European call option

A European call with maturity $N$ and strike $K$ pays $(S_N-K{)}^{+}$ at time $N$.

For $s\ge 0$ and $\epsilon >0$, integrate over $0\le K\le s$ to obtain the displayed identity. Tonelli theorem then expresses the moment of a nonnegative random variable as the same integral of expected call payoffs, even when the moment is infinite. This is a static representation across strikes rather than a dynamic replicating strategy in a restricted finite-asset market.

For $K>0$ and $s>K$, minimize $(s/K{)}^{1+\epsilon }/(s/K-1)$. Its minimum occurs at $s/K=(1+\epsilon )/\epsilon$, giving the sharp constant. Taking expected values yields a uniform bound on $K^{\epsilon }C(K)$ from a finite moment of order $1+\epsilon$.

For $C(K)=\mathbb{E}(S-K{)}_{+}$ and $\mathbb{E}S<\infty$, split the static call integral at a fixed positive strike. The first moment bounds the integrand near zero; polynomial call decay controls the integral at infinity for $0<\epsilon <\delta$. The endpoint can fail, as shown by a Pareto distribution with survival exponent $1+\delta$.

In a zero-interest one-period market with a continuous terminal stock law under a pricing measure, $C(K)={\mathbb{E}}_Q(S-K{)}^{+}$ implies $-C^{\prime }(K)=Q(S>K)$ and $C^{\prime \prime }(K)=f_Q(K)$. 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.

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 $2+\sqrt{2}$, and upper-state probability $3-2\sqrt{2}$ give $C(1)=\sqrt{2}-1$, 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.

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 $\int (u-K{)}^{+}{f}_p(u)du=C(K)$. It prices an integrable payoff by $\int g(u)f_p(u)du$ 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.

The midpoint butterfly buys half a call at each outer strike and sells one at the middle strike. Convexity of $K\mapsto (S-K{)}^{+}$ makes its terminal payoff nonnegative. A violation of the displayed price inequality makes its cost negative and creates arbitrage. The payoff is triangular between the outer strikes and zero elsewhere.

Buying the lower-strike call and selling the higher-strike call gives a strictly negative initial cost and terminal payoff $(S-K_1{)}^{+}-(S-K_2{)}^{+}\ge 0$. Hence an arbitrage-free call-price curve is decreasing in strike. The positive initial receipt may be consumed immediately or invested in cash.

A positive one-period martingale deflator applied to $C=(S-B{)}^{+}$ gives $(s-b{)}^{+}\le c\le s$. If both strict payoff regions $S>B$ and $S<B$ have positive probability, then $c>0$ and $c-(s-b)>0$, proving the strict lower bound. No normalization of the positive pricing variable is needed.

The Mellin transform of call prices can be inverted along a vertical contour with real part $x_0>1$: $C(k)=\frac{1}{2\pi i}{\int }_{x_0-i\infty }^{x_0+i\infty }\mathbb{E}{e}^{z\xi }/(z(z-1)e^{(z-1)k})dz$. A finite exponential moment at $x_0$ and the quadratic denominator give absolute convergence.

For a martingale with increments in $\{-1,0,1\}$, European call option prices satisfy $C(T+1,K)-C(T,K)=\frac{1}{2}{\sigma }^2(T,K)(C(T,K+1)-2C(T,K)+C(T,K-1))$. At positive-probability states, the variance and the upward/downward conditional transitions are determined by this identity; transitions at zero-probability states cannot be recovered.

For an integer-valued underlying with European call option prices $C(T,K)=\mathbb{E}(S_T-K{)}^{+}$, the second strike difference equals $\mathbb{P}(S_T=K)$. It is the discrete counterpart of recovering a payoff probability distribution from strike derivatives.

At a fixed maturity, no arbitrage makes a European call price nonincreasing in its strike because the lower-strike payoff dominates the higher-strike payoff state by state.

When terminal stock price takes finitely many ordered values, every payoff function of that price can be represented on the support by cash plus a linear combination of adjacent-strike call spreads.

The pathwise identity ${\sum }_{t=1}^T(\Delta S_t{)}^2=S_T^2-S_0^2-2{\sum }_{t=1}^T{S}_{t-1}\Delta S_t$ turns a squared-increment claim into a terminal-square claim and predictable stock gains. On integer terminal support $\{0,\dots ,N\}$, $S_T^2=S_T+2{\sum }_{K=1}^N(S_T-K{)}^{+}$, so a static call strip plus a dynamic stock hedge replicates the claim.

For European calls and puts with the same maturity and strike,

Conditioning on the first binomial move expresses a call with $N+1$ periods as a positive weighted sum of two $N$-period calls with rescaled strikes.

A forward-start call fixes its strike at a future time as a multiple of the stock price then. In a homogeneous binomial model, its time-zero price reduces to that of an ordinary call over the remaining periods.