The Journal of Finance

The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging

Published: 04/18/2012   |   DOI: 10.1111/j.1540-6261.1997.tb02721.x

EDUARDO S. SCHWARTZ

In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one‐factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.

DISCUSSION

Published: 05/01/1979   |   DOI: 10.1111/j.1540-6261.1979.tb02100.x

EDUARDO S. SCHWARTZ

DISCUSSION

Published: 05/01/1980   |   DOI: 10.1111/j.1540-6261.1980.tb02191.x

Eduardo S. Schwartz

The Pricing of Commodity‐Linked Bonds

Published: 05/01/1982   |   DOI: 10.1111/j.1540-6261.1982.tb03573.x

EDUARDO S. SCHWARTZ

Integration vs. Segmentation in the Canadian Stock Market

Published: 07/01/1986   |   DOI: 10.1111/j.1540-6261.1986.tb04521.x

PHILIPPE JORION, EDUARDO SCHWARTZ

This paper examines the issue of integration versus segmentation of the Canadian equity market relative to a global North American market. We compare the international and domestic versions of the CAPM, and find that integration, or the mean‐variance efficiency of the global market index, is rejected by the data. Segmentation is the preferred model, based on a maximum likelihood procedure correcting for thin trading. We further divide the sample into securities that are interlisted in Canada and the U.S., and those that are not. Integration is rejected for both groups, which indicates that the source of segmentation can be traced to legal barriers based on the nationality of issuing firms.

Stochastic Convenience Yield and the Pricing of Oil Contingent Claims

Published: 07/01/1990   |   DOI: 10.1111/j.1540-6261.1990.tb05114.x

RAJNA GIBSON, EDUARDO S. SCHWARTZ

This paper develops and empirically tests a two‐factor model for pricing financial and real assets contingent on the price of oil. The factors are the spot price of oil and the instantaneous convenience yield. The parameters of the model are estimated using weekly oil futures contract prices from January 1984 to November 1988, and the model's performance is assessed out of sample by valuing futures contracts over the period November 1988 to May 1989. Finally, the model is applied to determine the present values of one barrel of oil deliverable in one to ten years time.

Sovereign Debt: Optimal Contract, Underinvestment, and Forgiveness

Published: 07/01/1992   |   DOI: 10.1111/j.1540-6261.1992.tb04002.x

EDUARDO S. SCHWARTZ, SALVADOR ZURITA

In this paper we develop a time consistent rational expectations model which analyzes the equilibrium loan contract between a borrowing country and a foreign bank. The loan contract specifies both the amount of the loan and the promised interest payments, and rationally reflects the investment decisions of the country and the possibilities of renegotiation and repudiation of the debt. An important feature of the model is that at the initial negotiation of the loan there is uncertainty about whether the country will renegotiate for partial forgiveness in the future, and whether it will eventually repudiate the debt, even having successfully renegotiated. Moreover, the probabilities of renegotiation and repudiation, and the amount of possible forgiveness are endogenously determined. In the model the repudiation decision is directly related to the underinvestment problem; the objective of the renegotiation is precisely to alleviate this problem. The model is used to analyze the effects of four variables on both the optimal contract and the country's welfare: the degree of penalties that a bank can impose on a defaulting country, the uncertainty of production, the productivity of investments and the riskless interest rate. The analysis has policy implications as well as testable predictions.

Rights versus Underwritten Offerings: An Asymmetric Information Approach

Published: 03/01/1986   |   DOI: 10.1111/j.1540-6261.1986.tb04488.x

ROBERT HEINKEL, EDUARDO S. SCHWARTZ

By assuming asymmetric information between investors and firms seeking new equity, we derive a rational expectations, partially revealing information equilibrium in which three forms of equity financing are observed. The highest quality firms employ a standby rights offers, intermediate quality firms signal their true value in the choice of a subscription price in an uninsured rights offer, while low‐quality firms remain indistinguishable to investors by making fully underwritten issues. The model offers justification for many firms using apparently more costly underwritten offers, provides a reason why firms using uninsured rights offers do not set arbitrarily low subscription prices to ensure the success of the issue, and explains the simultaneous existence of the three financing vehicles.

Conditional Predictions of Bond Prices and Returns

Published: 05/01/1980   |   DOI: 10.1111/j.1540-6261.1980.tb02170.x

MICHAEL J. BRENNAN, EDUARDO S. SCHWARTZ

Time‐Dependent Variance and the Pricing of Bond Options

Published: 12/01/1987   |   DOI: 10.1111/j.1540-6261.1987.tb04356.x

STEPHEN M. SCHAEFER, EDUARDO S. SCHWARTZ

In this paper, we develop a model for valuing debt options that takes into account the changing characteristics of the underlying bond by assuming that the standard deviation of return is proportional to the bond's duration. The resulting model uses the bond price as the single state variable and thus preserves much of the simplicity and robustness of the Black‐Scholes approach. The paper provides comparisons between option prices computed using this model and those using the Black‐Scholes and Brennan and Schwartz models.

Regulation and Corporate Investment Policy

Published: 05/01/1982   |   DOI: 10.1111/j.1540-6261.1982.tb03551.x

MICHAEL J. BRENNAN, EDUARDO S. SCHWARTZ

Time‐Invariant Portfolio Insurance Strategies

Published: 06/01/1988   |   DOI: 10.1111/j.1540-6261.1988.tb03939.x

MICHAEL J. BRENNAN, EDUARDO S. SCHWARTZ

This paper characterizes the complete class of time‐invariant portfolio insurance strategies and derives the corresponding value functions that relate the wealth accumulated under the strategy to the value of the underlying insured portfolio. Time‐invariant strategies are shown to correspond to the long‐run policies for a broad class of portfolio insurance payoff functions.

Optimal Financial Policy and Firm Valuation

Published: 07/01/1984   |   DOI: 10.1111/j.1540-6261.1984.tb03647.x

MICHAEL J. BRENNAN, EDUARDO S. SCHWARTZ

Retractable and Extendible Bonds: The Canadian Experience

Published: 03/01/1980   |   DOI: 10.1111/j.1540-6261.1980.tb03469.x

A. L. ANANTHANARAYANAN, EDUARDO S. SCHWARTZ

Prepayment and the Valuation of Mortgage‐Backed Securities

Published: 06/01/1989   |   DOI: 10.1111/j.1540-6261.1989.tb05062.x

EDUARDO S. SCHWARTZ, WALTER N. TOROUS

This paper puts forward a valuation framework for mortgage‐backed securities. Rather than imposing an optimal, value‐minimizing call condition, we assume that at each point in time there exists a probability of prepaying; this conditional probability depends upon the prevailing state of the economy. To implement our valuation procedure, we use maximum‐likelihood techniques to estimate a prepayment function in light of recent aggregate GNMA prepayment experience. By integrating this empirical prepayment function into our valuation framework, we provide a complete model to value mortgage‐backed securities.

A Simple Approach to Valuing Risky Fixed and Floating Rate Debt

Published: 07/01/1995   |   DOI: 10.1111/j.1540-6261.1995.tb04037.x

FRANCIS A. LONGSTAFF, EDUARDO S. SCHWARTZ

We develop a simple approach to valuing risky corporate debt that incorporates both default and interest rate risk. We use this approach to derive simple closed‐form valuation expressions for fixed and floating rate debt. The model provides a number of interesting new insights about pricing and hedging corporate debt securities. For example, we find that the correlation between default risk and the interest rate has a significant effect on the properties of the credit spread. Using Moody's corporate bond yield data, we find that credit spreads are negatively related to interest rates and that durations of risky bonds depend on the correlation with interest rates. This empirical evidence is consistent with the implications of the valuation model.

LYON Taming

Published: 07/01/1986   |   DOI: 10.1111/j.1540-6261.1986.tb04516.x

JOHN J. McCONNELL, EDUARDO S. SCHWARTZ

A Liquid Yield Option Note (LYON) is a zero coupon, convertible, callable, puttable bond. This paper presents a simple contingent claims pricing model for valuing LYONS and uses the model to analyze a specific LYON issue.

Liquidity and the Law of One Price: The Case of the Futures‐Cash Basis

Published: 09/04/2007   |   DOI: 10.1111/j.1540-6261.2007.01273.x

RICHARD ROLL, EDUARDO SCHWARTZ, AVANIDHAR SUBRAHMANYAM

Deviations from no‐arbitrage relations should be related to market liquidity, because liquidity facilitates arbitrage. At the same time, a wide futures‐cash basis may trigger arbitrage trades and, in turn, affect liquidity. We test these ideas by studying the dynamic relation between stock market liquidity and the index futures basis. There is evidence of two‐way Granger causality between the short‐term absolute basis and liquidity, and liquidity Granger‐causes longer‐term absolute bases. Shocks to the absolute basis predict future stock market liquidity. The evidence suggests that liquidity enhances the efficiency of the futures‐cash pricing system.

Interest Rate Volatility and the Term Structure: A Two‐Factor General Equilibrium Model

Published: 09/01/1992   |   DOI: 10.1111/j.1540-6261.1992.tb04657.x

FRANCIS A. LONGSTAFF, EDUARDO S. SCHWARTZ

We develop a two‐factor general equilibrium model of the term structure. The factors are the short‐term interest rate and the volatility of the short‐term interest rate. We derive closed‐form expressions for discount bonds and study the properties of the term structure implied by the model. The dependence of yields on volatility allows the model to capture many observed properties of the term structure. We also derive closed‐form expressions for discount bond options. We use Hansen's generalized method of moments framework to test the cross‐sectional restrictions imposed by the model. The tests support the two‐factor model.

The Relative Valuation of Caps and Swaptions: Theory and Empirical Evidence

Published: 12/17/2002   |   DOI: 10.1111/0022-1082.00399

Francis A. Longstaff, Pedro Santa‐Clara, Eduardo S. Schwartz

Although traded as distinct products, caps and swaptions are linked by no‐arbitrage relations through the correlation structure of interest rates. Using a string market model, we solve for the correlation matrix implied by swaptions and examine the relative valuation of caps and swaptions. We find that swaption prices are generated by four factors and that implied correlations are lower than historical correlations. Long‐dated swaptions appear mispriced and there were major pricing distortions during the 1998 hedge‐fund crisis. Cap prices periodically deviate significantly from the no‐arbitrage values implied by the swaptions market.