Question

Show that in the absence of arbitrage ?_a ≥?_e where ?_a is the price of the American put option and ?_e is the price of the European put ? option. Assume that the underlying security, exercise time, and strike price are the same for both options.

Answer 1

First a small summary why American Put Option has more price than European Put Option:

An American option can be exercised at any time before expiration but European option can be exercisd only at expiration so American option gives more flexibility and hence it is more desirable and has higher price.

Value of an Option is: Intrinsic Value of Option + Time Premium of the Option

Now maximum Value of an American Put Option is

Pa = X

where X is nothing but the strike price.

Maximum value of a European Put option is:

Pe = X *e^-rt

where X is the Strike Price, r is the rate of interest and t is the time of expiration

Now Clearly , X > = X*e^-rt

Hence Pa is always greater than or equal to Pe.