Robert A. Jarrow

Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation

This paper presents a unifying theory for valuing contingent claims under a stochastic term structure of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processes for its subsequent movements. A no-arbitrage condition restricts this family of processes, yielding valuation formula for interest rate sensitive contingent claims that do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results. Copyright 1992 by The Econometric Society.

Approximate option valuation for arbitrary stochastic processes

Abstract We show how a given probability distribution can be approximated by an arbitrary distribution in terms of a series expansion involving second and higher moments. This theoretical development is specialized to the problem of option valuation where the underlying security distribution, if not lognormal, can be approximated by a lognormally distributed random variable. The resulting option price is expressed as the sum of a Black-Scholes price plus adjustment terms which depend on the second and higher moments of the underlying security stochastic process. This approach permits the impact on the option price of skewness and kurtosis of the underlying stocks distribution to be evaluated.

Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation

This paper studies the binomial approximation to the continuous trading term structure model of Heath, Jarrow, and Morton (1987). The discrete time approximation makes the original methodology accessible to a wider audience, and provides a computational procedure necessary for calculating the contingent claim values derived in the continuous time paper. This paper also extends and generalizes Ho and Lees (1986) model to include multiple random shocks to the forward rate process and to include an analysis of continuous time limits. The generalization provides insights into the limitations of the existing empirical implementation of Ho and Lees model.

The intersection of market and credit risk

Abstract Economic theory tells us that market and credit risks are intrinsically related to each other and not separable. We describe the two main approaches to pricing credit risky instruments: the structural approach and the reduced form approach. It is argued that the standard approaches to credit risk management – CreditMetrics, CreditRisk+ and KMV – are of limited value when applied to portfolios of interest rate sensitive instruments and in measuring market and credit risk. Empirically returns on high yield bonds have a higher correlation with equity index returns and a lower correlation with Treasury bond index returns than do low yield bonds. Also, macro economic variables appear to influence the aggregate rate of business failures. The CreditMetrics, CreditRisk+ and KMV methodologies cannot reproduce these empirical observations given their constant interest rate assumption. However, we can incorporate these empirical observations into the reduced form of Jarrow and Turnbull (1995b). Drawing the analogy. Risk 5, 63–70 model. Here default probabilities are correlated due to their dependence on common economic factors. Default risk and recovery rate uncertainty may not be the sole determinants of the credit spread. We show how to incorporate a convenience yield as one of the determinants of the credit spread. For credit risk management, the time horizon is typically one year or longer. This has two important implications, since the standard approximations do not apply over a one year horizon. First, we must use pricing models for risk management. Some practitioners have taken a different approach than academics in the pricing of credit risky bonds. In the event of default, a bond holder is legally entitled to accrued interest plus principal. We discuss the implications of this fact for pricing. Second, it is necessary to keep track of two probability measures: the martingale probability for pricing and the natural probability for value-at-risk. We discuss the benefits of keeping track of these two measures.

Market Manipulation, Bubbles, Corners, and Short Squeezes

This paper investigates market manipulation trading strategies by large traders in a securities market. A large trader is defined as any investor whose trades change prices. A market manipulation trading strategy is one that generates positive real wealth with no risk. Market manipulation trading strategies are shown to exist under reasonable hypotheses on the equilibrium price process. Sufficient conditions for their nonexistence are also provided.

Pricing foreign currency options under stochastic interest rates

AbstractIn this paper, we build a general framework to price contingent claims on foreign currencies using the Heath et al. (1987) model of the term structure. Closed form solutions are obtained for European options on currencies and currency futures assuming that the volatility functions determining the term structure are deterministic. As such, this paper provides an example of a bond price process (for both the domestic and foreign economies) consistent with Grabbes (1983) formulation of the same problem.

Forward contracts and futures contracts

Abstract This paper provides a detailed discussion of the similarities and differences between forward contracts and futures contracts. Under frictionless markets and continuous trading, simple arbitrage arguments are invoked to value forward contracts, to relate forward prices and spot prices, and to relate forward prices and futures prices. We also argue that forward prices need not equal futures prices unless default free interest rates are deterministic.

Default Parameter Estimation Using Market Prices

This article presents a new methodology for estimating recovery rates and the (pseudo) default probabilities implicit in both debt and equity prices. In this methodology, recovery rates and default probabilities are correlated and depend on the state of the macroeconomy. This approach makes two contributions: First, the methodology explicitly incorporates equity prices in the estimation procedure. This inclusion allows the separate identification of recovery rates and default probabilities and the use of an expanded and relevant data set. Equity prices may contain a bubble component—which is essential in light of recent experience with Internet stocks. Second, the methodology explicitly incorporates a liquidity premium in the estimation procedure—which is also essential in light of the large observed variability in the yield spread between risky debt and U.S. Treasury securities and the illiquidities present in risky-debt markets. The available models for pricing credit risk can be divided into two types—structural and reduced form. Structural models endogenize the bankruptcy process by explicitly modeling the asset and liability structure of the company. Reduced-form models exogenously specify an arbitrage-free evolution for the spread between default-free and credit-risky bonds. The two approaches seem to have partitioned the market data: Structural models use only equity prices, and reduced-form models use only debt prices. This partitioning is artificial and unnecessary. One particular parameterization of the structural approach—one of several that have been successfully implemented in professional software—uses only equity prices and balance sheet data to estimate the bankruptcy process parameters. The argument is that debt markets are too illiquid and debt prices too noisy to be useful; hence, they should be ignored. Unfortunately, this implementation of the structural approach ignores the possibility of stock price bubbles (e.g., as we have seen recently for Internet stocks) and the misspecification that such bubbles imply. At the same time, the existing literature on implementing reduced-form models concentrates on debt prices and ignores equity prices. Both markets provide relevant information about a companys default risk and parameters, however, and both should be used. The new methodology for implementing reduced-form models presented here includes both debt and equity prices in the estimation procedure. The methodology takes the approach of estimating recovery rates and the (pseudo) default probabilities implicit in debt and equity prices. The method is quite general; it allows default probabilities and recovery rates to be correlated and to be dependent on the state of the macroeconomy. This flexibility generates a reduced-form model that integrates market and credit risk with correlated defaults. My approach makes two contributions. First, the methodology explicitly incorporates equity prices in the reduced-form estimation procedure, which is unlike current models that use debt prices only. For a fractional recovery rate, the use of debt prices alone allows estimation of only the expected loss—that is, the multiplicative product of the recovery rate times the (pseudo) default probabilities. The introduction of equity prices enables one to separately estimate these quantities. Moreover, the procedure I use to include equity in the reduced-form model is one that is commonly used in portfolio theory literature. Simply stated, the equity price is viewed as the present value of future dividends and a resale value. The future resale value is consistent with the existence of equity price bubbles. In light of the recent market experience with Internet stocks, such an inclusion is necessary for accurate estimation of bankruptcy parameters using equity prices. The second contribution of the method is that it explicitly incorporates liquidity risk in the model and the estimation procedure. Debt markets are notoriously illiquid, especially in comparison with equity markets. Thus, a liquidity-risk adjustment is needed to accurately estimate the bankruptcy parameters from credit spreads. Liquidity risk introduces an important and necessary additional randomness in the yield spread between risky-bond prices and the prices of U.S. Treasury securities. In this methodology, liquidity risk is introduced through the notion of a “convenience yield,” a well-studied concept in the commodities pricing literature that is consistent with an arbitrage-free but incomplete debt market. Adding the randomness allows for the decomposition of the credit spread into a liquidity-risk component and a credit-risk component.

The Subprime Credit Crisis of 07

The current U.S. financial crisis began in 1997 when the market lost confidence in the creditworthiness of subprime mortgages and the structured financial products tied to them. In attempting to unravel the origins of the current problems, it has become clear that many factors and many market participants have played a role. In this article, the authors dissect the subprime mortgage crisis and clarify its many facets and offer a number of recommendations for changes to the system that could greatly reduce the probability of similar crises in the future.

Pricing Treasury Inflation Protected Securities and Related Derivatives Using an Hjm Model

AbstractThis paper uses an HJM model to price TIPS and related derivative securities. First, using the market prices of TIPS and ordinary U.S. Treasury securities, both the real and nominal zero-coupon bond price curves are obtained using standard coupon bond price stripping procedures. Next, a three-factor arbitrage-free term structure model is fit to the time-series evolutions of the CPI-U and the real and nominal zero-coupon bond price curves. Then, using these estimated term structure parameters, the validity of the HJM model for pricing TIPS is confirmed via its hedging performance. Lastly the usefulness of the pricing model is illustrated by valuing call options on the inflation index.