Joel M. Vanden

Exact Superreplication Strategies for a Class of Derivative Assets

A superreplicating hedging strategy is commonly used when delta hedging is infeasible or is too expensive. This article provides an exact analytical solution to the superreplication problem for a class of derivative asset payoffs. The class contains common payoffs that are neither uniformly convex nor concave. A digital option, a bull spread, a bear spread, and some portfolios of bull spreads or bear spreads, are all included as special cases. The problem is approached by first solving for the transition density of a process that has a two‐valued volatility. Using this process to model the underlying asset and identifying the two volatility values as σmin and σmax, the value function for any derivative asset in the class is shown to solve the Black–Scholes–Barenblatt equation. The subreplication problem and several related extensions, such as option pricing with transaction costs, calculating superreplicating bounds, and superreplication with multiple risky assets, are also addressed.

PORTFOLIO INSURANCE AND VOLATILITY REGIME SWITCHING

A new equilibrium model of portfolio insurance is presented in order to study the volatility effects of dynamic insurance strategies. While prior research has focused on the relationship between portfolio insurance and the overall level of market volatility, this article shows that the salient feature of portfolio insurance is volatility regime switching. Regime switching is shown to be a necessary condition for portfolio insurance, which provides a new explanation for the pervasive volatility clustering effect that is found in most equity markets. The equilibrium involves a free boundary and the local time of the equilibrium price process at the free boundary plays an important role in solving the model.

Asset Substitution and Structured Financing

This article shows how structured financing can be used to solve the asset substitution problem in a dynamic setting. Structuring induces the firm’s owner to optimally choose the first best operating strategy even though the owner’s value function might be locally convex (concave), which would ordinarily lead to overinvestment (underinvestment) in risky projects. This result is demonstrated in two different continuous time settings—one that is based on the risk-shifting framework of Leland (1998) and one that generalizes the scaled return model of Green (1984). It is shown that the contractual nature of the structuring is a key determinant of the issuing firm’s dynamic asset volatility. Furthermore, unlike nonstructured financing, the default (conversion) probability of a structured debt security may be increasing (decreasing) in the firm’s total assets. Structured securities are therefore hedge assets, which potentially explains the popularity of structured securities among investors and third-party issuers.

GENERAL PROPERTIES OF ISOELASTIC UTILITY ECONOMIES

This paper studies the class of single‐good Arrow–Debreu economies in which all agents have isoelastic utility functions and homogeneous beliefs, but have possibly different cautiousness parameters and endowments. For each economy in this class, the equilibrium stochastic discount factor is an exponential function of the inverse mapping of a completely monotone function, evaluated at the aggregate consumption. This fact allows for general properties of the class to be studied analytically in terms of known properties of completely monotone functions. For example, conditions are presented under which the agents’ cautiousness parameters and a distribution of initial wealth can be recovered from an equilibrium stochastic discount factor, even if nothing is known about the agents’ endowments. This is a multiagent inverse problem since information about economic primitives is extracted from equilibrium prices. Several example economies are used to illustrate the results.