Paper: arXiv 2609.06137

Authors: Evgeny Lakshtanov

Abstract

Pathwise differentiation of Monte Carlo estimators fails at payoff discontinuities, producing zero or biased sensitivities for barriers, autocallables, and digital options. The industry workaround — smoothing the indicator functions — introduces bias and requires per-product calibration. We derive a correction formula that restores unbiased Greeks without smoothing. For a payoff $F(Z,θ)$ that is piecewise smooth with discontinuities on surfaces ${g_i = 0}$, we show that the sensitivity decomposes into a pathwise term (computed by standard AAD) plus a sum of boundary corrections, each involving the payoff jump, the Gaussian density at the boundary, and the sensitivity of the boundary to the parameter. The correction is computed by Newton root-finding in the normal-random space, with the jump evaluated by two forward replays of the pricing kernel. The implementation uses AADC (\texttt{pip install aadc}), whose tape replay and automatic discontinuity tracking make the method fully automatic — the quant writes standard pricing code, and the correction driver identifies and handles all discontinuities. We prove the formula for arbitrary compositions of smooth functions and indicator functions (not just outer products), covering real autocallable payoff structures with recursive alive/dead logic. Benchmarks on QuantLib models (GBM, Heston, Hull-White) show all Greeks within 0.1–4% of analytic or bump-and-revalue references.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly novel and mathematically sophisticated method for computing unbiased Monte Carlo Greeks for discontinuous payoffs. The empirical validation against QuantLib models and comparison with industry workarounds demonstrates strong rigor. The clear exposition and explicit mention of an open-source library for implementation further enhance its value.

Research Flowchart

  flowchart TD
    A[Research Goal: Unbiased Monte Carlo Greeks for Discontinuous Payoffs] --> B{Problem: Pathwise Differentiation Fails at Discontinuities};
    B --> C[Methodology: Derive Correction Formula for Unbiased Greeks];
    C --> D{Computational Process: AADC for Automatic Discontinuity Tracking & Correction};
    D --> E[Correction Formula Inputs: Payoff Jump, Gaussian Density, Boundary Sensitivity];
    E --> F[Key Outcome: Unbiased Greeks for Discontinuous Payoffs (within 0.1-4% of references)];
    D --> G[Key Outcome: Automatic Implementation via AADC (Standard Pricing Code)];