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Feynman-Kac Derivatives Pricing on the Full Forward Curve

Kevin Mott

Abstract

This paper introduces a no-arbitrage, Monte Carlo-free approach to pricing path-dependent interest rate derivatives. The Heath-Jarrow-Morton model gives arbitrage-free contingent claims prices but is infinite-dimensional, making traditional numerical methods computationally prohibitive. To make the problem computationally tractable, I cast the stochastic pricing problem as a deterministic partial differential equation (PDE). Finance-Informed Neural Networks (FINNs) solve this PDE directly by minimizing violations of the differential equation and boundary condition, with automatic differentiation efficiently computing the exact derivatives needed to evaluate PDE terms. FINNs achieve pricing accuracy within 0.04 to 0.07 cents per dollar of contract value compared to Monte Carlo benchmarks. Once trained, FINNs price caplets in a few microseconds regardless of dimension, delivering speedups ranging from 300,000 to 4.5 million times faster than Monte Carlo simulation as the state space discretization of the forward curve grows from 10 to 150 nodes. The major Greeks-theta and curve deltas-come for free, computed automatically during PDE evaluation at zero marginal cost, whereas Monte Carlo requires complete re-simulation for each sensitivity. The framework generalizes naturally beyond caplets to other path-dependent derivatives-caps, swaptions, callable bonds-requiring only boundary condition modifications while retaining the same core PDE structure.

Feynman-Kac Derivatives Pricing on the Full Forward Curve

Abstract

This paper introduces a no-arbitrage, Monte Carlo-free approach to pricing path-dependent interest rate derivatives. The Heath-Jarrow-Morton model gives arbitrage-free contingent claims prices but is infinite-dimensional, making traditional numerical methods computationally prohibitive. To make the problem computationally tractable, I cast the stochastic pricing problem as a deterministic partial differential equation (PDE). Finance-Informed Neural Networks (FINNs) solve this PDE directly by minimizing violations of the differential equation and boundary condition, with automatic differentiation efficiently computing the exact derivatives needed to evaluate PDE terms. FINNs achieve pricing accuracy within 0.04 to 0.07 cents per dollar of contract value compared to Monte Carlo benchmarks. Once trained, FINNs price caplets in a few microseconds regardless of dimension, delivering speedups ranging from 300,000 to 4.5 million times faster than Monte Carlo simulation as the state space discretization of the forward curve grows from 10 to 150 nodes. The major Greeks-theta and curve deltas-come for free, computed automatically during PDE evaluation at zero marginal cost, whereas Monte Carlo requires complete re-simulation for each sensitivity. The framework generalizes naturally beyond caplets to other path-dependent derivatives-caps, swaptions, callable bonds-requiring only boundary condition modifications while retaining the same core PDE structure.
Paper Structure (15 sections, 1 theorem, 40 equations, 4 figures, 2 tables)

This paper contains 15 sections, 1 theorem, 40 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Heath-Jarrow-Morton Model Refresher
  4. Musiela Parameterization
  5. Standard Procedure for Computing the HJM Model
  6. Data and Forward Curve Construction
  7. Volatility Estimation
  8. Computing the No-Arbitrage Drift
  9. Local Volatility Specification
  10. Feynman-Kac Formula in the HJM Framework
  11. Application: Caplet Pricing
  12. Neural Network Strategy
  13. FINN Architecture and Training
  14. Results
  15. Conclusion

Key Result

Theorem 5.1

Consider a $K$-dimensional state vector $X(t) \in \mathbb{R}^K$ evolving according to the stochastic differential equation: where $\mu(t,X(t)) \in \mathbb{R}^K$ is the drift vector, $\sigma(t,X(t))\in\mathbb{R}^{K\times N}$ is the diffusion matrix, and $W_t\in\mathbb{R}^N$ is an $N$-dimensional Brownian motion. Let $r(t,X(t))$ denote a discount rate that may depend on the state. Define the pricin

Figures (4)

  • Figure 1: Mean Absolute Pricing Error vs. Discretization Level (1000-contract test set). All discretizations maintain error below 0.001, well within acceptable bounds for trading applications. The finest discretizations ($K=100$ and $K=150$) achieve the tightest error bounds around 0.0004. The non-monotonicity suggests complex interactions between discretization level, training dynamics, and network capacity.
  • Figure 2: Final Training Loss Across Discretizations. Both PDE residual (solid black) and boundary condition violation (dashed blue) remain stable across all values of $K$, with losses on the order of $10^{-7}$ to $10^{-8}$. The consistency of loss magnitudes demonstrates that the FINN training procedure scales robustly with dimension.
  • Figure 3: Pricing Speed: FINNs vs. Monte Carlo (1000-contract test set). Top panel: Monte Carlo time (solid) grows linearly from 1 to 10 seconds per contract as $K$ increases, while FINN time (dashed) remains near zero. Middle panel (log scale): FINN evaluation takes $\sim10^{-6}$ to $10^{-5}$ seconds (a few microseconds), while MC takes 1 to 10 seconds. Bottom panel: Speedup ratio grows from approximately 300,000× at $K=10$ to over 4.5 million× at $K=150$, representing a multi-million-fold computational advantage that increases with dimension.
  • Figure 4: FINN vs. Monte Carlo Prices Across Discretizations (1000-contract test set). Each panel shows test contracts for a different discretization level $K$, with points colored by strike price $L_E$. Points cluster tightly along the 45-degree line, with near-zero strike contracts (purple) showing the tightest agreement due to the analytical zero-strike anchoring. Higher-strike contracts (yellow/green) exhibit slightly more scatter, suggesting the FINN performs best near the analytical anchor and pointing to future work on dual anchoring from both extremes.

Theorems & Definitions (1)

  • Theorem 5.1: Multidimensional Discounted Feynman-Kac