Researchers present a new numerical method that makes it easier to compute option prices and the main Greeks (Delta and Gamma) under Bachelier-type stochastic volatility models. The Bachelier model treats asset prices as normally distributed and is useful when prices can be negative, a situation seen in some commodity markets. The new approach uses only elementary linear algebra and a finite set of precomputed expectations to produce option prices for as many strikes as you want, provided those strikes lie inside a model-dependent convergence range.

What the authors did: they build a two-scale expansion of the Bachelier price using Taylor series and Itô’s formula, and they extend earlier work to the case where the asset and volatility shocks are correlated. When the expansion is cut off at a chosen order, the result can be written as products of vectors and matrices. That algebraic form means you only need to evaluate a finite number of expectations (which depend on the volatility path), and then you obtain prices and Greeks across many strikes by simple matrix operations. The paper gives an explicit convergence range for the popular SABR stochastic volatility model and shows numerical examples for both SABR and the rough Bergomi model.

How it works at a high level: starting from a representation of the call price as an expectation of a Bachelier formula with a shifted spot (the Hull–White representation), the authors expand the price in powers of the shift and compute the expansion coefficients as expectations of derivatives of the Bachelier formula multiplied by powers of a random variable ξT. Applying Itô’s formula turns those coefficients into a small collection of expectations involving the volatility path (quantities related to the variance-swap and volatility-swap). Those expectations are assembled into matrices (denoted in the paper by Me and Mo) and fixed vectors of powers of (X0 − k). Once the expectations are computed for chosen truncation orders (hyper-parameters Mmax and Nmax), prices and Greeks for many strikes follow from matrix multiplications.