Rigidity of the Black–Scholes Valuation Operator

Overview This paper derives the Black–Scholes equation as a structural necessity rather than a model-based result.Instead of assuming stochastic price dynamics, valuation is classified directly under minimal invariance conditions required for coherence. Core Idea Valuation must be invariant under admissible transformations of a claim—specifically, transformations that: preserve terminal payoff identity are implementable via self-financing strategies introduce no external capital or information This requirement is formalized as standing preservation. Main Result Under the following minimal conditions: standing preservation locality (dependence only on current state) continuity scale invariance time homogeneity the admissible class of valuation rules collapses to a single operator form: → a constant-coefficient, second-order parabolic differential operator→ uniquely identified with the Black–Scholes equation What Is Eliminated The paper shows that alternative valuation rules fail under admissible transformations: First-order transport operators → representation-dependent valuation Stochastic valuation functionals → loss of pointwise identity Nonlocal operators → dependence on non-invariant states Nonlinear pricing rules → inconsistency under aggregation Path-dependent formulations → violation of state sufficiency Interpretation Black–Scholes is not one model among many It is the unique admissible valuation rule within this class Stochastic processes and martingale measures are therefore: coordinate representations of the valuation constraint, not its source Scope This is a classification result, not an empirical claim.Observed market features (e.g. volatility smiles) correspond to violations of specific admissibility conditions, not failures of the operator itself.