Cox-Ingersoll-Ross Research Articles

Using Interest Rate Models to Improve Mortality Forecast

Future evolution of mortality poses important challenges for life insurance, pension funds, public policy and fiscal planning. Indeed, when fair values, premium rates and risk reserves are computed, sound and accurate models to forecast stochastic longevity are needed. In this paper, we propose a methodological approach in order to improve the predictive accuracy of the existing survival models. The central idea is to model the ratio between the observed death rates and the corresponding fitted values obtained as outputs of a survival model we select, by means of the Cox-Ingersoll-Ross (CIR) model. For our numerical application, we choose to apply the CIR correction to the Cairns-Blake-Dowd (or M5) model. Using the Italian females mortality data and implementing the backtesting procedure, over both a static time horizon and fixed-length windows rolling one-year ahead through time, we empirically test the performance of the CBD model in forecasting death rates both for itself (CBD) and corrected by the CIR process (mCBD). On the basis of average measures of forecasting errors and information criteria we demonstrate that the mCBD model is a parsimonious model providing better results in terms of predictive accuracy than the CBD model.

An efficient algorithm for the valuation of a guaranteed annuity option with correlated financial and mortality risks

We introduce a pricing framework for a guaranteed annuity option (GAO) where both the interest and mortality risks are correlated. We assume that the short rate and the force of mortality follow the Cox–Ingersoll–Ross (CIR) and Lee–Carter models, respectively. Employing the change of measure technique, we decompose the pure endowment into the product of the bond price and survival probability, thereby facilitating the evaluation of the annuity expression. With the aid of the dynamics of interest and mortality processes under the forward measure, we construct an algorithm based on comonotonicity theory to estimate the quantiles of survival probability and annuity rate. The comonotonic upper and lower bounds in the convex order are used to approximate the annuity and GAO prices and henceforth avoiding the simulation-within-simulation problem. Numerical illustrations show that our algorithm gives an efficient and practical method to estimate GAO values.

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain D of the phase space R and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.

A Bayesian Pricing of Longevity Derivatives with Interest Rate Risks

Abstract Mortality-linked securities such as longevity bonds or longevity swaps usually depend on not only mortality risk but also interest rate risk. However, in the existing pricing methodologies, it is often the case that only the mortality risk is modeled to change in a stochastic manner and the interest rate is kept fixed at a pre-specified level. In order to develop large and liquid longevity markets, it is essential to incorporate the interest rate risk into pricing mortality-linked securities. In this paper we tackle the issue by considering the pricing of longevity derivatives under stochastic interest rates following the CIR model. As for the mortality modeling, we use a two-factor extension of the Lee-Carter model by noting the recent studies which point out the inconsistencies of the original Lee-Carter model with observed mortality rates due to its single factor structure. To address the issue of parameter uncertainty, we propose using a Bayesian methodology both to estimate the models and to price longevity derivatives in line with (Kogure, A., and Y. Kurachi. 2010. “A Bayesian Approach to Pricing Longevity Risk Based on Risk Neutral Predictive Distributions.” Insurance: Mathematics and Economics 46:162–172) and (Kogure, A., J. Li, and S. Y. Kamiya. 2014. “A Bayesian Multivariate Risk-neutral Method for Pricing Reverse Mortgages.” North American Actuarial Journal 18:242–257). We investigate the actual effects of the incorporation of the interest rate risk by applying our methodology to price a longevity cap with Japanese data. The results show significant differences according to whether the CIR model is used or the interest rate is kept fixed.

A Shannon Wavelet Method for Pricing Foreign Exchange Options under the Heston Multi-Factor CIR Model

We present a robust and highly efficient Shannon wavelet pricing method for plain-vanilla foreign exchange European options under the jump-extended Heston model with multi-factor CIR interest rate dynamics. Under a Monte Carlo and partial differential equation hybrid computational framework, the option price can be expressed as an expectation, conditional on the variance factor, of a convolution product that involves the densities of the time-integrated domestic and foreign multi-factor CIR interest rate processes. We propose an efficient treatment to this convolution product that effectively results in a significant dimension reduction, from two multi-factor interest rate processes to only a one-factor process. By means of a state-of-the-art Shannon wavelet inverse Fourier technique, the resulting convolution product is approximated analytically and the conditional expectation can be computed very efficiently. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.

LONG-TERM GROWTH RATE OF EXPECTED UTILITY FOR LEVERAGED ETFs: MARTINGALE EXTRACTION APPROACH

This paper studies the long-term growth rate of expected utility or expected return from holding a leveraged exchanged-traded fund (LETF), which is a constant proportion portfolio of the reference asset. We develop a martingale extraction approach to tackle the path-dependence in the expectation and determine the long-term growth rate through the eigenpair associated with the infinitesimal generator of a time-homogeneous Markovian diffusion. The long-term growth rates are derived explicitly under a number of models for the reference asset, including the geometric Brownian motion model, GARCH model, inverse GARCH model, extended CIR model, 3/2 model, quadratic model, as well as the Heston and [Formula: see text] stochastic volatility models. We also investigate the impact of stochastic interest rate such as the Vasicek model and the inverse GARCH short rate model. Additionally, we determine the optimal leverage ratio that maximizes the long-term growth rate, and examine the effects of model parameters.

Consistent recalibration of yield curve models

AbstractThe analytical tractability of affine (short rate) models, such as the Vasiček and the Cox–Ingersoll–Ross (CIR) models, has made them a popular choice for modeling the dynamics of interest rates. However, in order to properly account for the dynamics of real data, these models must exhibit time‐dependent or even stochastic parameters. This breaks their tractability, and modeling and simulating become an arduous task. We introduce a new class of Heath–Jarrow–Morton (HJM) models that both fit the dynamics of real market data and remain tractable. We call these models consistent recalibration (CRC) models. CRC models appear as limits of concatenations of forward rate increments, each belonging to a Hull–White extended affine factor model with possibly different parameters. That is, we construct HJM models from “tangent” affine models. We develop a theory for continuous path versions of such models and discuss their numerical implementations within the Vasiček and CIR frameworks.

Parametric inference of autoregressive heteroscedastic models with errors in variables

We propose a consistent and asymptotically normal parametric estimator for autoregressive heteroscedastic models with errors in variables based on contrast minimization and give an example for a discrete time observed CIR process with additive noises.

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

We consider a jump-type Cox–Ingersoll–Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak consistency and asymptotic normality, and, under an additional moment assumption, strong consistency as well. In the supercritical case, we prove strong consistency and mixed normal (but non-normal) asymptotic behavior, while in the critical case, weak consistency and non-standard asymptotic behavior are described. We specialize our results to so-called basic affine jump–diffusions as well. Concerning the asymptotic behavior of the MLE in the supercritical case, we derive a stochastic representation of the limiting mixed normal distribution, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case, where a diffusion-type critical CIR process plays a role.

A simple trinomial lattice approach for the skew-extended CIR models

In this paper, we introduce a simple and efficient trinomial lattice tree approach for the skew Cox-Ingersoll-Ross (CIR) model and the doubly skewed CIR model. Suffering from the terms of local times and non-constant volatility, we apply two transforms to the skew-extended CIR processes. Then we construct a modified trinomial tree for the transformed processes which are piecewise tractable diffusions with constant volatility. As a result, the tree for the original skew-extended CIR processes can be easily obtained by using the inverse transform. Results of applications to zero-coupon bonds, European and American options demonstrate that our simple tree approach is efficient and satisfactory.

Backward simulation methods for pricing American options under the CIR process

In this paper, we focus on backward simulation of the CIR process. The purpose is to solve the memory requirement issue of the Least Squares Monte Carlo method when pricing American options by simulation. The concept of backward simulation is presented and it is classified into two types. Under the framework of the second type backward simulation, we seek the solutions for the existing CIR schemes. Specifically, we propose forward–backward simulation approaches for Alfonsi’s two implicit schemes, the fixed Euler schemes and the exact scheme. The proposed schemes are numerically tested and compared in pricing American options under the Heston model and the stochastic interest rate model. Some numerical properties such as the convergence order of the explicit–implicit Euler schemes, the storage requirement estimation of the forward–backward exact scheme and its computing time comparison with the squared Bessel bridge are also tested. Finally, the pros and cons of the related backward simulation schemes are summarized.

Pricing American put option on zero-coupon bond under fractional CIR model with transaction cost

ABSTRACTThe motivation of this study is to evaluate the American put option on zero-coupon bond, when the interest rate model is governed by a fractional CIR (FCIR) interest rate model. Since the existence of fractional Brownian motion, leading to create the arbitrage, we employ the transaction cost for eliminating the arbitrage. We first of all apply the Leland's hedging strategy for a self-financing portfolio that contains an American option and zero-coupon bond and derive a formula for the transaction cost. We perform the least-square Monte Carlo (LSM) method for pricing American option under the proposed interest rate model.

Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling

We introduce a class of interest rate models, called the α-CIR model, which gives a natural extension of the standard CIR model by adopting the α-stable Levy process and preserving the branching property. This model allows to describe in a unified and parsimonious way several recent observations on the sovereign bond market such as the persistency of low interest rate together with the presence of large jumps at local extent. We emphasize on a general integral representation of the model by using random fields, with which we establish the link to the CBI processes and the affine models. Finally we analyze the jump behaviors and in particular the large jumps, and we provide numerical illustrations.

A Revised Approach to CIR Short-Term Interest Rates Model

It is well known that the Cox-Ingersoll-Ross (CIR) stochastic model to study the term structure of interest rates, as introduced in 1985, is inadequate for modelling the current market environment with negative short interest rates. Moreover, the diffusion term in the rate dynamics goes to zero when short rates are small; both volatility and long-run mean do not change with time; they do not fit with the skewed (fat tails) distribution of the interest rates, etc. The aim of the present work is to suggest a new framework, which we call the CIR # model, that fits well the term structure of short interest rates such that the market volatility structure is preserved as well as the analytical tractability of the original CIR model.

Pricing pension buy-outs under stochastic interest and mortality rates

Pension buy-out is a special financial asset issued to offload the pension liabilities holistically in exchange for an upfront premium. In this paper, we concentrate on the pricing of pension buy-outs under dependence between interest and mortality rates risks with an explicit correlation structure in a continuous time framework. Change of measure technique is invoked to simplify the valuation. We also present how to obtain the buy-out price for a hypothetical benefit pension scheme using stochastic models to govern the dynamics of interest and mortality rates. Besides employing a non-mean reverting specification of the Ornstein–Uhlenbeck process and a continuous version of Lee–Carter setting for modeling mortality rates, we prefer Vasicek and Cox–Ingersoll–Ross models for short rates. We provide numerical results under various scenarios along with the confidence intervals using Monte Carlo simulations.

Unspanned Stochastic Volatility in the Multi-Factor CIR Model

Empirical evidence suggests that fixed‐income markets exhibit unspanned stochastic volatility (USV), that is, that one cannot fully hedge volatility risk solely using a portfolio of bonds. While Collin‐Dufresne and Goldstein (2002, Journal of Finance, 57, 1685–1730) showed that no two‐factor Cox–Ingersoll–Ross (CIR) model can exhibit USV, it has been unknown to date whether CIR models with more than two factors can exhibit USV or not. We formally review USV and relate it to bond market incompleteness. We provide necessary and sufficient conditions for a multifactor CIR model to exhibit USV. We then construct a class of three‐factor CIR models that exhibit USV. This answers in the affirmative the above previously open question. We also show that multifactor CIR models with diagonal drift matrix cannot exhibit USV.

Analysis of non-negativity and convergence of solution of the balanced implicit method for the delay Cox–Ingersoll–Ross model

The delay Cox–Ingersoll–Ross (CIR) model is an important model in the financial markets. It has been proved that the solution of this model is non-negative and its pth moments are bounded. However, there is no explicit solution for this model. So, proposing appropriate numerical method for solving this model which preserves non-negativity and boundedness of the model's solution is very important. In this paper, we concentrate on the balanced implicit method (BIM) for this model and show that with choosing suitable control functions the BIM provides numerical solution that preserves non-negativity of solution of the model. Moreover, we show the pth moment boundedness of the numerical solution of the method and prove the convergence of the proposed numerical method. Finally, we present some numerical examples to confirm the theoretical results, and also application of BIM to compute some financial quantities.

Pricing CIR Yield Options by Conditional Moment Matching

We propose an approximation scheme for the pricing of yield options in the CIR model using conditional moment matching based on the gamma and lognormal distributions. This method is fast and simple to implement, and it shows a high degree of accuracy without being subject to the numerical instabilities that can be encountered with more sophisticated approaches.

Bond pricing under mixed generalized CIR model with mixed Wishart volatility process

This paper proposes an extended version of the Cox–Ingersoll–Ross (CIR) model with stochastic volatility and a pricing method on zero-coupon bond under this model. In this version, we replace the standard Brownian motion process with a semi-martingale process named the mixed fractional Brownian motion (mfBm) process which is a linear combination of a fractional Brownian motion (fBm) and a standard Brownian motion. We assume that the part of the volatility process follows a mixed Wishart process which defines by the square of the matrix-valued mfBm process. In order to evaluate the price of the zero-coupon bond under the proposed model we use Monte Carlo simulation method. The computed values of the zero-coupon bond compare with the other interest rate models.

An Improved CIR Tree

The Cox-Ingersoll-Ross (CIR)-process is a well-known model for interest rate or default rate modeling. Here we combine two existing tree approximations, each of them having some disadvantages, in order to obtain an improved tree implementation ensuring convergence for all CIR processes.