The Maximum Value of a Put

The Maximum Value of a Put

At expiration, the payoff from a European put is Max(0, X- S₀).  The best outcome that a put holder can expect is for the company to go bankrupt. In that case, the stock will be worthless and the put holder will be able to sell the shares to the put writer for X dollars. Thus the present value of the exercise price is the European pat's maximum possible value. Since an American option can be exercised at any time, its maximum value is the exercise price:

                                      P_e(S₀, T, X) ≥ X(1+r)^-T

                                      P_a(S₀, T, X) ≥ X

On the put's expiration date, no time value will remain. Expiring American puts       therefore are the same as European puts. The value of either type of put must be the intrinsic value:

                                      P(S₀, T₁, X) = Max(0, X – S₀)