Call-price density recovery ( $C (K) = f_{Q} (K)$ )

In a zero-interest one-period market with a continuous terminal stock law under a pricing measure,

C (K) = E_{Q} (S - K)^{+}

implies

- C^{'} (K) = Q (S > K)

and

C^{''} (K) = f_{Q} (K)

. Thus a full differentiable call curve determines its pricing density. A finite collection of strikes generally does not. With a deterministic nonunit discount factor, divide the call curve by that factor before recovering the probability density.

Table of contents
- Finite-strike nonidentification of a pricing density Call-price density recovery
- Power call-curve pricing density Call-price density recovery

Finite-strike nonidentification of a pricing density ( ${C (K_{i})}_{i = 1}^{n} \neq \Rightarrow a unique pricing density$ )

 0  0

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 + 2

, and upper-state probability

3 - 22

give

C (1) = 2 - 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.

Power call-curve pricing density ( $f_{p} (u) = (p - 1) u^{p - 2} (1 + u^{p})^{1/ p - 2}$ )

 0  0

For

p > 1

, the zero-interest curve

C (K) = (1 + K^{p})^{1/ p} - K

has positive second derivative

f_{p}

. Its mass and first moment are both one, and

\int (u - K)^{+} f_{p} (u) d u = C (K)

. It prices an integrable payoff by

\int g (u) f_{p} (u) d u

when this law is equivalent to the physical terminal stock law, or on a canonical model with this pricing law. Finite-strike consistency alone is insufficient for that equivalence.