Put payoffs, intrinsic value and time value
◈ 11 cardsA put is worth MAX(K − S_T, 0), its gain capped at K − premium; a premium today splits into intrinsic value and time value; and the four-position grid of max gain and max loss.
The right to sell
A Maritime Ferries put: strike 2.10. The holder has the right to sell Maritime at 45 — valuable when the share is below 45. At expiry:
Maritime at 38: the holder sells at 45 what is worth 38 — payoff =MAX(45-38,0) = 7.00; profit 7.00 − 2.10 = 4.90 a share, $490 a contract. Maritime at 49: the right to sell at 45 a share worth 49 is worthless — payoff 0, profit −2.10 (−210 a contract). The put's breakeven is the strike less the premium: 45 − 2.10 = 42.90 — the share must fall past the strike by the premium.
A capped gain
A call holder's gain is unbounded because a share can rise without limit. A put holder's gain is not: a share cannot fall below zero. The most a put can ever pay is the full strike, 45.00, when the share is worthless, so the holder's maximum gain is K − premium = 42.90 — the same number as the breakeven, by coincidence of the arithmetic, and the paper likes to ask both in one stem. The put writer is the mirror: maximum gain the premium (2.10), maximum loss 42.90.
Intrinsic value and time value — before expiry
The payoff formulas price an option at expiry. Before expiry the premium is more than the payoff would be today, and the difference has a name. Intrinsic value is what the option would pay if exercised now — =MAX(S-K,0) for a call, =MAX(K-S,0) for a put. Time value is the rest of the premium:
The Tamarack 50 call trades at 3.20 with the share at 52: intrinsic =MAX(52-50,0) = 2.00; time value 3.20 − 2.00 = 1.20. The Maritime 45 put trades at 2.10 with the share at 47: intrinsic =MAX(45-47,0) = 0; time value 2.10 — an out-of-the-money option's premium is entirely time value. Time value is the market's price for the chance that the option finishes further in the money before expiry; it rises with time to expiry and with the share's volatility, and it decays to zero at expiry, when premium = intrinsic value. Time value is never negative — an option cannot trade for less than its intrinsic value, because arbitrageurs would buy it and exercise.
The four positions
| Position | Payoff at expiry | Max gain | Max loss | Breakeven |
|---|---|---|---|---|
| Long call | MAX(S_T − K, 0) | unlimited | premium | K + premium |
| Short call | −MAX(S_T − K, 0) | premium | unlimited | K + premium |
| Long put | MAX(K − S_T, 0) | K − premium | premium | K − premium |
| Short put | −MAX(K − S_T, 0) | premium | K − premium | K − premium |
Two formulas and their negatives. A holder's maximum loss is always the premium; a writer's maximum gain is always the premium; the unlimited cell is the short call, and the short call alone.
Recompute
Maritime at 41: payoff =MAX(45-41,0) = 4.00; holder profit 1.90 (190 a contract); writer −1.90. Fill the grid cold: the only unlimited loss is the call writer's; a put's biggest number is K − premium.