Exercise 1

Consider a put option with maturity T and strike price K. Recall that this financial contract gives its holder the right to sell the stock at time t=T at K euro. Let ST denote the stock price at time t=T.

Exercise Ia: Argue that you do not exercise the option, i.e. sell the stock at price K, in case ST>K.

(answer Ia) A stock can be sold on the financial market at ST euro, while exercising the put would only yield K euro.

Exercise Ib: Argue that the payoff of the put, at time t=T, is equal to K-ST if ST≤K.

(answer Ib) Buy, at time t=T, a stock on the financial market. This costs you ST euro. Now exercise the option: you can sell the stock at K euro. Conclude that the net result, the payoff, equals K-ST euro.

Exercise Ic: Conclude that the payoff of the put, at maturity T, is given by max{ 0, K-ST } (max{a,b} is the maximum of a and b).

(answer Ic) Note that max{ 0, K-ST } = 0 if ST>K and max{ 0, K-ST } = K-ST if ST≤K.. A combination with (answer I)-(answer II) yields the result.

We thus have a simple formula, in terms of the stock price, for the value of the put option at maturity. But what is its current price, that is, at t=0?