Long put: a drop is not enough; it also has to come fast
See what this put costs in time each day at different prices, and where the European model and an American-style put part ways deep in the money.
All figures are Black-Scholes model estimates, not market quotes or guarantees.
US stock and ETF options are American-style and can be exercised at any time. This model treats every option as European and does not model early exercise or assignment.
P&L by price and day (model estimate, current price 100, IV 30%, 30 days)
| Underlying price | Day 0 | Day 7.5 | Day 15 | Day 22.5 | Day 27 | Day 30 |
|---|---|---|---|---|---|---|
| 90 | +$688 | +$676 | +$668 | +$667 | +$670 | +$674 |
| 95 | +$292 | +$261 | +$226 | +$190 | +$174 | +$174 |
| 100 | $0 | −$42 | −$92 | −$159 | −$219 | −$326 |
| 105 | −$179 | −$215 | −$255 | −$299 | −$322 | −$326 |
| 110 | −$270 | −$292 | −$311 | −$324 | −$326 | −$326 |
Structure and P&L at expiration: why max profit has a cap
A long put has one leg: you buy one put with a strike of 100 that expires in 30 days. Every number on this page assumes a current price of $100, implied volatility (IV) of 30%, a 4% risk-free rate, no dividends, 30 days to expiration, and one contract (100 shares).
- Net debit: $326.29 ($3.26 a share). This is the max loss. The premium is paid once, up front.
- Breakeven: $96.74, the strike minus the premium per share.
- Max profit: $9,673.71. The price can’t fall below zero, so the put is worth at most the $100 strike per share; subtract the premium paid.
Unlike a short stock position, the put needs no borrowed shares and has no open-ended loss if the price rallies. At entry the position has a Delta of −46.76 (it moves like about 47 shares sold short) and an instantaneous Theta of −$5.15 a day.
Time profile: five lines from $90 to $110
At the strike, the put behaves almost like a long call. The $100 row goes from $0 to −$42 on day 7.5, −$92 on day 15, −$159 on day 22.5, −$219 on day 27, and −$326 at expiration; 51% of that loss lands in the last 7.5 days.
Above the strike, time takes what is left of the premium: at $105 the position falls from −$179 to −$326. At $95 it falls from +$292 to +$174, the intrinsic value of $500 less the $326 paid.
The $90 row is different. The P&L slips from +$688 to a low of about +$666 near day 19.8 (+$668 on day 15, +$667 on day 22.5), then climbs back to +$674 at expiration. The last section explains why.
Not quite a mirror of the long call: $0.33 cheaper, $1.09 less decay a day
With the same strike, 30 days, and IV, the put costs $3.26 and the call $3.59. When the strike equals the current price, the call is worth more than the put by roughly the interest on the strike over 30 days, which at 4% is about $0.33 a share.
The same interest shows up in Theta. The put’s day-0 Theta is −$5.15 a day against −$6.24 for the call, a gap of $1.09 a day per contract: about the daily interest on $10,000 at 4%.
Sensitivity to IV, on the other hand, is identical. Both have a Vega of +$11.40 per point, and a 5-point IV drop on day 0 costs each of them $57.
Time helps deep in the money: the European model’s edge and American early exercise
In the model, the next-day change turns positive below a certain price: $85.51 on day 0, $88.65 on day 15, $94.19 on day 27, and $96.80 with 1 day left. A European put can only pay the strike at expiration, so deep in the money the model discounts it at 4% and values it below its intrinsic value. As expiration approaches, the discount shrinks and the value climbs toward intrinsic. At $90 on day 15, the model price is $9.94 against $10 of intrinsic value.
US stock options are American-style: the holder can exercise at any time and collect the intrinsic value immediately. So this “time helps” zone is a feature of the model, not money waiting to be collected. The Lab marks it with an early exercise notice and hatching on the time-effect map.
Try it in the Lab: three experiments
Drag the day slider and watch the $90 line dip and recover: +$688 on day 0, a low of about +$666 near day 20 (where the link opens), and +$674 at expiration.
Open in the LabOpen the time-effect map. At the bottom, below the time-neutral line, the next-day change is positive: below $88.65 on day 15. That is the zone where the Lab shows the early exercise notice.
Open in the LabThe link switches to the long call template in the same view. The call has no positive zone at any price or day, because a European call on a stock without dividends is never worth less than its intrinsic value.
Open in the Lab
Three common misconceptions
MisconceptionIf the price falls, you make money.
What the numbers showA drop to $97 (−3%) at expiration still shows −$26, because the breakeven is $96.74.
MisconceptionA put is just a call in reverse, symmetric in every other way.
What the numbers showWith both at the money and 30 days out, the put costs $3.26 and the call $3.59, and their day-0 Theta is −$5.15 and −$6.24 a day. The difference comes from the 4% interest rate.
MisconceptionOnce the put is deep in the money, time is on your side.
What the numbers showIn the model, time helps below $85.51 on day 0 and below $96.80 with 1 day left, but only because a European put can’t be exercised early. The holder of an American-style put can exercise at any time, and the Lab flags this zone.