Generalised Sharpe Ratios and Asset Pricing in Incomplete Markets *

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract. The paper presents an incomplete market pricing methodology generating asset price bounds conditional on the absence of attractive investment opportunities in equilibrium. The paper extends and generalises the seminal article of Cochrane and Saá-Requejo who pioneered option pricing based on the absence of arbitrage and high Sharpe Ratios. Our contribution is threefold: Permanent repository link:We base the equilibrium restrictions on an arbitrary utility function, obtaining the Cochrane and Saá-Requejo analysis as a special case with truncated quadratic utility. We extend the de…nition of Sharpe Ratio from quadratic utility to the entire family of CRRA utility functions and restate the equilibrium restrictions in terms of Generalised Sharpe Ratios which, unlike the standard Sharpe Ratio, provide a consistent ranking of investment opportunities even when asset returns are highly non-normal. Last but not least, we demonstrate that for Itô processes the Cochrane and Saá-Requejo price bounds are invariant to the choice of the utility function, and that in the limit they tend to a unique price determined by the minimal martingale measure.

The carry trade and fundamentals: Nothing to fear but FEER itself

“…The Sharpe ratio is a common measure of performance which has been found unreliable in some circumstances (see, e.g. Cochrane andSaá-Requejo, 2000, andČerný, 2003). The gain-loss ratio of Bernardo and Ledoit (2000) overcomes Pedroni (1999).…”

Measurement and Pricing of Risk in Insurance Markets

“…In a non-elliptical framework, there might be arbitrage opportunities that are not recognized as good deals [15]. To remedy this drawback, it was proposed to use Generalized Sharpe Ratios based on exponential utility functions [42], while this approach was extended to broader classes of utility functions in [14]. It is furthermore significant that when the valuation bounds implied by the utility function are made sharper, the good-deal price converges to the equilibrium price.…”

Dynamic no-good-deal pricing measures and extension theorems for linear operators on L ∞

“…Instead of selecting a single measure, one can restrict the set of equivalent martingale measures by characterizing those that are in some sense "reasonable". The approach suggested in [12] and [3] (see also [10] and [23]) is to rule out not only arbitrage opportunities, but also deals that are "too good to be true". In the same line, but with a different criterion, [1] and [15] suggest to restrict the set of equivalent martingale measures by choosing those with a density lying within pre-considered lower and upper bounds.…”