Cox-Ingersoll-Ross Research Articles

A nonlinear diffusion model for electricity prices and derivatives

In this paper, we first develop a one-factor diffusion model for electricity prices, which is based on a power transformation of CIR process. We show that the new model is tractable and we are able to derive the analytical solutions for future and future option prices. To enhance the model's ability to capture the prices spikes, we extend it to a time-changed model where the price is modelled by a nonlinear CIR process time changed by Levy subordinators. We employ the eigenfunction expansion methods to obtain the closed-form solutions for the derivatives. Our empirical study indicates the new models have the potential to capture the main features of electricity data better than the competing models.

A nonlinear diffusion model for electricity prices and derivatives

In this paper, we first develop a one-factor diffusion model for electricity prices, which is based on a power transformation of CIR process. We show that the new model is tractable and we are able to derive the analytical solutions for future and future option prices. To enhance the model's ability to capture the prices spikes, we extend it to a time-changed model where the price is modelled by a nonlinear CIR process time changed by Lévy subordinators. We employ the eigenfunction expansion methods to obtain the closed-form solutions for the derivatives. Our empirical study indicates the new models have the potential to capture the main features of electricity data better than the competing models.

Optimal strong approximation of the one-dimensional squared Bessel process

We consider the one-dimensional squared Bessel process given by the stochastic differential equation (SDE) \begin{align*} dX_t = 1\,dt + 2\sqrt{X_t}\,dW_t, \quad X_0=x_0, \quad t\in[0,1], \end{align*} and study strong (pathwise) approximation of the solution $X$ at the final time point $t=1$. This SDE is a particular instance of a Cox-Ingersoll-Ross (CIR) process where the boundary point zero is accessible. We consider numerical methods that have access to values of the driving Brownian motion $W$ at a finite number of time points. We show that the polynomial convergence rate of the $n$-th minimal errors for the class of adaptive algorithms as well as for the class of algorithms that rely on equidistant grids are equal to infinity and $1/2$, respectively. This shows that adaption results in a tremendously improved convergence rate. As a by-product, we obtain that the parameters appearing in the CIR process affect the convergence rate of strong approximation.

From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable

We study the Tauberian relations between the moment generating function (MGF) and the complementary cumulative distribution function of a random variable whose MGF is finite only on part of the real line. We relate the right tail behavior of the cumulative distribution function of such a random variable to the behavior of its MGF near the critical moment. We apply our results to an arbitrary superposition of a CIR process and the time-integral of this process.

Effects of Regime Switching on Pricing Credit Options in a Shifted CIR Model

Effects of Regime Switching on Pricing Credit Options in a Shifted CIR Model

Analytical Upper Bounds for American Exotic Currency Options with a Stochastic Skew Model

On the basis that most instruments traded on options markets are American-style ones, this paper develops the analytical upper and lower bounds of American cross-currency and quanto options under the stochastic skew model proposed by Carr and Wu (2007) when domestic risk free rates are higher or lower than the foreign risk free rates. The analytical bounds derived here are not only very tight and accurate for American option pricing, but also offer a quasi-closed form solution which is able to enhance evaluation and hedging efficiency in real world markets. We also acquire the analytical solutions for European cross-currency and quanto options given by applying two separate mean-reverting square-root processes to two separate time-changed Levy processes, consistent with the realistic phenomena of currency returns.

Uniform approximation of the Cox–Ingersoll–Ross process via exact simulation at random times

AbstractIn this paper we uniformly approximate the trajectories of the Cox–Ingersoll–Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view, the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact, simulation of the CIR dynamics at some deterministic time grid.

Can Exchange Rate Dynamics in Krugman's Target-Zone Model Be Directly Tested?

Despite Krugman's (1991) model being a benchmark for modelling target zones, empirical support has been sparse due to the subtle non-linear relationship between the observable exchange rate and underlying unobservable fundamental. This paper provides an alternative approach to derive explicit exchange rate dynamics by approximating a quadratic relationship between the exchange rate and fundamental through a power-series method. The exchange rate dynamics with a parametric class of drift terms of the stochastic fundamental including constant-trend, symmetric and asymmetric mean-reverting forces regarding how central banks intervene are ready for direct empirical tests. The empirical results demonstrate that the derived dynamics following a mean-reverting square-root or double square-root processes adequately fits the exchange rate data of various target-zone systems including the Exchange Rate Mechanism. The model parameters of the exchange rate dynamics under the asymmetric mean-reverting fundamental are shown to be associated with realignment of the currencies' target zones.

Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process

We study exponential integrability properties of the Cox--Ingersoll--Ross (CIR) process and its Euler--Maruyama discretizations with various types of truncation and reflection at $0$. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of stochastic differential equations arising in finance. We prove that both implicit and explicit Euler--Maruyama discretizations for the CIR process preserve the exponential integrability of the exact solution for a wide range of parameters, and find lower bounds on the explosion time.

A comparison of risk transfer strategies for a portfolio of life annuities based on RORAC

ABSTRACTThis paper aims to compare different reinsurance arrangements in order to reduce the longevity and financial risk originated by a life insurer while managing a portfolio of annuities policies. Linear and nonlinear reinsurance strategies as well as swap like agreements are evaluated via a discrete-time actuarial risk model. Specifically, longevity dynamics are represented by Lee–Carter type models, while interest rate is modeled by Cox–Ingersoll–Ross model. The reinsurance strategies effectiveness is evaluated according to the Return on Risk Adjusted Capital under a ruin probability constrain.

Exponential ergodicity of CIR interest rate model with random switching

In this paper, we consider ergodicity of Cox–Ingersoll–Ross (CIR) interest rate model with random switching. First, we show that the CIR model with switching has a unique stationary distribution. Next, we prove that the transition semigroup for the CIR model with switching converges to the stationary distribution at an exponential rate in the Wasserstein distance. Moreover, under two particular cases, the explicit expressions for stationary distributions are presented. Finally, the central limit theorem for the CIR model with random switching is established.

Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes

We investigate the asymptotic behavior as time goes to infinity of Hawkes processes whose regression kernel has $L^{1}$ norm close to one and power law tail of the form $x^{-(1+\alpha)}$, with $\alpha\in(0,1)$. We in particular prove that when $\alpha\in(1/2,1)$, after suitable rescaling, their law asymptotically behaves as a kind of integrated fractional Cox–Ingersoll–Ross process, with associated Hurst parameter $H=\alpha-1/2$. This result is in contrast to the case of a regression kernel with light tail, where a classical Brownian CIR process is obtained at the limit. Interestingly, it shows that persistence properties in the point process can lead to an irregular behavior of the limiting process. This theoretical result enables us to give an agent-based foundation to some recent findings about the rough nature of volatility in financial markets.

Maximum likelihood type estimation for discretely observed CIR model with small [formula omitted]-stable noises

A maximum likelihood type estimation of the drift and volatility coefficient parameters in the CIR type model driven by α-stable noises is studied when the dispersion parameter ε→0 and the discrete observations frequency n→∞ simultaneously.

Optimal mean–variance efficiency of a family with life insurance under inflation risk

We study an optimization problem of a family under mean–variance efficiency. The market consists of cash, a zero-coupon bond, an inflation-indexed zero-coupon bond, a stock, life insurance and income-replacement insurance. The instantaneous interest rate is modeled as the Cox–Ingersoll–Ross (CIR) model, and we use a generalized Black–Scholes model to characterize the stock and labor income. We also take into account the inflation risk and consider our problem in the real market. The goal of the family is to maximize the mean of the surplus wealth at the retirement or death of the breadwinner and minimize its variance by finding a portfolio selection. The efficient frontier and optimal strategies are derived through the dynamic programming method and the technique of solving associated nonlinear HJB equations. We also present a numerical illustration to explore the impact of economical parameters on the efficient frontier.

Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model

ABSTRACTMost of the empirical studies on stochastic volatility dynamics favour the 3/2 specification over the square-root (CIR) process in the Heston model. In the context of option pricing, the 3/2 stochastic volatility model (SVM) is reported to be able to capture the volatility skew evolution better than the Heston model. In this article, we make a thorough investigation on the analytic tractability of the 3/2 SVM by proposing a closed-form formula for the partial transform of the triple joint transition density which stand for the log asset price, the quadratic variation (continuous realized variance) and the instantaneous variance, respectively. Two distinct formulations are provided for deriving the main result. The closed-form partial transform enables us to deduce a variety of marginal partial transforms and characteristic functions and plays a crucial role in pricing discretely sampled variance derivatives and exotic options that depend on both the asset price and quadratic variation. Various applications and numerical examples on pricing moment swaps and timer options with discrete monitoring feature are given to demonstrate the versatility of the partial transform under the 3/2 model.

Pricing corporate bonds with interest rates following double square-root process

This paper develops a corporate bond pricing model following the structural approach in which the dynamics of the instantaneous risk-free interest rate is governed by the double square-root (DSR) process. Credit spreads generated from the pricing model depend explicitly upon the levels of interest rates via the non-linear effect arising from the DSR process. Given a positive correlation between the interest rates and leverage ratios, the credit spreads generated by the pricing model have negative relationship with the interest rates, that is consistent with empirical findings using bond market data during 2008–2013 when interest rates were low.

A mixed Monte Carlo and partial differential equation variance reduction method for foreign exchange options under the Heston–Cox–Ingersoll–Ross model

In this paper, we consider the valuation of European and path-dependent options in foreign exchange markets when the currency exchange rate evolves according to the Heston model combined with the Cox-Ingersoll-Ross (CIR) dynamics for the stochastic domestic and foreign short interest rates. The mixed Monte Carlo/partial differential equation method requires that we simulate only the paths of the squared volatility and the two interest rates, while an inner Black-Scholes-type expectation is evaluated by means of a partial differential equation. This can lead to a substantial variance reduction and complexity improvements under certain circumstances depending on the contract and the model parameters. In this work, we establish the uniform boundedness of moments of the exchange rate process and its approximation, and prove strong convergence of the latter in Lρ (ρ ⩾ 1). Then, we carry out a variance reduction analysis and obtain accurate approximations for quantities of interest. All theoretical contributions can be extended to multi-factor short rates in a straightforward manner. Finally, we illustrate the efficiency of the method for the four-factor Heston-CIR model through a detailed quantitative assessment.

A stable Cox–Ingersoll–Ross model with restart

We consider a stable Cox–Ingersoll–Ross model in a domain D=[0,∞) which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity λ>0 to a new point, according to a distribution ν. From this new point, it repeats the above behavior independently of what has transpired previously. We give the infinitesimal generator of this new process and prove that the corresponding semigroup is compact and thus the spectrum of the generator consists exclusively of eigenvalues. We further show that the principal eigenvalue gives the exponential rate of decay of not exiting the domain by time t. Finally, we study some properties of the principal eigenvalue.

Long-term behavior of stochastic interest rate models with Markov switching

In this paper, we consider the long time behavior of Cox–Ingersoll–Ross (CIR) interest rate model with Markov switching. Using the ergodic theory of switching diffusions, we show that CIR model with Markov switching has a unique stationary distribution. Furthermore, we prove that the sequence X¯t:=1t∫0tXsds converges almost surely. As a by-product, we find that the marginal stationary distribution for CIR model with Markov switching can be determined uniquely by its moments.

A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models

A novel time-stepping scheme, called transformed jump-adapted backward Euler method, is developed in this paper to simulate a class of jump-extended CIR and CEV models. The proposed scheme is able to preserve the positivity of the underlying problems. Furthermore, its strong convergence rate of order one is recovered for the considered models with non-Lipschitz diffusion coefficients. Numerical examples are finally reported to confirm our theoretical findings.