The Price of Convexity: A Unified Account of Option Premia, Portfolio Choice, and the Heuristics That Connect Them

An option premium is often described as the price of removing risk. This paper replaces that description with an accounting of payoff shape, valuation, replication, and portfolio choice. Relative to a selected stochastic discount factor, price decomposes algebraically into discounted physical expected payout and a pricing-kernel covariance. In incomplete markets that covariance is valuation-rule dependent rather than a unique market-wide charge. Dynamic manufacturing is a separate attainability question governed, under a chosen quadratic-hedging criterion, by a replicating projection and its orthogonal residual. Because internally issued derivative positions sum to zero, contracts redistribute statecontingent payoffs without changing aggregate endowment. Using the Breeden–Litzenberger–Carr–Madan representation, we show that the cost of a non-affine terminal profile equals its curvature integrated against the out-of-the-money option surface. Convexity therefore has a quoted market price. The same ledger extends to state-dependent feedback rules when the feedback signal is also the traded gains process. Constant-mix rebalancing admits an exact curvature ledger under lognormal dynamics; its excess-growth term is a pathwise accounting identity, not by itself an equilibrium premium.In an idealised complete static market, standard dual-integrability conditions yield an optimal payoff equal to inverse marginal utility evaluated at the state-price density per unit of physical probability. Under constant relative risk aversion and geometric Brownian motion it is a power claim whose exponent is the Merton fraction. Thus the Merton allocation, constant-mix curvature, and static option ledger are three representations of one benchmark payoff. Fully invested buy-and-hold is its nondegenerate convexity-neutral position; cash mixes are concave, whereas leveraged or short positions are convex. We also establish a Greek degeneracy for non-affine claims under a frozen flat implied-volatility surface and place common practitioner rules within controlled asymptotic expansions. The results separate what is valued, transferred, statically represented, and dynamically attainable, while making explicit the important estimation limits of each interpretation.