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Let a binomial options pricing model multiply the stock price by $u$ or $d$ in each period, with $0<d<R<u$ and bank account gross return $R$. The unique one-step risk-neutral probability is $q=(R-d)/(u-d)$. If $C_N(s,K)$ is the time-zero price of an $N$-period call option starting at $s$ with strike $K$, thenThis is conditional expectation after the first move, with one period of discounting. The terminal payoff's positive homogeneity gives $C_N(as,K)=a{C}_N(s,K/a)$ for $a>0$, henceThe coefficients are positive because the no-arbitrage inequalities put $q$ strictly between zero and one. This strike-rescaling recursion is useful when the shorter-maturity call curve is available. The physical up probability does not enter the recursion.

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.

Write $R=e^r$, $u=1.2$ and $d=0.9$. In a one-step binomial options pricing model, absence of arbitrage requires $d<R<u$. If $R\le d$, borrowing to buy the stock gives a nonnegative gain in either state and a positive gain in at least one. If $R\ge u$, shorting the stock and investing the proceeds does the same. Conversely, strict inequalities give the risk-neutral probability $q=(R-d)/(u-d)\in (0,1)$, under which the discounted stock is a martingale; this excludes arbitrage at each node. Hence

Write $R=e^r$, $u=4/3$ and $d=2/3$. Since both moves have positive physical probability, the binomial options pricing model is free of arbitrage exactly whenIf $R\le d$, borrowing the share price and buying one share gives terminal gain $S_t(u-R)$ or $S_t(d-R)$, both nonnegative and at least one positive. If $R\ge u$, shorting one share and investing its proceeds gives the opposite comparison, with gain $S_t(R-u)$ or $S_t(R-d)$. Equality at either endpoint still permits arbitrage because the other outcome has positive probability.

Conversely, strict inequalities give a risk-neutral probability $q=(R-d)/(u-d)\in (0,1)$, for which the discounted share is a martingale. Any zero-cost self-financing portfolio with nonnegative terminal value has zero discounted expectation under this equivalent positive-probability law and hence cannot have a strictly positive payoff in either state. Applying this argument conditionally at each node proves no-arbitrage over the tree.

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.