Interest Rate Process Research Articles

Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching

In this paper, we consider long time behavior of the Cox-Ingersoll-Ross (CIR) interest rate model driven by stable processes with Markov switching. Under some assumptions, we prove an ergodicity-transience dichotomy, namely, the interest rate process is either ergodic or transient. The sufficient and necessary conditions for ergodicity and transience of such interest model are given under some assumptions. Finally, an application to interval estimation of the interest rate processes is presented to illustrate our results.

A Linear Regression Approach for Determining Option Pricing for Currency-Rate Diffusion Model with Dependent Stochastic Volatility, Stochastic Interest Rate, and Return Processes

A three-factor exchange-rate diffusion model that includes three stochastically-dependent Brownian motion processes, namely, the domestic interest rate process, volatility process and return process is considered. A linear regression approach that derives explicit expressions for the distribution function of log return of foreign exchange rate is derived. Subsequently, a closed form workable formula for the call option price that has an algebraic expression similar to a Black-Scholes model, which facilitates easier study, is discussed.

Pricing catastrophe options with counterparty credit risk in a reduced form model

In this paper, we study the price of catastrophe options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model. We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.

OPTIMAL INVESTMENT AND OPTIMAL ADDITIONAL VOLUNTARY CONTRIBUTION RATE OF A DC PENSION FUND IN A JUMP-DIFFUSION ENVIRONMENT

This paper considers an optimal investment and an optimal additional contribution rate of a pension plan member (PPM) who faces both diffusion and jump risks in a defined contribution (DC) pension plan. We put into consideration three background risks which include interest rate, investment and salary risks. The stock prices, interest rate and salary process of a PPM are allowed to follow a jump-diffusion process. A PPM is expected to make two kind of contributions: compulsory and additional voluntary contributions. The compulsory one is a fixed proportion of a PPM's salary and the additional one is voluntary which is time and interest rate dependent. The aims of the investor is to determine the optimal investment and optimal contribution rate in a jump-diffusion environment. In order to obtain the optimal investment and optimal contribution rate, the resulting wealth process was transformed into Hamilton–Jacobi–Bellman equation by the method of dynamic programming. As a result, the optimal investment and optimal contribution rate of a PPM were obtained. Furthermore, some empirical analyses were conducted and results obtained. We found that the optimal investment ultimately depend on stocks diffusion and jump risks, interest rate and salary risks, optimal contribution rate and the salary process. The contribution rate of a PPM was found to depend on the investment strategies, salary process and interest rate risks, salary and its growth rate and CRRA coefficient. We also found that the contribution rate depends inversely on the salary process of a PPM over time.

Negative interest rates: why and how?

Abstract Interest rates (or nominal yields) can be negative, this is an unavoidable fact which has already been visible during the Great Depression (1929–39). Nowadays we can find negative rates easily by e.g. auditing. Several theoretical and practical ideas how to model and eventually overcome empirical negative rates can be suggested, however, they are far beyond a simple practical realization. In this paper we discuss the dynamical reasons why negative interest rates can happen in the second order differential dynamics and how they can influence the variance and expectation of the interest rate process. Such issues are highly practical, involving e.g. the banking sector and pension securities.

A Shannon Wavelet Method for Pricing Foreign Exchange Options under the Heston Multi-Factor CIR Model

We present a robust and highly efficient Shannon wavelet pricing method for plain-vanilla foreign exchange European options under the jump-extended Heston model with multi-factor CIR interest rate dynamics. Under a Monte Carlo and partial differential equation hybrid computational framework, the option price can be expressed as an expectation, conditional on the variance factor, of a convolution product that involves the densities of the time-integrated domestic and foreign multi-factor CIR interest rate processes. We propose an efficient treatment to this convolution product that effectively results in a significant dimension reduction, from two multi-factor interest rate processes to only a one-factor process. By means of a state-of-the-art Shannon wavelet inverse Fourier technique, the resulting convolution product is approximated analytically and the conditional expectation can be computed very efficiently. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.

FFT network for interest rate derivatives with Lévy processes

This paper extends the fast Fourier transform (FFT) network to interest derivative valuation under the Hull–White model driven by a Levy process. The classical trinomial tree for the Hull–White model is a widely adopted approach in practice, but fails to accommodate the change in the driving stochastic process. Recent finance research supports the use of a Levy process to replace Brownian motion in stochastic modeling. The FFT network overcomes the drawback of the trinomial approach but maintains its advantages in super-calibration to the term structure of interest rate and efficient computation to various kinds of interest rate derivatives under Levy processes. The FFT network only requires knowledge of the characteristic function of the Levy process driving the interest rate process, but not of the interest rate process itself. The numerical comparison between the closed-form solutions of interest rate caps and swaptions and those from FFT network confirms that the proposed network is accurate and efficient. We also demonstrate its use in pricing Bermudan swaptions and other American-style options. Finally, the FFT network is expanded to accommodate path-dependent variables, and is applied to interest rate target redemption notes and a range of accrual notes.

The Longstaff Schwartz Model of Yield Term Structure and its Expansion

The Longstaff – Schwartz model is considered both in the space of latent state variables and in the space of observable (or estimated) state variables. Analytical expressions for yield curves to maturity and forward curves are obtained in both cases. Based on the analysis of the Longstaff – Schwartz model, an extended version is proposed for an arbitrary number of state variables. Within the framework of this expansion, a method is proposed for obtaining analytical solutions of equations with respect to the functions of the yield term structure, when the initial equations for the dynamics of the short-term interest rate lead to non-linear systems of Riccati equations with respect to these functions, which do not allow an analytical solution to be obtained by known methods. This allows you to obtain analytical expressions for yield curves to maturity and forward interest rate curves. At the same time, the obvious principle is used: if the process of a short-term interest rate is specified, then the expressions for yield curves corresponding to this process do not depend on how the space of state variables of the financial market is described. It is shown that the application of this principle to the extension of the Longstaff – Schwartz model allows one to obtain analytical solutions of a system of nonlinear Riccati equations of arbitrary order, whose analytical methods are not described in the literature. The proposed extension of the Longstaff – Schwartz model is free from the drawback inherent in this model, which consists in the fact that in the transition from latent variables to real observable variables, the domain of definition of these real state variables is limited. In the expansion there is no such effect. Numerical examples are given.

Polynomial Models of Yield Term Structure

The possibility of representation of yield term structures in the form of polynomials or power series in models where short-term interest rate processes are described by stochastic differential equations is considered. In most diffusion models of short-term interest rate processes the functions of the drift and diffusion are polynomials. Among the well-known analytical form of yield term structures, corresponding to these models, there is a class of affine models in which term structure is also described by polynomials. The question therefore arises whether there are more such models short-term interest rate processes, for which the term structure are polynomials on the values of the interest rate. The paper shows that the answer to this question in the general case is negative. Such a representation takes place only in the case when the functions of drift and diffusion are polynomials not higher than of first order. Somewhat more complex to analyze is the assumption that in the present case, the term structure could be described by a power series on values of the interest rate. The problem of representation of the term structures by power series is connected with the solution of the infinite system of ordinary differential equations of first order for the coefficients of the series. This system of equations has features that do not allow to obtain the solution in analytical form in the general case. The conditions under which a power series could be a description of the term structure of yields are discussed. It is shown that there are models of the short rate for which the equations for the coefficients of the series are such that the subsequent coefficients are determined only by the previous one. Such models are, in particular, diffusion models of short-term interest rates CIR (1980) and Ahn–Gao. For them the system of differential equations is solved analytically and it is shown that a power series on values of the interest rate cannot be used as a model of term structure for these models. Unfortunately, the proof of this in the general case has not yet been found.

Contingent conversion convertible bond: New avenue to raise bank capital

This paper provides an in-depth analysis of the structuring and the pricing of an innovative financial market product. This instrument is called a contingent conversion convertible bond or “CoCoCo”. This hybrid bond is itself a combination of two other hybrid instruments: a contingent convertible (“CoCo”) and a convertible bond. This combination introduces more complexity in the structure but it also allows investors to profit from strong share price performances. This upside potential is added on top of the normal contingent convertible mechanics of CoCos, which expose the investors to mainly downside risk. First, we explain how the features of the contingent convertible bonds on one side and the features of the standard convertible bonds on the other side are combined. Thereafter, we propose a pricing approach which moves away from the standard Black[Formula: see text]Scholes setting. The CoCoCos are evaluated using the Heston process to which a Hull-White interest rate process has been added. We demonstrate the importance of using a stochastic interest rate when modeling this instrument. Finally we quantify the loss absorbing capacity of this instrument.

Analytical Pricing of European Bond Options within One-Factor Quadratic Term Structure Models

Jamshidian developed a model for pricing bond options within a Vasicek one-factor framework, with the very useful property that it allows an option on a coupon bond to be decomposed into a set of options on the individual coupons. In the Vasicek framework, the “Jamshidian trick” produces yields to maturity on the coupons that are linear functions of the underlying state variable. But it has not been clear whether this approach could extend to more complicated interest rate processes. In this article, Leblon and Moraux show how to extend the Jamshidian technique to quadratic interest rate processes and use it to derive analytic formulas for European call options on coupon bonds. Finally, they verify that the required conditions appear to hold in the real world.

Demand for longevity securities under relative performance concerns: Stochastic differential games with cointegration

This paper investigates the impact of relative performance concerns on the longevity risk transfer market. When an insurer concerns about the relative performance in a two-insurer economy, she maximizes the expected utility of her terminal wealth benchmarked against her competitor’s. The problem formulation for a general utility, a general interest rate process and cointegrated mortality rates uses a nonzero sum stochastic differential game approach. Explicit solution of the Nash equilibrium is derived for constant relative risk adverse insurers under the Vasicek-type stochastic interest and mortality rates. Existence and uniqueness of the Nash equilibrium are established for the CIR-type models, which rule out negative interest and mortality rates. While previous studies based on the single-agent approaches have shown a high investment demand in longevity bonds, the launch of it was unsuccessful in reality. Ours supplements that the demand is much lower subject to the relative performance concerns.

Adaptive Interest Rate Modelling

A good description of the dynamic process of interest rates is crucial to price derivatives and to hedge corresponding risk. An unstable macroeconomic context motivates the stochastic interest rate models with time‐varying parameters. In this paper, the local parameter approach is introduced to adaptively estimate interest rate models. This method can be generally used in time‐varying coefficient parametric models. It is used not only to detect jumps and structural breaks but also to choose the largest time homogeneous interval for each time point, such that in this interval the coefficients are statistically constant. We apply this adaptive approach in both simulations and real data analysis. Using the 3‐month Treasury bill rate as a proxy of the short rate, we find that our method can detect the structural breaks as well as the stable intervals for homogeneously modelling of the interest rate process. The time homogeneous interval cannot persist in an unstable macroeconomy. Furthermore, our approach performs well in long horizon forecasting. Copyright © 2016 John Wiley & Sons, Ltd.

CAN RBC MODELS EXPLAIN BUSINESS CYCLES IN KOREA?

I examine whether an RBC model can generate a higher volatility of consumption relative to output, a strong negative correlation between output and the trade balance, and a weak countercyclicality of the real interest rate, phenomena that have been observed in the business cycles of emerging economies, including Korea. From an RBC model with recursive utility, I show that it is not the degree of relative risk aversion, but the elasticity of intertemporal substitution (EIS), that governs the movements of the variables of the model in the log linearized environment. The Bayesian estimation results based on Korean data from the period 1987 to 2013 suggest that there are some elements of success in describing the Korean economy based on the simple RBC model both with the EIS larger than one and with an error term for the real interest rate equation. An EIS larger than one improves the performance of the simple RBC model mainly in the direction of raising the volatility of consumption relative to output. Simulation results show that the error term for the real interest rate process mostly reflects the endogenous channel of financial frictions where the domestic real interest rate depends negatively on the expected (transitory) productivity shock.

The End of the Month Option and Other Embedded Options in Futures Contracts

Futures contracts often contain several different kinds of embedded options related to the delivery of the underlying. The end of the month option allows the holder of the short end of a futures contract to deliver the underlying at any time during the last week of the contract period at a fixed price determined at the start of the last week. We derive a formula for this price in a general incomplete financial market, in which the process underlying the futures contract is a general adapted cadlag process and the risk free interest rate process is a general adapted process, both satisfying certain integrability conditions. In a similar setting we also derive a formula for the futures price process of a futures contract which first has no option active, then it has an active timing option (which lets the holder of the short end deliver at any time during the last month of the contract period) and lastly it has an active end of the month option. This combination of delivery options is present in the real-world financial markets. We show that this futures price process is dominated by a standard futures price process with maturity at the time of the activation of the end of the month option. We also show that if the underlying is an asset with a non-positive convenience yield and the interest rate is non-negative then it is optimal to deliver when the end of the month option becomes active. The main contribution of this paper is to properly define the end of the month option and to derive a formula for the futures price process of a futures contract with the combination of options described above.

Empirical Pricing Performance on Long-Dated Crude Oil Derivatives: Do Models with Stochastic Interest Rates Matter?

Does modelling stochastic interest rates beyond stochastic volatility improve pricing performance on long-dated crude oil derivatives? To answer this question, we examine the empirical pricing performance of two forward price models for commodity futures and options: a deterministic interest rate-stochastic volatility model and a stochastic interest rate-stochastic volatility model. Both models allow for a correlation structure between the futures price process, the futures volatility process and the interest rate process. By estimating the model parameters from historical crude oil futures prices and option prices, we find that stochastic interest rate models improve pricing performance on long-dated crude oil derivatives, with the effect being more pronounced when the interest rate volatility is relatively high. Several results relevant to practitioners have also emerged from our empirical investigations. With regards to balancing the trade-off between precision and computational effort, we find that two-factor models would provide good fit on long-dated derivative prices thus there is no need to add more factors. We also find empirical evidence for a negative correlation between crude oil futures prices and interest rates.

Value of the distant future: Model-independent results

Abstract This paper shows that the model-independent account of correlations in an interest rate process or a log-consumption growth process leads to declining long-term tails of discount curves. Under the assumption of an exponentially decaying memory in fluctuations of risk-free real interest rates, I derive the analytical expression for an apt value of the long run discount factor and provide a detailed comparison of the obtained result with the outcome of the benchmark risk-free interest rate models. Utilizing the standard consumption-based model with an isoelastic power utility of the representative economic agent, I derive the non-Markovian generalization of the Ramsey discounting formula. Obtained analytical results allowing simple calibration, may augment the rigorous cost-benefit and regulatory impact analysis of long-term environmental and infrastructure projects.

Pricing of Long-Dated Commodity Derivatives with Stochastic Volatility and Stochastic Interest Rates

Aiming to study pricing of long-dated commodity derivatives, this paper presents a class of models within the Heath, Jarrow, and Morton (1992) framework for commodity futures prices that incorporates stochastic volatility and stochastic interest rate and allows a correlation structure between the futures price process, the futures volatility process and the interest rate process. The functional form of the futures price volatility is specified so that the model admits finite dimensional realisations and retains affine representations, henceforth quasi-analytical European futures option pricing formulae can be obtained. A sensitivity analysis reveals that the correlation between the interest rate process and the futures price process has noticeable impact on the prices of long-dated futures options, while the correlation between the interest rate process and the futures price volatility process does not impact option prices. Furthermore, when interest rates are negatively correlated with futures prices then option prices are more sensitive to the volatility of interest rates, an effect that is more pronounced with longer maturity options.

Impact Study of Central Bank Communication to Money Market Benchmark Interest Rate

In the past two decades, central bank communication as a new type of monetary policy tool has continually received attention from central banks. With the gradual advance of the market-led interest rate process, most central banks establish policy operation framework which can effectively guide and regulate the market interest rate. This article is based on the implementation of China’s monetary policy tools framework, using EGARCH model to empirically study the impact of central bank communication as a new monetary policy tool to benchmark money market rates. The result indicates: the influence of central bank communication to SHIBOR is significant, the direction of impact is in accordance with the direction of monetary policy intentions; verbal communication is more remarkable than written communication in affecting the SHIBOR; increasing the central bank of communication can dampen short-term interest rate fluctuations and can play a positive role in the financial markets.

On Implied Binomial Trees With a Non Constant Interest Rate Dynamics

The paper examines a market for a stock, discount bonds of all maturities and European calls and puts on the stock of all strikes and all maturities. It derives a discrete time arbitrage free model. Said model is implemented in the binomial framework world in which both stock and bonds dynamics are determined so that risk neutral prices of the calls, exhibit the above smile and put prices are determined by the put-call parity. Fitting is done by deriving an algorithm which allows the interest rate process to be chosen specifically in order to generate fitting while keeping the stock’s volatility constant. As a result the initial market’s smile is allowed to be preserved forever, independently of time and state. The idea of using both stock and bonds in order to derive the local fit (and not viewing the bond dynamics as a given) is new. In this paper we extend the standards implied binomial models, to obtain a more flexible model which is calibrated with market data on European puts and calls. Constructing implied trees from a given smile along with a requirement for the model prices to be fair, generally involves solving a set of the well known CRR-equations under the restriction that the probability at each node is risk-neutral. The success of having a unique solution depends, among others, on the smile’s shape and can be easily violated by a quite steep one. Our model suggests a different approach by considering an optimization procedure. The model uses Brent method which incorporates other well known optimization methods. It is very stable for convergence and although complex it can easily be applied through Matlab program.