Interest Rate Model Research Articles

Coverings and nonlocal symmetries as well as fundamental solutions of nonlinear equations derived from the nonisospectral AKNS hierarchy

In the paper, we apply a new scheme to generate nonisospectral integrable hierarchies of evolution equations. First of all, we work out the positive-order and the negative-order nonisospectral AKNS hierarchies. Next we introduce a kind of loop algebra from which an expanding nonisospectral integrable model is derived. Specially, the expanding integrable model reduces to the well-known bond pricing equation and the interest rate modeling. By using the Lie group theory we investigate the Lie representations and characteristic solutions of a generalized bond pricing equation from the positive-order nonisospectral integrable equation, and with the help of the characteristic solutions and the Laplace transformation, we again obtain the fundamental solutions of the generalized bond pricing equation. Besides, we apply the Lie group analysis to deduce the invariant solutions and the nonlinear self-adjointness as well as conservation laws, non-invariant solutions of the generalized bond pricing equation. Some reductions of the negative-order nonisospectral integrable hierarchy are also produced. Two of them are the Sine-Gordan equation and the Sinh-Gordan equation. Finally, we investigate the coverings and the nonlocal symmetries of the generalized bond pricing equation and a generalized KdV equation, respectively, although the bond equation is linear which has many applications in the aspect of finance field, by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.

Agent-based model generating stylized facts of fixed income markets

We develop an agent-based model (ABM) of a financial market with multiple assets belonging either to the fixed income or equity asset classes. The aim is to reproduce the main stylized facts of fixed income markets with regards to the emerging dynamics of the yield curves. Our ABM is rooted in the market model of Kaizoji et al. (J Econ Behav Organ 112:289–310, 2015) formulated with two types of traders: the rational and risk-averse fundamentalist investors and the noise traders who invest under the influence of social imitation and price momentum. The investors involved in the present market model diversify their investments between a preferred stock equivalent to a perpetual bond and multiple bonds of selected maturities. Among those, a zero-coupon bond provides a constant rate of return, while the prices of the coupon-paying bonds are determined at each time step by the equilibrium between the investors’ demands and supplies. As a result, the ABM creates an evolving yield curve determined by the aggregate impact of the traders’ investments. In agreement with real markets, it also produces transient turbulent periods in the prices’ time series as well as a humped term structure of volatility. We compare the dynamics arising from different processes governing the risk-free rate with those of the historical US Treasury market. Introducing Vasicek’s model of interest rates to both synthetic and empirical rates demonstrates the capacity of our ABM in reproducing the main characteristics of the surface of autocorrelation of the volatilities of the yields to maturity of the US Treasury bonds for the selected time-frame.

One-Factor Keynesian Models of the Long-Term Interest Rate

There are several one-factor benchmark models of the long-term interest rate in quantitative finance. However, these models have yet to incorporate Keynes’s insights about interest rate dynamics. This article modifies several one-factor benchmark interest rate models to integrate Keynes’s insights.

Berry–Esseen inequalities for the fractional Black–Karasinski model of term structure of interest rates

Abstract The Black–Karasinski model is a one-factor non-affine interest rate model as it describes interest rate movements driven by a single source of randomness and the drift function is a nonlinear function of the interest rate. The drift parameters represent the level and the speed of mean reversion of the interest rate. It belongs to the class of no-arbitrage models. The paper introduces some new approximate minimum contrast estimators of the mean reversion speed parameter in the model based on discretely sampled data which are efficient and studies their asymptotic distributional properties with precise rates of convergence.

CALIBRATING LOCAL VOLATILITY MODELS WITH STOCHASTIC DRIFT AND DIFFUSION

We propose Monte Carlo calibration algorithms for three models: local volatility with stochastic interest rates, stochastic local volatility with deterministic interest rates and finally stochastic local volatility with stochastic interest rates. For each model, we include detailed derivations of the corresponding SDE systems and list the required input data and steps for calibration. We give conditions under which a local volatility can exist given European option prices, stochastic interest rate model parameters, and correlations. The models are posed in a foreign exchange setting. The drift term for the exchange rate is given as a difference of two stochastic short rates, domestic and foreign; each modeled by a Gaussian one-factor model with deterministic shift (G1[Formula: see text]) process. For stochastic volatility, we model the variance for the exchange rate by a Cox–Ingersoll–Ross (CIR) process. We include tests to show the convergence and the accuracy of the proposed algorithms.

Multifactor Keynesian models of the long-term interest rate

ABSTRACT This paper presents multifactor Keynesian models of the long-term interest rate. In recent years, there have been a proliferation of empirical studies based on the Keynesian approach to interest rate modelling. These studies evince the connection between the long-term interest rate and the short-term interest rate. However, standard multifactor models of the long-term interest rate in quantitative finance have not been yet incorporated Keynes’s insights about interest rate dynamics. Keynes’s insights are introduced in two different multifactor models of the long-term interest rate to illustrate how the long-term interest rate relates to the short-term interest rate, after controlling for the central bank’s policy rate, expected inflation, the central bank’s inflation target, volatility in financial markets, and Wiener processes.

Vulnerable European option pricing in a Markov regime-switching Heston model with stochastic interest rate

This paper considers pricing of European-style vulnerable options under the Heston stochastic volatility and stochastic interest rate model in which the mean-reversion levels of both variance and interest rate processes are modulated by a continuous-time Markov process with a finite state space. An analytical pricing formula is derived by using the Esscher transform, joint characteristic function and multivariate Fourier transform technique, where the closed-form solution of the characteristic function is obtained by the law of iterated expectation. Then we provide the efficient approximation to calculate the analytical pricing formula of option using the FFT approach and examine the accuracy of the approximation by Monte Carlo simulation. Finally, the sensitivity analysis of different parameters in the proposed model on the vulnerable call option price and its Delta value are provided, and the difference between the proposed model and the Heston and stochastic interest rate model with non-Markov regime-switching are presented by some numerical experiments, which shows the influence of introducing regime-switching into Heston model with stochastic interest rate.

PENENTUAN PREMI ASURANSI JIWA BERJANGKA SINGLE LIFE MENGGUNAKAN MODEL COX-INGERSOLL-ROSS (CIR)

Premium is an amount of money that must be paid by the customer at a certain time based on an insurance policy. This study aims to determine the value of single life term life insurance premiums using the Cox-Ingresoll-Ross (CIR) interest rate model. There are differences in the value of men's and women's premiums from the calculation of single life term life insurance premiums using the CIR interest rate model. The value of the premium paid by men is more expensive than the value of the premium paid by women. The premium value for men is greater than 1.5 times the premium value for women.Key words: premium, life insurance, CIR model

Vasicek interest rate model under Lévy process and pricing bond option

In this paper, we concentrate on the European option pricing problem in incomplete markets by taking Vasicek stochastic interest rate model into account and expanding it in accordance with a Lévy process named Variance Gamma (VG) process. In doing so, first, we prove the existence and uniqueness of the solution to the stochastic differential equation of this model. Then, by the means of the hedging and replicating techniques, pricing formulas for the bond and, in succession, for European options are established. Finally, from a numerical point of view, we will analyze the value of a bond based on our consideration model and also, we will study a European call option bond price. All these results are reported through the Monte-Carlo simulation method.

Pricing the Excess Volatility in Foreign Exchange Risk Premium and Forward Rate Bias

ABSTRACT We present the pricing of the documented excess volatility of the foreign exchange risk premium, relative to the interest rate differential. By specifying a term structure of interest rate model, the physical probability measure along with the pricing kernels or discount factors are used to derive a system for the expected future spot rate and the forward rate. The theoretical loads are found by solving the Riccati ordinary differential equations, and dynamic factors are captured to set up the global factors for both currencies. It shows that we prove the interest-rate surfaces are almost identical to the empirical ones, and the theoretical interest rates are guaranteed to be positive.

Optimal Asset Allocation for CRRA and CARA Insurers under the Vasicek Interest Rate Model

This paper considers the reinsurance‐investment problem with interest rate risks under constant relative risk aversion and constant absolute risk aversion preferences, respectively. Stochastic control theory and dynamic programming principle are applied to investigate the optimal proportional reinsurance‐investment strategy for an insurer under the Vasicek stochastic interest rate model. Solving the corresponding Hamilton‐Jacobi‐Bellman equation via the Legendre transform approach, the optimal premium allocation strategies maximizing the expected utilities of terminal wealth are derived. In addition, several sensitivity analyses and numerical illustrations are given to analyze the impacts of different risk preferences and interest rate fluctuation on the optimal strategies. We find that the asset allocation and reinsurance ratio of the insurer are correlated with risk preference coefficient and interest rate fluctuation, and the insurance company may adjust the reinsurance‐investment strategy to deal with interest rate risk.

Global attractivity for uncertain differential systems

<abstract><p>This paper studies global attractivity for uncertain differential systems, which are effective tools to solve the problems with uncertainty. And They have been applied in many areas. This article presents several global attractivity concepts. Based on the knowledge of uncertainty theory, some sufficient conditions of global attractivity for linear uncertain differential systems are given. In particular, the attractivity on the solutions and $ \alpha $-path of uncertain differential systems is studied. Last, as an application of attractivity, an interest rate model with uncertainty is shown.</p></abstract>

Pricing and hedging bond options and sinking-fund bonds under the CIR model

<abstract><p>This article derives simple closed-form solutions for computing Greeks of zero-coupon and coupon-bearing bond options under the CIR interest rate model, which are shown to be accurate, easy to implement, and computationally highly efficient. These novel analytical solutions allow us to extend the literature in two other directions. First, the static hedging portfolio approach is used for pricing and hedging American-style plain-vanilla zero-coupon bond options under the CIR model. Second, we derive analytically the comparative static properties of sinking-fund bonds under the same interest rate modeling setup.</p></abstract>

A three-factor hazard rate model for single-name credit default swap pricing

We propose a reduced-form model that integrates the modeling of interest rates, equity and default hazard rates. In this model the evolution of the risk-neutral hazard rate is driven by three risk factors: the reference firm’s equity price (in logarithmic form), the risk-free short rate and a latent risk premium factor that is orthogonal to the first two. The equity price and the short rate are considered to be the two primary risk factors, while the risk premium factor is introduced to capture the remaining risks for the hazard rate (such as the liquidity risk). The equity price follows a diffusion process with the possibility of a jump to default, the short rate is governed by a Vasicek model, and the dynamics of the risk premium factor are modeled as a Cox– Ingersoll–Ross process. Under the three-factor model, we derive computable pricing formulas for defaultable zero-coupon bonds, European call options and standard running credit default swaps (CDSs).We then discuss the calibration of the model using CDS and equity option data. Our numerical results reveal that the model is able to interpret a diverse set of CDS spread term structures.

Calibration of time-dependent volatility for European options under the fractional Vasicek model

<abstract><p>In this paper, we calibrate the time-dependent volatility function for European options under the fractional Vasicek interest rate model. A fully implicit finite difference method is applied to solve the partial differential equation of option pricing numerically. To find the volatility function, we minimize a cost function that is the sum of the squared errors between the theoretical prices and market prices with Tikhonov $ L_2 $ regularization and $ L_{1/2} $ regularization respectively. Finally numerical experiments with simulated and real market data verify the efficiency of the proposed methods.</p></abstract>

Строго конзистентне оцiнювання всiхпараметрiв моделi Васiчека задискретними спостереженнями

We construct estimators of all three unknown parameters in the Vasicek interest rate model based on discrete observations and prove their strong consistency. Theoretical results are illustrated by simulations.

Lévy Interest Rate Models with a Long Memory

This article proposes an interest rate model ruled by mean reverting Lévy processes with a sub-exponential memory of their sample path. This feature is achieved by considering an Ornstein–Uhlenbeck process in which the exponential decaying kernel is replaced by a Mittag–Leffler function. Based on a representation in term of an infinite dimensional Markov processes, we present the main characteristics of bonds and short-term rates in this setting. Their dynamics under risk neutral and forward measures are studied. Finally, bond options are valued with a discretization scheme and a discrete Fourier’s transform.

Law of interest rate changes in financial markets based on the differential equation model of liquidity

Abstract The paper establishes a related differential equation model about changes in financial interest rates. It uses information related to liquidity to feedback the law and stability of differential equations in interest rate changes. The article applies stochastic processes and partial differential equations to complex financial networks to confirm node yields in financial market networks. It confirms the existence of interest rate stickiness in Chinese financial markets. The advantage of this interest rate model is that when the external economic environment changes, the state of interest rates will also change accordingly.

Mathematical simulation experiment based on optimisation of heat treatment process of aluminium alloy materials

Abstract The paper establishes a related differential equation model about changes in financial interest rates. It uses information related to liquidity to feedback the law and stability of differential equations in interest rate changes. The article applies stochastic processes and partial differential equations to complex financial networks to confirm node yields in financial market networks. It confirms the existence of interest rate stickiness in Chinese financial markets. The advantage of this interest rate model is that when the external economic environment changes, the state of interest rates will also change accordingly.

Law of interest rate changes in financial markets based on the differential equation model of liquidity

Abstract The paper establishes a related differential equation model about changes in financial interest rates. It uses information related to liquidity to feedback the law and stability of differential equations in interest rate changes. The article applies stochastic processes and partial differential equations to complex financial networks to confirm node yields in financial market networks. It confirms the existence of interest rate stickiness in Chinese financial markets. The advantage of this interest rate model is that when the external economic environment changes, the state of interest rates will also change accordingly.