Cox-Ingersoll-Ross Research Articles

Risk‐neutral pricing techniques and examples

AbstractTractability and flexibility are among the two most attractive features of models in mathematical finance. For the pricing of derivative products, to avoid arbitrage opportunities, the fundamental theorem of asset pricing requires the existence of an equivalent martingale measure under which the discounted price process is a (local) martingale. In the semimartingale setting, the traditional approach of risk‐neutral valuation uses a change of measure invoking Girsanov's theorem. This approach often destroys the tractability of the process which is undesirable for quantitative finance applications. To overcome these limitations, we employ a transformation based on the concept of intertwining relationships that allows us to convert Markovian semigroups into a pricing semigroup while keeping its tractability. In order to illustrate the usefulness of this approach, we apply it to exponential Lévy, positive self‐similar, and generalized CIR processes that all fall within the class of polynomial processes introduced by Cuchiero et al. (2012). Furthermore, for the jump CIR class, relying on recent work of Patie and Savov (2019), we provide an eigenvalues expansion of the pricing semigroup and carry out some numerical experiments.

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.

English

The reverse mortgage market has been expanding rapidly in developed economies in recent years. Reverse mortgages provide an alternative source of funding for retirement income and health care costs. We often hear the phrase “house rich and cash poor” to refer the increasing number of elderly persons who hold a substantial proportion of their assets in home equity. Reverse mortgage contracts involve a range of risks from the insurer’s perspective. When the outstanding balance exceeds the housing value before the loan is settled, the insurer suffers an exposure to crossover risk induced by three risk factors: interest rates, house prices, and mortality rates. In this context, Covid-19 has occurred and the insurer is faced with this additional source of risk. We analyse the combined impact of these risks on the pricing and the risk profile of reverse mortgage loans. We consider a CIR process for the evolution of the interest rate, a Black & Scholes model for the dynamics of house prices and the Gompertz model for the trend in mortality Our results show that the decrease in the mortality curve due to Covid exposes the insurer to higher risks once the shock is reabsorbed. The risk is higher the higher the age of entry. Only a significant reduction of the shock adjustment coefficient will return the situation to normality.

Bayesian Estimation of CIR Model

This article concerns the Bayesian estimation of interest rate models based on Euler-Maruyama approximation. Assume the short term interest rate follows the CIR model, an iterative method of Bayesian estimation is proposed. Markov Chain Monte Carlo simulation based on Gibbs sampler is used for the posterior estimation of the parameters. The maximum A-posteriori estimation using the genetic algorithm is employed for finding the Bayesian estimates of the parameters. The method and the algorithm are calibrated with the historical data of US Treasury bills.

Bayesian Estimation of CIR Model

Bayesian Estimation of CIR Model

Pricing of volatility derivatives in a Heston–CIR model with Markov-modulated jump diffusion

This paper concerns the pricing of volatility and variance swaps with discrete sampling times using a hybrid Heston–CIR model with Markov-modulated jump–diffusion. We extend the regime-switching Heston stochastic volatility model by further considering the Cox–Ingersoll–Ross (CIR) stochastic interest rate with jump diffusion. The market parameters, including the jump–diffusion part, are modulated by a continuous-time Markov chain that represents different market states. The proposed hybrid model can capture the stochastic property of volatility rate, interest rate as well as the short-term and long-term effects on the financial market caused by unexpected events and structural changes of the macroeconomic environment respectively. The fair strike prices of variance swaps and volatility swaps based on four different formulae for the realized variance (volatility) are obtained by utilizing the characteristic function method. The accuracy and efficiency of our method are validated through the semi-Monte-Carlo simulation. Finally, numerical and sensitivity analysis are conducted to show the impact of the change of each market parameter. Overall, pricing results were greatly enhanced with a more realistic financial model, and more driving forces are considered to investigate the formation of the discrete variance and volatility swap prices more comprehensively.

Counterparty risk valuation on credit-linked notes under a Markov Chain framework

A credit-linked note (CLN) is a note paying an enhanced coupon to investors for bearing the credit risk of a reference entity. In this paper, we study the counterparty risk on CLNs under a Markov chain framework, and introduce a Markov copula model to describe joint defaults between the reference entity underlying the CLN and CLN issuer. Assuming that the respective default intensities are directly and inversely proportional to the interest rate, which follows a CIR process, we obtain the explicit formulae for CLN values through a PDE approach. Finally, credit valuation adjustment (CVA) formula is derived to price counterparty credit risk.

A catastrophe shock model and the bond pricing

In order to describe the memory effects of catastrophe events such as earthquake, typhoon and to transfer risk reasonably, we adopt a catastrophe impact model with power-law waiting time, so called the fractional compound Poisson process, to describe the underwriting risk of insurance companies and to study the pricing for catastrophe insurance-linked bonds. Firstly, we use the moment matching method to get the generalized Pareto-type approximation distribution for the cumulative losses. Then, the formula for bond price is derived under the CIR (Cox-Ingersoll-Ross) interest rate model. At last, numerical examples are used to verify the validity of the approximation for loss distribution. The results show that with the increase of memory parameter, the change trend of expected risk and bond price is opposite, and both of them may have a polymorphic trend of increasing, decreasing or both, which is closely related to the maturity level.

CVA AND VULNERABLE OPTIONS IN STOCHASTIC VOLATILITY MODELS

This work aims to provide an efficient method to evaluate the Credit Value Adjustment (CVA) for a vulnerable European option, which is an option subject to some default event concerning the issuer solvability. Financial options traded in OTC markets are of this type. In particular, we compute the CVA in some popular stochastic volatility models such as SABR, Hull et al., which have proven to fit quite well market derivatives prices, admitting correlation with the default event. This choice covers the relevant case of Wrong Way Risk (WWR) when a credit deterioration determines an increase in the claim value. Contrary to the structural modeling adopted in [G. Wang, X. Wang & K. Zhu (2017) Pricing vulnerable options with stochastic volatility, Physica A 485, 91–103; C. Ma, S. Yue & Y. Ma (2020) Pricing vulnerable options with Stochastic volatility and Stochastic interest rate, Computational Economics 56, 391–429], we use the reduced-form intensity-based approach to provide an explicit representation formula for the vulnerable option price and related CVA. Later, we specialize the evaluation formula and construct its approximation for the three models mentioned above. Assuming a CIR model for the default intensity process, we run a numerical study to test our approximation, comparing it with Monte Carlo simulations. The results show that for moderate values of the correlation and maturities not exceeding one year, the approximation is very satisfactory as of accuracy and computational time.

Valuing Multirisk Catastrophe Reinsurance Based on the Cox–Ingersoll–Ross (CIR) Model

Catastrophe risks lead to severe problems of insurance and reinsurance industry. In order to reduce the underwriting risk, the insurer would seek protection by transferring part of its risk exposure to the reinsurer. A framework for valuing multirisk catastrophe reinsurance under stochastic interest rates driven by the CIR model shall be discussed. To evaluate the distribution and the dependence of catastrophe variables, the Peaks over Threshold model and Copula function are used to measure them, respectively. Furthermore, the parameters of the valuing model are estimated and calibrated by using the Global Flood Date provided by Dartmouth College from 2000 to 2016. Finally, the value of catastrophe reinsurance is derived and a sensitivity analysis of how stochastic interest rates and catastrophe dependence affect the values is performed via Monte Carlo simulations. The results obtained show that the catastrophe reinsurance value is the inverse relation between initial value of interest rate and average interest rate in the long run. Additionally, a high level of dependence between catastrophe variables increases the catastrophe reinsurance value. The findings of this paper may be interesting to (re)insurance companies and other financial institutions that want to transfer catastrophic risks.

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

On The Closed Form Strategies of an Investor under the CEV and CIR Processes

In this paper, the explicit solutions of the optimal investment plans of an investor with exponential utility function exhibiting constant absolute risk aversion (CARA) under constant elasticity of variance (CEV) and stochastic interest rate is studied. A portfolio comprising of a risk-free asset modelled by the Cox-Ingersoll-Ross (CIR) process and two risky assets modelled by the CEV process is considered, where the instantaneous volatilities of the two risky assets form a 2 x 2 matrix n = {np,q}2x2 such that nnT is positive definite. Using the power transformation and change of variable approach with asymptotic expansion technique, explicit solutions of the optimal investment plans are found. Moreover, numerical simulations are used to study the effects of the interest rate, elasticity parameter, correlation coefficient and the risk averse coefficient on the optimal investment plans.

Research on Interest Rate Risk of Housing Mortgage Loan Based on Computer Simulation.

In recent years, with the rapid increase of the business volume of housing mortgage loans of commercial banks, the risk of prepayment is increasingly exposed. Prepayment will have a great impact on the duration and convexity of housing mortgage loans of commercial banks and then bring difficulties to the asset liability management of banks. Therefore, empirical research on the changes of duration and convexity of housing mortgage loans caused by prepayment when the market interest rate changes is of great significance for commercial banks to manage interest rate risk exposure. Based on the analysis of the option characteristics of prepayable housing mortgage loan, the CIR model with GARCH(1, 1) is selected to describe the interest rate change path, and the computer simulation method is used to calculate OAS and then calculate the effective duration and effective convexity of housing mortgage loan under different prepayment rates, so as to understand the interest rate risk of housing mortgage loan in the presence of embedded option.

Equilibrium short-rate models vs no-arbitrage models: Literature review and computational examples

In this paper equilibrium short-rate models are compared against no-arbitrage short-rate models. This article is composed of the introduction to this literature and a review, followed by numerical examples of one-factor short-rate models; the Cox-Ingersoll-Ross (CIR) model and the Vasicek model. No-arbitrage models were presented through the Hull-White (HW) model, the Binomial lattice model for bond pricing and interest rate modelling, the Black-Karasinski (BK) model, and the Heath-Jarrow-Morton (HJM) model. The results prove that no single interest rate model exists that can be used for all purposes. These models were compared in terms of volatility, mean reversion process and convergence. The end results confirm the dependence of volatility on the level rate as a determinant of the predictive success of these models.

Multilevel Monte Carlo Using Approximate Distribution of the CIR Process

Giles and Sheridan-Methven propose Multi-level Monte Carlo (MLMC) with approximate normal random variables. In this article, we develop this approach by embedding the approximate CIR process, which is approximated using the quantiles of the non-central chi-squared random variable. Under mild assumptions, we derive the convergence rate of the MLMC variance for the approximate CIR process, which recovers the corresponding rate without using approximation. The result applies for all parameter regimes. To illustrate its application, we apply it to a time-discrete scheme on the Heston model. The analysis is supported by numerical experiments

A Closed-Form Expansion for the Conditional Expectations of the Extended CIR Process

A Closed-Form Expansion for the Conditional Expectations of the Extended CIR Process

Pricing EIA with cliquet-style guarantees under time-changed Lévy models by frame duality projection

This paper presents an efficient and accurate method for pricing equity-indexed annuities (EIAs) with cliquet-style payoff structures under time-changed Lévy processes. We apply a square-root process to depict the activity rate process in stochastic time change, which enables us to use some Fourier transform methods to price EIAs. As the complexity of payoff structures, we make use of frame duality projection associated with the quadrature rule and the spectral filtering to realize our algorithm. Numerical experiments are reported to demonstrate the accuracy and efficiency of our approach.

Model Risk in Risk Analysis for No-Negative-Equity-Guarantees

Understanding the risk for No-Negative-Equity-Guarantees (NNEGs) requires the proper modeling of the housing return, interest rate, and mortality rate dynamics. This article investigates the model risk for the risk measures of NNEGs by calculating the Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) from the provider perspective, with an emphasis on the housing price return model. Therefore, we propose a jump ARMA-GARCH model, according to nationwide house price return data in the UK. Interest rate and mortality rate dynamics are assumed to follow the CIR model (Cox et al. 1985) and the CBD model (Cairns et al. 2006) respectively. Our numerical analyses reveal that the housing price risk, interest-rate risk, and longevity risk can affect the VaR and CTE of NNEGs, with the impact being as significant as that for housing risk. The VaR and CTE of NNEGs will be greater for female borrowers than for male borrowers, essentially because females have a longer life expectancy. The proposed framework can help financial institutions manage the major three risk factors for NNEGs and assist in meeting the regulator’s concerns. TOPICS:Quantitative methods, real estate, VAR and use of alternative risk measures of trading risk Key Findings ▪ We investigate model risk in risk analysis for No-Negative-Equity-Guarantees. ▪ Our numerical analyses reveal that the housing price risk, interest-rate risk, and longevity risk can affect the VaR and CTE of NNEGs, with the impact being as significant as that for housing risk.

Staffing many‐server queues with autoregressive inputs

AbstractRecent studies reveal significant overdispersion and autocorrelation in arrival data at service systems such as call centers and hospital emergency departments. These findings stimulate the needs for more practical non‐Poisson customer arrival models, and more importantly, new staffing formulas to account for the autocorrelative features in the arrival model. For this purpose, we study a multiserver queueing system where customer arrivals follow a doubly stochastic Poisson point process whose intensities are driven by a Cox–Ingersoll–Ross (CIR) process. The nonnegativity and autoregressive feature of the CIR process makes it a good candidate for modeling temporary dips and surges in arrivals. First, we devise an effective statistical procedure to calibrate our new arrival model to data which can be seen as a specification of the celebrated expectation–maximization algorithm. Second, we establish functional limit theorems for the CIR process, which in turn facilitate the derivation of functional limit theorems for our queueing model under suitable heavy‐traffic regimes. Third, using the corresponding heavy traffic limits, we asymptotically solve an optimal staffing problem subject to delay‐based constraints on the service levels. We find that, in order to achieve the designated service level, such an autoregressive feature in the arrival model translates into notable adjustment in the staffing formula, and such an adjustment can be fully characterized by the parameters of our new arrival model. In this respect, the staffing formulas acknowledge the presence of autoregressive structure in arrivals. Finally, we extend our analysis to queues having customer abandonment and conduct simulation experiments to provide engineering confirmations of our new staffing rules.

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.