This manuscript develops a structural account of derivative valuation in stochastic market models. The central question is when a derivative price is uniquely determined by the mathematical market structure, and when valuation remains dependent on the choice of filtration, martingale measure, admissible strategy class, calibration constraint, selection rule, stability topology, or execution convention. The work is theoretical and uses filtered probability spaces, martingales, local martingales, semimartingales, stochastic integration, stochastic differential equations, equivalent martingale measures, self-financing strategies, Black--Scholes replication, local and stochastic volatility, calibration maps, incomplete-market pricing, robust valuation, super-replication, hedging residuals, transaction costs, term-structure objects, singular payoffs, local time, and model-risk operators. The main output is a cumulative valuation architecture rather than a single closed pricing formula. The manuscript isolates the mathematical conditions under which pricing closes, and records the barriers that remain when volatility risk is unspanned, martingale measures are non-unique, calibration does not identify dynamics, hedging is discrete, execution is constrained, or model selection is external. The terminal conclusion is that a derivative price is well-defined only inside a declared valuation cell consisting of measure, filtration, tradable assets, admissibility, calibration, selection, stability, and execution structure.
Akmal Xodarev (Thu,) studied this question.