Portfolio Greeks and adjustments

OpeningThree “small” trades

At Friday's close your account ($100,000) holds three short-option positions: an SPY put credit spread, a short put on stock A and a short put on stock D. On the broker's page their Deltas are +60, +95 and +63 shares, and none of the lines looks big.

Together the three collect $65.6 of Theta a day. On Monday SPY opens 3% lower, stocks A and D fall 3.9% and 4.5% according to their β, and IV rises about 4 points across the board. The three lose $3,376 together, equal to 51 days of Theta. Each one on its own is “within expectations”; the trouble is that they all point the same way and they all hit at once.

You never added those three lines together. With the beta weighting from Chapter 4, the three add up to 94.5 shares of SPY, about $73,700 of long market exposure. And Gamma and Vega are both negative, so the further the market falls and the more IV rises, the faster you lose.

At Monday's close you open the position that lost the most, and the broker's “Roll” button says “Net credit available.” This chapter answers two questions: first, what is your portfolio actually betting on; second, when is an adjustment a repair, and when is it just placing the same bet again with a different date?

12.1Adding every position into one table

A portfolio's Greeks can be added directly, as long as you first convert them to the same unit. Chapter 4 already turned Delta into “SPY share equivalents”; this section converts the other three.

Why Gamma is multiplied by β squared

When SPY rises 1%, by the definition of β the stock rises β × 1%, that is, ΔS = β · S · 1%. The stock position's Delta therefore grows by Γ × 100 × q × ΔS shares. Those “stock shares” then have to be converted into SPY shares as in Chapter 4, which multiplies once more by S / SPY × β. Both steps carry a β, so you get:

With these two aggregates you can estimate the whole portfolio's P&L, as in Chapter 4. Let SPY's percentage move be x:

What you assume when you add up Theta and Vega

Theta added up is “how much more you'll collect by tomorrow if everything else stays the same.” That “everything else stays the same” is the key: it's only a reference value, and the market changes every day.

Adding Vega directly assumes every underlying's IV moves by the same number of points at the same time. In reality that rarely holds: when the market falls, single-stock IVs tend to rise by more points than SPY's IV, because they start out higher. In the opening portfolio, the two SPY legs' Vega totals −$111.0 and the A and D positions total −$75.2, for a simple sum of −$186.3. If you assume single-stock IVs move 1.5 times as many points as SPY's, then per 1-point move in SPY, the portfolio's Vega is really −111.0 + 1.5 × (−75.2) = −$223.9, 20% bigger than the headline number.

12.2The portfolio's shape: what you're betting on

Put the signs of Gamma, Theta and Vega together and you have the world you're betting on. Read the signs first, then decide whether to change anything.

For positions on the same underlying with the same expiration, Gamma and Theta almost always have opposite signs. That's what Chapter 4's Θ ≈ −½σ²S²Γ says: positive Theta is the reward for negative Gamma. Only structures that span expirations (calendars, diagonals) can produce a combination like “positive Theta, positive Vega,” and the price is negative Gamma in the front month.

The portfolio's “breakeven daily move”

A negative-Gamma portfolio collects Theta every day and also takes losses from price moves every day. Set ½ × |Γ_β,1%| × SPY × x²/1% equal to Theta, and you get the move at which that day's Theta is wiped out:

Start with size: the opening portfolio's beta-weighted Delta is +94.5 SPY shares, about $73,700, as if 74% of the account were invested in the index. If the plan in your head was “sell a little premium with small positions,” this number is the first alarm: the directional exposure is already close to that of an ordinary long stock account, and it comes with negative Gamma.

How to read the opening portfolio: SPY −5% loses $5,371, while +5% makes only $2,204; 7 days with nothing else changing makes $505; IV up 5 points loses $868. In one sentence, it's betting that “the market doesn't fall, volatility doesn't rise, and time passes slowly.” All three trades are betting on the same thing.

One more detail: when SPY falls 10%, rising IV actually makes the loss slightly smaller (−$12,023 vs. −$12,307). By then both legs of the SPY spread are in the money, and the spread's Vega has flipped from negative to positive. A portfolio's Vega changes with price, and the reading only holds near the current price.

12.3Fix Delta first, then look at Gamma and Vega

Stock or SPY can only bring Delta to zero; Gamma and Vega can only be changed with options. A hedge at the portfolio level is often cheaper than adjusting trade by trade, but you need to be clear about what it changes and what it doesn't.

Take the opening portfolio and compare two hedges. One: short 95 shares of SPY (about $74,100) to bring beta-weighted Delta to zero. Two: buy 2 30-day at-the-money SPY 780 puts (IV 17.5%, $14.34 each, $2,868 in total), then short 2 shares of SPY to flatten the remaining Delta (third row of the table). Theta is per day and Vega per IV point, both in dollars.

In the second row Delta is zero, but the portfolio loses on both sides: SPY up 5% loses $1,501, and down 5% loses $1,666. That's negative Gamma's true face. A Delta hedge just turns it from “lopsided” into “afraid of both directions.”

The third row uses options to hold down Delta, Gamma and Vega at once, and the curve is nearly flat. The cost shows up in the Theta column: daily income falls from $65.6 to $22.0, with two-thirds of it paid out for insurance. Whether the insurance is worth it is a separate question; at least it shows you that most of the original $65.6 was payment for bearing a market decline and a rise in IV.

12.4The three directions of rolling

A roll is two trades: buy back the old position and sell a new one. The three directions differ in what you give up in exchange for time and room.

• Rolling out: keep the strike and move to a later expiration. You sell more time and can usually collect a net credit; the price is that the risk lasts longer.
• Rolling down / up: keep the expiration and change the strike. Roll short puts down and short calls up, moving the strike away from the current price. Within the same expiration, a strike further away is cheaper, so this always costs money.
• Rolling out and down: both at once. The extra time value pays for moving the strike down, often at close to zero cost.

Example: 30 days ago, with the stock at $100, you sold 5 30-day 95 puts (IV 35%), collecting $1.80 per share, $900 in total. Now 18 days have passed, the stock has fallen to $92, IV has risen to 42%, and the put, with 12 days left, is worth $4.50. Buying back costs $2,252, an unrealized loss of $1,352. The table below assumes every strike has an IV of 42% (in practice, lower strikes tend to have higher IV).

To read this table, start with the “Best outcome” column:

• Out collects $1,030 and looks like the biggest “win.” It keeps the 95 strike, so Delta is still +276 shares, the breakeven only moves from $93.20 to $91.14, and you carry the risk for 28 more days. Daily Theta income actually falls from $51.0 to $28.5, because a 40-day option decays more slowly.
• Down costs $1,336, and Delta falls from +321 shares to +183 shares, with the same expiration. The cumulative premium is already negative, so this trade can't make its money back no matter what; even the best outcome is a $436 loss. It's essentially “closing part of the risk,” trading cash for distance.
• Out and down costs almost nothing, moves the strike below the current price and pushes the breakeven down to $88.78. The price is again 28 more days of risk, and the best outcome is only $611: what you're playing for is very little room to win back what you've lost.

Short calls: opposite direction, same logic

When a covered call is breached, it looks like a loss, but it isn't necessarily one. You hold 300 shares (cost $100) and sold 3 30-day 105 calls (IV 28%) for $1.46. Now the stock is at $109 with 10 days left, IV is 26%, and the call is worth $4.58, with a Delta of 0.82. Three rolls: out to the 38-day 105 call, net credit $491; up to the 10-day 110 call, net debit $933; up and out to the 38-day 110 call, net debit $357.

First work out what happens if you don't roll: you're assigned at expiry and sell the stock at 105, making (105 − 100 + 1.46) × 300 = $1,938 on the whole trade. The “loss” is only the gain above 105 that you miss. Rolling up spends money to buy that gain back, which amounts to going long part of the stock again at $109. The test is the same as before: if you held only the stock, without this call, would you sell a 38-day 110 call today? If you actually want to keep riding the rally, don't sell; if you're happy to sell at 110, then rolling makes sense.

12.5Without the old position, would you open this one?

Rolling doesn't repair a loss. The loss hit your account value the moment the price got there; buying back the old position just renames it from “unrealized” to “realized.” Only one question is left: is the new position worth opening on its own merits?

Marked to market, your account value doesn't change at the moment you roll: you pay $2,252 to buy back the old put and take in $1,963 for selling the new one (out and down), so the account sheds a $2,252 liability and takes on a $1,963 liability at the same time. The so-called “net credit” or “net debit” is just the cash difference between the two trades. It doesn't change the $1,352 you've already lost.

How the model sees the three choices

Use the Black-Scholes risk-neutral distribution (IV 42%) to work out the final outcome of each choice:

The four choices have nearly the same expected value, each roughly equal to the $1,352 you've already lost (the small differences come from interest). When options trade at fair prices, switching contracts doesn't create expected return out of thin air. What a roll does is swap a certain loss for a distribution: a little better than even odds of getting back to even, about a 40% chance of losing more, and when it loses, an average loss of more than $4,000.

Real markets have two factors the model doesn't. One is the volatility risk premium: on average IV runs a little above realized volatility, so sellers have a slight long-run edge. The other is trend: for a stock that has just broken support, next month's distribution may not be symmetric. The first makes rolling a little better; the second may make it worse. Neither changes the core conclusion: a roll opens a new trade, and it has to be judged by the standards for a new trade.

Run it through the standards for a new trade

Take the position after rolling out and down, and look at it as if you were seeing it for the first time today:

1. Direction: would I go long +199 shares today in this stock that just fell 8%?
2. Strike and expiration: I normally open by selling puts at Delta 0.20–0.30; this one is 0.40. Would I sell it?
3. Size: the notional exposure of five 90 puts is $45,000, 45% of the account. Under my rules (Chapter 13), is that size acceptable?
4. Reason: does the reason I opened it (say, “95 is support”) still hold? If 95 is the very level that just broke, the reason is gone.

If the answer to any of these is “no,” close, or keep only the part that passes the test.

12.6A credit spread is breached: four choices

Once a spread is breached, every choice trades one risk for another. To compare them, use the same yardstick: P&L from here. Money already lost isn't part of any of the choices.

The scenario: 35 days ago, with the stock at $100, you sold 10 95/90 put credit spreads (IV 30%, short-leg Delta −0.26) for $1.02, $1,021 in total. Now there are 14 days left, the stock is at $93 and IV has risen to 38%. The spread is worth $2.38, an unrealized loss of $1,363. If you do nothing, the most you can lose from here is another (5 − 2.384) × 1,000 = $2,616.

• A Close and take the loss: pay $2,384 to buy it back, and you're done.
• B Roll out and down: buy back the 95/90 and sell the 90/85 expiring in 42 days for $1.64. Net cost $745.
• C Add a leg to make an iron condor: keep the put spread and also sell 10 14-day 97/102 call spreads (short-leg Delta 0.31) for $0.91, $911 in total.
• D Convert to a calendar: sell the 90 puts you hold (collecting $1,428) and buy 42-day 95 puts (paying $5,641). The position becomes “short 14-day 95 put + long 42-day 95 put.”

How to read this table

• B Roll lowers Delta (+281 → +149, with Theta going from $5.9 a day to $10.4) and moves the loss zone to lower prices, but the maximum loss from here rises from $2,616 to $3,361, and you carry the risk 28 days longer. If you want to roll “for a credit,” you have to roll out at the same strikes (net credit $77, short-leg Delta −0.53, nearly at the money) or widen the spread to 90/80 (net credit $180, with the maximum loss becoming $7,436).
• C Iron condor cuts the maximum loss to $1,704, lowers Delta to +94 and raises Theta from $5.9 a day to $45.4. It requires you to be genuinely neutral right now: if the stock rebounds to 100, you make only $296, where you could have made $2,384.
• D Calendar has the smallest Delta (+64) and Theta of $38.8 a day, and does best with the stock sitting at 93–97. It spends $4,213 in cash on a back-month put and becomes a long back-month volatility position: if back-month IV falls back to 30% within 14 days, the profit with the stock unchanged drops from $1,023 to $210.
• A Close has a P&L from here of exactly zero. Its value is in freeing up margin and attention, so you can choose again with a clean account.

The order of choosing: first ask yourself what you now think of the stock, then find the row that matches that view. B fits “still bullish, I just need more room and time,” provided the new spread passes the test in Section 12.5. C fits “now neutral, and I think it'll struggle to get above 97.” D fits “I think it'll grind around 95, and back-month IV won't fall.” If you hold none of these views, choose A.

12.7When to just close

In three situations, don't adjust, just close: the thesis breaks, the loss hits your line, or the adjustment would make the maximum loss bigger. Closing doesn't need a reason; holding does.

1. The thesis breaks

When you open a trade, write down a one-sentence reason, such as “Earnings are out, 95 is support, and it won't break within 30 days.” If the stock breaks 95 and the company cuts guidance, that sentence is false. Any adjustment at that point just keeps you inside a story you no longer believe. The rule is simple: once the reason is proven wrong, close; if a new reason comes along later, open a new trade.

2. The loss hits your line

Two common lines: a loss of 2 times the premium collected (the spread's value rises to 3 times the premium), or a loss of half the maximum loss. The spread in Section 12.6 collected $1.02, so a 2× stop corresponds to a spread value of $3.06, which with 14 days to expiry and IV at 38% triggers at a stock price of about $90.6. Half the maximum loss corresponds to a value of $3.01, almost the same spot.

Where does “2 times the premium” come from? It ties the loss to the income, so that the win rate is enough to cover the losses. Let the fraction of the premium you keep when you win be g, and the loss when you lose be L times the premium. The breakeven win rate is:

So this rule only suits structures whose win rate is high to begin with, such as a short-leg Delta of around 0.15–0.20, and only if you really do take profits at 50%. If you're selling 0.30-Delta spreads, the win rate probably won't reach 80%, and this stop line is too loose.

Also remember that a stop line is a target, not a guarantee. If the stock gaps overnight from 93 to 88 and IV rises to 42%, the spread jumps straight to $3.68, a loss of $2,663, 2.6 times the premium. You never get the chance to exit at 2×.

3. The adjustment would make the maximum loss bigger

Choice B in Section 12.6 raises the maximum loss from here from $2,616 to $3,361; widening the spread to 90/80 raises it to $7,436; adding contracts to “average down” is worse still. The rule: any adjustment that makes the maximum loss bigger has to be approved again by the standards for a new trade, and if it doesn't pass, close.

12.8Timing and execution of adjustments

Many adjustments succeed or fail on timing and execution costs, and have little to do with your directional call.

Why people say “manage at 21 days”

This rule of thumb comes from a pattern that shows up again and again in sellers' backtests: the closer you get to expiration, the less premium is left to earn, while the loss from an ordinary shock changes little. In numbers: stock at $100, IV 30%, you sold a 95 put (Delta −0.28) 45 days ago for $1.92. Suppose the stock never moves:

At 21 days, with the stock unchanged, you've already earned half; the remaining $0.93 is exposed to $2.14 of ordinary shock risk. After that the ratio deteriorates fast, and once the stock falls near the strike, Gamma is two to four times what it was when you opened. That's where “manage at 21 days” comes from: before the ratio turns bad, close or move into a new cycle.

When it doesn't apply: if you opened with only 30 days left, 21 days is too early, and the common rule is “take 50% or reach 21 days, whichever comes first”; if you sold far-OTM options below 0.10 Delta, there's little premium left anyway; and with a very narrow defined-risk spread, the Gamma risk is already capped by the protective leg.

Placing the order: fill it in one go, and count the cost

• Use a combo order: roll with a single multi-leg order (rolling a spread into a spread is 4 legs), rather than closing first and opening afterward. If you place the orders separately, you're exposed for the few seconds in between, and if the stock moves you may pay more.
• Start at the mid: place the order at the mid price first, then move $0.01–0.02 toward the natural price each time. Don't pay the natural price right away.
• Count the cost: rolling 10 four-leg spreads with each leg filled $0.05 from the mid costs 4 × 0.05 × 100 × 10 = $200. The net credit for “rolling out at the same strikes” in Section 12.6 is only $77, so the cost is more than the income.
• Avoid the traps: don't roll into an earnings date; when selling calls, check the ex-dividend date; Gamma is highest in the last few days of expiration week, so adjust earlier if you can.

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