Profit and loss decomposition in continuous time and approximations

TL;DR

This paper introduces a new continuous-time profit and loss decomposition method for financial risk analysis, addressing limitations of discrete methods and proposing an approximation to handle high-dimensional risk factors.

Contribution

It develops a novel class of continuous-time decompositions based on an extended Itô's formula and identifies a unique, axiomatic decomposition, along with an approximation for high-dimensional cases.

Findings

Unique decomposition derived from axioms of exactness, symmetry, and normalization.

Decomposition as a stochastic limit of recursive Shapley values.

An approximation method that mitigates the curse of dimensionality when risk factors rarely jump simultaneously.

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 It\^o'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.