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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.

Determining the value of a European option or American option from a specified financial market model and its permitted exercise rules. Risk-neutral pricing, replicating strategies and optimal stopping address different parts of this problem.

An American option allows exercise at a stopping time $\tau \in [t,T]$. For its exercise financial payoff $H$, the pricing problem isThe discounted value is the Snell envelope of discounted exercise payoffs. Define the pricing operatorIn the continuation region the Black-Scholes equation holds, $\mathcal{L}V=0$. In the exercise region $V=H$, while the ability to wait and the supermartingale property give $\mathcal{L}V\le 0$. Together these conditions are the obstacle problemThus an unknown exercise boundary must be found alongside the value. At a regular boundary in the nondegenerate Itô diffusion, value matching is $V=H$, and smooth pasting is $V_S=H^{\prime }$ where the financial payoff is differentiable. A finite difference method or a binomial options pricing model steps backward taking the larger of continuation and immediate exercise values at each node.

For a non-dividend-paying American call option with $r\ge 0$, no early exercise of a call without dividends follows directly from the European lower boundAt every candidate exercise time, keeping the European call has value at least the immediate exercise financial payoff, so early exercise cannot improve the value; the American and European prices coincide. The first inequality follows from Jensen inequality or put-call parity and nonnegative put value. With $r>0$ an American put option can benefit from early receipt of its strike at sufficiently low stock prices, so its exercise boundary is generally nontrivial and its value is at least the European put value. Dividends can make call exercise worthwhile, and negative interest invalidates the stated no-early-exercise call argument.

For European options with the same strike and maturity, the terminal payoff identity isThus long the call and short the put has the same terminal payoff as one stock less $X$ units of the maturity-$T$ zero-coupon bond. With no dividends, the stock strategy has no intervening payments. The law of one price gives put-call parity:Equivalently the two nonnegative terminal-payoff portfolios, call plus strike bond and put plus stock, agree in every state. The parity proof does not require lognormal stock prices or positive interest.

For American options the exercise times can differ, so equality of their maturity payoffs is not an identity between their full rights. Under the nonnegative-interest, non-dividend assumptions of part (a), $C_A=C_E$ while $P_A=P_E+{\Pi }_P$ with early-exercise premium ${\Pi }_P\ge 0$. Hencewhich generally differs from European parity. For a concrete limiting state, if the stock is zero and remains zero and interest is positive, then $C_A=C_E=0$, $P_A=X$ and $P_E=XB(t,T)<X$. European parity is not a general American-option equality. It holds in a particular case only when the relevant early-exercise premiums cancel, or both vanish.

Put ${\mathcal{L}}_{r}V(s)=\frac{1}{2}{\sigma }^2{s}^2{V}^{\prime \prime }(s)+rs{V}^{\prime }(s)$ and define the nonnegative reserve rate $a(s)=rV(s)-{\mathcal{L}}_{r}V(s)$. The obstacle problem gives $V\ge g$ and $a\ge 0$. The American-option superhedge with a funded reserve invests the local surplus in the bond rather than consuming it.

For initial wealth $x\ge V(S_0)$, setand choose the stock and bond holdingsThus $D_t\ge 0$ and $F_t\ge V(S_t)\ge g(S_t)$, pathwise at every time. The Itô formula under the original drift givesAdding these equations yieldsThis is a self-financing strategy, and its nonnegative wealth makes it an admissible trading strategy. Continuity of the stock and local regularity of $V$ ensure local integrability of the holdings. The construction does not require $\mu =r$.

For a classical solution, the usual Itô formula applies directly. The smooth fit solution below is $C^1$ and piecewise $C^2$, with locally absolutely continuous first derivative. The generalized Itô formula applies with its almost-everywhere second derivative; the absence of a derivative jump means no boundary local time of a semimartingale term. This is the usual regularity interpretation of the perpetual American option obstacle equation.