Cox-Ingersoll-Ross Research Articles

Error bounds for the perturbation solution of the transition density under a multi-factor CIR term structure model with weak mean-reversion effect

We consider a multi-factor Cox-Ingersoll-Ross (CIR) model of the term structure of interest rates with weak mean-reversion effect. We use perturbation theory to analyze its conditional characteristic function illustrated by a system of Riccati equations and derive the error bounds for the perturbation approximations. Using the Fourier inversion theorem, we clarify that the perturbation approximation of the conditional characteristic function can be applied to estimate the transition density and likelihood function. We provide their error bounds and accuracy orders. Finally, we discuss the performance of the perturbation approximation in estimating the transition density via simulation.

Pricing American put options under stochastic volatility using the Malliavin derivative

The aim of this paper is to develop a methodology based on Malliavin calculus, in order to price American options under stochastic volatility. This leads to compute the conditional expectation E(Pt(Xt,Vt)∣(Xl,Vl)) for any 0≤l<t, where Vt is generated by the Cox-Ingersoll-Ross (CIR) process. Some simulations and comparisons are given.

SINH-ACCELERATION: EFFICIENT EVALUATION OF PROBABILITY DISTRIBUTIONS, OPTION PRICING, AND MONTE CARLO SIMULATIONS

Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around [Formula: see text]. The Fourier transform techniques reduce calculation of probability distributions and option prices in the evaluation of integrals whose integrands are analytic in domains enjoying these properties. In the paper, we suggest to use changes of variables of the form [Formula: see text] and the simplified trapezoid rule to evaluate the integrals accurately and fast. We formulate the general scheme, and apply the scheme for calculation probability distributions and pricing European options in Lévy models, the Heston model, the CIR model, and a Lévy model with the CIR-subordinator. We outline applications to fast and accurate calibration procedures and Monte Carlo simulations in Lévy models, regime switching Lévy models that can account for stochastic drift, volatility and skewness, the Heston model, other affine models and quadratic term structure models. For calculation of quantiles in the tails using the Newton or bisection method, it suffices to precalculate several hundreds of values of the characteristic exponent at points on an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf.

Perturbation solutions for bond-pricing equations under a multivariate CIR model with weak dependences

In this article, we derive the explicit perturbation solutions for bond-pricing equations under a multivariate Cox–Ingersoll–Ross model whose factors have weak dependences between each other. The error bounds of the perturbation approximations are demonstrated for the Riccati equation and zero-coupon bond price. These results provide the accuracy order for the perturbation approximation. Since the perturbation approximations are constructed based on the exact solution of the Riccati equation, this computational approach can be applied to deal with the stiffness problem. Numerical results are presented to show that this computation approach is tractable and accurate for evaluating the bond price.

On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case

We propose a positivity preserving implicit Euler-Maruyama scheme for a jump-extended Cox-Ingersoll-Ross (CIR) process where the jumps are governed by a compensated spectrally positive $\alpha$-stable process for $\alpha \in (1,2)$. Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV (Constant Elasticity Variance) process, the model considered here has infinite activity jumps. We calculate, in this specific model, the strong rate of convergence and give some numerical illustrations. Jump extended models of this type were initially studied in the context of branching processes and was recently introduced to the financial mathematics literature to model sovereign interest rates, power and energy markets.

A Feynman-Kac type formula for a fixed delay CIR model

Stochastic delay differential equations (SDDE’s) have been used for financial modeling. In this article, we study a SDDE obtained by the equation of a CIR process, with an additional fixed delay term in drift; in particular, we prove that there exists a unique strong solution (positive and integrable) which we call fixed delay CIR process. Moreover, for the fixed delay CIR process, we derive a Feynman-Kac type formula, leading to a generalized exponential-affine formula, which is used to determine a bond pricing formula when the interest rate follows the delay’s equation. It turns out that, for each maturity time T, the instantaneous forward rate is an affine function (with time dependent coefficients) of the rate process and of an auxiliary process (also depending on T). The coefficients satisfy a system of deterministic differential equations.

A note on ergodicity for CIR model with Markov switching

Recently, Zhang et al. show that if then the Cox-Ingersoll-Ross (CIR) model with Markov switching (see below, the SDE (1.2)) is ergodic in the Wasserstein distance if and only if In this article, we will show that if the Cox-Ingersoll-Ross (CIR) model with Markov switching is non-ergodic. Explicit expressions for the mean and variance of the CIR model with Markov switching are obtained. As a byproduct, the explicit expressions for mean of stationary distribution and second-order moments for such model are presented. Besides, we find the necessary and sufficient conditions of weak stationarity for CIR model with Markov switching.

Improving the Forecast of Longevity by Combining Models

Mortality is a dynamic process whose future evolution over time poses important challenges for life insurance, pension funds, public policy, and fiscal planning. In this paper, we propose two contributions: (1) a new dynamic corrective methodology of the predictive accuracy of the existing mortality projection models, by modeling a measure of their fitting errors as a Cox-Ingersoll-Ross process and; (2) various out-of-sample validation methods. Besides the usual static method, we develop a dynamic one allowing us to catch the change in behavior of the underlying data. For our numerical application, we choose the Cairns-Blake-Dowd (or M5) model. Using the Italian and French females mortality data and implementing the backtesting procedure, we empirically test the ex-post forecasting performance of the CBD model both for itself (CBD) and corrected by the CIR process (mCBD). We focus on age 65, but we show results for a wide range of ages, also much younger, and for cohort data. On the basis of average measures of forecasting errors and information criteria, we show that the mCBD model is parsimonious and provides better results in terms of predictive accuracy than the CBD model itself.

Performance Analysis of CIR and Path Loss Propagation Models in the Downlink of 3G Systems

Performance Analysis of CIR and Path Loss Propagation Models in the Downlink of 3G Systems

Dependence diagnosis for stationary stochastic processes based on both quantiles and copulas

Based on both quantiles and copulas, a new concept, the quantile-copula cross-covariance function, for stationary stochastic processes, is given. The function can capture many dependence characteristics that the traditional auto-covariance functioncannot cover, such as time reversibility, state dependence as well as long-range dependence when the auto-covariance function does not exist. A consistent estimator of the quantile-copula cross-covariance function is proposed based on discretely sampled stationary processes. By comparing the estimated quantile-copula cross-covariance functions, the dependence differences are distinguished for two types of processes, Gamma-OU processes and CIR models, which have the common one-dimensional stationary marginal distribution and the common auto-covariance function. By analyzing the estimated quantile-copula cross-covariance function of a ship vibration series, some suggestions for modeling the vibration process are provided.

Small sample properties of ML estimator in Vasicek and CIR models: a simulation experiment

In this paper we analyze small sample properties of the ML estimation procedure in Vasicek and CIR models. In particular, we consider short time series, with a length between 20 and 200, typically values observed in the field of survival data. We perform a simulation study in order to investigate which properties of the parameter estimators still remain valid and to evaluate the effect of a bootstrap bias correction method. The results show that the bias of the estimators can be really strong for small samples and the relative bias seems to be worse when the true parameters of the models are near to the nonstationarity case. The bootstrap bias correction is enough efficient in correcting the bias also for very small sample sizes, but the increase in RMSE of the estimator is greater as much as smaller is the bias in the ML estimator. Moreover, the bootstrap correction does not improve the performance of the tests on the parameters.

Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations

ABSTRACTWe consider a stable Cox–Ingersoll–Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. In all cases we prove strong consistency of the MLE in question, in the subcritical case asymptotic normality, and in the supercritical case asymptotic mixed normality are shown as well. In the critical case the description of the asymptotic behaviour of the MLE in question remains open.

A modified sequential Monte Carlo procedure for the efficient recursive estimation of extreme quantiles

AbstractMany applications in science involve finding estimates of unobserved variables from observed data, by combining model predictions with observations. The sequential Monte Carlo (SMC) is a well‐established technique for estimating the distribution of unobserved variables that are conditional on current observations. While the SMC is very successful at estimating the first central moments, estimating the extreme quantiles of a distribution via the current SMC methods is computationally very expensive. The purpose of this paper is to develop a new framework using probability distortion. We use an SMC with distorted weights in order to make computationally efficient inferences about tail probabilities of future interest rates using the Cox–Ingersoll–Ross (CIR) model, as well as with an observed yield curve. We show that the proposed method yields acceptable estimates about tail quantiles at a fraction of the computational cost of the full Monte Carlo.

Moments and ergodicity of the jump-diffusion CIR process

ABSTRACTWe study the jump-diffusion CIR process, which is an extension of the Cox-Ingersoll-Ross model and whose jumps are introduced by a subordinator. We provide sufficient conditions on the Lévy measure of the subordinator under which the jump-diffusion CIR process is ergodic and exponentially ergodic, respectively. Furthermore, we characterize the existence of the κ-moment () of the jump-diffusion CIR process by an integrability condition on the Lévy measure of the subordinator.

Markovian structure of the Volterra Heston model

We characterize the Markovian and affine structure of the Volterra Heston model in terms of an infinite-dimensional adjusted forward process and specify its state space. More precisely, we show that it satisfies a stochastic partial differential equation and displays an exponentially-affine characteristic functional. As an application, we deduce an existence and uniqueness result for a Banach-space valued square-root process and provide its state space. This leads to another representation of the Volterra Heston model together with its Fourier–Laplace transform in terms of this possibly infinite system of affine diffusions.

OPTIMAL INVESTMENT AND CONSUMPTION WITH STOCHASTIC FACTOR AND DELAY

We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results.

The Application of Affine Processes in Cohort Mortality Risk Models

This paper assesses and compares multi-factor continuous time affine mortality models applied to age-cohort mortality curves that are well suited for theoretical and practical application in finance and insurance. Models based on Gaussian distributed mortality rates, as well as the Cox-Ingersoll-Ross (CIR) process allowing for Gamma distributed mortality rates, are compared, also quantifying the probability of negative rates in the Gaussian models. In particular, we introduce the Gaussian Arbitrage-Free Nelson-Siegel (AFNS) mortality model incorporating level, slope and curvature factors. The models have appealing features including efficient estimation and computation. We estimate models using age-cohort data to capture cohort effects more effectively and in order to explain the variability in cohort mortality curves in the continuous time framework. The models allow for Poisson variation in the model estimation using the Kalman filter. The affine mortality models facilitate the derivation of closed-form survivor curves allowing for efficient valuation of mortality-linked claims. The models can also incorporate factor dependence allowing for age-dependence in the mortality curves. Importantly we show that the Gaussian independent factor AFNS model performs very well in explaining and forecasting cohort mortality.

A spectral approach to pricing of forward starting options

We propose a new class of models for pricing forward starting options. We assume that the asset price is a nonlinear function of a CIR process, time changed by a composition of a Levy subordinator and an absolutely ´ continuous process. The new models introduce the nonlinearity in both drift and diffusion components of the underlying process and can capture jumps and stochastic volatility in a flexible way. By employing the spectral expansion technique, we are able to derive the analytical formulas for the forward starting option prices. We also implement a specific model numerically and test its sensitivity to some of the key parameters of the model.

On non-negative modeling with CARMA processes

Two stationary and non-negative processes that are based on continuous-time autoregressive moving average (CARMA) processes are discussed. First, we consider a generalization of Cox–Ingersoll–Ross (CIR) processes. Next, we consider CARMA processes driven by compound Poisson processes with exponential jumps which are generalizations of Ornstein–Uhlenbeck (OU) processes driven by the same noise. The way in which the two processes generalize CIR and OU processes and the relation between them will be discussed. Furthermore, the stationary distribution, the autocorrelation function, and pricing of zero-coupon bonds are considered.

Exchange Rate Solutions With Currency Crashes

We present an exchange rate model in which a currency’s exchange rate is confined in a wide moving band and a currency crash occurs when the rate breaches the lower boundary. A solution is derived from the standard log exchange rate equation for the model with a smooth-pasting condition at the lower boundary. Using an asymmetric mean-reverting fundamental shock, the solution shows the exchange rate follows a mean-reverting square-root process, which is quasi-bounded at the boundary, and generates left-skewed exchange rate distributions consistent with empirical observations. The probability leakage for the exchange rate across the boundary increases with a weakened mean-reverting force for the exchange rate, suggesting an increase in currency crash risk. The empirical results show the exchange rates of nine major currencies against the US dollar can be calibrated according to the model, where the mean reversion is negatively cointegrated with the risk reversals in currency option markets, as expected by the model, and are consistent with the positive relationship between currency crash risk and risk reversals. The leakage condition for breaching the lower boundaries was met during the 2008 global financial crisis when most of the currencies were under the disaster shock.