Ross Model Research Articles

Some qualitative properties of the discrete models for malaria propagation

This paper addresses a reliable mathematical modeling of malaria propagation in infected societies for humans and mosquitoes with an extension of the basic Ross–Macdonald model. We analyze the extended Ross model numerically which is an initial value problem of a seven-dimensional system of the first-order ODEs. For this aim, the discretized scheme of the extended model is split into two parts. First, we apply the step-size functions to approximate the time derivatives with the first-order consistency. Then we use a nonlocal discretization of the standard θ-method to the right side of the system to obtain a linear system. We find the conditions for the step-size functions under which specific intervals are positively invariant for the total populations and each component of the solution of the extended Ross model. We suggest a step-size function for the extended Ross model to preserve the dynamical consistency of the model for any step size Δt. The numerical simulations confirm the theoretical results for the examples and we compare the efficiency of the nonstandard Runge–Kutta methods with the different step-size functions for sufficiently large step sizes.

Tail Distribution Estimates of Fractional CIR Model

The aim of this work is to study the tail distribution of the Cox–Ingersoll–Ross (CIR) model driven by fractional Brownian motion. We first prove the existence and uniqueness of the solution. Then based on the techniques of Malliavin calculus and a result established recently in [1], we obtain an explicit estimate for tail distributions.

The truncated Euler–Maruyama method for CIR model driven by fractional Brownian motion

Recently, Hong et al. (2020) established the strong convergence rate of the backward Euler scheme for the Cox–Ingersoll–Ross (CIR) model driven by fractional Brownian motion with Hurst parameter H∈(1/2,1), which may effect the efficiency of computation. Taking advantage of being explicit and easily implementable, a positivity preserving explicit scheme is proposed in this paper. For overcoming the difficulties caused by the unbounded diffusion coefficient, an auxiliary equation with a constant diffusion coefficient obtained by proper Lamperti transformation is used. By means of Malliavin calculus, we show that the truncated Euler–Maruyama scheme applied to this auxiliary equation not only ensures the positivity of the numerical solution, but also has the H-order rate of the root mean square error over a finite time interval. Furthermore, by transforming back, an explicit scheme for the original CIR model is obtained and has the same convergence order. Finally, some numerical experiments are provided to illustrate the theoretical results.

On the Deterministic-Shift Extended CIR Model in a Negative Interest Rate Framework

In this paper, we propose a new exogenous model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model with a perfect fit to the observed term-structure. We use the difference between two independent CIR processes and apply the deterministic-shift extension technique. To allow for a fast calibration to the market swaption surface, we apply the Gram–Charlier expansion to calculate the swaption prices in our model. We run several numerical tests to demonstrate the strengths of this model by using Monte-Carlo techniques. In particular, the model produces close Bermudan swaption prices compared to Bloomberg’s Hull–White one-factor model. Moreover, it finds constant maturity swap (CMS) rates very close to Bloomberg’s CMS rates.

The role of adaptivity in a numerical method for the Cox–Ingersoll–Ross model

We demonstrate the effectiveness of an adaptive explicit Euler method for the approximate solution of the Cox–Ingersoll–Ross model. This relies on a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach a neighbourhood of zero. The method is hybrid in the sense that a convergent backstop method is invoked if the timestep becomes too small, or to prevent solutions from overshooting zero and becoming negative. Under parameter constraints that imply Feller’s condition, we prove that such a scheme is strongly convergent, of order at least 1/2. Control of the strong error is important for multi-level Monte Carlo techniques. Under Feller’s condition we also prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. Numerically, we compare this adaptive method to fixed step implicit and explicit schemes, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive in a domain that extends beyond Feller’s condition, indicating suitability for the modelling of stochastic volatility in Heston-type asset models.

Assessment of PV Module Temperature Models for Building-Integrated Photovoltaics (BIPV)

This paper assesses two steady-state photovoltaic (PV) module temperature models when applied to building integrated photovoltaic (BIPV) rainscreens and curtain walls. The models are the Ross and the Faiman models, both extensively used for PV modules mounted on open-rack support structures in PV plants. The experimental setups arrange the BIPV modules vertically and with different backside boundary conditions to cover the mounting configurations under study. Data monitoring over more than a year was the experimental basis to assess each model by comparing simulated and measured temperatures with the help of four different metrics: mean absolute error, root mean square error, mean bias error, and coefficient of determination. The performance ratio of each system without the temperature effect was calculated by comparing the experimental energy output with the energy output determined with the measured temperatures. This parameter allowed the estimation of the PV energy with the predicted temperatures to assess the suitability of each temperature model for energy-prediction purposes. The assessment showed that the Ross model is the most suitable for predicting the annual PV energy in rainscreen and curtain-wall applications. Highlighted is the importance of fitting the model coefficients with a representative set of in situ monitored data. The data set should preferably include the inner (backside) temperature, i.e., the air chamber temperature in ventilated façades or the indoor temperature in curtain walls and windows.

Affine term structure models: A time‐change approach with perfect fit to market curves

AbstractWe address the so‐called calibration problem, which consists of fitting in a tractable way a given model to a specified term structure such as yield, prepayment or default probability curves. Time‐homogeneous affine jump diffusions (HAJD) are tractable processes but have limited flexibility; they fail to perfectly replicate actual market curves. Applying a deterministic shift to the latter is a simple but efficient solution that is widely used by both academics and practitioners. However, the shift approach may not be appropriate when positivity is required, a common constraint when dealing with credit spreads or default intensities. In this paper, we address this problem by adopting a time‐change technique. Specific attention is paid to the Cox–Ingersoll–Ross model with compound Poisson jumps (JCIR), which remains standard for modeling intensities. Our time‐changed JCIR (TC‐JCIR) is compared to the shifted JCIR (JCIR++) in various credit applications such as credit default swap (CDS), credit default swaption, and credit valuation adjustment (CVA) under wrong‐way risk (WWR). The TC‐JCIR model is able to generate much larger implied volatilities and covariance effects than JCIR++ under positivity constraints and represents an appealing alternative to the latter.

Lookback Option Pricing Problems of Uncertain Mean-Reverting Stock Model

A lookback option is a path-dependent option, offering a payoff that depends on the maximum or minimum value of the underlying asset price over the life of the option. This paper presents a new mean-reverting uncertain stock model with a floating interest rate to study the lookback option price, in which the processing of the interest rate is assumed to be the uncertain counterpart of the Cox–Ingersoll–Ross (CIR) model. The CIR model can reflect the fluctuations in the interest rate and ensure that such rate is positive. Subsequently, lookback option pricing formulas are derived through the α-path method and some mathematical properties of the uncertain option pricing formulas are discussed. In addition, several numerical examples are given to illustrate the effectiveness of the proposed model.

Copolymerization of Glycidyl Methacrylate and 1,1,1,3,3,3-Hexafluoroisopropyl Acrylate

The copolymerization of 1,1,1,3,3,3-hexafluoroisopropyl acrylate (HFIPA) and glycidyl methacrylate via reversible addition-fragmentation chain transfer (RAFT) process was investigated. 2-cyano-2-propyl dodecyl trithiocarbonate (CPDT) was used as chain transfer agent. It is turned out that CPDT and polymeric chain transfer agent obtained based on HFIPA and CPDT provide a good control over molar mass characteristic of copolymers (Đ = 1.05). Reactivity ratios were found to be r1(GMA) = 1.57 and r2(HFIPA) = 0.05 by Fineman–Ross model.

Convergence rates of large-time sensitivities with the Hansen–Scheinkman decomposition

This paper investigates the large-time asymptotic behavior of the sensitivities of cash flows. In quantitative finance, the price of a cash flow is expressed in terms of a pricing operator of a Markov diffusion process. We study the extent to which the pricing operator is affected by small changes of the underlying Markov diffusion. The main idea is a partial differential equation (PDE) representation of the pricing operator by incorporating the Hansen–Scheinkman decomposition method. The sensitivities of the cash flows and their large-time convergence rates can be represented via simple expressions in terms of eigenvalues and eigenfunctions of the pricing operator. Furthermore, compared to the work of Park (Finance Stoch 4:773–825, 2018), more detailed convergence rates are provided. In addition, we discuss the application of our results to three practical problems: utility maximization, entropic risk measures, and bond prices. Finally, as examples, explicit results for several market models such as the Cox–Ingersoll–Ross (CIR) model, 3/2 model and constant elasticity of variance (CEV) model are presented.

Interest rates forecasting: Between Hull and White and the CIR#—How to make a single‐factor model work

AbstractIn this work, we present our findings of the so‐called CIR#, which is a modified version of the Cox, Ingersoll, and Ross (CIR) model, turned into a forecasting tool for any term structure. The main feature of the CIR# model is its ability to cope with negative interest rates, cluster volatility, and jumps. By considering a dataset composed of money market interest rates during turmoil and calmer periods, we show how the CIR# performs in terms of directionality of rates and forecasting error. Comparison is carried out with a revamped version of the CIR model (denoted ), the Hull and White model, and the exponentially weighted moving average (EWMA) which is often adopted whenever no structure in data is assumed. To confirm the analysis, testing and validation is performed on both historical and ad hoc data with different metrics and clustering criteria.

Pricing foreign exchange options under a hybrid Heston-Cox-Ingersoll-Ross model with regime switching

Abstract In this paper, the pricing of foreign exchange options is considered under a modified Heston–Cox–Ingersoll–Ross hybrid model. This modified model reserves all the characteristics of the Heston–Cox–Ingersoll–Ross model and also additionally assumes regime switching in the key parameters of the volatility as well as the domestic and foreign interest rates. Even though complicated, we have derived a closed-form pricing formula for foreign exchange options after the affinity of this new model is verified. Various properties of the newly derived formula are also shown through numerical experiments. To show the performance of this newly proposed model, an empirical study is also conducted, the result of which suggests that our model is a good alternative to the Heston–Cox–Ingersoll–Ross model for practical purpose.

Optimal asset allocation under search frictions and stochastic interest rate

In this paper, we investigate an optimal asset allocation problem in a financial market consisting of one risk-free asset, one liquid risky asset and one illiquid risky asset. The liquidity risk stems from the asset that cannot be traded continuously, and the trading opportunities are captured by a Poisson process with constant intensity. Also, it is assumed that the interest rate is stochastically varying and follows the Cox–Ingersoll–Ross model. The performance functional of the decision maker is selected as the expected logarithmic utility of the total wealth at terminal time. The dynamic programming principle coupled with the Hamilton–Jacobi–Bellman equation has been adopted to solve this stochastic optimal control problem. In order to reduce the dimension of the problem, we introduce the proportion of the wealth invested in the illiquid risky asset and derive the semi-analytical form of the value function using a separation principle. A finite difference method is employed to solve the controlled partial differential equation satisfied by a function which is an important component of the value function. The numerical examples and their economic interpretations are then discussed.

Comparison of numerical methods to price zero coupon bonds in a two-factor CIR model

In this paper we price a zero coupon bond under a Cox–Ingersoll–Ross (CIR) two-factor model using various numerical schemes. To the best of our knowledge, a closed-form or explicit price functional is not trivial and has been less studied. The use and comparison of several numerical methods to determine the bond price is one contribution of this paper. Ordinary differential equations (ODEs) , finite difference schemes and simulation are the three classes of numerical methods considered. These are compared on the basis of computational efficiency and accuracy, with the second aim of this paper being to identify the most efficient numerical method. The numerical ODE methods used to solve the system of ODEs arising as a result of the affine structure of the CIR model are more accurate and efficient than the other classes of methods considered, with the Runge–Kutta ODE method being the most efficient. The Alternating Direction Implicit (ADI) method is the most efficient of the finite difference scheme methods considered, while the simulation methods are shown to be inefficient. Our choice of considering these methods instead of the other known and apparently new numerical methods (eg Fast Fourier Transform (FFT) method, Cosine (COS) method, etc.) is motivated by their popularity in handling interest rate instruments.
 Keywords: Cox–Ingersoll–Ross model; numerical methods; Runge–Kutta method; zero-coupon bonds; Alternating Direction Implicit method

China Markets and Global Stock Return Predictability

This paper explores the China equity market's impact on the global equity markets through the lens of asset pricing. We study the predictive properties of the lagged China returns for global stock returns and find that the lagged China returns can significantly predict other markets' stock returns, but not vice versa. We augmented three selected global equity pricing models (Chen, Roll, and Ross's model (CRR), Ang and Bekaert's model (AB), and Rapach, Strauss, and Zhou's model (RSZ)) with the lagged China returns respectively, and find that these popular global stock return predictors cannot explain the predictive power of the lagged China returns. Moreover, we find that only the lagged China returns can lead the international returns at the one-month horizon but not the lagged US returns. Further evidence reveals it is due to that the information diffusion of the China market occurs at the longer-horizons (e.g. one-month) while the information diffusion of the US market occurs at the shorter-horizons (e.g. one-week). Lastly, we documented that the lagged China returns during the low investor-attention period have stronger predictability compared to the predictive performance during the high attention period, which is in line with the information friction theory that when the attention-constrained investors make investment decisions they fail to allocate attention to certain economic state variables, which causes the slow information diffusion across markets and leads to the cross-market return predictability. Overall, our results indicate that the lagged China returns should be regarded as a global state variable that helps us predict future stock returns.

Positivity and convergence of the balanced implicit method for the nonlinear jump-extended CIR model

In this paper, we extend the balanced implicit method (BIM) to nonlinear jump-extended Cox–Ingersoll–Ross (CIR) model. Firstly, we construct the numerical method BIM for this nonlinear jump-extended CIR model and prove the positivity of the proposed method for the model. Furthermore the strong convergence of the BIM is proved in L1 and L2 sense. Finally, we give two examples to illustrate the positivity and convergence of the BIM; numerical results verify the theoretical findings.

Some characterizations for the CIR model with Markov switching

Recently, the Cox–Ingersoll–Ross (CIR) model with Markov switching has been discussed extensively. However, the covariance function and the [Formula: see text]th moment for this model are still open. In this paper, we consider some characterizations for the CIR model with Markov switching. First, the conditional moment generating functions for CIR model with Markov switching are given. Then, explicit expressions for the covariance function and moments of the CIR model with Markov switching are obtained. Finally, several examples have been presented to illustrate our results.

WHAT A DIFFERENCE ONE PROBABILITY MAKES IN THE CONVERGENCE OF BINOMIAL TREES

In the [Formula: see text]-period Cox, Ross, and Rubinstein (CRR) model, we achieve smooth convergence of European vanilla options to their Black–Scholes limits simply by altering the probability at one node, in fact, at the preterminal node between the closest neighbors of the strike in the terminal layer. For barrier options, we do even better, obtaining order [Formula: see text] convergence by altering the probability just at the node nearest the barrier, but only the first time it is hit. First-order smooth convergence for vanilla options was already achieved in Tian’s flexible model but here we show how second order smooth convergence can be achieved by changing one probability, leading to convergence of order [Formula: see text] with Richardson extrapolation. We illustrate our results with examples and provide numerical evidence of our results.

Semi-Closed Form Prices of Barrier Options in the Time-Dependent CEV and CIR Models

The authors continue a series of articles where prices of the barrier options written on the underlying, whose dynamics follow a one-factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form; see Carr and Itkin (2020) and Itkin and Muravey (2020). This article extends this methodology to the Cox–Ingersoll–Ross model for zero-coupon bonds and to the constant elasticity of variance model for stocks, which are used as the corresponding underlying for the barrier options. The authors describe two approaches. One is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so it is named “the method of Bessel potentials.” The authors also propose a general scheme for constructing the potential method for any linear differential operator with time-independent coefficients. The second approach is the method of generalized integral transform, which is also extended to the Bessel process. In all cases, a semi-closed solution means that first, we need to numerically solve a linear Volterra equation of the second kind, and then the option price is represented as a one-dimensional integral. The authors demonstrate that their method is computationally more efficient than both the backward and forward finite difference methods, while providing better accuracy and stability. Also, it is shown that these methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities. TOPICS:Options, statistical methods Key Findings • This article extends this methodology to the CIR model for zero-coupon bonds and to the CEV model for stocks, which are used as the corresponding underlying for the barrier options. • One approach is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so we call it “the method of Bessel potentials.” The second approach is the method of generalized integral transform, which is also extended to the Bessel process. • These methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities.

On the Convergence of a Crank–Nicolson Fitted Finite Volume Method for Pricing American Bond Options

This paper develops and analyses a Crank–Nicolson fitted finite volume method to price American options on a zero-coupon bond under the Cox–Ingersoll–Ross (CIR) model governed by a partial differential complementarity problem (PDCP). Based on a penalty approach, the PDCP results in a nonlinear partial differential equation (PDE). We then apply a fitted finite volume method for the spatial discretization along with a Crank–Nicolson time-stepping scheme for the PDE, which results in a nonlinear algebraic equation. We show that this scheme is consistent, stable, and monotone, and hence, the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. To solve the system of nonlinear equations effectively, an iterative algorithm is established and its convergence is proved. Numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the new numerical method.