General equivalence between score-based and Malliavin sensitivities

Establish a general theorem that relates the score-based sensitivity estimators derived from the score-shifted stochastic differential equation method (specifically, the sensitivity formula using the instantaneous score of the probability density) to Malliavin calculus sensitivities for stochastic differential equations beyond the Black–Scholes model, thereby extending the proven equivalence demonstrated for Black–Scholes to a broader class of SDEs.

Background

The paper introduces a score-shifted SDE framework that enables computation of sensitivities to perturbations in diffusivity by leveraging the instantaneous score (gradient of the log-density). This approach leads to a general sensitivity formula that can be evaluated using either exact scores (in equilibrium settings) or learned scores (in nonequilibrium contexts).

For the Black–Scholes model, the authors prove that the sensitivity formula derived via the score-shifted SDE exactly matches the classical Malliavin calculus vega. However, outside this special case, a general equivalence between the score-based sensitivity and Malliavin sensitivities is not established in the paper.

The authors explicitly note that deriving a more general statement linking these two sensitivity frameworks requires further investigation and is deferred to future work.