It can be proven that under non-negative interest rates, it is never optimal to exercise an American call option, such that:
We know, if R >= 0, the current price C of a Europen (and American) call option, with strike price K and time to expriry T, on a non- divided paying stock with current price S satisfies:
C >= max {S-exp(-rT)K, 0}
then, we also know that C >= 0
, otherwise buying the call would give a riskless profit now and no obligations later.
To prove that under non-negative interest rates, it is never optimal to exercise an american option we asssume that:
C < S-exp(-rT)K
The we get an arbitrage table like:
we have a non-negative return in all possible states of the world at expiry which has a positive current cash flow. This is clearly an arbitrage opportunity and hence the assumption is wrong.
Suppose now that the American call is exercised at some time t strictly less than expiry T , i.e. t < T . The financial agent thereby realises a cash-flow St − K. From the above proposition we know that the value of the call must be greater or equal to St − exp(−r(T − t))K, which is greater than St − K, if r ≥ 0. Hence selling the call would have realised a higher cash-flow and the early exercise of the call was suboptimal. In conclusion the price of an American call equals the price of an European call: AC = EC
I would like to do an analogous proof to show that it is never optimal to exercise an american put option on a non-dividend pying stock with r =< 0
: EP = AP
I am stuck with the arbitrage table.
- What does the portfolio consist of for an put call option ?
- Is there an easier way how to prove this?