Fundamental theorem of asset pricing

In a finite market, absence of arbitrage is equivalent to the existence of an equivalent martingale measure. If that measure is unique, every contingent claim has a unique no-arbitrage price given by its discounted expectation.

A complete market is a financial market in which every contingent claim in the chosen class can be replicated by an admissible self-financing portfolio. In an arbitrage-free finite market, completeness is equivalent to uniqueness of the equivalent martingale measure.

Two assets with linearly independent payoff vectors across two positive-probability states span all terminal payoffs. The state-price equations have a unique solution. A positive solution gives the state-price density after division by physical state probabilities. Arrow state prices sum to the riskless discount factor, while normalized Arrow state prices are risk-neutral probabilities; these three quantities must not be confused.

In a one-period complete market with deterministic initial holdings, two integrable positive pricing variables assigning the same prices to every traded asset are equal almost surely. Replicate indicator functions to obtain equality of their integrals on every event, then apply this to the event where one density exceeds the other.

With $n$ assets and a trivial initial sigma-field, market completeness forces at most $n$ positive-probability successors per current atom. Indicators of disjoint successor events are independent payoffs, whereas terminal holdings on a parent atom span at most $n$ asset-price functions. Iterating yields at most $n^t$ atoms at time $t$.

Let $P$ be the state-by-asset matrix of discounted gains. There exists $\theta$ with $Y-P\theta \ge 0$ exactly whenfor every $q\ge 0$ satisfying $P^{T}q=0$. This is the finite-dimensional cone-separation, or Farkas-duality, form of superhedging.

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.

A European put with maturity $N$ and strike $K$ pays $(K-S_N{)}^{+}$ at time $N$ and can be exercised only at maturity.

A contingent claim is a future payoff whose value depends on uncertain market outcomes. A replicable claim has the unique no-arbitrage price of its replicating portfolio.

A barrier contract paying a specified fixed amount if the underlying touches a specified barrier before expiry. Payment at touch and payment at expiry are different conventions. For payment at touch, valuation uses the truncated discounted Brownian first passage after converting a geometric stock barrier to a log-price level; discounting solely at expiry would price a different contract.

An agreement to exchange an underlying at a fixed delivery price on a specified maturity, without the futures convention of repeated settlement. Its fair delivery quote is ${\mathbb{E}}_Q[D_{tT}{S}_T\mid {\mathcal{F}}_t]/{\mathbb{E}}_Q[D_{tT}\mid {\mathcal{F}}_t]$. It may differ from a futures contract quote when rates are stochastic.

A standardized maturity-and-quantity agreement whose quote changes generate margin payments as gains and losses are marked to market. A newly settled contract has zero value, while its quoted delivery price is generally nonzero. In the continuous-settlement idealization, futures pricing uses a pricing-measure martingale quote.

With continuous cash settlement and a money-market risk-neutral measure, discounted futures gains have zero drift; under the necessary true-martingale integrability, the futures quote is a martingale ending at the terminal spot price. The forward contract discount-weighted expectation is a different valuation expression.

A futures contract delivery quote below current spot for the maturity considered, or a falling futures curve across maturities. This spot comparison should be distinguished from comparisons with the future physical-measure expected spot.

A futures contract delivery quote above the current spot price for the maturity considered; a rising quoted curve across maturities is also described as contango. This compares delivery prices rather than the current zero value of a newly settled contract.

The payout $\sqrt{S_T}$ is a concave stock payoff. Its conditional expected value has a discount-like dependence on future variance. The square-root stock supermartingale explains the negative variance drift.

If $\mathbb{E}[\sqrt{S_T}\mid {\mathcal{F}}_t]=\sqrt{S_t}\exp (-{\int }_t^T{f}_t(u)du)$ and $d{f}_t(T)=A_t(T)dt+B_t(T)d{W}_t$, the stochastic Fubini theorem and Itô formula give $f_t(t)={\sigma }_t^2/8$ and $A_t(T)=B_t(T)({\int }_t^T{B}_t(u)du-{\sigma }_t/2)$. The final term comes from the product cross-variation. Continuity extends the drift equality to the specified continuous versions.

In the joint filtration of Brownian history and an independent volatility history, conditioning on the entire volatility path and then using the tower property gives $\mathbb{E}[\sqrt{S_T}\mid {\mathcal{F}}_t]=\sqrt{S_t}\mathbb{E}[\exp (-\frac{1}{8}{\int }_t^T{\sigma }_u^{2}du)\mid {\mathcal{F}}_t]$. Removing the outer conditional expectation requires the integrated variance to be known at time $t$. Independence alone does not give this measurability; a volatility parameter disclosed later supplies a counterexample.

A contingent claim is replicated when an admissible self-financing portfolio has exactly its terminal payoff. In the absence of arbitrage, two replicating strategies for the same payoff have the same initial cost.

In a one-period complete market, an invertible payoff Gram matrix identifies the unique holdings replicating each claim. The pricing weight is $Z=P_0^T{Q}^{-1}{P}_1$ and the random portfolio kernel is $W=Q^{-1}{P}_1$. Positive definiteness gives uniqueness of holdings; positivity of $Z$ requires an additional no-arbitrage condition.

With a constant cash account, holding $T-t+1$ shares during interval $(t-1,t]$ and selling one share at each endpoint replicates ${\sum }_{t=1}^T{S}_t$. Each sale is retained in cash; the initial cost is $T{S}_0$. The proof is a pathwise telescoping identity.

An admissible self-financing portfolio superhedges a contingent claim when its terminal wealth is at least the claim payoff almost surely. The least permitted initial cost is the superhedging price. Exact claim replication requires equality.

The infimum of initial costs of admissible superhedging portfolios. Positive pricing kernels supply lower bounds: if $\mathbb{E}[ZP]=p$ and $H\cdot P\ge X$, then $H\cdot p\ge \mathbb{E}[ZX]$ whenever these expectations exist.

A replicating strategy is a self-financing portfolio whose terminal wealth equals the prescribed payoff of a contingent claim. Admissibility specifies the permitted wealth bounds and integrability of its holdings.

A European contingent claim pays a specified function of market variables at one fixed maturity. Under an equivalent martingale measure, an attainable claim is priced by the discounted conditional expectation of its payoff.

A maturity-$T$ claim is attainable if a predictable self-financing strategy with fixed initial capital has terminal wealth equal to the claim payoff almost surely. In discrete time its wealth is $X_t=x+{\sum }_{s=1}^t{H}_s\cdot (P_s-P_{s-1})$. The strategy replicates the claim; attainability is a statement about exact pathwise replication, not only equality of expectations.

An option whose payoff depends on an average of the underlying stock prices over time. The displayed arithmetic-average call differs from a geometric-average call, which uses the geometric mean. A convex payoff and a nondecreasing positive numéraire give an upper bound by the average costs of same-strike European call options at the averaging dates.

The holder chooses at a specified time between a European call option and a European put option with the same later maturity and strike. Put-call parity makes its value equal to a call of the original maturity plus a put of the choice-time maturity with appropriately discounted strike.

A power option is a European contingent claim whose payoff is a fixed power of the terminal underlying price. In the Black-Scholes model, its value is $S_t^p\exp ((p-1)r(T-t)+\frac{1}{2}p(p-1){\sigma }^2(T-t))$ when the required moments are finite.

A binary option pays one fixed amount when a specified event occurs at maturity and zero otherwise.

A digital call option with strike $K$ pays one unit if $S_T\ge K$ and zero otherwise.

A digital put option with strike $K$ pays one unit if $S_T<K$ and zero otherwise. With these complementary boundary conventions, one digital call plus one digital put equals a zero-coupon bond paying one unit at maturity.