Monte-Carlo on Inflection Quant

Monte Carlo Variance Reduction: What We Average, and How We Sample

Tue, 02 Jun 2026 00:00:00 +0000

Why This Matters

In the article on the Feynman-Kac theorem, we saw that the price of a derivative can be expressed equivalently as the solution to a deterministic PDE or as the expectation of a discounted payoff under the risk-neutral measure. This gives us two complementary numerical approaches to pricing. For low-dimensional problems with smooth payoffs, finite difference methods on the PDE side are efficient and accurate. For high-dimensional problems, path-dependent payoffs, or models where the PDE is hard to derive, Monte Carlo (MC) on the expectation side becomes the natural choice.

How Randomness Solves a Deterministic Equation: An Intuitive Look at the Feynman–Kac Theorem

Tue, 28 Apr 2026 00:00:00 +0000

Why This Matters

The first time I encountered the Feynman-Kac theorem, I found it fascinating but unintuitive. The theorem claims that a deterministic PDE and the expectation of a stochastic process are two representations of the same object. A PDE is smooth and deterministic. A stochastic expectation involves randomness, probability measures, and averaging over infinitely many paths. How could these be the same thing? I understood the steps of the proof, but I still didn’t have a clear intuition for why this equivalence should exist.