European calls arbitrage

Question

I have following question, it should be pretty easy, but this subject is still pretty new for me.

Given a market where calls of any strike price can be bought and sold. Assume that the interest rate for depositing or borrowing money is zero.

1. Show that if there is a call of strike price $K$, maturity $N$ and price $C$ such that $C > S_0$ (where $S_0$ represents the price of the underlying asset at time $0$), then this allows arbitrage.
2. Show that $C < (S_0-K)_+$ (positive part) also leads to an arbitrage opportunity.

Answer 1

Since interest is not an issue here, we have the following:

• If $C > S_0$, then I can sell a call for $C$ and immediately buy a share at $S_0$. At maturity I'll either get an additional $K$ or nothing, so I net at least $C - S_0$, so profit.
• If $C < S_0 - K$, then I can buy a call and short a share, which I immediately sell. At maturity, I buy the share for $K$ and settle my short. The net there is $S_0 - K - C > 0$, so profit.