European contingent claim

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.

An exchange option permits delivery of one asset in exchange for another at maturity, producing the positive-part payoff $(S_1(T)-S_2(T){)}^{+}$. In a complete finite market its price is the discounted expected value under the unique risk-neutral measure. For a two-period joint binomial model, evaluating this payoff at all four terminal paths and weighting by their risk-neutral path probabilities gives its price directly.

This compound European contingent claim gives the right to buy a later-expiring put option at time $T_1$ for $K_1$. Its payoff is $[p(T_1,S_{T_1};K_2,T_2)-K_1{]}^{+}$, with $T_1<T_2$. In the dividend-free Black-Scholes model with positive volatility, the underlying put price is continuous and strictly decreasing in positive spot. It approaches $K_2{e}^{-r(T_2-T_1)}$ as spot tends to zero and approaches zero as spot tends to infinity.

Thus a unique positive exercise threshold $s_{\ast }$ exists exactly when $0<K_1<K_2{e}^{-r(T_2-T_1)}$. Exercise is favorable for $S_{T_1}<s_{\ast }$, where the put value exceeds its purchase cost. If the cost is at least the upper limiting put value, exercise is never favorable. At zero exercise cost, the compound claim has exactly the underlying put's price by the tower property. These endpoint cases make the meaning of a nondegenerate threshold explicit.

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.

Let $c(u)$ be the Black-Scholes formula value divided by stock price for an at-the-money call of duration $u$. On a discrete set of possible entry dates, homogeneous function and the discounted stock martingale reduce the Snell envelope to $b_i=\max (c(T-T_i),b_{i+1})$, with zero terminal value. Thus the displayed maximum is exact. When $r>0$ and volatility is positive, $c$ increases with maturity, so immediate entry is optimal and the initial value equals the ordinary at-the-money European call option. The maximum recursion remains useful if a different parameter regime loses that monotonicity.

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.

Let $A_t=\exp (t^{-1}{\int }_0^t\log S_{u}du)$ for a positive geometric Brownian motion in the non-dividend-paying Black-Scholes model, and put $\tau =T-t$. Under the risk-neutral measure, the future log-average isThe final integral is a centred normal variable independent of the past, with variance ${\tau }^3/3$. Applying the moment-generating function of a normal distribution proves the displayed identity for every real $p$. Multiplication by $e^{-r\tau }$ prices the payoff $A_T^p$. The historical average must remain a state variable; terminal stock price alone does not determine this payoff.

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.

At its choice time this claim permits purchase of either of two later-expiring European options for an exercise cost $K$, or abandonment. Its payoff is $\max (p-K,c-K,0)$. For common strike $K_3$ and remaining time $\tau$, put-call parity places the branch crossing at $D=K_3{e}^{-r\tau }$. In the Black-Scholes model, both branch values there equal $D[2\Phi (\sigma \sqrt{\tau }/2)-1]$. An exercise cost above this minimum creates a central interval in which neither choice is worth exercising.

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.

In the Black-Scholes model, the claim paying $S_T^2$ has the displayed positive price $U$ and dynamics $dU/U=(2\mu -r)dt+2\sigma dW$. Its nonzero volatility lets it replace the stock as a risky hedging asset. A claim with ordinary stock-price function $F(t,s)$ is replicated by $\Gamma =s{F}_s/(2U)$ units of $U$ and bank-account value $F-s{F}_s/2$. Substitution in the self-financing equation proves the hedge.

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.

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.

For a European put option with deterministic discount factor $D$ and an integrable terminal stock price, $P(K)=D{\mathbb{E}}_{\mathbb{Q}}[(K-S_T{)}^{+}]$. If $\mathbb{Q}(S_T=K)=0$, bounded difference quotients and the dominated convergence theorem give the displayed derivative, the price of a digital put option. At an atom, the left derivative is $D\mathbb{Q}(S_T<K)$ and the right derivative is $D\mathbb{Q}(S_T\le K)$, so the ordinary derivative need not exist.