The Lower Bound of a European Put

The Lower Bound of a European Put

We showed that the minimum value of an American put is Max(0, X-S₀). This statement does not hold for a European put, because it cannot be exercised early. The price of a European put must at least equal the greater zero or the present value of the exercise price minus the stock price:

   P_e(S₀, T, X) = Max(0, X(1+r)^-T - S₀)

In the following figure, the curved line is the European put price, which must lie above the lower bound, As time to expiration gets smaller so the lower bound moves to the right, with the put price curve following it. However, as time goes by, the put price gradually declines. At expiration, the put price and the lower bound converge to Max(0, X - S₀).   

Finally we should note that if the stock pays dividends such that the stock price minus the present value of the dividends is

Then the lower bound is:

   P_e(S₀, T, X) = Max(0, X(1+r)^-T – S’₀)