Theta
Also written θ · Option theta
The option Greek that measures time decay — the change in an option's premium for a one-day decrease in time to expiry. It is negative for a long option, call or put alike.
In plain language
Theta answers a single question: what does one day cost me?
If an option has a theta of 1.2, the premium falls by Rs 1.20 for each day that passes, with everything else held still. The buyer pays that; the writer collects it. It is the only Greek whose direction is known in advance, because the one input that is certain about an option is that time will run out.
Theta is usually negative for a long option, whether it is a call or a put. Buying an option of any kind means being short time.
How it works
Theta measures the decay of time-value, so it inherits time value's shape:
- Largest at the money, where the whole premium is time value and the outcome is least decided.
- Smallest deep in the money, where most of the premium is intrinsic and immune.
- Accelerating into expiry. Theta is not a constant; it grows in magnitude as the contract ages, so the last few sessions destroy the most.
Among the five Greeks, theta is the one the writer wants on his side. A short-straddle, a short-strangle and a covered-call are all, mechanically, positive-theta positions: they earn if nothing happens.
The counterweight is gamma. A position with the largest positive theta also carries the largest negative gamma, so the seller who is being paid to wait is also the one who gets hurt worst by a sudden move. The workbook's risk table says the same thing in words: the writer's gain is limited to the premium and his loss is unlimited.
The formula
Theta = Change in option premium ÷ Change in time to expiry
Rupee decay on a position = Theta × Lot size × Number of lots × Days
Sign convention:
Long call, long put → theta negative (you pay)
Short call, short put → theta positive (you receive)
A worked example
A trader is short 4 lots of a call option with a theta of 1.2 and 5 days to expiry. The contract size is 50 and the premium received was Rs 95 a unit.
Premium collected = 95 × 50 × 4 lots = Rs 19,000
Daily theta gain = 1.2 × 50 × 4 = Rs 240 a day
Over 5 days = Rs 1,200
So Rs 1,200 of the Rs 19,000 — about 6.3% — accrues to the writer purely from the calendar, if the index does not move.
Now make the index move. Suppose the option's delta is 0.50. A 5-point rise in the index costs the short position
0.50 × 5 × 50 × 4 = Rs 500
and a 5-point rise is a move of less than 0.03% on an index near 17,562.
One day of theta buys the writer protection against roughly a 2.4-point move. Beyond that, delta wins. That single comparison is the whole risk profile of a premium-selling strategy: a small, certain daily income set against a large, uncertain, instantaneous loss — and it is why the exchange takes margin from the writer and nothing from the buyer.
Why NISM asks about it
Chapter 16.7 (Basics of Option Pricing and Option Greeks) defines theta and gives the "theta of 1.2 with 5 days to expiry means the price declines by Rs 1.20 each day" illustration. Expect a definition-matching question across the five Greeks, a sign question — theta is negative for a long option — and the straight arithmetic above.
Common exam traps
- Theta is negative for a long put as well as a long call. It is not a directional Greek.
- Theta is per day, not per year. Multiply by days and by the lot size before quoting a rupee figure.
- Theta is not constant. Treating it as a fixed daily rate understates the decay in the final week badly.
- Do not confuse theta with vega. Theta measures the passage of time; vega measures the level of volatility. Both sit inside the time value, and they can pull opposite ways on the same day.
- Positive theta is not a strategy. It is payment for carrying the unlimited-loss side of the contract.
- Deep ITM options have small theta, because there is little time value left to decay.
Where this is taught
- Series VIII · Chapter 4: Introduction to Optionsintroduced here
- Series V-D · Chapter 16: Introduction to Optionsintroduced here
- Series XVI · Chapter 4: Commodity Optionsintroduced here
- Series IV · Chapter 4: Exchange Traded Interest Rate Optionsintroduced here
- Series I · Chapter 4: Exchange Traded Currency Optionsintroduced here
- Series V-D · Chapter 21: Exchange Traded Interest Rate Options
Related terms
- DeltaThe change in an option's premium for a one-rupee change in the underlying — the first and most used Greek, and the hedge ratio that says how much underlying to hold against an option position.
- GammaThe rate at which an option's delta changes for a one-unit change in the underlying — the second-order Greek, and the reason a delta hedge stops working as soon as the market moves.
- RhoThe option Greek that measures interest rate sensitivity — the change in an option premium for a one percentage point change in the risk-free rate. It is positive for calls and negative for puts.
- Time decayThe steady loss of an option's time value as expiry approaches — the reason options are called wasting assets and the reason the workbook says option sellers hold a structural advantage.
- Time valueThe part of an option premium that is not intrinsic value — what the buyer pays for the possibility that the underlying moves further in his favour before expiry. It falls to zero on expiry day.
- VegaThe option Greek that measures sensitivity to volatility — how much an option premium changes for a one per cent change in the implied volatility of the underlying. It is positive for a long call and a long put alike.
- Covered callHolding the underlying in the cash market and writing a call against it — a way of earning premium income from a holding, at the cost of capping the gain above the strike.
- Wasting assetAn option, whose time value shrinks towards zero every day it is held and is worth nothing at expiry — so an option buyer loses money simply from the passage of time.
- At-the-moneyAn option whose strike price is closest to the spot price, so exercising it immediately would produce neither a gain nor a loss — the strike where the whole premium is time value and uncertainty peaks.