DEFAULT RISK AND DIVERSIFICATION: THEORY AND EMPIRICAL IMPLICATIONS

Recent advances in the theory of credit risk allow the use of standard term structure machinery for default risk modeling and estimation. The empirical literature in this area often interprets the drift adjustments of the default intensity's diffusion state variables as the only default risk premium. We show that this interpretation implies a restriction on the form of possible default risk premia, which can be justified through exact and approximate notions of "diversifiable default risk." The equivalence between the empirical and martingale default intensities that follows from diversifiable default risk greatly facilitates the pricing and management of credit risk. We emphasize that this is not an equivalence in distribution, and illustrate its importance using credit spread dynamics estimated in Duffee (1999) . We also argue that the assumption of diversifiability is implicitly used in certain existing models of mortgage-backed securities.     

A Reduced‐Form Model for Warrant Valuation

This paper studies warrant valuation using a reduced‐form model. Analogous to the credit risk literature, structural models require complete information about the asset value process and the firm’s liabilities. In contrast, reduced‐form models require only information about the firm’s stock price process. We introduce a reduced‐form model where the warrant holder is a price taker, and we relate our model to structural models appearing in the literature.     

The stop-loss start-gain paradox and option valuation: a new decomposition into intrinsic and time value

The downside risk in a leveraged stock position can be eliminatedby using stop-loss orders. The upside potential of such a positioncan be captured using contingent buy orders. The terminal payoffto this stop-loss start-gain strategy is identical to that ofa call option, but the strategy costs less initially. This articleresolves this paradox by showing that the strategy is not self-financingfor continuous stock-price processes of unbounded variation.The resolution of the paradox leads to a new decomposition ofan option's price into its intrinsic and time value. When thestock price follows geometric Brownian motion, this decompositionis proven to be mathematically equivalent to the Black-Scholes(1973) formula.     

Counterparty Risk and the Pricing of Defaultable Securities

Motivated by recent financial crises in East Asia and the United States where the downfall of a small number of firms had an economy-wide impact, this paper generalizes existing reduced-form models to include default intensities dependent on the default of a counterparty. In this model, firms have correlated defaults due not only to an exposure to common risk factors, but also to firm-specific risks that are termed "counterparty risks." Numerical examples illustrate the effect of counterparty risk on the pricing of defaultable bonds and credit derivatives such as default swaps.     

Liquidity risk and arbitrage pricing theory

Classical theories of financial markets assume an infinitely liquid market and that all traders act as price takers. This theory is a good approximation for highly liquid stocks, although even there it does not apply well for large traders or for modelling transaction costs. We extend the classical approach by formulating a new model that takes into account illiquidities. Our approach hypothesizes a stochastic supply curve for a securitys price as a function of trade size. This leads to a new definition of a self-financing trading strategy, additional restrictions on hedging strategies, and some interesting mathematical issues.Received: 1 November 2003, Mathematics Subject Classification: 60G44, 60H05, 90A09JEL Classification: G11, G12, G13Umut Çetin: This work was performed while Dr. Çetin was at the Center for Applied Mathematics, Cornell UniversityPhilip Protter: Supported in part by NSF grant DMS-0202958 and NSA grant MDA-904-03-1-0092 The authors wish to thank M. Warachka and Kiseop Lee for helpful comments, as well as the anonymous referee and Associate Editor for numerous helpful suggestions, which have made this a much improved paper.     

Tax liens: a novel application of asset pricing theory

Although relatively obscure, the market for distressed real estate tax liens exists in over 30 U.S. states, with a market size estimated to be around 20 billion dollars. While this niche asset class is relatively unknown to academics, internet advertising hypes tax liens to the populace as providing extraordinary returns. Not yet scientifically studied, this market provides a fertile and untouched arena for the application of asset pricing theory. Using insights from several areas of asset pricing, we formulate and test a pricing model for tax liens. The empirical evidence supports the pricing model, the (increasing) competitiveness of the tax lien market, and an unfair tax auction bidding mechanism for property owners that may provide extraordinary returns to investors, lending some credibility to the industry claims. We suggest avenues for extensions and further research.     

Information reduction via level crossings in a credit risk model

This paper provides an alternative credit risk model based on information reduction where the market only observes the firm’s asset value when it crosses certain levels, interpreted as changes significant enough for the firm’s management to make a public announcement. For a class of diffusion processes we are able to provide explicit expressions for the firm’s default intensity process and its zero-coupon bond prices.      

The valuation of a firm’s investment opportunities: a reduced form credit risk perspective

This paper develops a valuation model for a firm’s investment opportunities. Given standard market imperfections, we show that maximizing the firm’s equity value is consistent with the need to include a capital charge for an investment specific to a firm’s capital structure and in excess of the investment’s market determined risk. A reduced form credit risk perspective is taken to enable a continuous time implementation. This continuous time implementation is illustrated within the paper.      

A Critique of Revised Basel II

This paper critiques the revised Basel II capital requirements for banks. To provide a framework for analysis, the XYZ theory of regulatory capital is formulated. Independent of the XYZ theory, we argue that the revised Basel II capital rule for credit risk is not a good approximation to the ideal rule. Based on this, and using the XYZ theory, we argue that: (1) the revised Basel II rules should not replace the existing approaches for determining minimal capital standards, but should be used in conjunction with them, and (2) that calibrating the capital rules to maintain aggregate market capital is a prudent procedure.     

ESTIMATING THE VALUE OF DELIVERY OPTIONS IN FUTURES CONTRACTS

We analyze the effect various delivery options embedded in commodity futures contracts have on the futures price. The two embedded options considered are the timing and location options. We show that early delivery is always optimal when only a timing option is present, but not so when joint options are present. The estimates of the combined options are much smaller than the comparable estimates for the timing option alone. The average value of the joint option is about 5% of the average basis on the first day of the maturity month. This suggests that joint options can increase deliverable supplies while potentially having only a small effect on basis behavior.