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For an integrable terminal stock price and a deterministic positive discount factor $D$, a European call option curve has the form $C(K)=D{\mathbb{E}}_Q[(S_T-K{)}^{+}]$. Bounded difference quotients and dominated convergence give its one-sided derivatives:Thus the slope jump at a strike is $DQ(S_T=K)$. If the terminal probability distribution has a continuous probability density function $f_Q$, differentiation givesThe call curve's convexity makes this curvature nonnegative. The original construction obtains Arrow state prices from increasingly narrow call spreads.^[ref]

A continuous distribution need not have a density: a singular continuous law is a counterexample. In general the distributional second derivative is the measure $DQ(S_T\in dK)$. For a deterministic terminal value $s_0$, the curve $D(s_0-K{)}^{+}$ has a kink and second derivative $D{\delta }_{s_0}$, rather than an ordinary density function. The entire strike curve therefore records atoms as well as any density. Finitely many observed strikes usually leave many compatible terminal laws.

Douglas T. Breeden coauthored the extraction of state-contingent prices from option pricing with Robert H. Litzenberger. For sufficiently regular call prices across strikes, their second strike derivative identifies a discounted state-price density. The construction makes an entire option-price curve informative about the market's pricing of future outcomes, rather than only one payoff.^[ref]

Robert H. Litzenberger studied state-contingent claim prices with Douglas T. Breeden. Their call-spread argument relates derivatives of call option prices with respect to strike to state prices. Under suitable smoothness, the second derivative yields a discounted density, providing a bridge between observable option pricing and the pricing of narrow ranges of future asset values.^[ref]

Suppose the terminal stock price has support $s_1<\cdots <s_m$. For adjacent support points define the normalized call spread payoffAt a support point $s_j$, it equals zero for $j\le i$ and one for $j\ge i+1$. Consequently any specified terminal payoff values $h_j$ satisfyThe sum telescopes. Cash paying $h_1$ at maturity plus the displayed call spreads gives a static replicating strategy; the cash cost is discounted to today. With $m=3$ and payoffs $(2,5,1)$, use cash paying two, plus three units of $R_1$ minus four units of $R_2$.

The identity is required only on the stated support. It replicates every function of $S_T$, while a path-dependent contingent claim can distinguish histories with the same terminal price. Thus this spanning statement alone does not establish completeness for all history-dependent claims. The call-spread construction is the finite-support counterpart of strike-curvature recovery.^[ref]