Question

The value of a European put option must satisfy the following restriction: ?0 ≥ ??−?T− ?0 where ?0 is the current put price, ?0 is the current price of the underlying stock, ? is the exercise price, ? > 0 is the annualised continuously compounded risk-free rate, and ? is the time till expiration. Prove by contradiction that the above arbitrage restriction must hold, i.e. show that if the condition does not hold, there is an arbitrage opportunity.

Answer 1

Here, X is the strike price. Suppose this inequality is not satisfied and the put price is actually less than the RHS. So, we can start with buying the put. This put enables us to sell the underlying share at the price X if the price is less than X at maturity. If it is more than our put goes wasted. We buy the put at the price p, which is less than the RHS. We also buy the share at price S0 and borrow an amount equal to Xexp(-rT). Hence, at the time of maturity, we will have to pay back X for the borrowed amount. If the price of share rises above X, we will sell it at that price and from that amount payback the X amount we have to pay and keep the remaining amount. If it is less than X, then we use the put to sell it at X and pay back the borrowed amount. Hence, we see that we have made an arbitrage amount for ourselves.