Additive portfolio improvement and utility-efficient payoffs
This paper studies the notion of \textit{amelioration} (or additive improvement procedure) of payoffs. More specifically, an amelioration is a function \(A:L^1(\Omega,\mathcal{F},\mathbb{P})\rightarrow L^1(\Omega,\mathcal{F},\mathbb{P})\) satisfying: {\parindent=0.7cm\begin{itemize}\item[(1)] \(A(X+Y)=A(X)+A(Y)\); \item[(2)] \(X\geq c \Longrightarrow A(X)\geq c\), for all constants \(c\); \item[(3)] \(A(A(X))=A(X)\); \item[(4)] \(\mathbb{E}[A(X)^-]\leq\mathbb{E}[X^-]\). \end{itemize}} The additivity requirement has the important implication that \(A(-X)=-A(X)\), thus implying that both the seller and the buyer of \(X\) would agree on replacing \(X\) with its amelioration \(A(X)\). The authors prove that a function \(A\) is an amelioration if and only if it is a conditional expectation operator. Moreover, it is shown that ameliorations are the only additive improvement procedures that improve payoffs for every expected utility maximizer. Specific ameliorations can be chosen to achieve consistency with the pricing measure and with utility maximization. In general, an amelioration does not preserve the distribution of a payoff, but is consistent with risk-averse robust Savage preferences. Finally, the authors prove that an ameliorated payoff cannot represent a statistical arbitrage opportunity.
scientific article; zbMATH DE number 6692622
scientific article