Instantaneous Interest Rate Research Articles

Accuracy of analytical approximation formula for bond prices in a three-factor convergence model of interest rates

We consider a convergence model of interest rates, in which the behaviour of the domestic instantaneous interest rate (so called short rate) depends on the short rate in a monetary union that the country is going to join. The short rate in the monetary union is modelled by a two-factor model, which leads to a three-factor model for the domestic rate. In this setting, term structures of interest rates are computed from bond prices, which are obtained as solutions to a parabolic partial differential equation. A closed-form solution is known only in special cases. An analytical approximation formula for the domestic bond prices has been proposed, with the error estimate only for certain parameter values, when the solution has a separable form. In this paper, we derive the order of accuracy in the general case. We also study a special case, which makes it possible to model the phenomenon of negative interest rates that were observed in the previous years. It turns out that it leads to a higher accuracy than the one achieved in the general case without restriction on parameters.

A Closed-Form Pricing Formula for Log-Return Variance Swaps under Stochastic Volatility and Stochastic Interest Rate

At present, the study concerning pricing variance swaps under CIR the (Cox–Ingersoll–Ross)–Heston hybrid model has achieved many results; however, due to the instantaneous interest rate and instantaneous volatility in the model following the Feller square root process, only a semi-closed solution can be obtained by solving PDEs. This paper presents a simplified approach to price log-return variance swaps under the CIR–Heston hybrid model. Compared with Cao’s work, an important feature of our approach is that there is no need to solve complex PDEs; a closed-form solution is obtained by applying the martingale theory and Ito^’s lemma. The closed-form solution is significant because it can achieve accurate pricing and no longer takes time to adjust parameters by numerical method. Another significant feature of this paper is that the impact of sampling frequency on pricing formula is analyzed; then the closed-form solution can be extended to an approximate formula. The price curves of the closed-form solution and the approximate solution are presented by numerical simulation. When the sampling frequency is large enough, the two curves almost coincide, which means that our approximate formula is simple and reliable.

Equilibrium investment strategy for a DC pension plan with learning about stock return predictability

This paper investigates a time-consistent investment strategy under the mean-variance criterion for an investor who accumulates retirement savings through a defined contribution (DC) pension plan with stock and bond investment opportunities. The expected return rate on the stock is modulated by an unobservable predictor which follows a mean-reverting stochastic process. The evolution of the instantaneous interest rate is described by the Vasicek model. In addition, the contribution rate of the DC pension plan is stochastic and correlated with financial risks coming from the stochastic interest rate and stock price. In a game theoretic framework, we derive a closed-form equilibrium investment strategy and corresponding equilibrium value function for the mean-variance criterion by adopting the filtering technique and the stochastic control method. Furthermore, we provide an equilibrium investment strategy and equilibrium value function when the expected return rate of the stock is completely observable. Finally, some numerical examples are presented to demonstrate the sensitivity analysis of the equilibrium investment strategy and equilibrium efficient frontier. Numerical analysis confirms that there is non-negligible information loss on the equilibrium investment strategy and equilibrium value function due to partial observation in the stock price dynamics.

Country risk for emerging economies: a dynamical index proposal with a case study.

We introduce a dynamical country risk index for emerging economies. The proposal is based on the intensity approach of credit risk, i.e. the default is the first jump of a point process with stochastic intensity. Two different models are used to estimate the yield spread. They differ in the relationship between the default-free instantaneous interest rate process and the intensity process. The dynamics of the interest rates is modeled through a multidimensional affine model, and the Kalman filter with an Expectation-Maximization algorithm is used to calibrate it. The USD interest rates constitute part of the input of the model, while prices of relevant domestic bonds in the emerging market complete the input. For an application, we select the Uruguayan bond market as the emerging economy.

Toward pricing financial derivatives with an IBM quantum computer

Pricing interest-rate financial derivatives is a major problem in finance, in which it is crucial to accurately reproduce the time-evolution of interest rates. Several stochastic dynamics have been proposed in the literature to model either the instantaneous interest rate or the instantaneous forward rate. A successful approach to model the latter is the celebrated Heath-Jarrow-Morton framework, in which its dynamics is entirely specified by volatility factors. On its multifactor version, this model considers several noisy components to capture at best the dynamics of several time-maturing forward rates. However, as no general analytical solution is available, there is a trade-off between the number of noisy factors considered and the computational time to perform a numerical simulation. Here, we employ the quantum principal component analysis to reduce the number of noisy factors required to accurately simulate the time evolution of several time-maturing forward rates. The principal components are experimentally estimated with the $5$-qubit IBMQX2 quantum computer for $2\times 2$ and $3\times 3$ cross-correlation matrices, which are based on historical data for two and three time-maturing forward rates. This manuscript is a first step towards the design of a general quantum algorithm to fully simulate on quantum computers the Heath-Jarrow-Morton model for pricing interest-rate financial derivatives. It shows indeed that practical applications of quantum computers in finance will be achievable in the near future.

No-arbitrage one-factor term structure models in zero- or negative-lower-bound environments

One-factor no-arbitrage term structure models where the instantaneous interest rate follows either the process proposed by Vasicek (1977) or by Cox, Ingersoll, and Ross (1985), commonly known as CIR, are parsimonious and analytically tractable. Models based on the original CIR process have the important characteristic of allowing for a time-varying conditional interest rate volatility but are undefined in negative interest rate environments. A Shifted-CIR no-arbitrage term structure model, where the instantaneous interest rate is given by the sum of a constant lower bound and a non-negative CIR-like process, allows for negative yields and benefits from similar tractability of the original CIR model. Based on the U.S. and German yield curve data, the Vasicek and Shifted-CIR specifications, both considering constant and time-varying risk premia, are compared in terms of information criteria and forecasting ability. Information criteria prefer the Shifted-CIR specification to models based on the Vasicek process. It also provides similar or better in-sample and out-of-sample forecasting ability of future yield curve movements. Introducing a time variation of the interest rate risk premium in no-arbitrage one-factor term structure models is instead not recommended, as it provides worse information criteria and forecasting performance.

Accumulated (integrated) interest and the square root process

PurposeThe purpose of this study is to determine distributional properties of the accumulated rate of interest when the instantaneous rate of interest evolves in terms of the Cox et al. (1985) square root process.Design/methodology/approachThe law of iterated (or double) expectations is used to determine the mean and variance of the accumulated rate of interest on a cash management (or loan) account when interest accumulates at the instantaneous rates of interest implied by the square root process.FindingsThis study demonstrates how the accumulated rate of interest does not satisfy the strong mixing conditions necessary for convergence in distribution to the normal density function.Originality/valueThis study has strong educational value in determining distributional properties of the accumulated rate of interest when the instantaneous rate of interest evolves in terms of the Cox et al. (1985) square root process and demonstrating how the accumulated rate of interest does not satisfy the strong mixing conditions necessary for convergence in distribution to the normal density function.

Upper and lower bounds for annuities and life insurance from incomplete mortality data

ABSTRACT This study aimed to set upper and lower bounds for the expected present value of whole life annuities and whole life insurance policies from incomplete mortality data, generalizing previous results on life expectancy. Since its inception, in the 17th century, actuarial science has been devoted to the study of annuities and insurance plans. Thus, setting intervals that provide an initial idea about the cost of these products using incomplete mortality data represents a theoretical contribution to the area and this may have major applications in markets lacking historical records or those having little reliability of mortality data, as well as in new markets still poorly explored. For both the continuous and discrete cases, upper and lower bounds were constructed for the expected present value of whole life annuities and whole life insurance policies, contracted by a person currently aged x, based on information about the expected present value of these respective financial products subscribed to by a person of age x + n and the probability that an individual of age x survives to at least age x + n. Through the bounds of a continuous annuity, in an environment where the instantaneous interest rate is equal to zero, the results shown also set bounds for the complete life expectancy, which implies that the contribution of this research generalizes previous results in the literature. It was also found that, for both annuities and insurance plans, the length of constructed intervals increases as the data gap size increases and it decreases as the survival curve becomes more rectangular. Illustratively, bounds for life expectancy at 40 and 60 years of age, for the 10 municipalities showing the highest life expectancy at birth in Brazil in 2010, were constructed by using data available in the Atlas of Human Development in Brazil.

Optimal investment strategies with a minimum performance constraint

We consider the optimal investment problem of a fund manager in the presence of a minimum guarantee constraint on the fund performance. The manager receives a fee which is proportional to the liquidation value of the portfolio or of the surplus over the guarantee in case it is positive and zero otherwise, eventually augmented by a constant fee. Her remuneration is reduced through the application of a penalty if the value of the fund at maturity is below a specified-in-advance threshold (minimum guarantee). We deal with two different settings: a continuous time economy with constant instantaneous interest rate and the case where the interest rate evolves as the Vasicek model. Explicit formulas for the optimal investment strategy are presented. We compare our portfolio strategies to the Merton portfolio and to the Option Based Portfolio Insurance strategy.

A closed-form pricing formula for variance swaps under MRG–Vasicek model

In this paper, the pricing problems of variance swaps with discrete sampling times are studied, where the volatility of underlying assets follows a mean-reverting Gaussian (MRG in short) process, and the instantaneous interest rate is described by classical Vasicek model. By using measure transformation, Feynman–Kac formula and Fourier transform algorithm, a closed-form analytic pricing formula for variance swaps with the actual-return realized variance is presented.

Optimal investment and life insurance strategies in a mixed jump-diffusion framework

This article investigates an optimal investment and life insurance strategies in a mixed jump-diffusion framework. The individual life insurance policyholder who has CRRA preferences. The market consists of riskless asset, a zero-coupon bond, a stock and life insurance. The instantaneous interest rate is modeled as the O-U model, while a zero-coupon bond with credit risk follows a BSDE and a risky asset be driven by MJD-fBm model. The problem is solved by the mixed jump diffusion fractional HJB SDE which satisfied the admissible strategy, then the closed form solution and optimal strategies are derived and the simulation of the various parameters are also given.

A cross-sectional application of the Nelson-Siegel-Svensson model to several negative yield cases

The appearance of negative bond yields presents significant challenges for the fixed income markets, which mainly concern related forecasting models. The Nelson-Siegel-Svensson model (NSS) is one of the models that is most frequently used by central banks to estimate the term structure of interest rates. The objective of this study is to evaluate the application of the NSS model to fit the yield curve of a set of 20 countries, the majority from the Eurozone, which registered negative sovereign bond yields. We conclude that the model adjusted well for all countries’ yield curves, although no changes or constraints were introduced. In addition, a comparison was carried out between market instantaneous interest rate and the interest rate for the very distant future, which the model can predict, with good results for the instantaneous interest rate. An evaluation of the possible behaviour of shared debt securities (i.e. Eurobonds) was also analysed. In conclusion, the NSS model seems to remain a valuable, easy to use, and adaptable tool, to fit negative yield curves, for monetary policy institutions and market players alike.

A note on the worst case approach for a market with a stochastic interest rate

We solve robust optimization problem and show the example of the market model for which the worst case measure is not a martingale measure. In our model the instantaneous interest rate is determined by the Hull-White model and the investor employs the HARA utility to measure his satisfaction.To protect against the model uncertainty he uses the worst case measure approach. The problem is formulated as a stochastic game between the investor and the market from the other side. PDE methods are used to find the saddle point and the precise verification argument is provided.

Optimal mean–variance efficiency of a family with life insurance under inflation risk

We study an optimization problem of a family under mean–variance efficiency. The market consists of cash, a zero-coupon bond, an inflation-indexed zero-coupon bond, a stock, life insurance and income-replacement insurance. The instantaneous interest rate is modeled as the Cox–Ingersoll–Ross (CIR) model, and we use a generalized Black–Scholes model to characterize the stock and labor income. We also take into account the inflation risk and consider our problem in the real market. The goal of the family is to maximize the mean of the surplus wealth at the retirement or death of the breadwinner and minimize its variance by finding a portfolio selection. The efficient frontier and optimal strategies are derived through the dynamic programming method and the technique of solving associated nonlinear HJB equations. We also present a numerical illustration to explore the impact of economical parameters on the efficient frontier.

Negative real interest rates

Standard textbook general equilibrium term structure models such as that developed by Cox, Ingersoll, and Ross [1985b. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407], do not accommodate negative real interest rates. Given this, the Cox, Ingersoll, and Ross [1985b. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407] ‘technological uncertainty variable’ is formulated in terms of the Pearson Type IV probability density. The Pearson Type IV encompasses mean-reverting sample paths, time-varying volatility and also allows for negative real interest rates. The Fokker–Planck (i.e. the Chapman–Kolmogorov) equation is then used to determine the conditional moments of the instantaneous real rate of interest. These enable one to determine the mean and variance of the accumulated (i.e. integrated) real rate of interest on a bank (or loan) account when interest accumulates at the instantaneous real rate of interest defined by the Pearson Type IV probability density. A pricing formula for pure discount bonds is also developed. Our empirical analysis of short-dated Treasury bills shows that real interest rates in the UK and the USA are strongly compatible with a general equilibrium term structure model based on the Pearson Type IV probability density.

Value of the Distant Future: Model-Independent Results

If stochastic perturbation of the long-term discount factor by random shocks in instantaneous interest or consumption growth rates is weak, the approximate model-independent description of an effective discount rate is possible in terms of first and second cumulants of these processes. The derived simple analytical formulas describe the declining discount curves at times comparable with the correlation time of the relevant stochastic process. With a growth of time the relevant effective “certainty equivalent” discount rates converge towards their long-term time-invariant asymptotic corresponding to the conventional Markovian exponential decay of a discount factor. Comparison of the obtained results (approximate model-independent and exact model-dependent) with the outcome of the benchmark investment- and consumption-based models shows that the transient decline of a discount curve with a growth of time is related to serial correlations of shocks in the relevant rates. Account of fluctuations of real interest rates substantially reduces the annualized long-term discount rate, e.g., 0.5% for USA and UK. The model-independent character and simplicity of the obtained analytical results may warrant incorporation into the rigorous cost-benefit and regulatory impact analysis of long-term environmental and infrastructure projects.

Stabilizing Log-Normal Diffusion in View of Compounding Swaps Funding Risk Credit Valuation Adjustment (FRCVA)

We discuss several methodologies for stabilizing the diffusion of compounding instruments modeled by means of lognormal instantaneous interest rate models. We first delve into alternative mathematical and numerical methodologies before showing that the latter suffer from a high seed sensitivity which makes them difficult to apply in practice. We consequently resort to a fine-tune capping of the maximal value of the interest rate which ensures stabilization without altering the main diffusion moments characteristics. We illustrate our approach by examining the funding loss, the Funding Risk Adjustment (FRA) and the Funding Risk Credit Valuation Adjustment (FRCVA) of a compounding swap along the lines of a preceding paper where we advocate the benefit, when funding cannot be hedged, of examining funding risk in an actuarial setting.

Mean–variance efficiency of DC pension plan under stochastic interest rate and mean-reverting returns

This paper studies the optimization problem of DC pension plan under mean–variance criterion. The financial market consists of cash, bond and stock. Similar to Guan and Liang (2014), we assume that the instantaneous interest rate is an affine process including the Cox–Ingersoll–Ross (CIR) model and Vasicek model. However, we assume that the expected return of the stock follows a completely different mean-reverting process, which can well display the bear and bull features of the market, and the market price of the stock index is the Ornstein–Uhlenbeck process. The pension manager thus has to undertake the risks of interest rate and market price of stock index. Besides, a special stochastic contribution rate is formulated. The goal of the pension manager is to maximize the expected terminal value and minimize the variance of terminal value. We will use the technique developed by Guan and Liang (2014) to tackle this problem and derive the closed-forms of efficient frontier and strategies. Numerical analysis is given in the end of this paper to show the economic behavior of the efficient frontier and strategies.

Estimating the Short Rate from the Term Structures in the Vasicek Model

Abstract In short rate models, bond prices and term structures of interest rates are determined by the parameters of the model and the current level of the instantaneous interest rate (so called short rate). The instantaneous interest rate can be approximated by the market overnight, which, however, can be influenced by speculations on the market. The aim of this paper is to propose a calibration method, where we consider the short rate to be a variable unobservable on the market and estimate it together with the model parameters for the case of the Vasicek model

An Optimal Selection of the basis Functions for the Valuation of Interest Rate Structured Notes

This paper examines which basis functions are efficient to employ a combined method of Hull and White (1990) with the Monte Carlo simulation when we price a callable range note or a callable bond. We use the Huge and Rom-Poulsen (2007) method which has modified the least squared Monte Carlo simulation proposed by Longstaff and Schwartz (2001) to reduce the estimation errors of the continuation value or the underlying assets. To use Monte carlo Simulation for pricing the early exercise premium, it is essential to accurately estimate the continuation value, because the investors will choose the higher value between the exercise and the continuation value at the possible early exercise dates. The main purpose of this paper is to analyze the estimation errors originating from the choice of the basis functions for the underlying asset and the continuation value estimation. We choose the callable bond and the callable range accrual note to show which basis functions are reliable to reduce the estimation errors. For this purpose, we replicate the callable range accrual note with a portfolio of a fixed rate bond and a delayed digital option. We use several basis functions such as a constant, the instantaneous interest rates, and the range in order to see which basis function is efficient for our purpose. We examine several combinations of the basis functions depending on which basis functions will be used for the underlying asset or the continuation value estimation. We show that the range which is an important determinant of the callable range accrual note is an effective basis function to accurately determine the underlying asset and the continuation value for the pricing of the callable range accrual note.