= Finite-strike nonidentification of a pricing density
{title2=$\{C(K_i)\}_{i=1}^n\ \not\Rightarrow\ \text{a unique pricing density}$}
Finitely many call prices impose finitely many payoff-moment constraints. They do not require a continuous terminal law or determine prices of arbitrary new claims. For example, $S_0=1$, terminal values $1/2$ and $2+\sqrt2$, and upper-state probability $3-2\sqrt2$ give $C(1)=\sqrt2-1$, matching the $p=2$ power curve. A payoff vanishing on these two states must cost zero, whereas integration against the power curve's strictly positive density can assign it a positive cost. A pricing density intended for arbitrary claims must be compatible with an equivalent law on the actual state space.
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