Abstract

This paper proves a rigidity theorem for the Black–Scholes local valuation operator. Standard derivations begin with a stochastic model and use no-arbitrage replication to obtain the pricing PDE. Here the order of explanation is reversed. The question is which deterministic one-factor local valuation operators can assign a pointwise value to a terminal claim independently of payoffpreserving, self-financing redescriptions of that claim. The operative condition is quotient welldefinedness, stated as representative invariance. Within the class of payoff-linear, deterministic, one-state, local, continuous hedge-neutral, self-financing, scale-covariant, time-homogeneous valuation systems with a fixed risk-free numeraire, the only nondegenerate generator is the Black– Scholes operator. The proof is a normal-form exhaustion: any same-scope alternative must alter payoff identity, introduce an auxiliary selector, enlarge the state space, change financing, modify local continuation, break scale or time symmetry, add terminal or boundary data, or change the numeraire. Each route is bookkeeping, a declared enlargement, or a failure of representative invariance. Local volatility, stochastic volatility, jumps, stochastic rates, funding frictions, and incomplete-market measure selections are controlled scope changes, not counterexamples. This supplies a necessity result complementary to stochastic sufficiency derivations while preserving empirical model plurality within finance. The result clarifies stochastic representations as coordinates of a fixed valuation operator.