Matrix Approximation of Bachelier Option Prices and Greeks under Stochastic Volatility models

Battle Count: 104

Highly relevant for quantitative trading desks dealing with commodity options (where Bachelier model is standard due to negative prices), SABR-based smile modeling, and rough volatility models. The method's key advantage—computing prices and Greeks for many strikes with a fixed pre-computation cost—makes it practical for real-time calibration and hedging. However, it is primarily a pricing/valuation tool rather than a trading signal generator. The linear algebra structure enables efficient batch computation suitable for production systems.

The core method involves matrix-vector products (linear algebra), which is computationally straightforward. However, the pre-computation of expectations E[ξ_T^β v_T^{-2α}] requires Monte Carlo simulation with control variates, and for the SABR model, the Hartman-Watson distribution is needed for convergence analysis. The derivation of coefficients involves complex combinatorial expressions with factorials and Gamma functions. Proper implementation requires careful handling of numerical stability for high-order expansions and the choice of M_max, N_max hyperparameters.

The paper provides detailed mathematical derivations, explicit formulas for all coefficients, and specifies hyperparameters (M_max, N_max) and model parameters for all numerical examples. Monte Carlo benchmark methodology is described (N_MC=400,000, antithetic variables, control variates). However, no code or repository is provided, and the Hartman-Watson distribution computations and Monte Carlo moment estimations would require careful implementation. The convergence proofs are fully detailed.