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

2025-08-17

AI Summary

This paper addresses the parameter optimization challenge in arbitrage-free interpolation of option prices using randomly configured exponential quadratic splines. It systematically compares two strategies: (i) fixing ordinate values while optimizing abscissae (i.e., interpolation node locations), and (ii) fixing abscissae while optimizing B-spline coefficients. We propose a novel paradigm centered on node distribution optimization, jointly optimizing node positions within a constrained framework enforced via quadratic programming to rigorously satisfy no-arbitrage conditions—namely, monotonicity and convexity of the price curve. Compared with conventional fixed-grid B-spline methods, our approach achieves significantly higher interpolation accuracy and smoother implied volatility surfaces, while strictly preserving no-arbitrage constraints. This enhances market consistency and improves extrapolation stability. Empirical evaluation on real-world option data confirms the method’s robustness and practical applicability.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Optimize arbitrage-free option price interpolation
• Compare fixed ordinates vs abscissae in splines
• Evaluate B-spline parameter optimization methods

Innovation

• Exponential quadratic spline interpolation
• Optimizing abscissae versus ordinates
• B-spline parameter optimization