Construct portfolio given certain payoff function

Question

My question is similar to https://quant.stackexchange.com/questions/37419/replicate-a-portfolio-with-given-payoff but I am not quite sure how to apply this to my problem.

A portfolio of European call options on an asset $S$ has a payoff function given by $V_T$ where $$V_T = 0,\ S < A$$ $$V_T = S - A , \ A \leq S \leq B$$ $$V_T = B - A, \ S > B$$

(i) Construct a portfolio $H_1$ of European call options with this payoff function.

(ii) Use Put-Call parity to construct a portfolio $H_2$ of European put options with this payoff function.

Answer 1

This is known as a call spread. Buy one call at strike $A$ and sell one call at strike $B$ for part (i). Once you know (i), you should be able to get (ii). Also see the sections at this Wikipedia page.

Answer 2

A long European call with strike A has the payoff: $C(T)=(S(T)-A)^+$ such that, if it goes 'in the money', the profit is of $S-A$. A short European call with strike B has the payoff: $C(T)=-(S(T)-B)^=(S(T)-B)^-+$ such that if it goes in the money, its payoff is of $B-S$. Now, if you construct a portfolio made of this short and long call, given that $A<B$, at time T:

if $S<A$, we end up 'out of the money' so our profit is 0;

if $A\leq S \leq B$ (here, it would be better to leave out that equal sign to A), only the long call is 'in the money' so our profit is $S-A$

if $S > B$, both are 'in the money', so the profit we end up with is given by: $S-A-(S-B)=B-A$

This option strategy gives you the same payoff as the initial function as you can see.

Now, apply the Put-Call Parity noting that:

$P(T)=C(T)-S(T)+K$

Therefore, the long call becomes: $P(T)=(S(T)-A)^+ -S(T)+A=(A-S(T))^+$, i.e. a long put; the short call becomes: $P(T)=-(S(T)-B)^++B-S(T)=-(B-S(T))^+$, i.e. a short put. This comes as no surprise because their payoff diagrams are mirror images. So, noting that $A<B$, you will have replicated the portfolio with puts.

Hope this helps!