When Early Exercise at Option Value Preserves a Call’s Replication
Summary
This discussion considers a European call with an early-exercise feature that lets the holder exercise a portion and receive that portion’s current Black–Scholes value. The proposed replication is to hold the call and sell the corresponding fraction whenever the holder exercises. Under that interpretation, the feature does not change the initial value: the holder receives the value of the option being removed from the replicating position.
The caveat is how the contract defines the value paid at exercise. If the payout is calculated using the initial implied volatility rather than the then-current option value, replication becomes more difficult because future volatility is uncertain. A volatility term structure or smile therefore makes careful term-sheet interpretation important. The answer’s conclusion relies on receiving current option value and the relevant pricing value behaving as a martingale; it does not establish that every early-exercise contract is equivalent to a standard European option.
Key ideas
- A call can be replicated by holding the option and reducing the position when a portion is exercised at current market value.
- Under that payout rule, early exercise at the option’s value does not by itself change the initial price.
- Using initial implied volatility to determine later exercise payments introduces forward volatility risk.
- Contract wording determines whether the proposed replication applies.
- A volatility smile or term structure does not remove the need to identify the exercise valuation rule.
Tags
Full text
# Is this payoff an exotic option or a standard european?
# Is this payoff an exotic option or a standard european?
The writer is selling a european call option with $K=S_{0}$, $S=S_{0}$ ($payoff_{T} = (S_{T} -k)_{+}$), time to maturity $T$, with a twist:
With some probability, $Pr(l) \geq 0,$ $\forall t,$ $0 < t < T$ the option holder may 'exercise' some portion of the option $n$, $ 0 < n <= 1$ and receive $n \cdot BS(S,T-t, \sigma , k, r)$, where $BS(S,T-t, \sigma , k, r)$ is the B-S market value of a euro call option at the time of exercise.
The question I'm wrangling with is this, I believe (and please correct me if I am wrong) is that the Risk-Neutral price of the above option is $BS(S_{0},T, \sigma , k, r)$ regardless of $l$ because a replicating portfolio would be to purchase an option and liquidate $n$ of that option every time the holder exercised.
However, does this remain true when $\sigma_{T-t,Strike-Ratio}$ has a term structure and volatility smile?
My intuition says yes, because the replicating call portfolio still holds but I wanted to confirm.
Thanks!
## Answer by Frido (score 3, accepted)
https://quant.stackexchange.com/a/76329
If the 'exercise' means receiving the option and not its extrinsic value (such as in an American option) then yes I agree with you the replicating portfolio is just the call option. You have to be careful though that when exercised the option is not being exercised at the initial implied vol. Because if so, then in that case the replicating portfolio is not so easy as there is forward vol risk. In other words, the term sheet should be read very carefully.
## Answer by Arshdeep (score 0)
https://quant.stackexchange.com/a/76341
A martingale's immediate exercise value always equals the continuation value (=average value in the future) and therefore one is always indifferent to when it is stopped.
Since the BS price is a martingale this early exercise feature in not consequential.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.