Fundamental theorem of asset pricing with acceptable risk in markets with frictions

Abstract

We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given market. Trading is subject to nonproportional transaction costs and portfolio constraints, and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterisation requires an appropriate extension of the classical fundamental theorem of asset pricing where the role of arbitrage opportunities is played by good deals, i.e., costless investment opportunities with acceptable risk–reward tradeoff. In particular, we highlight the importance of scalable good deals, i.e., investment opportunities that are good deals regardless of their volume.

Sensitivity to large losses and ρ-arbitrage for convex risk measures

“…consists of all replicable payoffs that can be purchased at zero cost. Expected Shortfall is sensitive to large losses on S precisely when every nonzero replicable payoff with zero cost has a strictly positive risk, i.e., ES α (X) > 0 for every nonzero X ∈ S. This condition can be interpreted as the absence of a weak form of arbitrage opportunity, which could be termed good deal induced by Expected Shortfall in the language of, e.g., [2,5,17,35]. In a similar spirit, let N = d + 1 and assume the first asset is risk-free with relative return r ∈ (−1, ∞).…”

“…More recently, sensitivity to large expected losses was used in [34] to prove the existence of optimal portfolios in a mean-risk framework extending the classical mean-variance Markowitz's setting. On a side note, it is worth pointing out that the strategy of using recession functionals in our study of sensitivity to large losses for risk/utility functionals that are not positively homogeneous has been recently adopted in a number of papers focusing on different topics, ranging from pricing to portfolio selection and capital allocation, but with the common denominator of dealing with "nonconic" problems; see [2,5,34,43,44,45,51]. The remainder of this paper is organized as follows.…”