Theta and vega

OpeningA week when nothing happened

You and three friends each buy one call with a strike of 100 on the same day. The stock is at $100 and IV is 30%. The only difference is time to expiration: 7, 30, 90 and 365 days. A week later, the stock is still at $100.

The stock didn’t move a cent, and all four of you lost money. The 7-day call went straight to zero: at expiration an ATM option has no intrinsic value. Look only at the “IV unchanged” column and the shorter the term, the bigger the loss. That is Theta. Now look at the column where IV fell 5 points: the 365-day call lost an extra $1.90 per share, or $205 per contract, the most of the four; the 30-day call lost only $96. That is Vega.

In one sentence: short-dated options fear time, and long-dated options fear volatility. This chapter first takes these two sensitivities apart, then adds the interest-rate sensitivity, Rho, and a few second-order Greeks, and finally reads them all together in one complete table.

5.1Theta: the rent you pay every day

Theta is the change in an option’s price for each day that passes, with everything else held constant. Theta is negative for the buyer and positive for the seller. It eats only time value and never touches intrinsic value.

Break down a 30-day ATM call (stock and strike both 100, IV 30%): daily Theta is −0.0624, of which the volatility term is −0.0570 and the rate term −0.0054. For the put at the same strike, the volatility term is the same −0.0570, but the rate term becomes +0.0055, for a total of −0.0515. The put’s rate term is positive because when you exercise a put you receive K, and the closer you are to expiration, the higher the present value of that cash.

The volatility term is more than 90% of the total, so Theta is essentially “the fee you pay for movement.” As derived in Chapter 4, it is tied to Gamma by the Black-Scholes equation: Θ ≈ −½σ²S²Γ. Buy an option and you get positive Gamma, so you pay Theta. We won’t repeat that here; see Chapter 4, Section 4.6.

Three details that are easy to miss

• Theta is an instantaneous rate. A 30-day ATM call actually loses 0.0629 over one day, slightly more than the 0.0624 that Theta gives, because Theta itself grows over time. Estimate ten days of decay as 0.0624 × 10 = 0.62, and the actual figure is 0.68. The closer to expiration, the bigger this underestimate.
• The higher the IV, the higher the rent. The volatility term is proportional to σ. At IV of 15%, 30% and 60%, a 30-day ATM call has Theta of −0.034, −0.062 and −0.119, scaling up almost in proportion. A high-IV option isn’t just expensive; its daily rent is higher too.
• Theta can be positive. Chapter 3 noted that a deep ITM European-style put gets cheaper the longer its term. Seen another way, its Theta is positive: a 30-day European-style put with the stock at $100 and a strike of 150 has Theta of +0.0164/day. An American-style put would be exercised early, so this doesn’t happen.

5.2The shape of time decay

An ATM option’s time value decays with √T, falling faster the closer it gets to expiration. An OTM option starts losing big chunks early, and its Theta peaks while expiration is still far away.

The 0.4 rule from Chapter 3 tells us that an ATM call ≈ 0.4 × S × σ × √T. Price is proportional to √T, so daily decay is proportional to 1/√T: cut the time remaining to a quarter and Theta doubles. Theta is −0.038 at 90 days, −0.062 at 30 days, −0.124 at 7 days and −0.319 on the last day.

The same relationship gives a mental-math rule: an ATM option loses about 1 / (2 × days remaining) of its value each day. Price is proportional to √T, and its relative rate of change with respect to T is 1/(2T). With 30 days left that is about 1/60 ≈ 1.7% a day (Black-Scholes gives 1.74%); with 7 days left, about 7.1% (actual 7.3%); with 365 days left, about 0.14% (actual 0.15%). The actual values run slightly higher because of the rate term.

Split the 90 days into “the first 60 days” and “the last 30 days” and see how much time value each period eats:

• 100 call (ATM): loses 44% in the first 60 days and 56% in the last 30. The later it gets, the faster it falls.
• 105 call: loses 61% in the first 60 days and 39% in the last 30.
• 110 call (about one standard deviation OTM): loses 76% in the first 60 days alone, leaving only a quarter for the last month.

An OTM option’s value comes from “the chance that the stock reaches the strike.” As time runs out, that chance falls much faster than it does for an ATM option, so its value evaporates in big chunks early on. Near expiration it is already close to 0, so there is little left to lose each day.

When an OTM option’s theta peaks

There is a rule you can derive: an OTM option’s Theta is largest when “the strike is about one standard deviation from the current price.”

How to use it

• Buyers: try to keep your holding period out of the steepest stretch of Theta. For ATM options, avoid the last two or three weeks; when buying OTM options, pick an expiration somewhat beyond the point where “the strike is one standard deviation away.”
• Sellers: selling ATM options in their last few weeks collects accelerating Theta, but also carries the most Gamma (Chapter 4, Case 1). If you sell OTM options, you actually collect the most Theta between 30 and 60 days out.
• In percentage terms: in the final week, OTM options lose the largest share of their value each day. With 7 days left, the 105 call loses 24% a day and the ATM call only 7%. A small dollar loss doesn’t mean a small percentage loss.

5.3Weekends and holidays: two clocks

Black-Scholes has a single time variable. Count calendar days and Theta is charged for both weekend days; count trading days and the weekend is free, but each trading day is charged more. Over a week the total decay is the same; only its distribution differs.

This course uses calendar days, 365 to a year. The other convention is trading days, about 252 to a year, with weekends and holidays not counted as time. The same price corresponds to slightly different IV numbers under the two clocks: to keep total variance σ²T equal, trading-day IV = calendar-day IV × √(calendar days / 365) ÷ √(trading days / 252).

Example: it is Friday’s close, and the option expires on the Friday 4 weeks from now, with 28 calendar days or 20 trading days left. Priced at a calendar-day IV of 30%, it has an IV of 29.49% under the trading-day convention, and under both today’s price is $3.464. With the stock not moving, here is the price at each close over the next week:

Which clock the market actually uses

Stock prices move almost entirely during trading hours; over a weekend there is news risk but no continuous price movement. So for describing volatility, the trading-day clock is closer to reality; for computing interest, calendar days are correct. Many market makers build the weekend’s decay into their quotes as early as Friday, typically by lowering the volatility they price with on Friday, or by simply measuring time in trading hours. The result:

• Option quotes on Friday afternoon often look “cheap” by a calendar-day model, and Monday’s open doesn’t show the big drop the calendar-day model expects.
• As for the idea of “sell Friday, buy back Monday, and pocket two days of weekend Theta for free”: most of that profit is already priced in before Friday’s close. What’s left is the insurance premium you collect for weekend gap risk.
• Holidays work the same way: market holidays such as Thanksgiving and Christmas don’t count as time on the trading-day clock, and options price them in ahead of the holiday.

Distortion in very short-dated options

The shorter the term, the bigger the gap between the two clocks. Suppose a stock’s true volatility, measured in trading days, is 30%. On Friday you buy an option expiring Monday: 1 day left on the trading-day clock, 3 days on the calendar-day clock. If your platform backs out IV using calendar days, it will show 30% × √(1/252) ÷ √(3/365) ≈ 20.8%. Buy an option on Monday that expires Tuesday, and it will show 30% × √(365/252) ≈ 36.1%. Same stock, same true volatility: very short-dated IV looks low on Friday and high midweek. That comes from how time is counted, not from market sentiment. When comparing IV on short-dated options, first check which clock your platform uses.

5.4Vega: what one vol point is worth

Vega is the change in an option’s price when IV moves by 1 percentage point. The longer the term and the closer to the money, the larger the Vega; but as a percentage of price, OTM options are more sensitive to IV.

Four observations:

1. Vega is proportional to √T. The 365-day ATM Vega (0.383) is 7 times the 7-day figure. This is exactly the reverse of Theta, which is proportional to 1/√T. The longer the term, the more the position behaves like a volatility position; the shorter the term, the more it behaves like a time position (Case 1).
2. The peak sits slightly above the current price. The strike with the largest Vega is Se^(r+σ²/2)T: 100.7 at 30 days and 108.9 at one year. So the one-year 110 call has a larger Vega (0.399) than the ATM call (0.383).
3. In dollars, look at ATM; in percentage terms, look at OTM. The 30-day 110 call has a Vega of only 0.068, but that is 10.3% of its price. When IV rises from 30% to 35%, it goes from 0.655 to 1.022, up 56%; over the same move the ATM call goes from 3.591 to 4.161, up only 16%. If you want to bet a small amount on rising IV, OTM options give you the most leverage; by the same token, they also shrink the most when IV falls.
4. ATM Vega barely changes with IV. At IV of 15%, 30% and 60%, a 30-day ATM call has Vega of 0.1138, 0.1140 and 0.1137. The OTM 110 call, by contrast, goes from 0.012 to 0.068 to 0.103. That difference is Volga, covered in Section 5.7.

The first thing to do with Vega is to ask, before you buy: “If IV goes back to this stock’s normal level, how much do I lose?” Suppose a 30-day ATM call is at IV 45% now and the stock’s usual IV is 30%. IV reverting alone would cost about 15 × 0.114 = 1.71/share (a full repricing also gives 1.71), equal to 19 days of this option’s Theta (−0.091 a day at IV 45%). Weigh this potential loss together with your view on direction.

An IV point is not the same across terms

Vega answers “what is a 1-point move in IV worth,” but IV at different expirations doesn’t move by the same amount. When the market gets a scare, 30-day IV might jump 10 points while one-year IV moves only three or four: short-term IV is like the waves, long-term IV like the tide. Comparing the one-year Vega of 0.383 directly with the 30-day 0.114 overstates the IV risk of long-dated options.

A common fix is “time-weighted Vega”: Vega × √(30 / days), which assumes that IV at a T-day term moves √(30/T) times as much as 30-day IV. On this basis, ATM options at 7, 30, 90 and 365 days come out at 0.114, 0.114, 0.113 and 0.110, almost the same. In other words, when 30-day IV rises 10 points (ATM call +1.14) and one-year IV rises only 2.9 points, the one-year ATM call gains +1.10, a similar dollar impact; if one-year IV also rose the full 10 points, it would gain 3.83. The square-root relationship is only a rule of thumb, and how closely short- and long-term IV move together varies a lot from period to period. We look at this more closely with term structure in Chapter 6.

Vega of a position

Position Vega = per-option Vega × 100 × number of contracts × direction. Long options have positive Vega; short options, negative. The two legs of a spread partly offset: a 30-day 100/110 call spread (buy the 100, sell the 110) has a Vega of 0.114 − 0.068 = 0.046, only 40% of a single long ATM call. That is why many traders use spreads instead of single legs before an event: they keep part of the directional exposure and cut more than half of the IV risk.

5.5IV crush

IV gets bid up before an event and collapses back to normal as soon as the event passes. If the collapse is big enough, a buyer who got the direction right still loses money.

Take an earnings scenario (the IVs are hypothetical): the stock is at $100 and reports earnings tonight. ATM options expiring in 7 days are at IV 80%: the call is $4.454, the put $4.378 and the straddle $8.832. Tomorrow, with 6 days left, IV drops back to the stock’s usual 35%.

First, read the price the market puts on earnings

Variance adds up day by day. Total variance over 7 days = the variance of 6 ordinary days + the variance of earnings day:

Reading it row by row:

• Up 3%, and the call still loses 16%. With IV unchanged, the same rise would have made $1.41/share. The call’s breakeven is +3.90%: any smaller rise loses money even though the direction was right.
• Breaking down the loss with the stock unchanged. −2.632 = one day of time decay, −0.333, + the IV drop, −2.299. The latter ≈ the Vega with 6 days left, 0.051 × 45 points = 2.30. Multiplying by the Vega at entry (0.055) would overstate it a little, because Vega also shrinks over time.
• The straddle needs a move of more than 8.7% to break even. The breakevens are −8.83% and +8.65%, close to the 8.1% mean absolute move the market priced. Buying the straddle is a bet that “this move will be even bigger than the market expects.”

IV creep before an event, and deferred theta

IV climbs all the way into an event, partly out of fear and partly from pure arithmetic. The event-day variance is a fixed chunk; the fewer days remain, the more of that chunk falls on each day, so annualized IV naturally rises. Keeping the setup above (usual IV 35%, earnings-day standard deviation 10.1%), take an ATM call that expires 3 days after earnings and assume the stock never moves:

From 30 days before earnings to 1 day before, the option loses only 1.69 over 29 days, an average of 0.058 a day, far less than the “constant IV” Theta the platform shows. Once earnings pass, it drops 2.97 overnight, 70% of its remaining value. The time value of the event day doesn’t bleed away early; it is “stored” in IV and paid out all at once when the event happens. Seen this way, IV crush is simply deferred Theta.

Two corollaries: the high Theta readings you see before an event overstate daily decay, because they assume IV stays constant; and if you buy an event option a week or two early, you pay little day-to-day Theta. The real cost is that last night.

What to do about it

• Use a spread: the two legs’ Vega partly offsets (Section 5.4), so IV crush does much less damage. The price is a capped payoff.
• Go further out in time: the event’s variance is spread over more days, so a longer-dated option’s IV is lifted by only a few points. In the same setup, a 60-day ATM call has a pre-event IV of about 42.8% and loses 18% the next day if the stock doesn’t move; the 7-day call loses 59%.
• Use a different instrument: if your view is on direction rather than volatility, consider the stock itself, or enter after the event.
• Take the other side: sellers collect exactly this IV premium, but they carry the gap risk. Chapter 10 is devoted to choosing structures for event trades.

5.6Rho, and LEAPS when rates are high

Rho is the change in an option’s price when interest rates move by 1 percentage point. It is proportional to term: negligible for a 30-day option, but for a two-year LEAPS it is worth about two months of Theta.

• Short-dated options: when rates rise from 4% to 6%, a 30-day call gains only 0.082, a little more than one day of Theta. You can safely ignore it.
• LEAPS: for the same change in rates, a 2-year call gains 1.84 and the put loses 1.78. A 2-year ATM call has Rho of $90.6 per contract and Theta of $1.57 a day: a 1-percentage-point move in rates equals 58 days of time decay.
• Low-rate vs. high-rate environments: at a 1% rate, a 2-year ATM call and put differ by only $2; at 6%, by $11. The same LEAPS call costs noticeably more when rates are high, and the put noticeably less.

Dividends: the other “interest rate” in LEAPS

Dividends push option prices the opposite way from interest rates: shareholders receive the dividend, call holders don’t. For a 30-day option, a 2% dividend yield makes the ATM call only 0.09 cheaper (Chapter 3, Table 3-2); for a 2-year ATM call, as the dividend yield goes from 0 to 1% to 2%, the price falls from 20.28 to 19.00 to 17.78, while the put rises from 12.59 to 14.01. When you use a LEAPS call in place of a high-dividend stock, the two years of dividends you give up are already deducted from the premium; if the company unexpectedly raises or cuts its dividend, the LEAPS price changes with it.

Falling rates while you hold LEAPS cost you money: if rates drop from 6% back to 4%, a 2-year ATM call loses $184 per contract without the stock or IV moving at all. When you hold LEAPS for a long time, Rho is a risk to watch, just like Delta and Vega.

5.7Second-order Greeks

The first-order Greeks themselves change. Vanna links Delta and Vega, Charm links Delta and Theta, and Volga describes how Vega itself changes with IV.

Vanna: two readings of one number

Chapter 4 showed that the Delta of a 30-day 110 call rises with IV, by about 0.009 for each 1-point rise. Look at it the other way: when the stock goes from 100 to 101, this option’s Vega goes from 0.0675 to 0.0763, an increase of 0.0087. Computed by central differences, both numbers are 0.00873, exactly equal. This is no coincidence: both are the cross second derivative of price with respect to S and σ, and the order of differentiation can be swapped.

In practice: as the stock moves toward an OTM option’s strike, the option’s IV exposure grows. An ATM option’s Vanna is close to 0; the ITM 90 call has negative Vanna (−0.0069), so its Vega actually shrinks as the stock rises.

Charm: theta and delta tell the same story

After one day, the 110 call (30 days) sees its Delta fall from 0.152 by 0.0047. At the same time, for every $1 the stock rises, its daily Theta takes an extra 0.0047. Again, this is the same cross derivative. The ITM 90 call is the mirror image: its Delta drifts toward 1 by 0.0033 a day. Chapter 4 covered the expiration-day hedging flows that Charm creates; what we add here is its link to Theta: the more sensitive a position’s Theta is to the stock price, the faster its Delta drifts over time.

Volga: OTM options gain “extra” when IV spikes

Vega extrapolates linearly from a 1-point change in IV. An ATM option’s price is almost a straight line in IV, so the extrapolation is accurate; an OTM option’s price curves upward in IV, so when IV jumps, the actual gain far exceeds what Vega predicts.

When IV doubles from 30% to 60%, the 85 put’s actual price is 2.5 times the Vega estimate. This is one reason far-OTM puts serve as crash insurance: in a crisis they benefit from Delta, Gamma, Vega and Volga all at once. It is also why sellers of far-OTM options lose far more than expected in a crisis: negative Volga means that the more IV rises, the bigger your Vega exposure gets. To put a number on it: sell 10 contracts of the 30-day 85 put and collect $81; with the stock unchanged and IV doubling from 30% to 60%, the loss is $1,328, while the Vega estimate is only 0.0164 × 30 × 1,000 = $492. Stress-testing a short position with Vega understates tail risk by more than half. You have to reprice it in full.

Veta: vega ages too

A 30-day ATM call’s Vega falls from 0.1140 to 0.1121 in one day. If you hold a long-term positive-Vega position, your IV exposure shrinks day by day even if IV doesn’t change. That is why Section 5.5 used “Vega with 6 days left” to estimate the IV-crush loss.

5.8Reading a full Greeks table

Read a Greeks table in three steps: first use the signs to judge the shape of the position, then turn each number into a dollar scenario, and finally use the second-order terms to check how those numbers will change.

The example is a common diagonal spread (often called a “poor man’s covered call”): with the stock at $100 and IV 30%, buy 5 contracts of the 365-day 85 call (in place of the stock) and sell 5 contracts of the 30-day 105 call (to collect rent).

Line by line

1. Delta +241: directionally, this is like holding 241 shares, with $10,343 of capital tied up. Buying 241 shares would take $24,100.
2. Gamma −15.9: for every $1 the stock rises, Delta drops by 15.9 shares, so the more it rises, the less the position acts like stock; on the way down Delta grows instead, so the further it falls, the faster you lose. Negative Gamma means big moves in either direction are unwelcome. It comes from the short-dated call you sold.
3. Theta +$18/day: you net $18 a day. The short leg pays $27 a day; the long leg costs $9 a day.
4. Vega +$91/point: each 1-point drop in IV loses $91, about 5 days of Theta. The long leg’s Vega is far larger than the short leg’s (Section 5.4: Vega is proportional to √T), so this “rent-collecting” position is actually long volatility.
5. Rho +$274/1%: a 1-percentage-point rise in rates makes $274, about 15 days of Theta. The reason is that the long leg is a one-year option (Section 5.6).
6. Vanna −5.9: for every $1 the stock rises, position Vega falls by $5.9. As the stock climbs toward 105, the short leg gets closer to the money and its negative Vega grows.
7. Charm +2.0: with nothing else changing, Delta rises by about 2 shares a day. The short leg is an OTM option whose Delta drifts toward 0 every day, so it hedges away less and less.

Turning the numbers into scenarios

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