Cox-Ingersoll-Ross Research Articles

Generación de covarianzas con el modelo multifactorial CIR

This paper presents a general framework of how to generate covariances between riskless interest rate r(t) instruments, and financial instruments with intensity of default λ(t) , in Cox, Ingersoll, Ross (CIR), or in the extended multifactor CIR model.

Explicit formulas for pricing credit-linked notes with counterparty risk under reduced-form framework

A credit-linked note (CLN) is a type of credit derivative, constructed with a bond and an embedded credit default swap, which allows the issuer to transfer a specific credit risk to credit investors. In this paper, we model CLNs with and without counterparty risk in the reduced-form framework. For CLNs with counterparty risk, we consider two different scenarios, i.e. the issuer of CLNs and reference assets have either positive correlation or negative correlation. Assuming the interest rate follows the Cox–Ingersoll–Ross (CIR) model (Cox et al., 1985) and the default events mainly depend on the interest rate, we model the two different correlations. Explicit formulas for value functions are obtained through a partial differential equation approach. In addition, counterparty valuation adjustment and the dependence on related parameters are also investigated.

How Long Does the Market Think You Will Live? Implying Longevity from Annuity Prices

We use life annuity prices to extract information about human longevity by formulating a model that links the term structure of mortality and interest rates. We then invert the model and perform nonlinear least squares to obtain implied longevity forecasts. Methodologically, we assume a Cox-Ingersoll-Ross (CIR) model for the underlying yield curve and a cohort-varying Gompertz-Makeham (GM) law for mortality. Our main result is that over the last decade markets have been implying an improvement in longevity of an average of 6 weeks per year for males and 3 weeks for females. In the year 2004 market prices implied a 40.1% probability of survival to the age 90 for a 75-year old male (51.2% for a female) annuitant. By the year 2013 the implied survival probability had increased to 46.1% (and 53.1%). The corresponding implied life expectancy has increased (at the age of 75) from 13.09 years for males (15.08 years for females) to 14.28 years (and 15.61 years). Although these values are implied directly from markets, they are consistent with demographic projections. Similar to implied volatility in option pricing, we believe that our implied survival probabilities (ISP) and implied life expectancy (ILE) are relevant for the financial management of assets post-retirement and very important for the optimal timing and allocation to annuities; procrastinators are swimming against an uncertain but rather strong longevity trend.

Ergodic properties for $\alpha$-CIR models and a class of generalized Fleming-Viot processes

We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.

Quantitative Models for Financial Markets

The study analyses quantitative models for financial markets by starting from geometric Brown process and Wiener process by analyzing Ito’s lemma and first passage model. Furthermore, it is analyzed the prices of the options, Vanilla & Exotic, by using the expected value and numerical model. From contingent claim approach ALM strategies are also analyzed so to get the effective duration measure of liabilities. Furthermore, the study analyses interest rate models by showing that the yield curve is given by the average of the expected short rates & variation of GDP with the liquidity risk, but in the case we have crisis it is possible to have risk premium as well, the study is based on simulated modelisation by using the drift condition in combination with the inflation models as expectation of the markets. Moreover, the CIR process is considered as well by getting with modification of the diffusion process the same result of the simulated modelisation but we have to consider that the CIR process is considered in the simulated environment as well. The credit risk model is considered as well in intensity model & structural model by getting the liquidity and risk premium and the PD probability from the Rating Matrix as well by using the diagonal. Furthermore, the systemic risk is considered as well by using a deco relation concept. Moreover, along the equilibrium condition between financial markets is achieved the equity pricing with implications for the portfolio construction in simulated environment. Finally, the VaR model is presented in combination with a percentile approach by using a simulated approach consistent with back test as well in the case we consider the percentile of the simulated distribution.

Optimal starting–stopping and switching of a CIR process with fixed costs

This paper analyzes the problem of starting and stopping a Cox-Ingersoll-Ross (CIR) process with fixed costs. In addition, we also study a related optimal switching problem that involves an infinite sequence of starts and stops. We establish the conditions under which the starting-stopping and switching problems admit the same optimal starting and/or stopping strategies. We rigorously prove that the optimal starting and stopping strategies are of threshold type, and give the analytical expressions for the value functions in terms of confluent hypergeometric functions. Numerical examples are provided to illustrate the dependence of timing strategies on model parameters and transaction costs.

Optimal Investment for Insurers with the Extended CIR Interest Rate Model

A fundamental challenge for insurance companies (insurers) is to strike the best balance between optimal investment and risk management of paying insurance liabilities, especially in a low interest rate environment. The stochastic interest rate becomes a critical factor in this asset-liability management (ALM) problem. This paper derives the closed-form solution to the optimal investment problem for an insurer subject to the insurance liability of compound Poisson process and the stochastic interest rate following the extended CIR model. Therefore, the insurer’s wealth follows a jump-diffusion model with stochastic interest rate when she invests in stocks and bonds. Our problem involves maximizing the expected constant relative risk averse (CRRA) utility function subject to stochastic interest rate and Poisson shocks. After solving the stochastic optimal control problem with the HJB framework, we offer a verification theorem by proving the uniform integrability of a tight upper bound for the objective function.

An Empirical Analysis of Yield Curves Across Euro and Non-Euro Countries Using Interbank Interest Rates

This paper studies the interrelations among yield curve factors, market expectations and monetary policy rates using interbank interest rates across Euro- and non-Euro countries. The term structure of interest rates can be summarized by the level and slope factor, whereas curvature is not a common feature of interbank rates. Interest rates are first modeled in an equilibrium framework using a two-factor CIR (1985) model, and Kalman filter is used to extract the two factors under the no-arbitrage restriction.Impulse response analysis shows that German factors and forecast errors have the biggest influence on those factors from other markets, and that yield slope is a useful variable for capturing market expectations. Based on the estimated factors, theoretical yields implied by the Expectations hypothesis match remarkably well the movements of monetary policy rates, providing a consistent link between yield curve factors and macro-economic variables and thus integrating the approaches between no-arbitrage yield curve modeling and macro-economic based Expectations Hypothesis Approach.

Term Structure Modeling and Forecasting of Government Bond Yields

Accurate modelling and precise estimation of the term structure of interest rate are of crucial importance in many areas of finance and macroeconomics as it is the most important factor in the capital market and probably the economy. This study compares the in‐sample fit and out‐of‐sample forecast accuracy of the Cox–Ingersoll–Ross (CIR) and Nelson–Siegel models. For the in‐sample fit, there is a significant lack of information on the short‐term CIR model. The CIR model should also be considered too poor to describe the term structure in a simulation‐based context. It generates a downward slope average yield curve. Contrary to CIR model, Nelson–Siegel model is not only compatible to fit attractively the yield curve but also accurately forecast the future yield for various maturities. Furthermore, the non‐linear version of the Nelson–Siegel model outperforms the linearised one. In a simulation‐based context, the Nelson–Siegel model is capable to replicate most of the stylised facts of the Japanese market yield curve. Therefore, it turns out that the Nelson–Siegel model (non‐linear version) could be a good candidate among various alternatives to study the evolution of the yield curve in Japanese market.

SIMPLE SIMULATION SCHEMES FOR CIR AND WISHART PROCESSES

We develop some simple simulation algorithms for CIR and Wishart processes. We investigate rigorously the square of a matrix valued Ornstein–Uhlenbeck process, the main idea being to split the generator and to reduce the problem to the simulation of the square of a matrix valued Ornstein–Uhlenbeck process to be added to a deterministic process. In this way, we provide a weak second-order scheme that requires only the simulation of i.i.d. Gaussian r.v.'s and simple matrix manipulations.

TRANSITION DENSITY FOR CIR PROCESS BY LIE SYMMETRIES AND APPLICATION TO ZCB PRICING

Using a Lie algebraic approach we explicitly compute the transition density function for the solution of the stochastic differential equation defining the CIR process. Moreover we show how to use such a derivation to recover the well-known formula for the asscoiated ZCB fair price.

Efficient Estimation Using the Characteristic Function

The method of moments proposed by Carrasco and Florens (2000) permits to fully exploit the information contained in the characteristic function and yields an estimator which is asymptotically as efficient as the maximum likelihood estimator. However, this estimation procedure depends on a regularization or tuning parameter ∝ that needs to be selected. The aim of the present paper is to provide a way to optimally choose ∝ by minimizing the approximate mean square error (AMSE) of the estimator. Following an approach similar to that of Newey and Smith (2004), we derive a higher-order expansion of the estimator from which we characterize the finite sample dependence of the AMSE on ∝. We provide a data-driven procedure for selecting the regularization parameter that relies on parametric bootstrap. We show that this procedure delivers a root T consistent estimator of ∝. Moreover, the data-driven selection of the regularization parameter preserves the consistency, asymptotic normality and efficiency of the CGMM estimator. Simulation experiments based on a CIR model show the relevance of the proposed approach.

Recovering default risk from CDS spreads with a nonlinear filter

We propose a nonlinear filter to estimate the time-varying default risk from the term structure of credit default swap (CDS) spreads. Based on the numerical solution of the Fokker–Planck equation (FPE) using a meshfree interpolation method, the filter performs a joint estimation of the risk-neutral default intensity and CIR model parameters. As the FPE can account for nonlinear functions and non-Gaussian errors, the proposed framework provides outstanding flexibility and accuracy. We test the nonlinear filter on simulated spreads and apply it to daily CDS data of the Dow Jones Industrial Average component companies from 2005 to 2010 with supportive results.

Positive Harris recurrence of the CIR process and its applications

Positive Harris recurrence of the CIR process and its applications

The improved split-step θ methods for stochastic differential equation

Two improved split-step θ methods, which, respectively, named split-step composite θ method and modified split-step θ-Milstein method, are proposed for numerically solving stochastic differential equation of Ito type. The stability and convergence of these methods are investigated in the mean-square sense. Moreover, an approach to improve the numerical stability is illustrated by choices of parameters of these two methods. Some numerical examples show the accordance between the theoretical and numerical results. Further numerical tests exhibit not only the Hamiltonian-preserving property of the improved split-step θ methods for a stochastic differential system but also the positivity-preserving property of the modified split-step θ-Milstein method for the Cox–Ingersoll–Ross model. Copyright © 2013 John Wiley & Sons, Ltd.

On a discrete version of the CIR process

We consider the so-called gambler's ruin problem for a discrete-time Markov chain that converges to a Cox–Ingersoll–Ross (CIR) process. Both the probability that the chain will hit a given boundary before the other and the average number of transitions needed to end the game are computed explicitly. Furthermore, we show that the quantities that we obtained tend to the corresponding ones for the CIR process. A real-life application to a problem in hydrology is presented.

Exotic Geometric Average Options Pricing under Stochastic Volatility

This article derives semi-analytical pricing formulae for geometric average options (GAOs) within a stochastic volatility framework. Assuming a general mean reverting process for the underlying asset and a square-root process for the volatility, the cross-moment generating function is derived and the cumulative probabilities are recovered using the Gauss–Laguerre quadrature rule. Fixed and floating strikes as well as other exotic GAO on different assets such as stocks, currency exchange rates and interest rates are derived. The approach is found to be very accurate and efficient.

An alternative approach to the calibration of the Vasicek and CIR interest rate models via generating functions

We propose a new method to calibrate the Vasicek and Cox--Ingersoll--Ross interest rate models from bond prices. We define an appropriate generating function and derive recursive relations between the derivatives of the generating function and the bond prices. The parameters of the Vasicek and CIR models are then obtained by solving a system of linearly independent equations arising from the recursive relations. We include numerical results that show the method’s accuracy when bond prices generated from the exact formulas are used.

Rare Shock, Two-Factor Stochastic Volatility and Currency Option Pricing

In this paper, we develop an option valuation model where the dynamics of the spot foreign exchange rate is governed by a two-factor Markov-modulated jump-diffusion process. The short-term fluctuation of stochastic volatility is driven by a Cox–Ingersoll–Ross (CIR) process and the long-term variation of stochastic volatility is driven by a continuous-time Markov chain which can be interpreted as economy states. Rare events are governed by a compound Poisson process with log-normal jump amplitude and stochastic jump intensity is modulated by a common continuous-time Markov chain. Since the market is incomplete under regime-switching assumptions, we determine a risk-neutral martingale measure via the Esscher transform and then give a pricing formula of currency options. Numerical results are presented for investigating the impact of the long-term volatility and the annual jump intensity on option prices.

Long-term behavior of stochastic interest rate models with jumps and memory

The long-term interest rates, for example, determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option. In this paper, for a one-factor model, we reveal that the long-term return t−μ∫0tX(s)ds for some μ≥1, in which X(t) follows an extension of the Cox–Ingersoll–Ross model with jumps and memory, converges almost surely to a reversion level which is random itself. Such a convergence can be applied in the determination of models of participation in the benefit or of saving products with a guaranteed minimum return. As an immediate application of the result obtained for the one-factor model, for a class of two-factor model, we also investigate the almost sure convergence of the long-term return t−μ∫0tY(s)ds for some μ≥1, where Y(t) follows an extended Cox–Ingersoll–Ross model with stochastic reversion level −X(t)/(2β) in which X(t) follows an extension of the square root process. This result can be applied to, e.g., how the percentage of interest should be determined when insurance companies promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on.