THE SURVIVAL ZONE FOR A BOND WITH BOTH CALL AND PUT OPTIONS EMBEDDED

We present a numerical method to price bonds that have multiple embedded options with an emphasis on the case with both long call and short put options. The valuation framework is a one-factor model for the term structure of interest rates, where the instantaneous interest rate is allowed to follow a fairly general stochastic process. The equilibrium interest rates that define the free boundaries for the embedded call and put options are given. We demonstrate the survival zone within which a bond with both long call and short put options remains afloat. We show that even moderate levels of transaction costs can have a significant effect on exercise of options.     

Catastrophe Risk Bonds

Abstract

This article examines the pricing of catastrophe risk bonds. Catastrophe risk cannot be hedged by traditional securities. Therefore, the pricing of catastrophe risk bonds requires an incomplete markets setting, and this creates special difficulties in the pricing methodology. The authors briefly discuss the theory of equilibrium pricing and its relationship to the standard arbitrage-free valuation framework. Equilibrium pricing theory is used to develop a pricing method based on a model of the term structure of interest rates and a probability structure for the catastrophe risk. This pricing methodology can be used to assess the default spread on catastrophe risk bonds relative to traditional defaultable securities.     

On the term structure of interest rates

The paper presents a valuation formula for default free bonds for a certain class of tastes when the instantaneously riskfree rate of interest follows a geometric Wiener process. Properties of the resulting term structure of interest rates are studied, and an application of the analysis to the pricing of Treasury Bills is proposed.     

An Asymptotic Expansion Approach to Pricing Financial Contingent Claims

We propose a new methodology for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Itô processes. Our method can be applicable to a wide range of valuation problems including contingent claims associated with stocks, foreign exchange rates, the term structure of interest rates, and even their combinations. We illustrate our method by discussing the Black-Scholes economy when the underlying asset prices follow the continuous diffusion processes, which are not necessarily time-homogeneous. The standard Black-Scholes model on stocks and the Cox-Ingersoll-Ross model on the spot interest rate are simple examples. Then we shall give a series of examples on the valuation formulae including plain vanilla options, average options, and other contingent claims. We shall also give some numerical evidence of the accuracy of the approximations we have obtained for practical purposes. Our approach can be rigorously justified by an infinite dimensional mathematics, the Malliavin-Watanabe-Yoshida theory recently developed in stochastic analysis.     

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.     

Corporate Debt Value,Bond Covenants,and Optimal Capital Structure

This article examines corporate debt values and capital structure in a unified analytical framework. It derives closed-form results for the value of long-term risky debt and yield spreads, and for optimal capital structure, when firm asset value follows a diffusion process with constant volatility. Debt values and optimal leverage are explicitly linked to firm risk, taxes, bankruptcy costs, risk-free interest rates, payout rates, and bond covenants. The results elucidate the different behavior of junk bonds versus investment-grade bonds, and aspects of asset substitution, debt repurchase, and debt renegotiation.     

Arbitrage bounds in markets with noisy prices and the puzzle of negative option prices implicit in bonds

The term structure of interest rates has occupied economists for many years. This is testimony to the importance of the term structure (or, equivalently, valuation operator) and to the difficulty of obtaining reliable estimates of it. Until recently, it was not possible to reconcile the theoretical existence of a multiplicity of valuation operators in an incomplete bond market and the empirical nonexistence of even a single one. MacKay and Prisman [Estimating valuation operators in incomplete market with noises: Can noise complete the market. Working paper, 1996] tackle this problem. In this paper, an amendment to and a generalization of their methodology is presented. The amendment preserves the linearity of valuation operators and allows use of efficient linear programming techniques. Further, it facilitates an admissible consideration of the puzzle of negative option prices embedded in bonds. The empirical results presented in this paper were obtained using data on extendable Government of Canada bonds. The results indicate that although the gap between the lowest and the highest prices assigned to a cash flow by different valuation operators is significant, it is not sufficient to resolve the above-mentioned puzzle.     

Actuarial pricing of deposit insurance

Using a pricing formula for options on coupon bonds (Jamshidian [1989], El Karoui and Rochet [1990]) we are able to compute the actuarial pricing of deposit insurance for a commercial bank. Our formula takes into account the maturity structure of the bank's balance sheet, as well as market parameters such as the term structure of interest rates and the volatilities of zero coupon bonds. The relation with asset liability management methods is explored.     

The Effect of Interest Rates on the Value of Corporate Assets and the Risk Premia of Corporate Debt

A growing number of papers have applied option pricing techniques to the valuation of risky debt. This paper deals directly with how a firm's relationship to interest rates affects its debt. A sequential binomial model is used to price the zero-coupon bonds of a firm whose value is related to interest rate changes.The results show that the strength of the relationship between firm value and interest rates (interest-rate risk) can have a significant impact on the value of a firm's debt. The model produces its most powerful results when the volatility of firm value is high and the term structure has a steep (negative or positive) slope; there is no impact when the term structure is flat. Our results indicate that empirical studies of yield spreads may have severe shortcomings if the relationship of firm value to interest rate changes is ignored.     

A simple robust model for Cat bond valuation

This note provides a simple closed form solution for valuing Cat bonds. The formula is consistent with any arbitrage-free model for the evolution of the Libor term structure of interest rates. The crucial inputs to the valuation formula are the likelihood of the catastrophe event, per unit time, and the percentage loss rate realized if an event occurs. The pricing methodology is based on the reduced form models used to price credit derivatives.     

When can you immunize a bond portfolio?

This paper presents a condition equivalent to the existence of a Riskless Shadow Asset that guarantees a minimum return when the asset prices are convex functions of interest rates or other state variables. We apply this lemma to immunize default-free and option-free coupon bonds and reach three main conclusions. First, we give a solution to an old puzzle: why do simple duration matching portfolios work well in empirical studies of immunization even though they are derived in a model inconsistent with equilibrium and shifts on the term structure of interest rates are not parallel, as assumed? Second, we establish a clear distinction between the concepts of immunized and maxmin portfolios. Third, we develop a framework that includes the main results of this literature as special cases. Next, we present a new strategy of immunization that consists in matching duration and minimizing a new linear dispersion measure of immunization risk.     

The Value of Deposit Insurance in the Presence of Interest Rate and Credit Risk

In this paper we employ the theory of the term structure of interest rates and the pricing of interest contingent contracts to determine the fair value of insurance for depository institutions. The balance sheet of a bank is taken to consist of long and short positions in various fixed income securities. Deposit insurance for the bank is a put option on the value of the assets. The value of deposits, assets, the implied exercise price of the put and the value of the put are all determined simultaneously as part of the same valuation solution. The approach is developed initially for a single‐state term structure. It is extended to incorporate credit risk on bank assets.
The most important policy implication is that for a bank whose assets are longer term than its liabilities and whose borrowers are not excessively leveraged the properly calculated, risk‐adjusted deposit insurance premia are increasing functions of the level of interest rates. Sensitivity analyses also treat such factors as the bank's deposit to asset ratio, duration gap, interest volatility, the volatility of assets backing the bank loans, and the bank's borrowers' debt to equity ratio.     

Pricing and Hedging of Contingent Claims in Term Structure Models with Exogenous Issuing of New Bonds

If calibrated to an observed term structure of interest rates that only covers a finite range of times-to-maturity an HJM-model of the term structure of interest rates will eventually die out in finite time as bonds reach maturity. This poses problems for the pricing and hedging of certain contingent claims. Therefore, we extend the HJM-model in such a way that it lives on an arbitrary time horizon and possesses term structures that cover a constant finite interval of times-to-maturity. We consider the pricing and hedging of contingent claims in this framework.     

Decreasing Yield Curves in a Model with an Unknown Constant Growth Rate

The effect of incomplete information on the term structure ofinterest rates is examined in the framework of a pure exchangeeconomy under uncertainty where aggregate output grows at aconstant rate. If the growth rate is known, the term structure isflat. In contrast, the term structure is a decreasing curve whenagents do not know the growth rate. Long term yields are less thanthe short rate and the yield of long term bonds is determined bythe worst possible realizations of future short rates.     

Pricing pension buy-outs under stochastic interest and mortality rates

Pension buy-out is a special financial asset issued to offload the pension liabilities holistically in exchange for an upfront premium. In this paper, we concentrate on the pricing of pension buy-outs under dependence between interest and mortality rates risks with an explicit correlation structure in a continuous time framework. Change of measure technique is invoked to simplify the valuation. We also present how to obtain the buy-out price for a hypothetical benefit pension scheme using stochastic models to govern the dynamics of interest and mortality rates. Besides employing a non-mean reverting specification of the Ornstein–Uhlenbeck process and a continuous version of Lee–Carter setting for modeling mortality rates, we prefer Vasicek and Cox–Ingersoll–Ross models for short rates. We provide numerical results under various scenarios along with the confidence intervals using Monte Carlo simulations.     

The high-frequency impact of news on long-term yields and forward rates: Is it real?

Macroeconomic news announcements move yields and forward rates on nominal and index-linked bonds and inflation compensation. This paper estimates the reactions using high-frequency data on nominal and index-linked bond yields, allowing the effects of news announcements on real rates and inflation compensation to be parsed far more precisely than is possible using daily data. Long-term nominal yields and forward rates are very sensitive to macroeconomic news announcements. Inflation compensation is sensitive to announcements about price indices and monetary policy. However, for news announcements about real economic activity, such as nonfarm payrolls, the vast majority of the sensitivity is concentrated in real rates. Accordingly, most of the sizeable impact of news about real economic activity on the nominal term structure of interest rates represents changes in expected future real short-term interest rates and/or real risk premia rather than changes in expected future inflation and/or inflation risk premia. Such sensitivity of real rates to macroeconomics news is hard to rationalize within the framework of existing macroeconomic models.     

A Re-examination of Traditional Hypotheses about the Term Structure of Interest Rates

The term structure of interest rates is an important subject to economists, and has a long history of traditions. This paper re-examines many of these traditional hypotheses while employing recent advances in the theory of valuation and contingent claims. We show how the Expectations Hypothesis and the Preferred Habitat Theory must be reformulated if they are to obtain in a continuous-time, rational-expectations equilibrium. We also modify the linear adaptive interest rate forecasting models, which are common to the macroeconomic literature, so that they will be consistent in the same framework.     

The Variation of Economic Risk Premiums in Real Estate Returns

We examine the predictable components of returns on stocks, bonds, and real estate investment trusts (REITs). We employ a multiple-beta asset pricing model and find that there are varying degrees of predictability among stocks, bonds, and REITs. Furthermore, we find that most of the predictability of returns is associated with the economic variables employed in the asset pricing model. The stock market risk premium is highly important in capturing the predictable variation in stock portfolios, and the bond market risk premiums (term and risk structure of interest rates) are important in capturing the predictable variation in bond portfolios. For REITs, however, both the stock and bond market risk premiums capture the predictable variation in returns. REITs have comparable return predictability to stock portfolios. We conclude that there is an important economic risk premium for REITs that are not captured by traditional multiple-beta asset pricing models.     

人口学特征与利率期限结构:老年社会平缓的收益率曲线

本文将刻画人口结构的生命周期模型引入消费-资本资产定价模型,考察人口结构对利率期限结构的影响。模型表明,人口结构及其家庭生命周期特征不仅决定利率水平,而且将通过人对债券期限的不同偏好,影响利率期限结构。少年人口占比对利率期限结构的影响为正,中年和老年人口占比的影响为负。相比少年人口,中老年人口更偏好长期债券,使长期收益率下降,期限结构的斜率更为平缓。基于全球数据的经验研究验证了这一结论。少年人口占比增加期限利差,中老年人口占比则起反向作用。因此,在年长的经济体中,期限利差更小,呈现更平缓的收益率曲线特征。在更换人口结构变量、期限利差变量、估计方法、赋权样本和处理遗漏变量后,结果表现稳健。本文从人口学视角拓宽了利率期限结构的决定因素,揭示了老年经济体可能面临一个平缓的收益率曲线,而这说明老龄化还可能通过抑制短期投机和促进长期投资来提高长期经济发展质量。     

Bond yields compared to amortizing yields

A large amount of theoretical research has been conducted concerning the impact of debt instrument features such as callability, tax treatment, and default risk upon valuation and yield. Almost all this research has been performed on bond-like instruments; this is unfortunate since a large proportion of debt instruments is amortizing. This research develops a model for the spread between bonds and amortizing instruments with otherwise identical features. The shape of the term structure is crucial. Furthermore, the behavior of the spread is systematically related to the level of interest rates, tax rates, degree of default risk, and maturity where the maturity relation changes sign at intermediate maturities.