We will prove the following fact arising from finance.
- American call and European call with the same strike and maturity has the same price given that their underlying assets are the same non-dividend stock.
Of course, this fact shall not be generalized to put or some other scenarios.
- the stock price by random process ,
- and the strike by ,
- the maturity by ,
- interest rate by .
The mathematical counterpart of the claim, which we want to prove below, is that
We set . Because
- is convex;
- is a martingale;
being a submartingale. This implies that
which in turn gives that
for any . The other direction of the inequality is obvious from operator.