Interest rate dynamics and the term structure: A note

The Brennan and Schwartz two-factor model of the term structure is re-examined using constant-duration long-term yields in place of the fluctuating-duration consol yields, and additional stability is achieved in the empirical estimations. The model is then reformulated to examine more closely the relationship between changes in long and short rates of interest. It is found that movements in the short rate of interest are related to movements in the long rate, the spread between the long and short rates, and the prior movement in the long rate. The model is then tested for its out-of-sample predictive ability and found to possess a modest amount of predictive power.     

Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation

In this article we provide an asymptotic distribution theoryfor some nonparametric tests of the hypothesis that asset priceshave continuous sample paths. We study the behaviour of thetests using simulated data and see that certain versions ofthe tests have good finite sample behavior. We also apply thetests to exchange rate data and show that the null of a continuoussample path is frequently rejected. Most of the jumps the statisticsidentify are associated with governmental macroeconomic announcements.     

Nonparametric estimation of scalar diffusion models of interest rates using asymmetric kernels

This paper proposes an asymmetric kernel-based method for nonparametric estimation of scalar diffusion models of spot interest rates. We derive the asymptotic theory for the asymmetric kernel estimators of the drift and diffusion functions for general and positive recurrent processes and illustrate the advantages of the Gamma kernel for bias correction and efficiency gains. The finite-sample properties and the practical relevance of the proposed nonparametric estimators for bond and option pricing are evaluated using actual and simulated data for U.S. interest rates.     

An empirical investigation of the two-factor Brennan-Schwartz term structure model

This paper studies the pricing behaviors of default-free bonds based on the two-factor model by Brennan and Schwartz (1979), where a short-term spot rate and a long-term consol rate are the state variables. The logarithm of these two factors is assumed to follow a linear transformation of an Ornstein-Uhlenbeck process. An exact discrete time model is derived to estimate the parameters in the process. The model prices are then numerically solved. The sensitivity analysis indicates that the long-rate process, especially the long-rate volatility parameter, is important in characterizing the term structure of interest rates.     

Optimal investment strategies for general utilities under dynamic elasticity of variance models

This paper studies the optimal investment strategies under the dynamic elasticity of variance (DEV) model which maximize the expected utility of terminal wealth. The DEV model is an extension of the constant elasticity of variance model, in which the volatility term is a power function of stock prices with the power being a nonparametric time function. It is not possible to find the explicit solution to the utility maximization problem under the DEV model. In this paper, a dual-control Monte-Carlo method is developed to compute the optimal investment strategies for a variety of utility functions, including power, non-hyperbolic absolute risk aversion and symmetric asymptotic hyperbolic absolute risk aversion utilities. Numerical examples show that this dual-control Monte-Carlo method is quite efficient.     

Consistent estimation of cross-sectional models in event studies

Event studies often include cross-sectional regressions of announcementeffects on exogenous variables. If the event is voluntary andinvestors are rational, then standard OLS and GLS estimatorsare inconsistent. Consistent ML estimators are constructed fora cross-sectional model of horizontal mergers relating announcementeffects to exogenous characteristics of firms and industries.The OLS and ML estimates differ dramatically for bidders butnot for targets. The evidence suggests that manager of bidders,but not targets, have valuable private information about thepotential synergies from proposed mergers.     

Estimating the Gerber–Shiu function by Fourier–Sinc series expansion

In this paper, we consider the nonparametric estimation of the Gerber–Shiu function in a compound Poisson risk model perturbed by diffusion. We present a more efficient estimator based on Fourier–Sinc series expansion. Our estimator is easily computed and has a faster convergence rate. Some simulation examples are provided to show that the estimator performs well when the sample size is finite.     

Valuing debt options: Empirical evidence

A two-factor model using the instantaneous rate of interest and the return on a consol bond to describe the term structure of interest rates — the Brennan-Schwartz model — is used to derive theoretical prices for American call and put options on US government bonds and treasury bills. These model prices are then compared with market prices. The theoretical model used to value the dept options also provides hedge ratios which may be used to construct zero-investment portfolios which, in theory, are perfectly riskless. Several trading strategies based on these ‘riskless’ portfolios are examined.     

Nonparametric density estimation and tests of continuous time interest rate models

I study the finite sample distribution of one of Ait-Sahalia's(1996c) nonparametric tests of continuous-time models of theshort-term riskless rate. The test rejects true models too oftenbecause interest rate data are highly persistent but the asymptoticdistribution of the test (and of the kernel density estimatoron which the test is based) treats the data as if it were independentlyand identically distributed. To attain the accuracy of the kerneldensity estimator implied by its asymptotic distribution with22 years of data generated from the Vasicek model in fact requires2755 years of data.     

Confidence intervals of the premiums of optimal bonus malus systems

In view of the economic importance of motor third-party liability insurance in developed countries the construction of optimal BMS has been given considerable interest. However, a major drawback in the construction of optimal BMS is that they fail to account for the variability on premium calculations which are treated as point estimates. The present study addresses this issue. Specifically, nonparametric mixtures of Poisson laws are used to construct an optimal BMS with a finite number of classes. The mixing distribution is estimated by nonparametric maximum likelihood (NPML). The main contribution of this paper is the use of the NPML estimator for the construction of confidence intervals for the premium rates derived by updating the posterior mean claim frequency. Furthermore, we advance one step further by improving the performance of the confidence intervals based on a bootstrap procedure where the estimated mixture is used for resampling. The construction of confidence intervals for the individual premiums based on the asymptotic maximum likelihood theory is beneficial for the insurance company as it can result in accurate and effective adjustments to the premium rating policies from a practical point of view.     

Is the Short Rate Drift Actually Nonlinear?

Aït-Sahalia (1996) and Stanton (1997) use nonparametric estimators applied to short-term interest rate data to conclude that the drift function contains important nonlinearities. We study the finite-sample properties of their estimators by applying them to simulated sample paths of a square-root diffusion. Although the drift function is linear, both estimators suggest nonlinearities of the type and magnitude reported in Aït-Sahalia (1996) and Stanton (1997). Combined with the results of a weighted least squares estimator, this evidence implies that nonlinearity of the short rate drift is not a robust stylized fact.     

Nonparametric correlation models for portfolio allocation

This article proposes time-varying nonparametric and semiparametric estimators of the conditional cross-correlation matrix in the context of portfolio allocation. Simulations results show that the nonparametric and semiparametric models are best in DGPs with substantial variability or structural breaks in correlations. Only when correlations are constant does the parametric DCC model deliver the best outcome. The methodologies are illustrated by evaluating two interesting portfolios. The first portfolio consists of the equity sector SPDRs and the S&P 500, while the second one contains major currencies. Results show the nonparametric model generally dominates the others when evaluating in-sample. However, the semiparametric model is best for out-of-sample analysis.     

Aggregation of Nonparametric Estimators for Volatility Matrix

An aggregated method of nonparametric estimators based on time-domainand state-domain estimators is proposed and studied. To attenuatethe curse of dimensionality, we propose a factor modeling strategy.We first investigate the asymptotic behavior of nonparametricestimators of the volatility matrix in the time domain and inthe state domain. Asymptotic normality is separately establishedfor nonparametric estimators in the time domain and state domain.These two estimators are asymptotically independent. Hence,they can be combined, through a dynamic weighting scheme, toimprove the efficiency of volatility matrix estimation. Theoptimal dynamic weights are derived, and it is shown that theaggregated estimator uniformly dominates volatility matrix estimatorsusing time-domain or state-domain smoothing alone. A simulationstudy, based on an essentially affine model for the term structure,is conducted, and it demonstrates convincingly that the newlyproposed procedure outperforms both time- and state-domain estimators.Empirical studies further endorse the advantages of our aggregatedmethod.     

Efficient discretization of stochastic integrals

Sharp asymptotic lower bounds on the expected quadratic variation of the discretization error in stochastic integration are given when the integrator admits a predictable quadratic variation and the integrand is a continuous semimartingale with nondegenerate local martingale part. The theory relies on inequalities for the kurtosis and skewness of a general random variable which are themselves seemingly new. Asymptotically efficient schemes which attain the lower bounds are constructed explicitly. The result is directly applicable to a practical hedging problem in mathematical finance; for hedging a payoff which is replicated by a continuous-time trading strategy, it gives an asymptotically optimal way to choose discrete rebalancing dates and portfolios with respect to transaction costs. The asymptotically efficient strategies in fact reflect the structure of the transaction costs. In particular, a specific biased rebalancing scheme is shown to be superior to unbiased schemes if the transaction costs follow a convex model. The problem is discussed also in terms of exponential utility maximization.     

Sensitivity analysis of VaR and Expected Shortfall for portfolios under netting agreements

In this paper, we characterize explicitly the first derivative of the Value at Risk and the Expected Shortfall with respect to portfolio allocations when netting between positions exists. As a particular case, we examine a simple Gaussian example in order to illustrate the impact of netting agreements in credit risk management. Collateral issues are also dealt with. For practical purposes we further provide nonparametric estimators for sensitivities and derive their asymptotic distributions. An empirical application on a typical banking portfolio is finally provided.     

Estimating Systematic Risk with Daily Security Returns: A Note on the Relative Efficiency of Selected Estimators

This paper utilizes asymptotic analysis and daily security returns to examine the estimation efficiency of two unbiased robust estimators compared with ordinary least squares. Our results demonstrate a relative efficiency gain for a nonparametric rank estimator and a relative efficiency loss for the minimum absolute deviation estimator when estimating the systematic risk of securities using daily security returns.     

A bridge from ruin theory to credit risk

The structural model uses the firm-value process and the default threshold to obtain the implied credit spread. Merton’s (J Finance 29:449–470, 1974) credit spread is reported too small compared to the observed market spread. Zhou (J Bank Finance 25:2015–2040, 2001) proposes a jump-diffusion firm-value process and obtains a credit spread that is closer to the observed market spread. Going in a different direction, the reduced-form model uses the observed market credit spread to obtain the probability of default and the mean recovery rate. We use a jump-diffusion firm-value process and the observed credit spread to obtain the implied jump distribution. Therefore, the discrepancy in credit spreads between the structural model and the reduced-form model can be removed. From the market credit spread, we obtain the implied probability of default and the mean recovery rate. When the solvency-ratio process in credit risk and the surplus process in ruin theory both follow jump-diffusion processes, we show a bridge between ruin theory and credit risk so that results developed in ruin theory can be used to develop analogous results in credit risk. Specifically, when the jump is Logexponentially distributed, it results in a Beta distributed recovery rate that is close to market experience. For bonds of multiple seniorities, we obtain closed-form solutions of the mean and variance of the recovery rate. We prove that the defective renewal equation still holds, even if the jumps are possibly negative. Therefore, we can use ruin theory as a methodology for assessing credit ratings.      

Online Kernel estimation of stationary stochastic diffusion models

Nonparametric regression has recently become important in quantitative finance due to its distribution-free property. However, this advantage does not come without any cost. As large sample sizes are always required to adequately estimate local structures, nonparametric regression is computationally intensive in real applications. This paper proposes an online method to decrease the computational cost of nonparametric regression for estimating stationary stochastic diffusion models. We establish asymptotic behaviours of the proposed estimators under appropriate conditions. Numerical examples and an empirical study of US 3-month treasury bill rates are illustrated. The application to financial risk management is also taken into consideration.