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Profit and loss decomposition in continuous time and approximations

Author

Listed:
  • Gero Junike

    (Carl von Ossietzky Universität)

  • Hauke Stier

    (Carl von Ossietzky Universität)

  • Marcus Christiansen

    (Carl von Ossietzky Universität)

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.

Suggested Citation

  • Gero Junike & Hauke Stier & Marcus Christiansen, 2025. "Profit and loss decomposition in continuous time and approximations," Finance and Stochastics, Springer, vol. 29(4), pages 1075-1107, October.
    • Handle: RePEc:spr:finsto:v:29:y:2025:i:4:d:10.1007_s00780-025-00571-7
    DOI: 10.1007/s00780-025-00571-7
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    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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