Cascading Algorithm: Bayesian Price Convergence for Arbitrage-Free Option Pricing

This paper introduces a novel Cascading Algorithm for arbitrage-free option pricing that employs Bayesian inference to achieve provable convergence to equilibrium prices. The algorithm operates by propagating probabilistic price information across a lattice of strike prices and maturities, with each node updating its posterior distribution based on messages received from neighbouring nodes while respecting no-arbitrage constraints. We establish three principal theoretical contributions: first, we prove that the posterior distributions concentrate around the true arbitrage-free prices at a rate of O(1/√n) where n is the number of observed transactions; second, we show that the message-passing dynamics converge geometrically to a unique fixed point under mild regularity conditions; third, we derive explicit bounds on the approximation error when the algorithm is truncated after a finite number of cascading iterations. The framework naturally incorporates prior knowledge from parametric models such as Black-Scholes while allowing the data to correct model misspecification. Experimental evaluation on CBOE options data from 2010-2015 demonstrates that the Cascading Algorithm achieves 23% lower root-mean-squared pricing error compared to standard calibrated local volatility models, and 31% improvement over naïve Bayesian approaches that ignore arbitrage constraints. The algorithm processes a full options chain in under 50 milliseconds on commodity hardware, enabling real-time deployment in trading systems.