Pricing Options under Stochastic Interest Rates: A New Approach

We will generalize the Black-Scholes option pricing formula by incorporating stochastic interest rates. Although the existing literature has obtained some formulae for stock options under stochastic interest rates, the closed-form solutions have been known only under the Gaussian (Merton type) interest rate processes. We will show that an explicit solution, which is an extended Black-Scholes formula under stochastic interest rates in certain asymptotic sense, can be obtained by extending the asymptotic expansion approach when the interest rate volatility is small. This method, called the small-disturbance asymptotics for Itô processes, has recently been developed by Kunitomo and Takahashi (1995, 1998) and Takahashi (1997). We found that the extended Black-Scholes formula is decomposed into the original Black-Scholes formula under the deterministic interest rates and the adjustment term driven by the volatility of interest rates. We will illustrate the numerical accuracy of our new formula by using the Cox–Ingersoll–Ross model for the interest rates.     

Corporate Bond Valuation and Hedging with Stochastic Interest Rates and Endogenous Bankruptcy

This paper analyzes corporate bond valuation and optimal calland default rules when interest rates and firm value are stochastic.It then uses the results to explain the dynamics of hedging.Bankruptcy rules are important determinants of corporate bondsensitivity to interest rates and firm value. Although endogenousand exogenous bankruptcy models can be calibrated to producethe same prices, they can have very different hedging implications.We show that empirical results on the relation between corporatespreads and Treasury rates provide evidence on duration, andwe find that the endogenous model explains the empirical patternsbetter than do typical exogenous models.     

Pricing Commodity Spread Options with Stochastic Term Structure of Convenience Yields and Interest Rates

The purpose of this research is to provide a valuation formula for commodity spread options. Commodity spread options are options written on the difference of the prices (spread) of two commodities. From the aspect of commodity contingent claims, it is considered that commodity spread options are difficult to evaluate with accuracy because of the existence of the convenience yield. Hence, the model of the convenience yield is the key factor to price commodity spread options. We use the concept of future convenience yields to develop the model that enriches the stochastic behavior of convenience yield. We also introduce Heath-Jarrow-Morton interest rate model to the valuation framework. This general model not only captures the mean reverting feature of the convenience yield, but also allows us to handle a very wide range of shape that the term structure of convenience yield can take. Therefore our model provides various types of models. The numerical analysis presented in this paper provides some unique features of commodity spread options in contrast to normal options. These characteristics have never been addressed in previous studies. Moreover, it suggests that the existing model overprice commodity spread options through neglecting the effect of interest rates.     

The Valuation of Interest Rate Digital Options and Range Notes Revisited

The aim of this paper is to value interest rate structured products in a simpler and more intuitive way than Turnbull (1995). Considering some assumptions with respect to the evolution of the term structure of interest rates, the price of a European interest rate digital call option is given. Recall it is a contract designed to pay one dollar at maturity if a reference interest rate is above a prespecified level (the strike), and zero in all the others cases. Combining two options of this type enables us to value a European range digital option. Then using a one factor linear gaussian model and the new well‐known change of numeraire approach, a closed‐form formula is found to value range notes which pay at the end of each defined period, a sum equal to a prespecified interest rate times the number of days the reference interest rate lies inside a corridor.     

The Stochastic Volatility of Short-Term Interest Rates: Some International Evidence

This paper estimates a stochastic volatility model of short-term riskless interest rate dynamics. Estimated interest rate dynamics are broadly similar across a number of countries and reliable evidence of stochastic volatility is found throughout. In contrast to stock returns, interest rate volatility exhibits faster mean-reverting behavior and innovations in interest rate volatility are negligibly correlated with innovations in interest rates. The less persistent behavior of interest rate volatility reflects the fact that interest rate dynamics are impacted by transient economic shocks such as central bank announcements and other macroeconomic news.     

An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston [(1993) The Review of Financial studies, 6, 327–343], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.     

A Technique for Reducing Discretization Bias from Monte Carlo Simulations: Option Pricing under Stochastic Interest Rates

Abstract

Control variates are often used to reduce variability in Monte Carlo estimates and their effectiveness is traditionally measured by the so-called speed-up factor. The main objective of this paper is to demonstrate that a control variate can also be applied to reduce the bias stemming from the discretization of the state variable dynamics. This is particularly valuable when stochastic interest rate models are discretized, since bias reduction through more grid points is computationally expensive.     

Valuation of American Futures Options: Theory and Empirical Tests

This paper reviews the theory of futures option pricing and tests the valuation principles on transaction prices from the S&P 500 equity futures option market. The American futures option valuation equations are shown to generate mispricing errors which are systematically related to the degree the option is in-the-money and to the option's time to expiration. The models are also shown to generate abnormal risk-adjusted rates of return after transaction costs. The joint hypothesis that the American futures option pricing models are correctly specified and that the S&P 500 futures option market is efficient is refuted, at least for the sample period January 28, 1983 through December 30, 1983.     

International Interest Rates,Exchange Rates,and the Stochastic Structure of Supply

In a dual-currency, flexible exchange rate model, both nominal and real foreign exchange premia depend on investor risk attitudes, consumption parameters, and the stochastic structure of currency and commodity supplies. When supplies are random, their joint correlation structure determines the sign of the premia. If the money supplies are identically distributed, then all foreign exchange premia, regardless of the currency of denomination, are zero. A positive correlation between the value of a country's currency and its nominal interest rate need not indicate real interest rate movements. Relative bond prices can be negatively correlated with the terms of trade.     

Terms Structure of Interest Rates and Implicit Options: The Case of Japanese Bond Futures

The quality option for Japanese Government Bond Futures contracts is analysed using a term structure approach based upon a two-factor Heath, Jarrow and Morton (1990b) model. The option value is found to be 0.12%–0.2% of par three months prior to delivery. Also, analysis of variance confirms that the quality option has a negative theta .     

Real Options Valuation: A Case Study of an E‐commerce Company

This paper presents a real options valuation model with original solutions to some issues that arise frequently when trying to apply these models to real‐life situations. The authors build on existing models by introducing an innovative and intuitive risk neutral adjustment that allows us to work with all the simulated paths. The problem of incorporating real options into each path is solved with a “nearest neighbors” technique, and uncertainty is simulated using a beta distribution that adapts better to company‐specific information. The model is then applied to a real life e‐commerce company to produce the following insights: the expanded present value is higher than the traditional present value; the presence of several real options make them interact so that their values are nonadditive; and part of the expanded present value is explained by the presence of “Jensen's inequality” that stems from the “convexity” between the value of each year's cash flow and the uncertain variables.     

The Effective Duration and Convexity of Liabilities for Property-Liability Insurers Under Stochastic Interest Rates

Managing interest rate risk for property-liability insurers requires appropriate measurement of the sensitivity of liabilities to movements in interest rates. Most prior studies have assumed that interest rates shift in a parallel fashion and that the cash flows from liabilities are unaffected by interest rate changes. This article recognizes that unpaid property-liability (P-L) insurance losses are inflation-sensitive, that movements in interest rates will affect future claim payouts due to the correlation between interest rates and inflation and that interest rates are stochastic. The effective duration and convexity of P-L insurance liabilities calculated based on this approach are substantially lower than those measured using traditional approaches, which has important implications for asset-liability management by P-L insurers.     

The Valuation of Semiannual Bonds between Interest Payment Dates: A Correction

This note provides a correction to Taylor's 1988 work on the valuation of semiannual coupon bonds between interest payment dates. It shows that the discrepancy in values between Taylor's model and the standard Wall Street pricing formula is much smaller than indicated by Taylor and is unlikely to generate opportunities for arbitrage profits.     

The Pricing of Options on Assets with Stochastic Volatilities

One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity.     

Pricing Contingent Claims on a Risky Asset and Stochastic Interest Rates: A New Discrete Time Approach

This paper presents a new discrete time approach to pricing contingent claims on a risky asset and stochastic interest rates. The term structure of interest rates is modeled so that arbitrage-free bond prices depend on an observable initial forward rate curve rather than an exogenously specified market price of risk. A restricted binomial process is employed to model both interest rates and an asset price. As a result, a complete market valuation formula obtains. By choosing the parameters of the discrete joint distribution such that, in the limit, the discrete model converges to the continuous one, a model is obtained that requires the estimation of only three parameters. The approach is parsimonious with respect to alternative models in the literature and can be used to price contingent claims on any two state variables. The procedure is used to numerically analyze the effects of the volatility of interest rates on the determination of mortgage contract rates.     

Financial and Demographic Risks of a Portfolio of Life Insurance Policies with Stochastic Interest Rates

Abstract

We refer to a recent paper by G. Parker (1997) in which the risk of a portfolio of life insurance policies (namely the risk related to the entire contractual life) is studied by separating the demographic component from the financial component. In our paper, after making a brief summary of Parker’s model, we propose two additional contributions: 1. We first give the problem a different formalization, thus allowing a portfolio risk analysis by management periods and a study of the risk due to the interactions among years;

2. We elaborate on a powerful and flexible algorithm for calculating the probability distribution of the sum of random variables that proves useful to solve not only the problems discussed in this paper concerning the risk analysis but also various other problems.

In the paper, we also show, for both contributions, some applications made under the same financial and demographic assumptions taken by Parker; we also compare our results with Parker’s results.     

Macroeconomic Implications of Term Structures of Interest Rates Under Stochastic Differential Utility with Non-Unitary EIS

This paper proposes a continuous-time term-structure model under stochastic differential utility with non-unitary elasticity of intertemporal substitution (EIS, henceforth) in a representative-agent endowment economy with mean-reverting expectations on real output growth and inflation. Using this model, we make clear structural relationships among a term structure of real and nominal interest rates, utility form and underlying economic factors (in particular, inflation expectation). Notably, we show that, if (1) the EIS is less than one, (2) the agent is comparatively more risk-averse relative to time-separable utility, (3) short-term interest rates are pro-cyclical, and (4) the rate of expected inflation is negatively correlated with the rate of real output growth and its expected rate, then a nominal yield curve can have a low instantaneous riskless rate and an upward slope.