The option Greeks

The Greeks measure how an option price responds to the things that move it: the stock, the clock, volatility and rates. Almost every explanation is written for buyers, so every sign has to be inverted by the people who sell. Twelve articles below, all of them taking apart the same short covered call.

• Black-Scholes, and what it actually assumesThe model behind every option price on your screen takes five inputs and returns one number. What each input does, worked by moving them one at a time, and the six assumptions it makes that are false on a real American equity option.
• Delta, read three waysDelta is the rate an option price moves against the stock, the share-equivalent exposure of the position, and a rough probability of finishing in the money. All three readings of 0.33 on one real covered call.
• Delta is not the probability, and the difference has a shapeDelta approximates the odds an option finishes in the money, and overstates them by a predictable amount. The real number is N(d2), the gap on one real chain is 3.8 points, and the probability that actually decides your trade is a third one nobody quotes.
• Gamma, and the last week of a short optionGamma is the rate delta changes. It is small and harmless on a 42-day option and enormous in the final week, which is why the last few days of a short position carry risk out of all proportion to the premium left in it.
• Theta, and the word incomeTheta is the dollars an option loses per day from the passage of time alone. On one real short call it says $3.45 a day. What the position actually made over seven days, across four stock paths, was between plus $85 and minus $122.
• The decay curve bends two waysThe standard claim that time decay accelerates into expiry is true at the money and false out of the money. One chain, two strikes, two opposite curves, and what each one says about when to close.
• Vega, and losing money on a flat dayVega is what an option gains or loses when implied volatility changes by one point. It is why a short position can lose money on a day the stock did not move, and why a 90-day contract is three times as exposed to volatility as a 7-day one.
• Rho, and why you can mostly ignore itRho measures sensitivity to interest rates and is the least useful Greek for almost everyone. On a 42-day contract a full point of rate change is worth $2. On a two-year LEAP it is worth $61, and that is the whole story.
• Vanna, charm and vomma, and whether you need themVanna, charm and vomma measure how the Greeks themselves move. Each one checked against what the position actually repriced at, plus a straight answer on whether a retail premium seller needs any of them.
• When you sell, every sign flipsSelling an option inverts every Greek. What that means position by position, why short gamma and long theta always arrive together, and the composite Greeks of a covered call worked out in full.
• Adding up delta across a bookAdding delta across positions converts a list of trades into one number: your true share-equivalent exposure. A four-position book worked in full, plus the two things the total quietly hides.
• Put-call parity, and what it forces to be trueCall minus put equals stock minus the discounted strike. It holds by arbitrage rather than by assumption, it is why the put and call at one strike must share an implied volatility, and it is how synthetic positions are built.

Read them in this order

If you sell premium and want the four that matter: delta, then theta, then gamma, then the page that flips every sign for you. That is the working set.

The pages that change decisions rather than vocabulary: why the decay curve bends two different ways depending on your strike, the 59 percent nobody quotes, and how long your book actually is.

The decay one is also a video: one chain, two strikes, the two curves side by side (6:37 on YouTube).

One position, twelve articles

Everything here runs on a single illustrative trade: short one October 17 $72.50 call against 100 shares of UBER at $68.40, 42 days out, at 32 percent implied volatility, for $153 of credit. Delta 0.331, gamma 0.0488, theta $3.45 a day, vega $8.41 a point. Every Greek on every page was computed from that one Black-Scholes run rather than asserted, and the composite covered call works out to 66.9 share-equivalents.

What this series will not tell you

That more Greeks make you a better trader. The four in the working set describe your position. The rest explain effects you have already felt, and the outcome of a premium trade is decided by strike selection, position size and whether you picked something you can hold. The second-order page says so on the page rather than selling you a dashboard.

It also will not tell you that theta is income. Theta is a partial derivative that assumes nothing else moves. Over one ordinary week the worked position ranged from plus $85 to minus $122 against a theta prediction of $24 in every case.

And it will not explain how the OptionsKing confidence score is computed. Several of its inputs share names with pages here. What the engine does with them stays private: the score feeds a ranking and the app shows you the highest-ranked handful of what it found, and how it works covers what that guarantees.

Questions people actually ask

Run your own numbers

• Black-Scholes calculator price and the full set of Greeks from five inputs.
• Assignment probability calculator delta to keep-odds, and probability of touch.
• Implied volatility calculator the one input you have to solve for.
• Covered call calculator the position every page here takes apart.