Asymptotic Loss-Gain Ratio Analysis

• Asymptotic loss-gain ratio is a metric defined as the ratio of expected positive outcomes to expected negative outcomes in limit settings, showcasing properties like monotonicity, law invariance, and scale invariance.
• The concept incorporates dual representations linked to martingale measures and optimization frameworks, emphasizing its role in evaluating risk-adjusted performance under market constraints.
• While scale invariance offers mathematical elegance, it also leads to challenges such as non-attainment of optimal values and sensitivity to extreme positions in continuous-time or infinite-dimensional markets.

The asymptotic loss-gain ratio is a concept that arises in the analysis of performance measures, dynamical stability, and optimization in stochastic, financial, or control-theoretic systems, especially when the system dimension grows large or the time horizon becomes infinite. It reflects the behavior of measures, indices, or quantities used to balance "positive" (gain) and "negative" (loss) outcomes as one pushes system parameters or modeling assumptions to their limit. The mathematical and conceptual structure of such ratios is deeply influenced by properties like scale invariance, duality, and the topology of the underlying optimization space.

1. Definition and Normative Properties

The classical gain-loss ratio for a random variable $X$ (interpreted as a payoff or portfolio value) is given by

$$\alpha(X) = \frac{\mathbb{E}[X^+]}{\mathbb{E}[X^-]},$$

for any integrable $X^+$ or $X^-$, where $X^+ = \max\{X, 0\}$ and $X^- = \max\{-X, 0\}$.

Key normative properties:

• Monotonicity: If $X$ increases, then $\alpha(X)$ does not decrease.
• Law invariance: If $X, Y$ are identically distributed, $\alpha(X) = \alpha(Y)$.
• Fatou-continuity: Stability under limit passages.
• Scale invariance: For any $c > 0$, $\alpha(c X) = \alpha(X)$.

While these axioms give the ratio mathematical elegance and certain intuitive appeal (e.g., focus on the direction rather than magnitude), they also critically impact the ratio’s asymptotic and optimization properties, particularly in infinite-dimensional (continuous time or infinite-sample) models.

2. Optimization and Acceptability Index

To account for the effects of external claims (random endowments) $B$ and market constraints, the optimized gain-loss ratio is introduced:

$$\alpha^*(B) = \sup_{K \in \mathcal{K},~B + K \neq 0}~ \alpha(B + K),$$

where $\mathcal{K}$ is the cone of gains (e.g., achievable by admissible trading strategies).

$\alpha^*(B)$ qualifies as an acceptability index:

• Always lies in $[\alpha^*(0), +\infty]$.
• Monotone and quasi-concave.
• Scale invariant.
• Continuous from below.

Thus, $\alpha^*(B)$ generalizes the notion of risk-adjusted performance and evaluates how acceptable a claim $B$ is under optimal hedging. However, the scale invariance property ensures that the index is insensitive to the size of $B$, only to its "direction," which has profound implications in infinite-dimensional optimization.

3. Dual Representation and FTAP Equivalence

A central aspect is the dual representation, connecting the asymptotic loss-gain ratio to martingale measures with "nice" densities. Without a random endowment:

$$\alpha^* = \sup_{K \in \mathcal{K},~K \neq 0} \alpha(K).$$

The Fundamental Theorem of Asset Pricing (simple form) states the following equivalence:

• The market is $\lambda$ gain-loss free if and only if there exists an equivalent martingale measure $Q$ whose density $Z$ is both bounded and bounded away from zero: $c \leq Z \leq C$ for some $c, C > 0$.
• The dual representation is

$$\alpha^* = \min_{Q \in \mathcal{M}_\infty}\, \frac{\operatorname{ess\,sup} Z}{\operatorname{ess\,inf} Z},$$

where $\mathcal{M}_\infty$ denotes such "good" martingale measures.

For claims $B$, the extension is:

$$\alpha^*(B) = \min_{Q \in \mathcal{M}_\infty,~\mathbb{E}_Q B \leq 0} \frac{\operatorname{ess\,sup} Z}{\operatorname{ess\,inf} Z},$$

with further minimization possible under certain additional constraints.

This duality underlines the financial interpretation: limitations on the best gain-loss ratio correspond to the existence and properties of viable pricing kernels.

4. Asymptotic Behavior and Limitations in Infinite-Dimensional Models

In finite $\Omega$ (finite probability spaces, discrete-time models), the supremum in $\alpha^*$ is typically attained and the optimization problem is well-posed.

For infinite $\Omega$ (general measure spaces, continuous-time models), several pathologies emerge:

• Blow-up/infinite value: In many models (e.g., Black-Scholes), the only equivalent local martingale measure has a density $Z$ unbounded or not bounded away from zero. The set $\mathcal{M}_\infty$ is empty or trivial, so $\alpha^* = +\infty$.
• Non-attainment: Even when $\alpha^*$ is finite, there is generally no gain $K$ that actually attains the supremum. This stems from the non-compactness of the $L^1$ unit sphere when $\Omega$ is infinite, permitting maximizing sequences with no convergent point.
• Counter-intuitive economics: For random endowments $B$, if only a unique pricing kernel exists, $\alpha^*(B)$ may become infinite for nonzero positive expectation, or reduce to 1 for negative expectation, regardless of whether any economic improvement is possible.

These effects collectively render the asymptotic (optimized) loss-gain ratio a poor or even misleading performance measure in most continuous time or infinite market settings.

5. The Double-Edged Role of Scale Invariance

Scale invariance, while normatively attractive for focusing on the "direction" of returns, has several negative consequences in the asymptotic regime:

• Any maximizing sequence can be scaled to have unit $L^1$ norm without changing the performance, but weak compactness fails in $L^1$ over infinite-dimensional spaces. Thus, optimization sequences may not have accumulation points, undermining practical attainability.
• Measures based solely on scale-invariant ratios ignore the total size of exposure, which is crucial for risk management and practical investment decisions.
• The measure does not penalize over-leveraging: an arbitrarily large position in a "good" direction is considered as acceptable as a small position.

Practical performance measures ought to reward or penalize both direction and size, penalizing unbounded risk exposures, but scale invariance prevents this.

6. Overall Evaluation and Impact

The core outcomes and implications are as follows:

• The asymptotic (best) gain-loss ratio $\alpha^*$ is either infinite or not attained in typical infinite-dimensional or continuous-time markets.
• For random endowments, the acceptability index $\alpha^*(B)$ can behave in highly counter-intuitive ways: conferring acceptability on any claim regardless of economic content.
• Scale invariance, while mathematically convenient and normatively justified in some contexts, subverts the measure's ability to reflect practical risk/return trade-offs when asymptotic regimes (large time horizons, unbounded sample spaces) are considered.
• The optimization underlying $\alpha^*$ is ill-posed in the asymptotic sense, so the measure loses its discriminatory and evaluative effectiveness.

In summary, despite possessing several attractive axiomatic and duality-based properties, the asymptotic loss-gain (gain-loss) ratio is fundamentally unsuited as a performance measure in general stochastic, especially continuous-time, models. Its behavior under limits exposes deep issues linked to the topology of function spaces, the non-compactness of relevant sets, and the economic illogic induced by scale invariance, making it unreliable and poorly informative for real-world performance assessment in infinite-dimensional markets (Biagini et al., 2012).