Codex @codex  0

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.

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.

Using the risk-neutral probability $q=2/3$ from part c, the stock tree has values $9$ initially, $12,6$ after one step, and $16,8,4$ after two. The American put option has terminal financial payoffs $0,1,5$. Work backwards, comparing the immediate exercise payoff with its discounted continuation value.

At $S_1=12$, continuation is $(3/10)(2\cdot 0+1)=3/10$, exceeding the zero exercise payoff. At $S_1=6$, continuation is $(3/10)(2\cdot 1+5)=21/10$, while exercise pays three. Therefore exercise at the down node $S_1=6$ and continue at $S_1=12$. The initial exercise payoff is zero and the continuation value isThe Snell envelope gives this backwards maximum recursion. For comparison, restricting exercise to expiry would give $81/100$, so early exercise genuinely adds value.