American Call Early Exercise, Interest Cost, and Put-Call Parity
Summary
The document clarifies the opportunity cost behind early exercise of an American call. In the example discussed, the amount used to estimate forgone interest is the call’s intrinsic value—the difference between spot and strike—not the option’s market premium. Exercising converts the call into ownership of the underlying by paying the strike; the alternative is to retain or sell the call and hold a replicating position.
Put-call parity provides that replication: a call can be represented using the underlying asset and a put, adjusted for the present value of the strike. Comparing this with early exercise highlights the interest earned by delaying payment of the strike. The answer states that buying the asset and put creates the same structure at a lower cost, and relates the savings to the gap between the strike and its discounted value. This is an explanatory response to a specific example, not a general proof covering dividends, transaction costs, or other market conditions that can affect early exercise decisions.
Key ideas
- The interest opportunity cost of delaying exercise is based on the call’s intrinsic value in the example.
- Early exercise of a call means paying the strike to acquire the underlying asset.
- Put-call parity expresses a call through the underlying, a put, and the discounted strike.
- The comparison attributes the cost difference to the benefit of delaying payment of the strike.
- The example does not address dividends, transaction costs, or other conditions affecting exercise.
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# Ideas behind early exercise of American Option
# 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.
- 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.
- 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.
- 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.
## Answer by KaiSqDist (score 4, accepted)
https://quant.stackexchange.com/a/79750
- 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.
- 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:
$$C_t = S_t - Ke^{-r(T-t)} + P_t$$
- 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.
$$Amt \: Saved = K - Ke^{-r(T-t)} > 0.3$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.