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.     

Interest Rate Futures Options and Interest Rate Options

This paper derives pricing models of interest rate options and interest rate futures options. The models utilize the arbitrage-free interest rate movements model of Ho and Lee. In their model, they take the initial term structure as given, and for the subsequent periods, they only require that the bond prices move relative to each other in an arbitrage-free manner. Viewing the interest rate options as contingent claims to the underlying bonds, we derive the closed-form solutions to the options. Since these models are sufficiently simple, they can be used to investigate empirically the pricing of bond options. We also empirically examine the pricing of Eurodollar futures options. The results show that the model has significant explanatory power and, on average, has smaller estimation errors than Black's model. The results suggest that the model can be used to price options relative to each other, even though they may have different expiration dates and strike prices.     

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.     

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 .     

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.     

黄金——全球高负债、低利率背景下的避险选择

虽然黄金在布雷顿森林体系崩溃之后远离了国际货币体系的中心,但是在世界范围内黄金仍被公认为是一种地位特殊的资产。目前黄金仍然是国际上可以接受的继美元、欧元、英镑、日元之后的第五大硬通货。用黄金进行清偿结算仍然是公认的唯一可以代替货币进行结算的方式。商品期货交易所甚至允许交易商将黄金作为所有商品交易的保证金抵押物。     

Interest Rates and Credit Risk

This paper explores the effects of shifts in interest rates on corporate leverage and default in the context of a dynamic model in which the link between leverage and default risk comes from the lower incentives of overindebted entrepreneurs to guarantee firm survival. The need to finance new investment pushes firms' leverage ratio above some state‐contingent target toward which firms gradually adjust through earnings retention. The response to interest rate rises and cuts is both asymmetric and heterogeneously distributed across firms. Our results help rationalize some of the evidence regarding the risk‐taking channel of monetary policy.     

The Behavior of Interest Rates

The evidence in Fama and Bliss (1987) that forward interestrates forecast future spot interest rates for horizons beyonda year repeats in the out-of-sample 1986–2004 period.But the inference that this forecast power is due to mean reversionof the spot rate toward a constant expected value no longerseems valid. Instead, the predictability of the spot rate capturedby forward rates seems to be due to mean reversion toward atime-varying expected value that is subject to a sequence ofapparently permanent shocks that are on balance positive tomid-1981 and on balance negative thereafter.     

Modeling International Long-Term Interest Rates

This study investigates the relationship among interest rates on the long-term government bonds of five industrialized countries. Both standard and new unit root tests are applied, all of which confirm the presence of exactly one unit root. New cointegration tests are also applied to these data. In contrast to previous research on short-term bonds, stock prices, and exchange rates, these results find little evidence of cointegration among the five long-term interest rate series. Thus, when modeling or forecasting these central government long-term bond yields, one may assume separate sets of fundamentals and difference the data to achieve stationarity. An error correction model may not be appropriate.     

利率市场化背景下的利率衍生产品发展策略

利率市场化是相对于利率管制而言,是指政府部分或完全放弃对利率的管制,使利率由市场的资金供求关系决定,按价值规律自发调节,包括利率决定、利率传导、利率结构和利率管理的市场化,核心内容是利率形成机制的市场化。     

The Valuation of American Options with Stochastic Interest Rates: A Generalization of the Geske—Johnson Technique

The Geske–Johnson approach provides an efficient and intuitively appealing technique for the valuation and hedging of American-style contingent claims. Here, we generalize their approach to a stochastic interest rate economy. The method is implemented using options exercisable on one of a finite number of dates. We illustrate how the value of an American-style option increases with interest rate volatility. The magnitude of this effect depends on the extent to which the option is in the money, the volatilities of the underlying asset and the interest rates, as well as the correlation between them.     

The Role of Interest Rates in Influencing Long-Run Homeownership Rates

As a stated policy objective, the U.S. Department of Housing and Urban Development (HUD) seeks to boost the national homeownership rate to 70 percent by 2006. To accomplish this goal, they estimate that 3.8 million additional families be added to the ranks of U.S. homeowners. Furthermore, HUD estimates that the homeownership gap between minority and nonminority families must be reduced by a full 15 percent. Many policy instruments—both targeted and otherwise—have been suggested to increase homeownership. These range from low downpayment loans, greater access to credit in underserved areas, and interest rates subsidies. However, little is know about the efficacy of these measures to raise long-term homeownership rates. In this analysis, we focus on the role of interest rates on homeownership rates and the housing stock. In particular, we provide a critical review of the literature on the relationship between housing and interest rates in contrast to other determinants of homeownership and changes in housing supply. We then present our own estimates of the influence of interest rates on homeownership and housing starts. We find that interest rates play little direct role in changing homeownership rates. While changes in interest rates may affect the timing of changes in tenure status from renter to owner, the long-run ownership rate appears independent of interest rates. We find housing starts are, however, sensitive to changes in the interest rate. This implies that housing supply, or at least the timing of changes in housing supply, is sensitive to interest rates. It is though this mechanism that the stock of owner-occupied housing expands, though household formation and immigration may leave the ownership rate unchanged. We conclude by discussing whether other instruments, such as low down payment loans and improved technology for assessment of credit risk, may potentially be better suited to increasing long-term homeownership rates.     

The Cyclical Behavior of Interest Rates

This article investigates the behavior of the term structure of interest rates over the business cycle. In contrast to prior studies that measure the business cycle by the simple growth in aggregate economic activity, we consider the deviation of aggregate economic activity from its potentially stochastic trend. We show that incorporating both an independent trend and cyclical component in consumption improves the efficiency in estimating consumption-based asset pricing models. We also find that the term spread is more informative about future changes in stochastically detrended real gross domestic product (GDP) than future growth rates in real GDP.     

Interest Rates in Trade Credit Markets

All things equal, interest rates should increase with the borrower's risk. And yet, Klapper, Laeven, and Rajan (2012) cannot find such a positive relation in a broad sample of trade credit contracts. We shed some light on this puzzle by arguing that competition between informed and uninformed suppliers weakens the link between the trade credit cost and the borrower's creditworthiness. Our model implies that trade credit rates are more likely to increase with the borrower's risk if suppliers are less profitable, have high cost of funds, or sell inputs to firms plagued by moral hazard and financial distress.     

Nominal Interest Rates and the News

This paper provides new estimates of the impact of monetary policy actions and macroeconomic news on the term structure of nominal interest rates. The key novelty is to parsimoniously capture the impact of news on all interest rates using a simple no‐arbitrage model. The different types of news are analyzed in a common framework by recognizing their heterogeneity, which allows for a systematic comparison of their effects. This approach leads to novel empirical findings. First, monetary policy causes a substantial amount of volatility in both short‐term and long‐term interest rates. Second, macroeconomic data surprises have small and mostly insignificant effects on the long end of the term structure. Third, the term‐structure response to macroeconomic news is consistent with considerable interest‐rate smoothing by the Federal Reserve. Fourth, monetary policy surprises are multidimensional while macroeconomic surprises are one‐dimensional.     

Accounting-Based Valuation with Changing Interest Rates

This paper generalizes Ohlsons [Contemporary Accounting Research Vol. 11 No. 2. 661–687 (1995)] equity valuation framework to allow for stochastic interest rates. Much of this analysis initially deals with the specialized setting in which earnings suffice for cum-dividend value. In such a case, the beginning-of-period (lagged) rate determines the capitalization factor, not the current rate. The underlying earnings dynamic modifies the traditional random walk model via an additional term, namely current earnings multiplied by the percentage change in interest rates. The general model retains these basic aspects of the earnings-sufficiency setting. Empirical implications bear on the returns-to-earnings regression: The earnings-response coefficient decreases as the beginning-of-period rate increases.JEL Classification: M41, G12     

Interest Tax Shields: A Barrier Options Approach

There is a link between barrier options and tax shields of interest expense. We combine this link with a traditional valuation approach, to present practical valuation formulas for interest tax shields in three debt scenarios with risk of default: (1) constant debt, (2) delayed debt, and (3) debt refinancing. In all cases, default and refinancing are contingent on the random evolution of the income of the firm. For each scenario, we work out sensitivity analysis of the value of tax shields with respect to income, growth, systematic and business risk, risk-free interest rate, interest coverage ratio covenant, and the firm??s refinancing strategy.     

Inflation,Taxation, and Interest Rates

This paper demonstrates that the response of nominal interest rates to changes in inflationary expectations should lie between that predicted by the “Fisher” and “Darby” effects. The exact nature of the response will depend on the relative size of the income and capital gains tax rates, and the relative size of the derivatives of investment and savings to their respective after-tax real rates. The other major conclusion of this paper is that capital gains taxation offsets the negative effect on investment produced by treating depreciation on a historic rather than a replacement cost basis.