Option Pricing

One of the most sacred assumptions in financial modeling is the existence of a traded risk-free asset. It anchors discounting, defines arbitrage boundaries, and supports the edifice of Black–Scholes. But what happens when you remove this pillar? Can we still price options, hedge risk, or extract information about funding conditions? In a striking extension of the Lindquist–Rachev (LR) framework, Ziyao Wang shows that not only is it possible — it may reveal financial dynamics that conventional models obscure. ...

In financial engineering, pricing exotic options often boils down to estimating one number: the expected payoff under a risk-neutral measure. But what if we’re asking the wrong question? That’s the provocative premise of a recent study by Ahmet Umur Özsoy, who reimagines option pricing as a distributional learning problem, not merely a statistical expectation problem. By combining insights from Distributional Reinforcement Learning (DistRL) with classical option theory, the paper offers a fresh solution to an old problem: how do we properly account for tail risk and payoff uncertainty in path-dependent derivatives like Asian options? ...

What if the hardest part of finance isn’t prediction, but precision? Behind every real-time portfolio adjustment or split-second options quote lies a giant math problem: solving Ax = b, where A is large, sparse, and often very poorly behaved. In traditional finance pipelines, iterative solvers like GMRES or its flexible cousin FGMRES are tasked with solving these linear systems — be it from a Markowitz portfolio optimization or a discretized Black–Scholes PDE for option pricing. But when the matrix A is ill-conditioned (which it often is), convergence slows to a crawl. Preconditioning helps, but tuning these parameters is more art than science — until now. ...