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On Stochastic Partial Differential Equations and their applications to Derivative Pricing through a conditional Feynman-Kac formula

Kaustav Das, Ivan Guo, Grégoire Loeper

TL;DR

This work develops a conditional Feynman-Kac framework linking derivative pricing to backward SPDEs governed by a backward filtration and a backward Brownian motion. It proves existence and regularity of a well-posed SPDE and demonstrates that its solution represents the conditional expectation of a payoff given the forward state and the future of an auxiliary process. The authors then propose a mixed Monte-Carlo PDE method that leverages the SPDE representation to achieve variance and dimensionality reduction in pricing, and they illustrate the approach with a European put in the Inverse-Gamma model, including numerical schemes (semi-implicit and Crank-Nicolson) and comparisons to a Mixing Solution benchmark. The methodology extends to multivariable settings and suggests broad applicability to American-style derivatives via continuation-value frameworks, offering new tools for integrating SPDEs into quantitative finance. Overall, the paper advances both the theoretical foundation of backward SPDEs in finance and their practical numerical application for derivative pricing in models with stochastic volatility.

Abstract

The price of a financial derivative can be expressed as an iterated conditional expectation, where the inner term conditions on the future of an auxiliary process. We show that this inner conditional expectation solves an SPDE (a 'conditional Feynman-Kac formula'). The problem requires conditioning on a backward filtration generated by the noise of the auxiliary process and enlarged by its terminal value, leading us to search for a backward Brownian motion here. This adds a source of irregularity to the SPDE which we tackle with new techniques. Lastly, we establish a new class of mixed Monte-Carlo PDE numerical methods.

On Stochastic Partial Differential Equations and their applications to Derivative Pricing through a conditional Feynman-Kac formula

TL;DR

This work develops a conditional Feynman-Kac framework linking derivative pricing to backward SPDEs governed by a backward filtration and a backward Brownian motion. It proves existence and regularity of a well-posed SPDE and demonstrates that its solution represents the conditional expectation of a payoff given the forward state and the future of an auxiliary process. The authors then propose a mixed Monte-Carlo PDE method that leverages the SPDE representation to achieve variance and dimensionality reduction in pricing, and they illustrate the approach with a European put in the Inverse-Gamma model, including numerical schemes (semi-implicit and Crank-Nicolson) and comparisons to a Mixing Solution benchmark. The methodology extends to multivariable settings and suggests broad applicability to American-style derivatives via continuation-value frameworks, offering new tools for integrating SPDEs into quantitative finance. Overall, the paper advances both the theoretical foundation of backward SPDEs in finance and their practical numerical application for derivative pricing in models with stochastic volatility.

Abstract

The price of a financial derivative can be expressed as an iterated conditional expectation, where the inner term conditions on the future of an auxiliary process. We show that this inner conditional expectation solves an SPDE (a 'conditional Feynman-Kac formula'). The problem requires conditioning on a backward filtration generated by the noise of the auxiliary process and enlarged by its terminal value, leading us to search for a backward Brownian motion here. This adds a source of irregularity to the SPDE which we tackle with new techniques. Lastly, we establish a new class of mixed Monte-Carlo PDE numerical methods.

Paper Structure

This paper contains 14 sections, 8 theorems, 122 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Informal derivation of conditional Feynman-Kac formula
  3. Preliminaries
  4. Model framework
  5. The SPDE
  6. Main results
  7. Proofs of main results
  8. Multivariable setting
  9. Numerical analysis
  10. Numerical SPDE schemes
  11. Numerical implementation
  12. Conclusion
  13. Some content on backward stochastic calculus
  14. Numerical results

Key Result

Theorem 3.1

There exists a unique solution $u(t,x)$ to the SPDE eqn:spdewellposed, adapted to $(\bar{\mathcal{F}}_{t,T}^{V, B})_{t\in[0,T]}$. Moreover, $t \mapsto u(t,x)$ belongs to $L^2(\varepsilon, T ; H^1(\mathbb{R})) \cap C([\varepsilon, T]; L^2(\mathbb{R}))$ for all $\varepsilon > 0$, $\mathbb{Q}$ a.s.

Figures (1)

  • Figure 1: The implied volatility curve in the Inverse-Gamma model. The number of Monte-Carlo paths for the Monte-Carlo Mixing Solution, Full Monte-Carlo and mixed Monte-Carlo PDE methods are $10 \times 10^5, 15 \times 10^5, 10 \times 10^4$ respectively, whereas the number of time steps are $24, 48$ and $1$ per day respectively.

Theorems & Definitions (30)

  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.1: Backward stochastic integral
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6: Variational formulation
  • Theorem 3.1
  • Proposition 3.1
  • ...and 20 more