A NOTE ON UTILITY INDIFFERENCE PRICING

Utility-based valuation methods are enjoying growing popularity among researchers as a means to overcome the challenges in contingent claim pricing posed by the many sources of market incompleteness. However, we show that under the most common utility functions (including CARA and CRRA), any realistic and actually practicable hedging strategy involving a possible short position has infinitely negative utility. We then demonstrate for utility indifference prices (and also for the related so-called utility-based (marginal) prices) how this problem leads to a severe divergence between results obtained under the assumption of continuous trading and realistic results. The combination of continuous trading and common utility functions is thus not justified in these methods, raising the question of whether and how results obtained under such assumptions could be put to real-world use.

CONFRONTING MODEL MISSPECIFICATION IN FINANCE: TRACTABLE COLLECTIONS OF SCENARIO PROBABILITY MEASURES FOR ROBUST FINANCIAL OPTIMIZATION PROBLEMS

Despite the widespread realization that financial models for contingent claim pricing, asset allocation and risk management depend critically on their underlying assumptions, the vast majority of financial models are based on single probability measures. In such models, asset prices are assumed to be random, but asset price probabilities are assumed to be known with certainty, an obviously false assumption. We explore practical methods to specify collections of probability measures for an assortment of important financial problems; we provide practical methods to solve the robust financial optimization problems that arise and, in the process, discover "dangerous" measures.

LOCAL SCALE INVARIANCE AND CONTINGENT CLAIM PRICING

Prices of tradables can only be expressed relative to one another at any instant of time. This fundamental fact should therefore also hold for contingent claims, i.e. tradable instruments, whose prices depend on the prices of other tradables. We show that this property induces a local scale invariance in the problem of pricing contingent claims. Due to this symmetry we do not require any martingale techniques to arrive at the price of a claim. If the tradables are driven by Brownian motion, we find, in a natural way, that this price satisfies a PDE. Both possess a manifest gauge invariance. A unique solution can only be given when we impose restrictions on the drifts and volatilities of the tradables, i.e. the underlying market structure. We give some examples of the application of this PDE to the pricing of claims. In the Black–Scholes world we show the equivalence of our formulation with the standard approach. It is stressed that the formulation in terms of tradables leads to a significant conceptual simplification of the pricing-problem.

LOCAL SCALE INVARIANCE AND CONTINGENT CLAIM PRICING II: PATH-DEPENDENT CONTINGENT CLAIMS

This article is the second one in a series on the use of local scale invariance in finance. In the first [6], we introduced a new formalism for the pricing of derivative securities, which focuses on tradable objects only, and which completely avoids the use of martingale techniques. In this article we show the use of the formalism in the context of path-dependent options. We derive compact and intuitive formulae for the prices of a whole range of well-known options such as arithmetic and geometric average options, barriers, rebates and lookback options. Some of these have not appeared in the literature before. For example, we find rather elegant formulae for double barrier options with exponentially moving barriers, continuous dividends and all possible configurations of the barriers. The strength of the formalism reveals itself in the ease with which these prices can be derived. This allowed us to pinpoint some mistakes regarding geometric mean options, which frequently appear in the literature. Furthermore, symmetries such as put-call transformations appear in a natural way within the framework.