Stochastic Interest Rate Model Research Articles

Interest rates risk-premium and shape of the yield curve

We apply the general theory of pricing in incomplete markets, due to the author, on the problem of pricing bonds for theHull-White stochastic interest rate model. As pricing in incomplete markets involves more market parameters than the classical theory, and as thederived risk premium is time-dependent, the proposed methodology might offer a better way for replicating different shapes of the empirically observedyield curves. For example, the so-called humped yield curve can be obtained from a normal yield curve by only increasing the investors risk aversion.

Pitfalls of the Fourier Transform Method in Affine Models, and Remedies

ABSTRACTWe study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models and explain that, in many cases, a calibration procedure based on such a realization will be able to find a “correct parameter set” only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time-consuming. We explain how to construct more accurate and faster pricing and calibration procedures. An important ingredient of our method is the study of the analytic continuation of the solution of the associated system of generalized Riccati equations, and contour deformation techniques. As a byproduct, we show that the straightforward application of the Runge–Kutta method may lead to sizable errors, and suggest certain remedies. In the paper, the method is applied to a wide class of stochastic volatility models with stochastic interest rate and interest rate models of An(n) class. The methodology of the paper can be applied to other models (e.g., quadratic term structure models, Wishart dynamics, 3/2-model).

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.

Explicit Solution of Reinsurance-Investment Problem for an Insurer with Dynamic Income under Vasicek Model

Unlike traditionally used reserves models, this paper focuses on a reserve process with dynamic income to study the reinsurance-investment problem for an insurer under Vasicek stochastic interest rate model. The insurer’s dynamic income is given by the remainder after a dynamic reward budget being subtracted from the insurer’s net premium which is calculated according to expected premium principle. Applying stochastic control technique, a Hamilton-Jacobi-Bellman equation is established and the explicit solution is obtained under the objective of maximizing the insurer’s power utility of terminal wealth. Some economic interpretations of the obtained results are explained in detail. In addition, numerical analysis and several graphics are given to illustrate our results more meticulous.

Stability of some stochastic interest rate models

In this paper, we study exponential mean-square stability, boundedness in probability uniformly in t and pth-moments of the solution process of well-known interest rate models such as Vasicek, BrennanSchwartz, Hull-White and Dothan. Some of the results obtained here appear to be new.

Econophysics, Statistical Mechanics for Financial Applications, and Financial Mathematics

Econophysics, statistical mechanics, and financial mathematics are interdisciplinary study that combines economics, statistical physics, stochastic analysis, stochastic processes, optimization, numerical methods, and quantitative finance. This special issue presents five high-quality researches in this interdisciplinary area. A. Ishikawa et al. analytically show that the logarithmic average sales of firms first follow power-law growth and subsequently follow exponential growth, if the growth-rate distributions of the sales obey the extended-Gibrat’s property and law. They analytically show that the parameter of the extended-Gibrat’s property is identical to the powerlaw growth exponent and that it decides the parameter of the exponential growth. Y. H. Shin and H.-S. Lee propose a regime switching model of schooling choice as a job search process. They adopt a two-state Markov process and the derived coupled Bellman equations and suggest some numerical examples in this framework. D.-L. Sheng focuses on a reserve process with dynamic income to study the reinsurance-investment problem for an insurer under Vasicek stochastic interest rate model. C. Nie and X. Jin effectively predict the long-term time series data to schedule an investment strategy and obtain higher profit, considering that a stock price is a typical but complex type of time series data. They argue that their prediction method on the longterm stock price dynamics can be more precise and prevent the development of cumulative errors compared to the other methodologies. C. Xie et al. test the existence of systematic risk contagion within the Chinese interbank market with a particular data set from 2005 to 2013. By building the networks of the Chinese interbank market for each year and using the measure of mutual information, they quantitatively detect the changes of interbankmarket networks and observe that the correlations between banks become increasingly tighter in recent years. By compiling these interdisciplinary and quantitative papers, the editors hope to enrich the readers and academic researchers knowledge with respect to the econophysics, statistical mechanics, and quantitative economics and finance.

Stock loan valuation under a stochastic interest rate model

Stock loans are loans collateralized by stocks. They are modern financial products designed for investors with large equity positions. Mathematically, stock loans can be regarded as American call options with a time-dependent strike price. This study is the first in the literature that considers the valuation of stock loans in a stochastic interest rate framework. Based on portfolio analysis, a partial differential equation (PDE) governing the value of stock loans is derived. A set of appropriate boundary conditions, particularly in the interest rate direction, are also proposed to close the pricing system. A sound justification is provided for the proposed boundary conditions mathematically as well as financially. To solve the proposed nonlinear PDE system, a predictor–corrector finite difference method is adopted. Moreover, an alternating direction implicit (ADI) method is used to improve the computational efficiency. Numerical results suggest that the current method is reliable and the stochastic interest rate leads to a higher optimal exercise price of the stock loan in comparison with that calculated from the Black–Scholes model.

Valuation of interest rate ceiling and floor in uncertain financial market

The interest rate ceiling and floor are the popular interest rate derivatives in a financial market. In this paper, the valuation of interest rate ceiling and floor is investigated by using uncertainty theory. Different from the classical stochastic interest rate models, the uncertain interest rate model is used in this paper as the basis of evaluating the interest rate ceiling and floor. Based on the assumption that the short interest rate follows uncertain differential equations, the price formulas of interest rate ceiling and floor are derived.

Time-consistent reinsurance–investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk

In this paper, we consider the time-consistent reinsurance–investment strategy under the mean–variance criterion for an insurer whose surplus process is described by a Brownian motion with drift. The insurer can transfer part of the risk to a reinsurer via proportional reinsurance or acquire new business. Moreover, stochastic interest rate and inflation risks are taken into account. To reduce the two kinds of risks, not only a risk-free asset and a risky asset, but also a zero-coupon bond and Treasury Inflation Protected Securities (TIPS) are available to invest in for the insurer. Applying stochastic control theory, we provide and prove a verification theorem and establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) equation. By solving the extended HJB equation, we derive the time-consistent reinsurance–investment strategy as well as the corresponding value function for the mean–variance problem, explicitly. Furthermore, we formulate a precommitment mean–variance problem and obtain the corresponding time-inconsistent strategy to compare with the time-consistent strategy. Finally, numerical simulations are presented to illustrate the effects of model parameters on the time-consistent strategy.

Option Pricing with Transaction Costs and Stochastic Interest Rate

In the case when transaction costs are associated with trading assets the option pricing problem is known to lead to solving nonlinear partial differential equations even when the underlying asset is modelled using a simple geometric Brownian motion. The nonlinear term in the resulting PDE corresponds to the presence of transaction costs. We generalize this model to a stochastic one-factor interest rate model. We show that the model follows a nonlinear parabolic type partial differential equation. Under certain assumption we prove the existence of classical solution for this model.

The pricing of credit risky securities under stochastic interest rate model with default correlation

In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R.A. Jarrow and F.Yu (2001), S.Y.Leung and Y.K.Kwok (2005), A.Wang and Z.Ye (2011)). By using the method of change of measure and the technology (H. S.Park (2008), R.Hao and Z.Ye (2011)) of dealing with jump diffusion processes, we obtain the analytic pricing formulas of defaultable zero-coupon bonds. Moreover, by the “total hazard construction”, we give the analytic pricing formulas of credit default swap (CDS).

A stochastic volatility model and optimal portfolio selection

In this paper, first we study a stochastic volatility market model for which an explicit candidate solution to the problem of maximizing the utility function of terminal wealth is obtained. Applying this result, we present a complete solution for the Heston model, which is a particular case of the general model. A verification result and a martingale representation of the solution are provided for the Heston model. Finally, the same techniques are used to study a stochastic interest rate model and a necessary and sufficient condition for exploding growth is presented.

A dimension-reduction algorithm for the valuation of surrender options in EIA contracts with stochastic interest rates

This paper proposes a fast algorithm for the fair valuation of a ratchet-type equity-indexed annuity (EIA) endowment contract with surrender options under Vasicek stochastic interest rate models. Traditionally, the valuation for the indexed equity and interest rate of an American-type surrender option is performed under two-dimensional tree models, which is time-consuming for computation. This paper first applies the Black–Scholes method for ratchet-type options to reduce the two-dimensional tree structure to single one. Next, to overcome the path dependent problem inherent in the ratchet option, we also propose a recursive formula to implement the backward computation. By using the proposed algorithm, we are able to perform numerical analysis to verify that surrender options are more valuable with the increase of interest rates. High interest rate volatility enhances both the bonus and surrender option values entitled to the policyholder. A numerical experiment also shows that increasing interest rates may decrease the bonus option value but increase the surrender option value. These results can provide suggestions for insurance companies regarding the issue of EIA policies.

Long-term behavior of stochastic interest rate models with jumps and memory

The long-term interest rates, for example, determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option. In this paper, for a one-factor model, we reveal that the long-term return t−μ∫0tX(s)ds for some μ≥1, in which X(t) follows an extension of the Cox–Ingersoll–Ross model with jumps and memory, converges almost surely to a reversion level which is random itself. Such a convergence can be applied in the determination of models of participation in the benefit or of saving products with a guaranteed minimum return. As an immediate application of the result obtained for the one-factor model, for a class of two-factor model, we also investigate the almost sure convergence of the long-term return t−μ∫0tY(s)ds for some μ≥1, where Y(t) follows an extended Cox–Ingersoll–Ross model with stochastic reversion level −X(t)/(2β) in which X(t) follows an extension of the square root process. This result can be applied to, e.g., how the percentage of interest should be determined when insurance companies promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on.

Managing Life Insurer Risk and Profitability: Annuity Market Development Using Natural Hedging Strategies

Changing demographics creates the potential for the expansion of existing and new products to manage longevity risk. Life annuities address this risk, yet these annuity product markets are thin. Insurers are concerned about the long term risks associated with these longevity products and capital requirements. Life insurers also offer life insurance products, whole-of-life and term, that provide an opportunity to offset longevity risks. This can allow capital efficient longevity risk products to be sold as part of a product portfolio. Natural hedging, or the offsetting of risks in life insurance and annuity business, provides a way of managing capital efficiently as well as improving profitability. This paper uses stochastic mortality and interest rate models to assess life and annuity capital requirements and quantify the benefits of natural hedging taking into account relative profit loadings on products. The benefits of offering longevity products, in terms of capital requirements, as well as the importance of the type of life insurance products offered are illustrated using standard life and annuity products. The impact of capital requirements, such as solvency II with a one-year horizon, are considered and compared to multiple period risk measures to confirm the results hold for regulatory capital requirements.

Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching

In this paper, we investigate the pricing of variance swaps under a Markovian regime-switching extension of the Schöbel–Zhu–Hull–White hybrid model. The parameters of this model, including the mean-reversion levels and the volatility rates of both stochastic interest rate and volatility, switch over time according to a continuous-time, finite-state, observable Markov chain. By utilizing techniques of measure changes, we separate the interest rate risk from the volatility risk. The prices of variance swaps and related fair strike values are represented in integral forms. We illustrate the practical implementation of the model by providing a numerical analysis in a two-state Markov chain case, which shows that the effect of both stochastic interest rate and regime-switching is significant in the pricing of variance swaps.

Longevity bond pricing under stochastic interest rate and mortality with regime-switching

We develop a flexible model to value longevity bonds which incorporates several important sources of risk, namely, interest rate risk, mortality risk and the risk due to structural changes in economic and environmental conditions. In particular, Markov, regime-switching, jump-diffusion models are used to describe stochastic movements of short-term interest rate and force of mortality. These models capture jumps in short rate and mortality rate and the impacts of economic and environmental fundamentals on their movements over time. Using the concept of stochastic flows, we derive an exponential affine form of the longevity bond price in the proposed joint stochastic interest rate and mortality models. In particular, a representation for the exponential affine form of the longevity bond price is obtained in terms of fundamental matrix solutions of linear, matrix-valued, ordinary differential equations.

A Stochastic Volatility Model and Optimal Portfolio Selection

In this paper, first we study a stochastic volatility market model for which an explicit candidate solution to the problem of maximizing utility function of terminal wealth is obtained. Applying this result, we present a complete solution for the Heston model which is a particular case of the general model. A verication result and a martingale representation of the solution are providedfor the Heston model. Finally, the same techniques are used to study a stochastic interest rate model and a necessary and sufficient condition for exploding growth is presented.

European Option Pricing Formula Under Stochastic Interest Rate

Abstract: This paper reviews the option pricing model and its application, on the basis of former studies, we assume that the interest rate satisfy a given Vasicek stochastic di erential equation, using option pricing by martingale method to study the stochastic interest rate model of European option pricing and obtain its pricing formula. Finally, we compare the di erences between the standard European option pricing formula and European option pricing formula under stochastic interest rate. Key words: Option Pricing; Stochastic Interest Rates; Vasicek model; Brownian motions

Option Pricing Under Stochastic Interest Rates

This paper reviews the research history of option pricing, then our model assumes that the interest rate subject to a given Vasicek stochastic differential equations, using option pricing by martingale method to study the stochastic interest rate model of European option pricing and obtain the pricing formula. Finally, we compare the differences between the standard European option pricing formulas and European option pricing formula under stochastic interest rate.