Valuing a Put When Its Premium Is Paid at Exercise
Summary
The note explains how to value a put whose premium is deferred until exercise. It rejects treating the arrangement as a standard put with a reduced strike, because that changes the payoff: a conventional option buyer can lose the upfront premium even when the put finishes slightly in the money. Instead, it treats the deferred premium as the future value of an otherwise equivalent premium paid at inception, using an interest rate over the period to exercise. The resulting expiry payoff is the ordinary put payoff less that accumulated premium.
For a European put exercisable only at maturity, the note applies this adjustment to the Black–Scholes put price. It also points out that the rate used to accumulate the deferred payment may differ from the risk-free rate used in option valuation, since deferral is economically a loan of the premium. The discussion is limited to this setup and does not fully specify how to price American exercise or determine a market borrowing rate.
Key ideas
- Deferring an option premium changes its value through the time value of money.
- A reduced strike does not reproduce the payoff of a standard put with a deferred premium.
- The exercise-date premium can be represented as the initial premium accumulated at an agreed rate.
- A European put valuation can incorporate the deferred payment by applying this adjustment at maturity.
- The rate for the premium deferral may differ from the risk-free rate in the option model.
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Full text
# Put option price where premium is paid at exercise
# Put option price where premium is paid at exercise
Let's say there is a put option but the premium is paid at exercise.
This means the strike must be seen as the strike + premium.
How would you go about solving this?
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/71352
For a put option the modified strike would be ‘strike’ - premium. But, there is no way to solve for the premium, because the whole thing represents a free option to the holder. (Proof: there is no scenario where the holder loses money).
## Answer by D Stanley (score 1)
https://quant.stackexchange.com/a/71353
I don't think your modelling of "strike - premium" is correct.
If I bought a "normal" put option with a strike of 10 for 2, and the underlying was at 9 at expiry, I would have lost 1 (made 1 on the option but paid 2 upfront). That's a different payoff than adjusting the option to have a strike of 8.
The only difference between this option and a "standard" option is that the premium deferred until exercise ($P_E$) would just be the future value (at some interest rate) of the present normal premium ($P_0$):
$$ P_E = P_0 \times e^{rt_E} $$
Where $r$ and $t_E$ are the interest rate and time from initiation to exercise, respectively.
The payoff at expiry would then be:
$$ \max(0, K-S) - P_0 \times e^{rt_E} $$
If this were a European option, and could only be exercised at expiry, you could apply that to the black-scholes formula for a put option (setting $t_E = T$), and would get:
$$ \begin{eqnarray*} P_E &=& (KN(−d2)e^{−rT} − S(0)N(−d1)) \times e^{rT}\\ &=& KN(−d2) − S(0)N(−d1)e^{rT} \end{eqnarray*} $$
Note that it's possible for a different interest rate to be used to calculate the premium future value than the present value of the option, since the rate used in the option value is a "risk-free" rate, while the option holder is essentially "loaning" you the premium, and may charge a different interest rate for such a benefit.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.