Stochastic Interest Rate Model Research Articles

Testing Models of the Spot Interest Rate Volatility

This paper estimates discrete-time stochastic volatility models of the short-term spot interest rate for Germany and France, 1981-1997. Stochastic volatility interest rate models with both news and square root level effects seem to dominate alternative specifications. We extend previous results by using an exponential functional form for volatility, which is able to mitigate the impact of very large realizations and yield a better fit. Interest rate volatility shows a less persistent behavior than in stock returns and US Treasury bill yields volatility. Positive innovations in interest rates have a greater impact on conditional volatility than negative innovations.

A Class of Stochastic Volatility HJM Interest Rate Models

This paper considers a class of stochastic volatility HJM term structure models with explicit finite dimensional realisations. The resulting bond market is arbitrage free but incomplete resulting in a non-unique martingale measure. Nevertheless, the market price of risk is partially determined by the forward rate drift and volatility. Numerical simulation for bond and bond option prices are included to illustrate the effect of stochastic volatility on these prices.

Fitting Yield Curve Models Using the Kalman Filter

AbstractWe consider several stochastic interest rate models relying on the classes of Gaussian and square root processes. In particular, we introduce both, the Vasicek and the Cox‐Ingersoll‐Ross model, with k factors. Kalman filter methods are applied in order to estimate the parameters of these models. A detailed empirical analysis based on data from the German debt securities market is performed. The models are compared with respect to the number of factors and the different model classes. As a result, we suggest the application of the Vasicek model with at least two factors and the Cox‐Ingersoll‐Ross model with two factors.

Regularity of finite-dimensional realizations for evolution equations

We show that a continuous local semiflow of C k -maps on a finite-dimensional C k -manifold M with boundary is in fact a local C k -semiflow on M and can be embedded into a local C k -flow around interior points of M under some weak assumption. This result is applied to an open regularity problem for finite-dimensional realizations of stochastic interest rate models.

Testing a three-state model in currency derivative markets

This paper examines the ability of the jump diffusion models to explain systematic deviations in implicit distributions from the benchmark assumption of lognormality. Jumps that occur in the spot exchange rate due to supply and demand fluctuations in the currency market impose distributions for spot and futures prices that have degrees of skewness and kurtosis different from those of the lognormal distribution. Merton (1976)model allows for diversifiable jump risk. Bates' (1991, 1996) model allows the jump exchange risk to be systematic and derives the correct functional form of the market price of risk. Recent transactions data on futures and futures options are used to test the jump diffusion stochastic interest rates model developed by Doffou and Hilliard (2001a), as well as Bates' (1991, 1996)and Black's (1976) models to price out - of - sample options in a British pound, German mark,and Japanese yen futures market. The test results show that the jump diffusion stochastic interest rates model performs better than Bates' model which in turn performs better than Black's model.

Implied volatilities, stochastic interest rates, and currency futures options valuation: an empirical investigation

Different models of pricing currency call and put options on futures are empirically tested. Option prices are determined using different models and compared to actual market prices. Option prices are determined using historical as well as implied volatility. The different models tested include both constant and stochastic interest rate models. To determine if the model prices are different from the market prices, regression analysis and paired t-tests are performed. To see which model misprices the least, root mean square errors are determined. It is found that better results are obtained when implied volatility is used. Stochastic interest rate models perform better than constant interest rate models.

CONDITIONAL RUIN PROBABILITY WITH STOCHASTIC INTEREST RATE

In this paper, we use a diffusion process to model the risk reserve of an insurance company. By using some stochastic calculus techniques, a partial differential equation satisfied by the finite time horizon conditional ruin probability is obtained. Some special cases are studied. We also incorporate the stochastic interest rate model in our setup.

American option valuation under stochastic interest rates

By applying Ho, Stapleton and Subrahmanyam's (1997, hereafter HSS) generalised Geske–Johnson (1984, hereafter GJ) method, this paper provides analytic solutions for the valuation and hedging of American options in a stochastic interest rate economy. The proposed method simplifies HSS's three-dimensional solution to a one-dimensional solution. The simulations verify that the proposed method is more efficient and accurate than the HSS (1997) method. We illustrate how the price, the delta, and the rho of an American option vary between the stochastic and non-stochastic interest rate models. The magnitude of this effect depends on the moneyness of the option, interest rates, volatilities of the underlying asset price and the bond price, as well as the correlation between them.

Quadratic Interest Rate Models as Approximations to Effective Rate Models

Many of the classic stochastic interest rate models use the instantaneous spot rate as the fundamental sate variable. For both practical and theoretical reasons, it is sometimes convenient to base the model on the effective interest rate over some period such as three months. One disadvantage of this procedure is that the prices of some basic securities and derivatives do not have closed-form solutions. Under some circumstances, accurate values for these instruments can be obtained using an approximation to the true model. The approximation here is based on a quadratic interest rate model. Examples illustrate the accuracy of the approximation.

Valuing Federal Disaster Loans for Investment

MARK POTTER is an assistant professor of finance at Babson College in Babson Park, Massachusetts. T he Small Business AdministraŽ . tion SBA makes loans to owners of businesses and residences damaged in formally declared disasters, such as floods or hurricanes. Although these loans are now held within the SBA, the President’s 1999 budget submission proposes that SBA loans be sold to private investors. Anticipating new markets for these types of loans, this article takes a fresh look at government loan valuation. Using data on over 130,000 flood and hurricane disaster loans disbursed between 1980 and 1990, we estimate loan write-offs and early repayments using a discrete proportional hazards model. Then, using a stochastic interest rate model, we simulate loan termination probabilities to calculate the expected present value of the loans’ cash flows. Our analysis indicates an implied subsidy rate in the neighborhood of 26% for the 1990 loan cohort. Adding administrative costs and transactions fees to this rate defines the ‘‘true’’ cost to the government and, therefore, to private investors, of this particular disaster loan program. In the following section, the SBA’s Disaster Loan Program and related research in mortgage loan valuation is discussed. The data and methodology, including details of the valuation of disaster loans, are then presented. Empirical results as well as loan valuation simulations follow. We conclude with a discussion of the implications of our results for potential investors to these and other similar U.S. government loan programs.

Valuing Federal Disaster Loans: A Stochastic Model Approach

Between 1986 and 1990, the Small Business Administration made approximately $800 million in hurricane and flood disaster loans. The President's 1999 budget submission proposes that these loans be sold to private investors. Using data on over 130,000 disaster loans, and employing a stochastic interest rate model, we estimate the probability of a loan prepaying or being written off. We find that interest rate subsidy costs are significant components of the overall subsidy rate - for every dollar in loans granted, the government recovers between seventy and seventy-four cents, in present value terms. The valuation approach used in this paper can be applied to other government programs that repackage and sell loans to private investors.

Pricing by Arbitrage Under Arbitrary Information

A substantial applications literature on pricing by arbitrage has effectively restricted information to that arising solely from securities markets; return distributions are then governed solely by past prices. We reconsider pricing by arbitrage in markets rendered incomplete by arbitrary information, which, moreover, may influence required returns. We show that contingent claims depending solely on securities’ normalized price histories are priced by arbitrage if and only if all risk‐adjusted probabilities agree upon the law of primitive securities’ normalized prices. We show how existing diffusion‐based results can be preserved, and resolve an issue relating to Merton's (1973) stochastic interest rate model.

Long-term returns in stochastic interest rate models: different convergence results

In this paper, we focus on different convergence results of the long-term return (1/t)∫t0rudu, where the short interest rate r follows an extension of the Cox–Ingersoll–Ross (1985) model. Using the theory of Bessel processes, we proved the convergence almost everywhere of (1/t)∫t0Xudu, where (Xu)u⩾0 denotes a generalization of a Besselsquare process with drift. We also studied the convergence in law of the long-term return in order to make some approximations. We observed the convergence in law of the sequence of processes (Yn)n⩾1 with (Ynt)t⩾0=[(−2β3 δ¯n)1/2∫nt0(Xu+δu 2β) du]t⩾0 By the Aldous criterion, this sequence converges in law to a Brownian motion. These convergence results have some immediate applications. © 1998 John Wiley & Sons, Ltd.

Long‐term returns in stochastic interest rate models: different convergence results

In this paper, we focus on different convergence results of the long-term return (1/t)∫t0rudu, where the short interest rate r follows an extension of the Cox–Ingersoll–Ross (1985) model. Using the theory of Bessel processes, we proved the convergence almost everywhere of (1/t)∫t0Xudu, where (Xu)u⩾0 denotes a generalization of a Besselsquare process with drift. We also studied the convergence in law of the long-term return in order to make some approximations. We observed the convergence in law of the sequence of processes (Yn)n⩾1 with (Ynt)t⩾0=[(−2β3 δ¯n)1/2∫nt0(Xu+δu 2β) du]t⩾0 By the Aldous criterion, this sequence converges in law to a Brownian motion. These convergence results have some immediate applications. © 1998 John Wiley & Sons, Ltd.

Long-term returns in stochastic interest rate models: convergence in law

Using an extension of the Cox-Ingersoll-Ross [1] stochastic model of the short interest rate r, we study the convergence in law of the long-term return in order to make some approximations. We use the theory of Bessel processes and observe the convergence in law of the sequence with X a generalized Besselsquare process with drift with stochastic reversion level. By Aldous' criterion, we are able to prove that this sequence converges in law to a Brownian motion

Long-term returns in stochastic interest rate models

In this paper, we observed the convergence of the long-term return, using an extension of the Cox-Ingersoll-Ross (1985) stochastic model of the short interest rate r. Using the theory of Bessel processes, we were able to prove the convergence almost everywhere of (1/ t) f 0 t X s d s with X a generalized Bessel-square process with drift to a stochastic reversion level.

Mortgages and Markov Chains

This paper has two purposes. The first is purely expository: to introduce stochastic interest-rate models and security-evaluation methods in a simple mathematical setting. Specifically, we assume t...

Some remarks concerning stochastic interest rates in relation to long term insurance policies

Abstract This paper starts by considering the calculation of the moments of the present values of the profit on a long term sickness insurance policy using a relatively simple stochastic model for interest rates. It then considers the covariance between the present value of the profit on two policies for more general long term insurance policies using more general stochastic interest rate models.

An Empirical Investigation of the Market for Comex Gold Futures Options

ABSTRACTOption‐pricing models that assume a constant interest rate may misprice futures options if the interest rate fluctuates significantly or if the price of the underlying asset is correlated with the interest rate. The futures option‐pricing model of Ramaswamy and Sundaresan allows for a stochastic interest rate and correlation of the underlying asset's price with the interest rate. Using a data set of daily closing prices for Comex gold futures options, this paper tests the Ramaswamy and Sundaresan model against a constant interest rate model. Results indicate that the stochastic interest rate model is a superior predictor of market prices.