Revisiting Stochastic Collocation with Exponential Splines for an Arbitrage-Free Interpolation of Option Prices

ArXiv ID: 2508.12419 “View on arXiv”

Authors: Fabien Le Floc’h

Abstract

We revisit the stochastic collocation method using the exponential of a quadratic spline. In particular, we look in details whether it is more appropriate to fix the ordinates and optimize the abscissae of an interpolating spline or to fix the abscissae and optimize the parameters of a B-spline representation.

Keywords: stochastic collocation, exponential quadratic spline, B-spline representation, interpolation, numerical methods, Derivatives (General)

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper involves dense and advanced mathematics, including stochastic collocation, exponential splines, B-spline representations, and complex integral formulas, warranting a high math score. It demonstrates empirical rigor with backtest-ready calibration on real market data (TSLA options), regularization, and sensitivity analysis to initial guesses, though the excerpt lacks full implementation details like code or performance metrics.

  flowchart TD
    A["Research Goal<br>Fix ordinates optimize abscissae vs. Fix abscissae optimize B-spline?"] --> B{"Evaluate Two Approaches"}
    B --> C["Approach 1<br>Fix Ordinates, Optimize Abscissae"]
    B --> D["Approach 2<br>Fix Abscissae, Optimize B-spline Weights"]
    C --> E["Computational Process<br>Minimize L-infinity distance<br>between implied volatilities"]
    D --> E
    E --> F["Data Input<br>Discrete market option prices<br>and implied volatilities"]
    F --> G["Key Findings & Outcomes"]
    G --> H["Approach 1 (Variable Abscissae)"]
    G --> I["Approach 2 (B-spline Optimization)"]
    H --> J["Simplifies inverse problem<br>but requires node sorting<br>and produces valid CDF"]
    I --> K["Better computational stability<br>via linear system solve<br>ensures arbitrage-free prices"]