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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.

In a finite-state, finite-horizon financial market with frictionless trading, a strictly positive bank account $B$, and unrestricted predictable self-financing strategies, absence of arbitrage is equivalent to existence of an equivalent martingale measure $Q$. Discounted asset prices $S_t/B_t$ must be martingales under $Q$. Equivalence means that $Q$ and the physical probability measure have the same null events; it does not mean their probabilities agree.^[ref]

One direction follows immediately: a zero-cost self-financing gain has zero discounted expectation under $Q$, so it cannot be nonnegative everywhere and positive with positive probability. Conversely, Hahn-Banach separation theorem produces strictly positive Arrow state prices annihilating the attainable zero-cost gains; normalizing by the bank account gives $Q$. The more general infinite-state and continuous-time statements require attention to closure and admissibility.

With trivial initial information and terminal information equal to the chosen final sigma-algebra, a finite arbitrage-free market is a complete market exactly when $Q$ is unique. Every terminal contingent claim $H$ is then replicable, with value $V_t=B_t{\mathbb{E}}_Q[H/B_T\mid {\mathcal{F}}_t]$.^[ref]

For example, take $B_0=B_1=1$, $S_0=1$, and terminal stock prices $0.8$ and $1.2$. The martingale equation forces both $Q$-probabilities to be $1/2$. A strike-one call option pays zero or $0.2$, so its price is $0.1$. Holding half a share and borrowing $0.4$ replicates those two payoffs. The physical up probability could instead be $3/4$: the pricing weights express attainable-payoff consistency, rather than a forecast of frequencies.

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.

For deterministic square-integrable volatility, apply Itô's formula to the exponential to obtainThe stochastic integral is centered Gaussian with variance ${\int }_0^t\sigma (u{)}^{2}du$.

Under the risk-neutral measure the drift is $r$. Conditional on $S_t=s$, put $\tau =T-t$ and $v={\int }_t^T\sigma (u{)}^{2}du$. ThenIntegrating the discounted call option against this normal distribution yieldsThus the total remaining variance replaces ${\sigma }^2\tau$ throughout the ordinary formula; the corresponding effective constant volatility is $\sqrt{v/\tau }$. If $v=0$, use the limiting deterministic value $(s-K{e}^{-r\tau }{)}_{+}$.

put-call parity and nonnegativity of the European put option giveFor $r\ge 0$ this is at least $S-K$, and the call value is also nonnegative, so $C^E$ dominates the immediate call exercise contingent claim payoff. Its discounted price is a martingale. At any admissible exercise time, optional stopping theorem therefore bounds the expected discounted call option by the European call option price. Exercising at maturity attains that price. Hence American and European calls coincide for this non-dividend model with nonnegative rates.

For the strict put comparison assume $r>0$, $\sigma >0$, $S>0$ and $t<T$. Choose $u$ strictly between $t$ and $T$. A European put option at $u$ has value at most $K{e}^{-r(T-u)}$. On the eventimmediate exercise gives $K-S_u>K{e}^{-r(T-u)}\ge P^E(u,S_u)$. This event has positive probability because the lognormal stock density is positive on $(0,\infty )$. Exercise there at time $u$, and otherwise hold to maturity. This strategy strictly improves the European put option expected value, provingunder these positive-rate, pre-maturity assumptions.

The source's strict statement needs those qualifications. At maturity the prices agree, and at $r=0$ the put option is a convex function of the martingale stock and therefore is a submartingale, so waiting to maturity is optimal and the American and European puts coincide. Negative rates also invalidate the nonnegative-rate call argument. These are mathematical assumptions of the comparison, rather than changes to the printed question.

A down-and-out option with no rebate pays $(S_T-K{)}_{+}$ only if the stock has stayed above the lower barrier $B$ up to maturity; if the barrier is hit, it expires worthless. For an unknocked-out contract currently at $S>B$, apply part (a) to the ordinary European call option price:Both terms solve the PDE. At $S=B$ they cancel, enforcing the absorbing boundary. At maturity and $S>B$, the reflected stock value is $B^2/S<B\le K$, so the reflected call option is zero and the required terminal contingent claim payoff remains. Standard growth and uniqueness conditions for the killed pricing problem identify this solution as the option value.

If the contract has already hit its barrier, its value is zero regardless of a later hypothetical stock recovery. Thus the formula is conditional on survival to the current time; the barrier history is part of the contract state.

Robert H. Litzenberger studied state-contingent claim prices with Douglas T. Breeden. Their call-spread argument relates derivatives of call option prices with respect to strike to state prices. Under suitable smoothness, the second derivative yields a discounted density, providing a bridge between observable option pricing and the pricing of narrow ranges of future asset values.^[ref]