Faster Forward Sensitivities: Reduced stochastic hedge ratios from pathwise algorithmic differentiation

Abstract

Monte-Carlo valuation engines can generate pathwise sensitivities of a derivative value with respect to a high-dimensional vector of model primitives. Hedge ratios with respect to market instruments are then linked to these primitive sensitivities by a pathwise linear relation. Solving this relation independently on every simulated path may be expensive, unstable, and unnecessarily high-dimensional. This paper studies reduced stochastic hedge ratios of the form $\phi_j^r=\sum_{q=1}^r\xi_j^qX_q$, where the number of solution basis functions is much smaller than the number of Monte-Carlo paths. The hedge-instrument sensitivity tensor is not replaced by its own basis expansion; it is retained through empirical averages over the simulated paths. The basis ansatz alone does not determine the coefficients, so two coefficient criteria are distinguished. The first minimizes the full empirical pathwise residual $\sum_\ell\|A_\ell\phi_\ell^r-b_\ell\|_2^2$. The second is a projected moment equation requiring $\langle A\phi^r-b,Y_s\rangle_N=0$ for selected test functions. The special case $Y_s=X_s$ is the usual Galerkin choice; different test functions give a Petrov--Galerkin formulation. The criteria coincide in special cases but differ when the hedge-instrument sensitivities are path-dependent. The paper gives the tensor and matrix forms of both reductions, discusses regularization and conditioning, and records implementation considerations. The constructions are motivated by sensitivity-based margin valuation adjustment and replication-consistent liquidity forecasting, where pathwise primitive sensitivities have to be converted into hedge ratios with respect to market instruments.