Profit and loss decomposition in continuous time and approximations

Gero Junike (), Hauke Stier () and Marcus Christiansen ()
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Gero Junike: Carl von Ossietzky Universität
Hauke Stier: Carl von Ossietzky Universität
Marcus Christiansen: Carl von Ossietzky Universität

Finance and Stochastics, 2025, vol. 29, issue 4, No 4, 1075-1107

Abstract: Abstract Financial institutions and insurance companies that analyse 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 rearranged version of Itô’s formula, and uniquely identify a preferred decomposition from the axioms of exactness, symmetry and normalisation. 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.

Keywords: Profit and loss attribution; Sequential decompositions; Change analysis; Risk decomposition; Itô’s formula; 60H05; 60H30; 91G10; 91G60; 91G30; 91G40 (search for similar items in EconPapers)
JEL-codes: C02 C30 C63 G10 G12 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00780-025-00571-7

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