Contingent claim

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 financial option grants a right, without a corresponding obligation to exercise, to transact at specified terms. A call option is a right to buy and a put option is a right to sell. The contract specifies its permitted exercise times and strike price; the value of that right differs from the exercise payoff at any particular time.

A financial option granting the right to buy its underlying financial asset at a specified strike price $K$. Its exercise payoff on one stock share is the positive part $(S-K{)}^{+}$. A European call option permits exercise at expiry, while an American call option also permits earlier exercise. A forward-start call option determines the strike at a future fixing date.

The monetary amount delivered by a contingent claim at its payment time, represented by a random variable measurable with respect to the information then available. For a European call option it is $(S_T-K{)}^{+}$, and for a European put option it is $(K-S_T{)}^{+}$. A payoff is the delivered amount, whereas the claim is the contract and its current price is the cost of acquiring or replicating that contract.

The financial payoff obtained by exercising a financial option at a permitted stopping time $\tau$. An American option compares this immediate payoff with the value of keeping the exercise right, whereas a European option restricts exercise to expiry.

A contingent claim whose payoff depends on the underlying price history, rather than only its terminal price. Asian options use averages, barrier options use whether a threshold has been crossed, and lookback options use running extrema. In the Black-Scholes model, the Brownian Martingale representation theorem still gives a replicating strategy; its portfolio wealth need not be a function of the current stock price alone. Adding a running integral or extremum can give a finite-dimensional state for pricing.

A path-dependent contingent claim whose payoff depends on the running minimum or maximum of the underlying stock. Floating-strike calls and puts pay $S_T-{\min }_{u\le T}{S}_u$ and ${\max }_{u\le T}{S}_u-S_T$, respectively. Fixed-strike versions use positive parts of the difference between a running extremum and a specified strike. A sufficient pricing state contains the current price and the relevant running extremum.

For running maximum $M_t={\max }_{u\le t}{S}_u$, a smooth price $v(t,S_t,M_t)$ has an Itô formula term $v_{m}d{M}_t$. The maximum increases only when $S_t=M_t$. A self-financing portfolio in stock and bank account cannot supply an extra singular finite-variation term there, so the smooth price must satisfy the displayed boundary condition. Inside $0<s<m$, it obeys the Black-Scholes equation in the price variable. The condition need only hold before maturity, where terminal corners may destroy differentiability.

A put option gives its holder the right to sell an underlying asset at a specified strike price. A European put option may be exercised only at maturity, whereas an American put option permits earlier exercise.

A power put has financial payoff $(K-S{)}^n$ below the strike price $K$ and zero at or above it, where $n$ is a positive integer. For $n=1$ this is the ordinary put option. For every $n\ge 1$ the payoff is decreasing and is a convex function of the stock price. A European option permits exercise only at expiry; a perpetual power put option permits exercise at any finite time without a fixed expiry.

For a dividend-free Black-Scholes model with $r>0$, $\sigma >0$, $K>0$ and positive integer $n$, the optimal stopping value of the power put option isIn the continuation region the stationary Black-Scholes equation has powers $S$ and $S^{-\gamma }$. Boundedness at infinity removes $S$. Value matching gives the coefficient $(K-b{)}^n{b}^{\gamma }$, maximized at the displayed $b$; equivalently this is the smooth fit boundary.

For a verification beyond constant-trigger policies, let $g(S)=(K-S{)}_{+}^n$. The maximum property of $b^{\gamma }(K-b{)}^n$ gives $V\ge g$. For the infinitesimal generator $\mathcal{L}=rS{\partial }_S+{\sigma }^2{S}^2{\partial }_{SS}/2$, $(\mathcal{L}-r)V=0$ above $b$. Below $b$, set $y=S/(K-S)\le \gamma /n$; thenThe candidate is continuously differentiable across $b$, so the generalized Itô formula has no derivative-jump local time of a semimartingale term. The discounted candidate is a nonnegative supermartingale, bounding every stopping reward by optional stopping. First hitting of $b$ attains the bound; the remainder at infinite time vanishes since $V\le K^n$ and $r>0$. For $n=1$ this recovers the perpetual put option. At $r=0$ the value is $K^n$ for $S>0$, approached but not attained at a finite exercise time. At $r<0$ the perpetual value is infinite, because exercise at deterministic times tending to infinity has unbounded discounted expected value.

In the dividend-free Black-Scholes model, let $\tau =T-t>0$, $K>0$, $S>0$, $\sigma >0$ andThen the displayed finite sum prices the European option with power put option payoff. Here $\Phi$ is the standard normal distribution function. Indeed, $\log S_T$ conditionally has a normal distribution with mean $\log S+(r-{\sigma }^2/2)\tau$ and variance ${\sigma }^2\tau$. Expand $(K-S_T{)}^n$ by the binomial theorem on $S_T<K$ and apply the lower truncated lognormal moment to each term. Equivalently,This expectation representation proves monotonicity and convexity in $S$. It also gives $P_n(0,t)=K^n{e}^{-r\tau }$ and $P_n(S,t)\to 0$ as $S\to \infty$ by the dominated convergence theorem. For positive $S$ the lognormal distribution smoothing makes the price smooth and strictly positive.

A European put option gives its holder the right, without an obligation, to sell one unit of an underlying financial asset at the strike price $K$ only on the expiry date $T$. For one stock share, its terminal financial payoff is the positive partFor strike price $K=100$, terminal stock prices $80$, $100$, and $120$ give financial payoffs $20$, $0$, and $0$, respectively. At $S_T=80$, selling for $100$ rather than the market price $80$ supplies the difference $20$; at $S_T=120$, the holder can let the right expire. Equality at $S_T=100$ gives zero net exercise value. The European call option example plots both financial payoffs on the same axes.

Combining a European put option with the underlying stock share produces the terminal amount $S_T+(K-S_T{)}^{+}=\max (S_T,K)$. Its floor at the strike price is why a European put option can describe protection against a fall in terminal stock value. The protection has an initial cost and lasts to the specified expiry; the terminal floor alone says nothing about the financial return. Put-call parity replicates this same amount with a European call option and a zero-coupon bond.

Using the put-call parity example's price $P_0=10+100e^{-0.05}-100\approx 5.12294245$, the cost financed for one year at interest rate $r=0.05$ grows to $P_0{e}^{0.05}=100-90e^{0.05}\approx 5.38560133$. The terminal net amount is therefore approximately $14.61439867$ when $S_T=80$, and $-5.38560133$ when $S_T=100$ or $120$. These computations use the unrounded price and distinguish the financial payoff from the financed gain. Unlike an American put option, a European put option supplies no right to exercise before $T$.

An American put has exercise payoff $(K-S_n{)}^{+}$. In a finite binomial market, backward option pricing compares that payoff with the discounted risk-neutral continuation value at every node.

A put with a prescribed exercise rule: exercise at the first touch of $B<K$, or at expiry if no touch occurs. Its value is bounded above by the American put option, which optimizes over exercise times. With zero interest rate, its value equals the down-and-out put value plus $(K-B)$ times a cash-at-expiry one-touch option value. At positive interest rate the rebate is paid at touch, so replacing it by an expiry payment would generally be incorrect.

For an alive barrier-triggered put, the difference from the corresponding European put option is the displayed discounted conditional expectation, evaluated under the risk-neutral measure. To prove it, condition the European payoff on the first hitting time or expiry and use the Strong Markov property. At a hit, continuation of the European claim is worth $P(B,{\tau }_B;K)$, whereas the fixed exercise rule pays $K-B$ immediately. On survival to expiry the two payoffs coincide. The relevant stopped quantities are bounded, so the conditional-expectation argument is justified.

For risk-neutral stock dynamics $dS=(r-\delta )Sdt+\sigma SdW$, with $r>0$ and $\sigma >0$, let ${\beta }_{-}<0$ solve ${\sigma }^2\beta (\beta -1)/2+(r-\delta )\beta -r=0$. A fixed downward trigger $L\in (0,X)$ gives continuation value $(X-L)(S/L{)}^{{\beta }_{-}}$ for $S>L$. Optimizing over the trigger, or imposing smooth fit, gives the displayed boundary. The optimal value equals $X-S$ below $S^{\ast }$ and the continuation expression above. A verification argument uses its payoff majorization and nonpositive discounted generator. At $r=0$ and $\delta \ge 0$, the perpetual value is $X$ for positive spot, approached by triggers tending to zero rather than attained at a positive finite trigger.

Here $\Delta =\sqrt{(r-\delta -{\sigma }^2/2{)}^2+2r{\sigma }^2}$. Implicit differentiation of the negative-root equation gives $d{\beta }_{-}/d\delta =-{\beta }_{-}/\Delta >0$. Differentiating $S^{\ast }=X{\beta }_{-}/({\beta }_{-}-1)$ gives the displayed negative derivative. A higher proportional dividend yield lowers risk-neutral stock growth and makes waiting for lower stock prices more attractive to the put holder.

A barrier option activates or expires according to whether the underlying price reaches a prescribed level during its lifetime. Its state must include whether the barrier has already been reached; the current price alone generally does not determine its remaining payoff.

A digital put option extinguished if the stock reaches an upper barrier $B$. Under the dividend-free Black-Scholes model, conditional on not having hit the barrier before the current time, let $0<S<B$, $\tau =T-t>0$, $m=\min (K,B)$, $\gamma =1-2r/{\sigma }^2$, andThen its price isThe Black-Scholes reflection symmetry proves the pricing equation, and the two terms cancel at $S=B$. At expiry the reflected argument is above $B\ge m$, so its payoff is zero, while $S<m$ is equivalent to $S<K$ inside the alive region. Thus the correct free strike is capped at the barrier. For $K\ge B$, the value is the discounted upper-barrier survival probability, independent of $K$, as for a barrier digital put. For $K>B$ its terminal data have a corner discontinuity at the absorbing barrier. Set the price to zero after a hit; the reflected expression on the other side is not a valid contract value.

A contract paying a fixed amount at expiry if specified barriers have not been touched. Its complement, with the same expiry payment convention, is a one-touch option. Paying upon first touch rather than at expiry changes the discounting and gives a different contract.

An up-and-in claim activates when the stock reaches an upper barrier $b>S_0$ during its lifetime. After activation its terminal payoff is the prescribed function $f(S_T)$; without activation it pays zero. Under continuous stock paths the barrier event is described by the displayed running maximum.

For an up-and-in claim in the dividend-free Black-Scholes model, set $\nu =(\rho -{\sigma }^2/2)/\sigma$, $\kappa =(S_0/b{)}^2$ and assume finite absolute payoff expectations. Its initial price equals the risk-neutral pricing value of the displayed European option. To prove this, express the log-price as $\sigma (W_t+\nu t)$ and apply the joint endpoint and maximum law for drifted Brownian motion. Above the logarithmic barrier $a=\log (b/S_0)/\sigma$, use the unrestricted endpoint density. Below it, the density of paths which hit the barrier is $e^{2a\nu }{\varphi }_T(y-2a-\nu T)$. Substitution $z=y-2a$ gives the scaled payoff, the threshold $\kappa b$, and the factor ${\kappa }^{-\nu /\sigma }$. This is an initial-price identity, not a pathwise equality of terminal payoffs.

An up-and-out power claim pays a power of the terminal stock price provided an upper barrier has never been reached. In the Black-Scholes model, set $A=\log (c/S_0)/\sigma$ and $m_p=(\rho +(p-\frac{1}{2}){\sigma }^2)/\sigma$, where $\sigma >0$, $c>S_0$ and $T>0$. The risk-neutral pricing value isHere $\Phi$ is the standard normal distribution function. Weighting the Brownian motion endpoint by $e^{p\sigma W_T-p^2{\sigma }^{2}T/2}$ changes the logarithmic drift to $m_p$; the finite-horizon maximum of Brownian motion with drift then supplies the survival factor.

A down-and-out claim expires worthless if the stock reaches a lower barrier before maturity. A matching down-and-in claim plus a down-and-out claim pays the ordinary terminal claim on every path; this is in-out parity.

A digital call option that pays its terminal indicator only if a lower barrier was avoided. When its strike equals the lower barrier, survival already implies the terminal indicator is one. It is then a no-touch option, with alive price $C_d(s,t;B)-(s/B{)}^{1-2r/{\sigma }^2}{C}_d(B^2/s,t;B)$ in the dividend-free Black-Scholes model.

A European put option that becomes worthless when its continuously monitored lower barrier is touched. The alive state has $s>B$ and no earlier hit. A terminal building block for the Black-Scholes reflection symmetry is $F=C(K)-C(B)+(K-B)C_d(B)$ for $0<B<K$. Its terminal payoff is $(K-S{)}^{+}{1}_{S\ge B}$. The alive price is $F(s,t)-(s/B{)}^{1-2r/{\sigma }^2}F(B^2/s,t)$ and is set to zero after any hit.

A down-and-in claim with lower barrier $b$ pays a specified terminal payoff only if the stock reaches $b$ before maturity. Before activation its value solves a killed-domain pricing problem; at activation its value equals the ordinary terminal-payoff claim value.

For strike $c>b$, write $C(s,c,\tau )$ for the ordinary European call option value. Before activation the down-and-in claim is $I(s,\tau )=(b/s{)}^{\alpha }C(b^2/s,c,\tau )$, where $\alpha =2\rho /{\sigma }^2-1$. Consequently its left-limit option delta at the barrier is $-\alpha C(b,c,\tau )/b-C_s(b,c,\tau )$, whereas after activation it is $C_s(b,c,\tau )$. Their difference is strictly positive for $\tau >0$: differentiating the killed normal transition density at its boundary gives the integral of a strictly positive function against the positive call payoff. The value is continuous, so changing the bank holding finances the stock rebalance.

For initial spot $s>b$ in the Black-Scholes model, put $\nu =(\rho -{\sigma }^2/2)/\sigma$ and $\kappa =(s/b{)}^2$. A down-and-in claim with terminal payoff $f$ has the same initial price as the terminal claim $g(x)=f(x)1_{x\le b}+{\kappa }^{-\nu /\sigma }f(x/\kappa )1_{x>\kappa b}$. Reflection of the log-price endpoint above the lower barrier changes its normal probability density to $e^{2\nu \ell }{\varphi }_T(y-2\ell -\nu T)$, where $\ell =\log (b/s)/\sigma$. Substitution $z=y-2\ell$ gives the payoff identity. This is price equality, not pathwise equality of payments.

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 prepaid forward pays for delivery of an asset at a future time entirely in advance. In the Black-Scholes model with deterministic continuous dividend yield $q$, one share delivered at $T$ costs the displayed amount at $t$. Reinvesting the dividends on $e^{-q(T-t)}$ shares replicates one terminal share. An ordinary forward contract instead has no initial payment and a delivery payment fixed in advance; at constant interest rate $r$ its fair delivery price is $S_t{e}^{(r-q)(T-t)}$.

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.

For a square-root stock claim priced by $C(t,T)=\mathbb{E}[\sqrt{S_T}\mid {\mathcal{F}}_t]$, the constant-volatility geometric Brownian motion price is $\sqrt{S_t}\exp (-{\sigma }^2(T-t)/8)$. If $t<T$, $S_t>0$ and $0<C(t,T)\le \sqrt{S_t}$, its unique nonnegative inverse volatility is the displayed $\Sigma (t,T)$. For bounded predictable volatility in $dS=S\sigma dW$, bounds $0\le a\le \sigma \le b$ give $a\le \Sigma \le b$, without independence from the Brownian motion. At zero stock price the claim price is also zero and no volatility is identified by this equation; a separate convention is needed.

For bounded predictable $\sigma$ on $[0,T]$, the stochastic exponential $Z_s=\exp (\frac{1}{2}{\int }_0^s{\sigma }_{u}d{W}_u-\frac{1}{8}{\int }_0^s{\sigma }_u^{2}du)$ is a positive martingale by the Novikov condition. Under $dQ/dP=Z_T$, the Bayes formula for conditional expectation givesIndeed, the square-root stock ratio is $(Z_T/Z_t)\exp (-\frac{1}{8}{\int }_t^T{\sigma }_u^{2}du)$. This exact identity gives the bounds on square-root stock implied volatility by bounding integrated variance pathwise. The Girsanov theorem gives $W^Q=W-\frac{1}{2}\int \sigma du$. The measure is an auxiliary pricing identity for this payoff, not a claim that it is an equivalent martingale measure for the original stock.

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.

A one-period quadratic hedge minimizes $\mathbb{E}[(\xi -\varphi R-\pi \cdot S_1{)}^2]$ over deterministic holdings, where $R\ne 0$ is the riskless gross return, $\mu =\mathbb{E}{S}_1$, and the covariance matrix $V$ is invertible. For a square-integrable claim, the unique solution is ${\pi }^{\ast }=V^{-1}\mathrm{Cov}(S_1,\xi )$ and ${\varphi }^{\ast }=(\mathbb{E}\xi -{\pi }^{\ast }\cdot \mu )/R$. Centering separates the squared bias from the quadratic error. Completing the square gives the excess error as $(\pi -{\pi }^{\ast }{)}^{T}V(\pi -{\pi }^{\ast })$ after the bias is minimized. Unlike claim replication, the residual need not vanish.

For a discounted contingent claim $H$ and gain vector $Y$, fix initial capital $v$ and minimize $\mathbb{E}(H-v-{\theta }^{T}Y{)}^2$. Differentiation gives the least-squares normal equations $G\theta =\mathbb{E}[Y(H-v)]$, where $G=\mathbb{E}[Y{Y}^T]$ is assumed invertible. This gives the displayed hedge. Optimizing capital as well instead centers the gains and uses their covariance matrix: ${\theta }^{\ast }=\mathrm{Cov}(Y{)}^{-1}\mathrm{Cov}(Y,H)$ and $v^{\ast }=\mathbb{E}H-({\theta }^{\ast }{)}^T\mathbb{E}Y$. Fixing the initial capital is therefore a different quadratic problem.

Let $m=\mathbb{E}Y$ and $C=\mathrm{Cov}(Y)$ be an invertible covariance matrix of discounted gains. The displayed signed martingale measure leaves unchanged the mean-zero random variables orthogonal to the martingale part $Y-m$: if $\mathbb{E}L=0$ and $\mathbb{E}[(Y-m)L]=0$, then $\mathbb{E}[Z_{\ast }L]=0$. Conversely, preservation of all these orthogonal directions forces the density to lie in the span of $1,Y-m$, and its pricing constraints give the displayed formula. If $Z_{\ast }>0$ it is an equivalent martingale measure; if $Z_{\ast }\ge 0$ it is a dominated martingale measure. Positivity is an extra condition, not a consequence of absence of arbitrage alone. In one period this density also minimizes $\mathbb{E}{Z}^2$ among square-integrable signed pricing densities, because every other such density differs by a vector orthogonal to $1,Y-m$. The initial capital in the unrestricted one-period quadratic hedge of a discounted payoff $H$ is $\mathbb{E}[Z_{\ast }H]$.

Let a discounted gain $Y$ take values $-1,1,10$ with physical probabilities $1/10,4/5,1/10$. Then $m=17/10$ and $C=801/100$, so the minimal martingale measure in a one-period market has signed density $Z_{\ast }(10)=-610/801$. Nevertheless $(109/200,89/200,1/100)$ is a strictly positive pricing probability with zero mean gain. Thus absence of arbitrage does not ensure positivity of the minimal density. The minimum-norm identity holds for signed pricing densities; interpretation as a dominated martingale measure requires nonnegativity.

A finite signed measure with total mass one under which discounted asset gains have zero integrals. Its density can be negative, so it need not be a probability measure or give arbitrage-free prices to every nonnegative contingent claim. A square-integrable signed density is useful in one-period least-squares hedging, even when it is not an equivalent martingale measure.

If $S_1$ has a multivariate normal distribution with mean $\mu$ and invertible covariance $V$, and $g$ has bounded gradient, Gaussian integration by parts gives $\mathrm{Cov}(S_1,g(S_1))=V\mathbb{E}[\nabla g(S_1)]$. Consequently the risky holdings of its one-period quadratic hedge equal the expected payoff gradient. The initial cost is $R^{-1}\mathbb{E}[g(S_1)]+(S_0-R^{-1}\mu )\cdot \mathbb{E}[\nabla g(S_1)]$. Bounded gradient ensures linear growth and square integrability, even if the payoff is not bounded.

The initial cost of the one-period quadratic hedge is $\mathbb{E}[{\rho }^{\ast }\xi ]$. This pricing weight satisfies $\mathbb{E}[{\rho }^{\ast }R]=1$ and $\mathbb{E}[{\rho }^{\ast }{S}_1]=S_0$. Among square-integrable weights satisfying these equations it has the smallest squared norm:For another admissible weight $\rho$, its difference from ${\rho }^{\ast }$ is orthogonal to $1$ and all coordinates of $S_1$, hence to ${\rho }^{\ast }$. The Pythagorean theorem in an inner-product space gives $\mathbb{E}[{\rho }^2]=\mathbb{E}[({\rho }^{\ast }{)}^2]+\mathbb{E}[(\rho -{\rho }^{\ast }{)}^2]$. This weight can be signed; positivity, and hence interpretation as a state-price density, requires additional assumptions.

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 financial option whose strike or other exercise terms are fixed at a specified future date, with expiry later still. A forward-start call option pays the positive part of terminal stock price minus that future strike. Specifying the rule for fixing the strike is part of defining the contingent claim.

A forward start option whose call option strike is fixed at time $T_1$ as $\lambda S_{T_1}$, with expiry $T_2>T_1$ and $\lambda >0$. Its terminal financial payoff depends on the intermediate stock price as well as the final price. Two paths ending at the same price can therefore have different payoffs, so a recombining stock tree must retain enough history to value the contract.

In a homogeneous dividend-free Black-Scholes model, let $c(1,\lambda ,T_2-T_1)$ be the European call option price with current stock one and strike $\lambda$ over the remaining duration. Homogeneity gives value $S_{T_1}c(1,\lambda ,T_2-T_1)$ when the strike is fixed. The discounted stock is a martingale, so risk-neutral pricing before that date givesThe same argument holds in a homogeneous binomial options pricing model with the corresponding remaining-step call value. It depends on the normalized future pricing rule being independent of the stock level at the fixing date.

A European option paying the positive part of the logarithm of terminal stock price divided by a positive strike $K$. In the dividend-free Black-Scholes model, that logarithm has a normal distribution under the risk-neutral measure. The Gaussian positive-part expectation gives $V=e^{-r\tau }[m\Phi (m/a)+a\varphi (m/a)]$, where $m=\log (S/K)+(r-{\sigma }^2/2)\tau$ and $a=\sigma \sqrt{\tau }$. This differs from the untruncated logarithmic payoff and from the payoff $S_T\log S_T$.

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.

An Asian option whose averaging uses the geometric mean. In the constant-coefficient Black-Scholes model, the logarithm of the continuous geometric average is Gaussian with mean $\log S_0+(r-{\sigma }^2/2)T/2$ and variance ${\sigma }^{2}T/3$, because $\mathrm{Var}({\int }_0^T{W}_{t}dt)=T^3/3$. Its forward mean is $S_0{e}^{rT/2-{\sigma }^{2}T/12}$. The discounted lognormal call formula gives its price. It is strictly cheaper than the terminal European call when $\sigma ,K,T>0$ and $r\ge -{\sigma }^2/6$, since both its forward mean and log-variance are no larger, with strictly smaller variance. No unconditional comparison holds for arbitrary negative interest rates: for $r<-{\sigma }^2/6$, sufficiently small positive strikes reverse the inequality.

For $I_t={\int }_0^t\log S_{u}du$ and $G_T=\exp (T^{-1}{I}_T)$ in the Black-Scholes model, the conditional law of $\log G_T$ under the risk-neutral measure has a normal distribution with mean $m_t$ and variance $q_t$. The future Brownian integral equals ${\int }_t^T(T-u)d{W}_u$, so the Itô isometry gives the variance. For strike $K>0$ and $t<T$, the call price is $e^{-\rho (T-t)}[e^{m_t+q_t/2}\Phi (d_1)-K\Phi (d_2)]$, where $d_2=(m_t-\log K)/\sqrt{q_t}$ and $d_1=d_2+\sqrt{q_t}$. This follows by completing the square in the truncated normal exponential moment; the known past integral must be included in the mean.

For sampling times $t_i$ and $\bar{t}=n^{-1}{\sum }_i{t}_i$, the geometric mean $A=({\prod }_i{S}_{t_i}{)}^{1/n}$ has $\log A$ normally distributed under the stock-numeraire measure in the Black-Scholes model. Its mean is $\log S_0+(\rho +{\sigma }^2/2)\bar{t}$, and its variance is $v={\sigma }^2{n}^{-2}{\sum }_{i,j}\min (t_i,t_j)$. For descending times the sum is ${\sum }_i(2i-1)t_i$. The drift shift comes from exponential tilting, and must be included when the delivered payment contains a factor $S_T$.

A delivery $S_T\max (r,c/A)$ has value $S_0[r\Phi (d)+c{e}^{-m+v/2}\Phi (-d+\sqrt{v})]$, where $m,v$ are the mean and variance of $\log A$ under the stock-numeraire measure in the Black-Scholes model, and $d=(m-\log (c/r))/\sqrt{v}$. Split the normal integral at $\log (c/r)$ and complete the square in $e^{-y}$ times its normal probability density.

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 cash-at-maturity upper-barrier digital option pays a fixed amount if the stock touches an upper barrier by maturity. In the Black-Scholes model, write $a=\log (c/S_0)/\sigma$ and $b=-(r-{\sigma }^2/2)/\sigma$. Its unit-payoff price is $e^{-rT}$ times the linear-boundary Brownian first-passage distribution at $T$. This differs from paying at the hitting time, when the discount factor is random, and from an ordinary terminal-price digital option.

A contingent claim paying one at maturity if a specified price path has remained below an upper barrier, and zero otherwise. This survival payoff differs from a terminal digital put option, which tests only the terminal price. In the dividend-free Black-Scholes model, its value is the discount factor times the risk-neutral probability of avoiding the barrier, obtainable from the linear-boundary Brownian first-passage distribution.

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

A barrier claim paying a fixed cash amount at the first time the stock reaches a higher level, provided this happens before expiry. In the Black-Scholes model, take $a=\log (c/S_0)/\sigma$ and $b=({\sigma }^2/2-\rho )/\sigma$. Its unit-cash price is ${\mathbb{E}}_Q[e^{-\rho T_{a,b}}{1}_{\{T_{a,b}\le T\}}]$, given by truncated discounted Brownian first passage. The effective square root $\sqrt{b^2+2\rho }=|\rho +{\sigma }^2/2|/\sigma$ is real for every real interest rate.