Profit and loss decomposition in continuous time and approximations

Paper: arXiv 2212.06733

Authors: Gero Junike, Hauke Stier, Marcus C. Christiansen

Abstract

Financial institutions and insurance companies that analyze the evolution and sources of profits and losses often look at risk factors only at discrete reporting dates, ignoring the detailed paths. Continuous-time decompositions avoid this weakness and also make decompositions consistent across different reporting grids. We construct a large class of continuous-time decompositions from a new extended version of Itô’s formula and uniquely identify a preferred decomposition from the axioms of exactness, symmetry and normalization. This unique decomposition turns out to be a stochastic limit of recursive Shapley values, but it suffers from a curse of dimensionality as the number of risk factors increases. We develop an approximation that breaks this curse when the risk factors almost surely have no simultaneous jumps.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly novel and mathematically rigorous continuous-time P&L decomposition framework using an extended Itô formula and axiomatic Shapley values. However, it is primarily a theoretical contribution with minimal empirical backtesting or real-world data validation presented in the excerpt.

Research Flowchart

  flowchart TD
    A[Research Goal: Continuous-Time Profit & Loss Attribution] --> B[Inputs: P&L Paths & Multi-Asset Risk Factors]
    B --> C[Derive Extended Itô's Formula]
    C --> D[Apply Axioms: Exactness, Symmetry, Normalization]
    D --> E[Outcome 1: Preferred Decomposition via Stochastic Shapley Values]
    E --> F[Identify Issue: Curse of Dimensionality]
    F --> G[Outcome 2: No-Simultaneous-Jumps Approximation Scheme]