proposed. The pricing partial differential equations are discretized using second-order finite difference approximations and an exponential time integration scheme combined with best rational approximations based on the Carathéodory–Fejér procedure is employed for solving the resulting semi-discrete equations. The algorithm has a linear computational complexity and provides accurate bond and European bond option prices. We give several numerical results which illustrate the computational efficiency of the algorithm and uniform second-order convergence rates for the computed bond and bond option prices.

Tail return analysis of Bear Stearns' credit default swaps

We compare several models for Bear Stearns' credit default swap spreads estimated via a Markov chain Monte Carlo algorithm. The Bayes Factor selects a CKLS model with GARCH–EPD errors as the best model. This model best captures the volatility clustering and extreme tail returns of the swaps during the crisis. Prior to November 2007, only four months ahead of Bear Stearns' collapse though, the swap spreads were indistinguishable statistically from the risk-free rate.

Tail Return Analysis of Bear Stearns Credit Default Swaps

We compare several models for Bear Stearns' credit default swap spreads estimated via a Markov chain Monte Carlo algorithm. The Bayes Factor selects a CKLS model with GARCH-EPD errors as the best model. This model captures the volatility clustering and extreme tail returns of the swaps during the crisis. Prior to November 2007, only four months ahead of Bear Stearns' collapse though, the swap spreads were indistinguishable statistically from the risk free rate.

Crude Oil Price Dynamics Considering Structure Change Using CKLS Stochastic Models

ABSTRACT Nine stochastic processes was applied to WTI crude oil prices, following the CKLS model suggested by Chan et al. (1982). In particular, the study period was divided into two parts for representing structural change in the oil price. The estimation results shows that the crude oil price volatilities follow different stochastic processes in two sub-periods. We can conclude that the main characteristics of the crude oil price dynamics do change over time. Thus, considering multiple stochastic processes for long-term analyses of crude oil price series is important. Also, option values were calculated for comparing the effects of contingent claims depends on periods.

Implied derivative security prices based two-factor interest model: a UK application

In this paper the extended Box Method recently introduced to finance is used to value bond and option prices based on the two-factor CKLS interest rate model. The two-factor CKLS model is estimated using the one-year Eurodollar rate for the UK as the long rate and either the one-week, or one-month Euro dollar rate for the UK as the short rate. Overall, it is found that both and option prices are sensitive to the model used.

Derivative prices from interest rate models: results for Canada, Hong Kong, and United States

In this paper, we compute implied bond and contingent claim prices from the CKLS, Vasicek, CIR, and BS interest rate models using historical estimates for Canada, Hong Kong, and the United States. We find that default-free bond prices and contingent claim prices are sensitive to the assumed model used for these currencies, and that for Canada the CIR is the best, for Hong Kong the Vasicek and CIR models, and for the US the BS model.

Korean Treasury Bond Futures Pricing Model

This paper proposes two alternative methods which are used for pricing the theoretical value of the KTB futures on the non-traded underlying asset; first method is to use the CKLS model, under which the volatility of interest rate changes is highly sensitive to the level of the interest rate, and then employ binomial trees to compute the theoretical value of futures, second one is to use the multifactor Vasicek model considering correlations between yields-to-maturity and then employ the Monte Carlo simulation to compute it. In the empirical study on KTB303 and KTB306, an CKLS methodology is superior to the conventional KORFX method based on the cost-of-carry model in terms of the size of difference between market price and theoretical price. However, the phenomena, the price discrepancy using the KOFEX methodology is very small for all test perlod, implies that the KOFEX one is being used for the most market participants. The reasons that an multifactor Vasicek methodlogy is performed poorly in comparison to another methods are 1) the Vasicek model might be not a good model for explaining the level of interest rates, or 2) the important point considered by the most market participants may be on the volatility or interest rate, not on the correlations between yields-to-maturity.

Valuation of Common Two-Factor Interest Rate Contingent Claims

Numerous two-factor interest rate models have been proposed that have attempted to overcome the limitations of one-factor interest rate models. These two-factor interest rate models lead to complex valuation equations for interest rate contingent claims. These valuation equations can be expressed in the form of partial differential equations that can only be solved numerically. Sorwar and Barone-Adesi (2002) have recently extended the Box Method to solve second order partial differential equations. They find that the Box Method outperforms the existing finite difference scheme. In this paper we use their approach to value interest rate contingent claims based on a wide range of two-factor interest rate models. Our approach is to treat the first factor as the short term interest rate and the second factor as the long term interest rate. In this paper we use the Markov Chain Monte Carlo approach to estimate parameters based on two-factor, Vasicek, CIR and CKLS models. We find wide variation in parameter values across different models. We also find that these parameters have a major impact on contingent claim values. In particular we find a wide variation in contingent claim price across different models.

A Note on Gaussian Estimation of the CKLS and CIR Models with Feedback Effects for Japan

In this note we extend the Gaussian estimation of two factor CKLS and CIR models recently considered in Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34) to include feedback effects in the conditional mean as was originally formulated in general continuous time models by Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) with constant volatility. We use the exact discrete model of Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) to estimate the parameters which was first used by Brennan, M. J. and Schwartz, E. S. (1979, A continuous time approach to the pricing of bonds, J. Bank. Financ. 3, 133–155) to estimate their two factor interest model but incorporating the assumption of Nowman, K. B. (1997, Gaussian estimation of single-factor continuous time models of the term structure of interest rates, J. Financ. 52, 1695–1706; 2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34). An application to monthly Japanese Euro currency rates indicates some evidence of feedback from the 1-year rate to the 1-month rate in both the CKLS and CIR models. We also find a low level-volatility effect supporting Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34).

Robust GMM analysis of models for the short rate process

We re-examine the empirical evidence concerning a well-known class of one-factor models for the short rate process (cf. Chan et al. [Journal of Finance 47 (1992) 1209] (CKLS)) and some recent extensions allowing for a nonlinear drift and for changing parameters with a new statistical methodology based on robust statistics, the Robust Generalized Method of Moments (RGMM). We find that standard GMM model selection procedures are highly unstable in these applications. When testing the CKLS models with the RGMM we find that they are all clearly misspecified and we identify a clustering of influential observations in the 1979–1982 subperiod, a time span that is well known to coincide with a temporary change in the monetary policy of the Federal Reserve. This clustering of influential observations does not disappear when we introduce a non-linearity in the drift and allow for a parameter shift during the 1979–1982 period. Moreover, a Cox–Ingersoll–Ross model (selected by the RGMM) might offer a satisfactory data description for the period after 1982, since there only a few isolated outliers are found. Comparable results are obtained for the Euro-mark case.

Implied option prices from the continuous time CKLS interest rate model: an application to the UK

In this paper a numerical procedure recently applied in finance is used to compute implied bond and contingent claim prices starting from the CKLS interest rate model. The CKLS model is estimated using a range of maturities from the UK interbank market including the one week and one, two, three, six and twelve month rates. It is found that the implied default free bond prices and contingent claim prices vary across models and maturities for the UK.

Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom

In this paper we extend the exact discrete model of Bergstrom (1966) first used in empirical finance by Brennan and Schwartz (1979) to estimate their two-factor term structure model to estimate other two-factor term structure models using the recent assumption in Nowman (1997) for single factor models. Following Nowman (1997) we use the exact Gaussian estimation methods of Bergstrom (1983–1986, 1990) to estimate two-factor CKLS, Vasicek and CIR models. We estimate the models using monthly UK and Japanese interest rate data and our results indicate that the estimation method works well in practice.

Pricing UK and US securities within the CKLS model Further results

In a recent article, Byers and Nowman obtained estimates of the CKLS interest rate model based on weekly Euro-currency data for the UK and US over a range of maturities. In this article we apply the Box numerical method for valuing default free bonds and cointegent claims using these historical UK and US estimates to compare implied bond and contingent claim prices. Our results indicate that default free bond prices and contingent claim prices are sensitive to the underlying interest rate model used.