# Ideas behind early exercise of American Option

In Dynamic Hedging by Taleb, there is an example at pag 24-25 about early exercise of American options, already present here, but without a clear explanation, at least for me, about the cost/opportunity considered.

Suppose that an asset trade at 100\$, interest rate is 6% and volatility is at 15.7%. Assume that the 3 month call is worth 20\$, at least if it is American.

1. What is the 20\$used in the below formula? The premium (option price) or the difference between the asset and strike price obtained by option exercise and immediate selling of the underlying? Forgoing early exercise would create an opportunity cost of 20 x 90/360 x .06 = .30 cents. 1. Is the intelligent operator exercising the call by paying 80\$ for the asset? Or, is the operator selling the call and using the money to buy the asset and the put?

The intelligent operator can swap the call into the underlying asset and buy the put, whose price is close to zero because of put/call parity, to replicate the same initial structure at a better cost.

1. What are the cost opportunity considered by the operator to come to this conclusion? I can't figure them out.

Please, let me know if something is not clear. Thanks for the help.

1. What is the 20$used in the below formula? The premium (option price) or the difference between the asset and strike price obtained by option exercise and immediate selling of the underlying? • The difference. The author clearly states "early exercise", which refers to the difference between the spot and the strike. 1. Is the intelligent operator exercising the call by paying 80$ for the asset? Or, is the operator selling the call and using the money to buy the asset and the put?
• As the operator is trying to replicate the structure by using put-call parity, we can replicate the call by selling the call and buying the asset + put:

$$C_t = S_t - Ke^{-r(T-t)} + P_t$$

1. What are the cost opportunity considered by the operator to come to this conclusion? I can't figure them out.
• You mean the opportunity cost from question (1.)? If you forgo early exercise, you lose out on investing the cash at the current interest rate. That is why he uses the difference, TTM and the interest rate of 6%.

UPDATE

If we replicate the call by buying the asset and put, we have a cheaper option:

$$S_t - K + P_t < S_t - Ke^{-r(T-t)} + P_t$$

Therefore, the amount saved must be greater than 30 cents.

$$Amt \: Saved = K - Ke^{-r(T-t)} > 0.3$$

• thank you for the clear answer. For question 3. I would like to know the cost/opportunity of the "asset + put" vs "call", since he decide to have his position at the end. In principle, the 30cent are less than this other cost opportunity, that I can't figure out. Am I right? Commented Jun 16 at 7:43
• I am not quite sure what you are asking. Commented Jun 16 at 8:18
• It seems to me that the 30cents are too few to early exercise the call. So he decides to keep the position but by replicating it with the put and the asset. This position has to give to the investor more than 30cent I suppose. This is what I'm trying to figure out. Is it more clear now? Thanks for the patience. Commented Jun 16 at 8:28
• Hi, I updated my solution to reflect the answer to your question. Commented Jun 16 at 11:02
• The inequality states that $C_t$ should be greater than the value of a portfolio formed by the asset, bought by borrowing $K$, and a put, right? Basically, the sell of the call and the cash K should be able to pay for the asset and the put, at least. The 30 cents are compared with what of two terms in the inequality? Sorry if my question is trivial. I suppose that the difference of the RHS-LHS ,invested at the risk free rate up to maturity, should be more than 30 cents. Correct? Commented Jun 16 at 11:34