Paper: arXiv 2502.08613

Authors: Paul McCloud

Abstract

As operators acting on the undetermined final settlement of a derivative security, expectation is linear but price is non-linear. When the market of underlying securities is incomplete, non-linearity emerges from the bid-offer around the mid price that accounts for the residual risks of the optimal funding and hedging strategy. At the extremes, non-linearity also arises from the embedded options on capital that are exercised upon default. In this essay, these convexities are quantified in an entropic risk metric that evaluates the strategic risks, which is realised as a cost with the introduction of bilateral margin. Price is then adjusted for market incompleteness and the risk of default caused by the exhaustion of capital. In the complete market theory, price is derived from a martingale condition. In the incomplete market theory presented here, price is instead derived from a log-martingale condition: \begin{equation} p=-\frac{1}α\log\mathbb{E}\exp[-αP] \notag \end{equation} for the price $p$ and payoff $P$ of a funded and hedged derivative security, where the price measure $\mathbb{E}$ has minimum entropy relative to economic expectations, and the parameter $α$ matches the risk aversion of the investor. This price principle is easily applied to standard models for market evolution, with applications considered here in model risk analysis, deep hedging and quantum information.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical foundation for option pricing in incomplete markets using relative entropy and log-martingale conditions. However, it is purely theoretical in this excerpt with no backtests, empirical market data, or statistical validations included.

Research Flowchart

  flowchart TD
    A[Goal: Quantify Non-Linear Derivative Pricing & Residual Risks in Incomplete Markets] --> B[Inputs: Undetermined Settlement Payoff, Economic Expectations & Risk Aversion Parameter]
    B --> C[Method: Apply Entropic Risk Metric to Account for Funding, Hedging & Default Convexities]
    C --> D[Process: Minimize Relative Entropy & Evaluate Log-Martingale Condition]
    D --> E[Outcome: Non-Linear Price Adjusted for Bilateral Margin, Market Incompleteness & Default Risk]
    E --> F[Applications: Model Risk Analysis, Deep Hedging & Quantum Information]