Paper: arXiv 2605.06677

Authors: Tristan Guillaume

Abstract

Barrier derivatives depend on extrema and first-passage events and are therefore highly sensitive to volatility dynamics – especially to the instantaneous return-volatility correlation $ρ$, often called ``leverage’’. This sensitivity makes accurate and fast pricing under realistic stochastic-volatility specifications difficult: two-dimensional PDE solvers are expensive inside calibration loops, while Monte Carlo methods converge slowly when barrier hits are rare and discretely monitored. In equity markets in particular, the pronounced implied-volatility skew motivates factoring in a negative return-volatility correlation. We study a class of continuous-path stochastic-clock volatility models in which the log-price is represented as a Brownian motion run on a random increasing clock. In the baseline independent-clock case (ρ=0), a broad family of barrier-relevant objects-maximum distributions, survival probabilities, and killed joint laws-reduces to one-dimensional quantities determined by the Laplace transform of the terminal clock. This yields transform-only pricing formulas for single- and double-barrier contracts that are fast and numerically stable once the clock transform is available, notably for affine and quadratic clocks. To incorporate leverage without forfeiting tractability, we develop a systematic small-ρexpansion around the ρ=0 backbone. The expansion produces a hierarchy of forced problems whose forcing terms are semi-analytic and computable from baseline barrier objects. We provide two implementable leverage-correction routes,: forced PDEs and a Duhamel-type Monte Carlo representation, and we show how Pad{é} acceleration can extend practical accuracy to equity-like correlations. Calibration then proceeds by,: (i) fitting clock parameters from vanillas using only one-dimensional transforms, (ii) precomputing the ρ=0 barrier backbone once, and (iii) iterating on ρ(and any remaining parameters) using the fast semi-analytic corrections-optionally Pad{é}-accelerated-inside a standard least-squares loop.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper demonstrates high mathematical sophistication by leveraging stochastic calculus, Laplace transforms, and perturbative expansions to handle barrier options. It simultaneously maintains strong empirical focus by providing specific, implementable numerical routines (forced PDEs, Monte Carlo, and Padé acceleration) targeted directly at faster calibration in financial markets.

Research Flowchart

  flowchart TD
    Goal[Goal: Fast, Accurate Pricing of Barrier Options with Leverage] --> Inputs[Inputs: Stochastic-Clock Volatility Models & Market Vanillas]
    Inputs --> Baseline[Step 1: Solve ρ=0 backbone using 1D Laplace Transforms]
    Baseline --> Correction[Step 2: Apply Small-ρ expansion for Leverage-Correction]
    Correction --> Ops[Computational Methods: Forced PDEs & Duhamel Monte Carlo]
    Ops --> Pade[Step 3: Padé Acceleration for Equity-like Correlations]
    Pade --> Outcomes[Outcomes: Efficient Calibration & Stable Pricing]

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0 of 27 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.