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Option Pricing & The Greeks

Option Pricing & The Greeks

Option prices are not arbitrary — they are the output of a model (Black-Scholes and its descendants) fed by a small set of inputs, and the Greeks are simply the partial derivatives of that model: each one tells you how much an option's price moves when one input changes and everything else is held still. This section builds the intuition behind the pricing model, defines delta, gamma, theta, vega, and rho the way active option sellers teach them, and then connects the Greeks into the single most consequential idea in the premium-selling playbook: the theta-versus-gamma tradeoff across an option's life that motivates entering trades near 45 days to expiration (DTE).

This section is theory that supports the mechanics. The "why 45 DTE / why manage at 21 DTE" rules built on top of it live in 05_trade_management; the volatility inputs (IV, IV Rank, IV Percentile) live in 03_implied_volatility.

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1. Black-Scholes intuition and its inputs

The Black-Scholes-Merton model (1973) is the foundation of modern option pricing. Its practical value is not the formula itself but what it reveals: an option's fair value boils down to a handful of measurable variables.

The five inputs to a (non-dividend) Black-Scholes price:

The key intuition: of these five inputs, four are observable facts. The underlying price, the strike, the time remaining, and the interest rate are all known with certainty at any moment. Volatility is the only input that must be estimated — it is the trader's forecast of how much the underlying will move.

Because the other four are fixed, the market does not really "agree on volatility" — it agrees on a price, and then the volatility consistent with that price is backed out of the model. That solved-for number is the implied volatility (IV). This is the single most important conceptual hinge for a premium seller: when you sell an option you are, in model terms, selling volatility — taking the position that realized movement will come in below what the price implies.

Black-Scholes is a model, not reality. It assumes constant volatility, lognormal returns with no jumps, and European exercise — assumptions real markets violate, most visibly through the volatility skew (OTM puts priced at higher IV than a single σ allows) and through gaps. The premium-selling emphasis on selling across many occurrences is partly a practical hedge against any single price being only as good as a flawed model.

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2. The Greeks — definitions and behavior

The Greeks are "mathematical measures that describe how an option's price responds to changes in market conditions such as price movements, time, and volatility." Each Greek isolates one input from Section 1.

Platform note (sign and scaling). On most broker platforms, Greeks quote with a long bias — they show your exposure as if you bought the option. Delta, gamma, and vega are multiplied by 100 to express dollar exposure per contract; theta is the exception and is not multiplied by 100. Selling an option flips the sign of the displayed Greek.

Delta (Δ) — directional exposure

Delta "expresses the change in an option's price for every +$1 change in the underlying asset's price."

Gamma (Γ) — the rate of change of delta

Gamma is "the rate of change of Delta with respect to changes in the underlying asset's price" — the theoretical change in delta after a +$1 move.

Theta (θ) — time decay

Theta is "the one-day rate of decline of an option's extrinsic or time value."

Vega (ν) — volatility sensitivity

Vega "measures the change of an option's price after a 1% change in implied volatility in the expiration they're trading."

Rho (ρ) — interest-rate sensitivity

Rho measures the change in an option's price for a 1-percentage-point change in the risk-free interest rate. Premium sellers list it among the Greeks but devote little attention to it, because its effect is small for the short-dated (≈45 DTE and in) options they trade. Rho matters more for LEAPS and long-dated options, and rises in relevance during large rate-regime shifts; for the typical short-premium time frame it is the least important Greek.

Greeks at a glance

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3. Decay, gamma risk, and the option life-cycle

This is where the individual Greeks combine into the core premium-selling thesis.

Extrinsic value and accelerating theta decay near expiration

Every option's price = intrinsic value + extrinsic value. Theta only erodes the extrinsic (time) component. Extrinsic value is greatest for at-the-money options and shrinks as the option moves deep ITM or far OTM.

Theta decay is non-linear and accelerates as expiration approaches — and this is taught directly in the options-education literature, whose theta write-ups state that Theta "accelerates near expiration (especially for at-the-money options)." An ATM option sheds time value slowly when far-dated and rapidly in the final stretch; the precise shape of that curve is often described as a "square-root-of-time" decay, which is a standard Black-Scholes property rather than a published house figure.

What is and isn't sourced. The acceleration rule itself is well-backed on two independent fronts: it is a standard Black-Scholes property and it is stated explicitly in the source material, whose theta page says Theta "accelerates near expiration (especially for at-the-money options)" and reinforces it visually by contrasting a 3-DTE option against a 129-DTE option (showing far larger theta near expiry). What remains unverified is only the precise quantitative profile — specific "half-life" or decay-percentage figures (e.g., "an ATM option loses X% of its value in the final week"). No such numbers come from a published study, so treat any specific percentage as a general-knowledge approximation, not house canon.

Rising gamma risk near expiration

The flip side of accelerating decay is exploding gamma. As expiration nears, gamma concentrates sharply around the money: a near-dated short option's delta can swing violently on a small underlying move, so a position that looked safely OTM can become deeply ITM almost instantly. Premium sellers frame this as the reason 0DTE/near-expiry trades offer fast profits but carry "much less time to be right."

So the premium seller faces a genuine tension in the final weeks: theta is biggest there (good), but gamma risk is also biggest there (bad). The house position is that in the last week the marginal gamma risk tends to outweigh the marginal theta you collect.

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4. The theta/gamma tradeoff that motivates ~45 DTE entry

Put the two curves together over an option's life:

The published study "Comparing 30 and 60 DTE" (industry research, 2016-07-22) is the canonical backing: testing entries on either side of the middle, 45 DTE emerged as the balance point — capturing the benefits without the drawbacks of going shorter (more gamma risk in the ATM strikes) or longer (slower, more capital-tied-up decay and more vega/path risk).

This is why the firm's default is to open trades at the expiration closest to 45 DTE, and — to avoid the late-cycle gamma spike — to manage the position at 21 DTE or 50% of max profit, rather than carrying short premium into the high-gamma expiration week.

Scope honesty. The 45 DTE entry and 21 DTE management rules are defaults for short-premium, non-event trades, not universal laws. Calendars, diagonals, earnings plays, and explicitly directional debit trades deliberately deviate (see 12_calendar_spreads and 13_diagonals).

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Key Takeaways

Common Misconceptions

Review Questions

1. Of the five Black-Scholes inputs, which is the only one that must be estimated, and why does that make implied volatility a solved-for number rather than an input the market agrees on?

2. A short $105 call has a delta of −0.20 and a gamma of 0.05 (long-biased gamma 0.05). The underlying rises $1. What is the new delta, and is that change helping or hurting the short seller?

3. Explain in one sentence why a long option has negative theta while the same option, sold, has positive theta.

4. A trader is short a strangle with −0.30 net vega heading into an earnings announcement. Directionally, what does the trader want IV to do after the event, and what is that effect called?

5. State the theta/gamma tradeoff in your own words and explain why it points to ~45 DTE rather than 7 DTE or 120 DTE for a premium seller.

6. Which published study is the primary citation for the 45 DTE preference, and what did it compare?

Sources

_Note on sourcing: the educational Learn pages above were fetched and quoted directly. The industry research episode pages are real, search-indexed URLs whose video/transcript bodies did not render to the text fetcher (HTTP 404 on automated fetch); they are cited as named show segments with their canonical URLs, and their conclusions are corroborated by the Learn pages on gamma and by repeated house teaching. No URL here is fabricated._

_Evidence-labeled per the Project Charter. Education only, not financial advice._