Theta and Vega: The Two Leaks

Theta is rent, and it is not linear

Theta is the daily cost of holding an option rather than the shares. On the worked position it is $0.032 a share a day, which is $9.60 across three contracts and $48 over a five-session week in which nothing happens — and since the premium is 7.08% of the share price, the position begins each week a week of decay behind. The useful reframing is that theta is not a fee the market levies arbitrarily: it is the rent on the floor and the leverage. A long option cannot lose more than its premium and controls more shares than the premium could buy, and the two advantages are paid for daily. A trader who resents theta has misunderstood the instrument; a trader who ignores it has mispriced the trade. The important subtlety is that theta is not constant, and it accelerates. Time value decays slowly when a contract has months to run and increasingly quickly as expiry approaches, because the remaining time is what the extrinsic value is pricing and there is progressively less of it. The shape is roughly the square root of time: a contract with four days left loses its value four times faster per day than one with sixteen. That is why a long option position feels comfortable for weeks and then alarming in its final fortnight, and it is why the same position can be held patiently in one month and not the next. It is also asymmetric around the strike: the decay is fastest at the money and slower for options that are far in or out of the money, because the in-the-money intrinsic part is not decaying at all and the far out-of-the-money part has little left to lose. Two practical consequences follow. The first is that a long option position has a *time budget* rather than a price target: it needs the movement to happen before the decay consumes the edge, which is a different and stricter requirement than being right eventually. The second is that rolling to a later expiry is never free — the new contract prices the same decay, simply over a longer window, and the roll usually pays a spread on both legs. Rolling is therefore a decision to buy more time at the market’s price for it, which is exactly the trade the seller on the other side is making. A week of nothing — Three contracts at $3.40: $1,020 of premium at risk · Theta: $0.032 a share a day: $9.60 a day, $48 a week ← · Vega: $0.11 a share per point: $33 a point across the position · Implied volatility falls 5 points: −$165, or more than three weeks of decay in one session ← Decay accelerates roughly with the square root of time remaining: four days left is about four times the daily decay of sixteen days left. That is why the last fortnight of a long option feels different from the first.

Vega is the price of movement, and events are where it shows

Vega is what the position is worth per point of implied volatility, and it is the cleanest way to see that an option is a trade in volatility rather than in direction. The worked position has vega of $0.11 a share per point, so $33 a point across three contracts: a two-point rise in implied volatility is worth $66, and a five-point fall costs $165. Now put a calendar beside it, because that is where the number becomes concrete. Earnings are a dated, uncertain event, so the implied volatility of options expiring after the print contains an event component, and the option is priced as if a larger move were coming. The market is right on average and wrong on the day, and the important part is mechanical rather than predictive: once the print is out, the uncertainty is resolved and that component drains out of the premium. An option that was paying for a 9% expected move can be left holding a 3% one, and the difference is handed back immediately, which is the “implied volatility crush” that costs holders money precisely when their direction call was correct. That gives the honest reading of the good-day loss, and it generalises past earnings. A long option position is profitable when the movement that arrives exceeds the movement that was priced — not when the direction is right, and not even when the stock moves a lot in the right direction. The premium carries a forecast of the size of the move (the expected move from O8) and a price for the uncertainty (vega), and if the realised move comes in under the forecast, or if the uncertainty was resolved without a proportional change in price, the position can be right and lose. The most common retail version is buying options shortly before a known event at an elevated implied volatility, which stacks both leaks against the holder: the highest theta available and the largest vega exposure to a decline that is scheduled. Two structural facts make the asymmetry worse than it sounds. Implied volatility usually exceeds subsequent realised volatility — the variance risk premium — which is compensation to the seller for accepting the tails and a headwind for a systematic buyer of options. And vega is itself concentrated: it is largest for at-the-money options and for longer expiries, where the price of the uncertainty has the most time to work, and smallest for short-dated far out-of-the-money contracts, where the whole premium is a lottery and the vega is correspondingly small. That is why the same volatility view has to be expressed with the right expiry as well as the right strike, and why a trader who believes volatility is cheap should be buying time rather than the nearest contract. • Vega is the price of movement, and events are where it shows: earnings premium drains when the print lands. • A long option pays when realised movement beats the movement that was priced, not when direction is right. • Implied usually exceeds realised — the variance risk premium — which is a headwind for a systematic buyer. • Vega is largest at the money and for longer expiries, smallest in short-dated lottery tickets. • Buying options into a scheduled event at elevated implied volatility stacks the highest theta against the holder. The trap is buying the calendar rather than the thesis. A trader who expects a large move because earnings are coming has made the market’s own observation: the expected move is already in the price, and the only way the position pays is if the realised move exceeds the implied one. The question to ask before the trade is not “will the stock move” but “will it move more than the band the premium is charging, and will the volatility hold long enough for the direction to matter”.

Theta against vega: the break-even move

Theta and vega are usually taught as two separate sensitivities, and their ratio is the number that describes a position: how far the underlying has to move, per day, for the passage of time to be paid for by the movement. Computing it converts the two Greeks into a single question with a numerical answer. The arithmetic comes out of the hedge in the delta lesson. A long option position that is delta-hedged earns approximately half its gamma times the square of the day’s move, and it loses its theta over the same day. Those two terms balance at a specific move — square root of twice the daily theta divided by gamma — and that move is the position’s break-even for the day. Compare it with what the stock has actually been doing and the position’s economics become visible: an option whose break-even move is half a percent a day, bought on a stock that typically moves one percent, is being paid for; the same option on a stock that moves a quarter of a percent is renting a decay it cannot recover. The same comparison can be made without the Greeks, and the second version is the one that survives a quick glance. The **implied move** for a period is read straight from the at-the-money straddle price: the market’s own estimate of the distance the stock needs to travel, in either direction, to make owning the option worthwhile. Comparing that figure with the distance the stock has historically travelled over the same period is the same test as the Greek ratio, expressed in units a learner already has. When the implied move is much larger than recent realised movement, the option market is charging for something the stock has not been delivering, and the seller of the option is being paid for that gap. Two refinements matter because they explain why the answer changes with the calendar. The first is that theta is not linear: the decay of the remaining time value accelerates as expiry approaches, so a break-even move computed from today’s theta understates what the same position will need next week. A position that pays for itself this month can be impossible to fund in the final week, which is the mechanism behind the changes in behaviour that surround expiration. The second is that implied volatility has its own calendar: it rises into scheduled events and falls after them, so a vega exposure held through an earnings date is a position on the event rather than on the trend. The pattern also has a weekly rhythm, since options cannot decay over a closed market — which is why the decay of a Friday-to-Monday holding is compressed into fewer sessions than its price suggests. The practical use of the ratio is as a filter rather than a signal. Before buying time, compute the break-even move and check it against the recent realised range: if the market is implying more movement than the stock has recently delivered, the option is expensive and a seller’s position on that gap is well paid. Then check the calendar for the events that will change both terms, and decide whether the exposure is to the trend or to the event. Those two checks take a minute and they replace the vague sense that options “lose money over time” with a number that says whether this one does. • A position’s break-even move is the daily theta set against the gamma it collects. • The implied move from the at-the-money straddle does the same job in familiar units. • Theta accelerates into expiry, so this week’s break-even understates next week’s. • Implied volatility has a calendar of its own: events and weekends move both terms. One line that makes it usable: divide the at-the-money straddle price by the spot and annualise it. The result is the market’s expected move for the period, and it is directly comparable with the stock’s realised volatility over the same window — which is the whole question in one division.

Sources

• Theta, vega and the term structure of volatilityNatenberg, “Option Volatility and Pricing”
• Earnings implied moves and the volatility crushCBOE Education — Trading Earnings with Options
• The variance risk premium: why implied usually exceeds realisedCarr & Wu, “Variance Risk Premia” (Review of Financial Studies)