Week 29: The Greeks - Delta, Gamma, Theta, Vega, Rho

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1. Why This Is Important

Weeks 25-28 taught you the mechanics of options: what a call is, what a put is, how to sell a covered call against stock you own, how to sell a cash-secured put against cash you would deploy anyway. You can already open positions and you understand the payoff diagrams at expiration. What you cannot yet do is answer the questions a position asks of you before expiration: the underlying moved $2 today, why did my call only go up $0.40? I sold a put two weeks ago, the stock has not moved, why am I up half my premium already? The S&P is flat and the VIX collapsed five points, why did my long call get crushed? Each of these is a Greek question, and the Greeks have closed-form answers.

The Greeks are the partial derivatives of the Black-Scholes premium with respect to each of its inputs. That is a sentence calculus students recognise; for everyone else the practical translation is: *how much does the option's price change when one input moves and everything else stays put?* Five inputs matter (spot, time, vol, rate, and the strike, which does not move), so there are five Greeks. They are not optional reading - by Week 30 we layer multiple options into spreads and condors, and a spread is exactly a *combination of Greek exposures* designed to isolate one risk while neutralising another. You cannot think about a vertical or an iron condor coherently without delta, gamma, theta, and vega in your head.

Four reasons to put the work in:

Greeks turn confusion into accounting. When a position behaves

in a way you did not expect, the Greeks decompose the surprise into line items: *delta P&L plus gamma P&L plus theta P&L plus vega P&L plus rho P&L equals what your account did today.* That decomposition is the difference between "options are weird" and "options are measured."

Greeks let you size by risk, not by ticket count. "I sold three

calls" is not a position description. "I am short 150 deltas, long $9 of theta per day, short $40 of vega per vol-point" is. The first tells you nothing about how the position breathes; the second lets you compare the trade to every other trade in your book on the same dimensions.

Greeks expose the second-order risk that kills option sellers.

Theta-collectors (covered calls, cash-secured puts, condors) sound like an income strategy and feel like an income strategy right up until volatility expands. The position is short vega and short gamma; both work against you when realised volatility ticks higher. The vol tail wags the equity dog, and that statement is fundamentally about gamma and vega risk - the Greek decomposition makes the mechanism literal.

Greeks are the language of every advanced lesson that follows.

Spreads (Week 30), LEAPS (Week 38), the VIX (Week 40), volatility surfaces, and the entire side25 lesson on second-order Greeks all assume you can think natively in Δ, Γ, Θ, ν. This week is the foundation those lessons build on.

We will derive each Greek from Black-Scholes, work a single example end-to-end on a $100 stock with a 30-day at-the-money call, look at how each Greek moves with spot and with time, and then translate the Greek profiles into how the four conservative strategies from Weeks 25-28 actually breathe in your account.

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2. What You Need to Know

2.1 The Five Greeks at a Glance

There are five primary Greeks. Each is a partial derivative of the option premium $P$ with respect to one input, holding the other four fixed. The standard symbols and what they measure:

$$ \Delta = \frac{\partial P}{\partial S} \qquad \Gamma = \frac{\partial^2 P}{\partial S^2} \qquad \Theta = \frac{\partial P}{\partial t} \qquad \nu = \frac{\partial P}{\partial \sigma} \qquad \rho = \frac{\partial P}{\partial r} $$

Plain English, in order: delta is the option's speedometer (how much premium changes per $1 of spot), gamma is its acceleration (how much delta itself changes per $1 of spot), theta is its time-decay clock (how much premium melts per calendar day), vega is its vol-sensitivity (how much premium changes per one percentage-point move in implied volatility), and rho is its rate-sensitivity (how much premium changes per one percentage- point move in the risk-free rate).

There is a caution about units before we go further. Brokerage platforms quote theta as per calendar day (annual theta divided by 365), vega as per one volatility point (vega divided by 100), and rho as per one percentage-point of rate (also divided by 100). Black-Scholes-the-formula gives them in continuous-time, per-unit form; everything in this lesson is in the broker-quoted, practical form because that is what your trading screen will show you.

2.2 The Black-Scholes Engine, in One Box

For a non-dividend-paying European call with spot $S$, strike $K$, time-to-expiry $T$ (in years), implied vol $\sigma$, and risk-free rate $r$:

$$ d_1 = \frac{\ln(S/K) + (r + \tfrac{1}{2}\sigma^2)T}{\sigma \sqrt{T}}, \qquad d_2 = d_1 - \sigma \sqrt{T} $$

$$ C = S \, N(d_1) - K e^{-rT} N(d_2) $$

where $N(\cdot)$ is the standard-normal CDF. Each Greek then has a closed form:

$$ \Delta_{\text{call}} = N(d_1), \qquad \Gamma = \frac{\varphi(d_1)}{S \sigma \sqrt{T}} $$

$$ \Theta_{\text{call}} = -\frac{S \varphi(d_1) \sigma}{2\sqrt{T}}

• r K e^{-rT} N(d_2)

$$

$$ \nu = S \varphi(d_1) \sqrt{T}, \qquad \rho_{\text{call}} = K T e^{-rT} N(d_2) $$

Here $\varphi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ is the standard- normal density. For puts, delta becomes $N(d_1) - 1$, the rho term flips sign and uses $-N(-d_2)$, and the second term in theta flips sign; gamma and vega are identical for calls and puts at the same strike (this is one of the cleanest facts in option pricing - any asymmetry between calls and puts at the same strike has to live in delta, theta-via-rate, or rho).

You do not need to memorise these. You need to recognise that every Greek is just one input changing and the rest held fixed - and that the closed forms make every panel of every chart in this lesson a two-line Python function.

2.3 The Worked Example - $100 Stock, 30-Day ATM Call

Spot $S = \$100$, strike $K = \$100$, expiry in 30 calendar days ($T = 30/365 \approx 0.0822$ years), implied vol $\sigma = 20\%$, risk-free rate $r = 4\%$. Plugging into the formulas:

Quantity	Value
$d_1$	$0.086$
$d_2$	$0.029$
$N(d_1)$	$0.534$
$N(d_2)$	$0.511$
Call premium $C$	$\$2.45$
Delta	+0.534
Gamma	+0.069
Theta / day	-$0.044
Vega / 1 vol-pt	+0.114
Rho / 1% rate	+0.042

Read those rows like a position briefing. The call costs $2.45. For every $1 the stock moves up, the call gains roughly $0.53 (delta). For every additional $1 of upside after that first dollar, the option gains another ~$0.07 more than the previous dollar (gamma). One day passing costs you 4.4 cents per share even if nothing else changes (theta). If implied vol ticks up from 20% to 21%, you make 11 cents (vega). If the Fed surprised the market by hiking rates 100 basis points overnight (it does not), you would make about 4 cents (rho). Multiply every dollar by 100 because one contract controls 100 shares: this $2.45 option is a $245 ticket with a $53 / $1-stock-move directional exposure.

2.4 Delta - Direction and "Probability ITM"

Delta is the most-used Greek. It tells you the *equivalent stock position* the option behaves like, and it doubles as a quick-and- dirty estimate of the probability the option finishes in the money.

Three ways to read the same number:

• As a hedge ratio. A call with $\Delta = 0.50$ moves like

half a share of stock. To hedge ten such calls (1,000 share- equivalents) to delta-neutral, you sell 500 shares against them.

• As a directional exposure. A portfolio of options with net

delta $+150$ behaves like 150 shares of the underlying for small moves. If the stock moves up $1 you make ~$150; if it moves down $1 you lose ~$150.

• As a probability proxy. Under Black-Scholes, $N(d_1)$ and

$N(d_2)$ are not the same probability; the true risk-neutral probability of finishing ITM is $N(d_2)$, not delta. But the two are close for ATM options - close enough that traders quote "30-delta put" and "30% chance of expiring ITM" interchangeably, which is fine for quick decisions and wrong for anything requiring three decimals.

Delta runs from 0 to +1 for calls and 0 to -1 for puts. ATM options have $|\Delta| \approx 0.5$. Deep ITM calls converge to +1 (they behave like the underlying). Deep OTM calls converge to 0 (they behave like nothing).

2.5 Gamma - The Curvature That Bites Sellers

Gamma is the rate of change of delta per $1 move in the underlying. Two facts about gamma drive the bulk of Greek-aware risk management:

Gamma peaks at-the-money. Delta is most rapidly changing where

the option is closest to flipping between "expiring in" and "expiring out". Far ITM and far OTM options have stable deltas (near 1 and near 0 respectively); ATM options have delta that ricochets with every dollar of spot.

Gamma explodes near expiry. As $T \to 0$, the gamma of an

ATM option diverges. A 30-day ATM call has gamma ~0.07; a 1-day ATM call can have gamma north of 1.0. This is why the last week before expiry is qualitatively different from any earlier week and why option sellers either close or roll before they carry a short ATM strike into expiration week.

Long gamma (you bought the option) is good news on a move - your delta moves with you. Stock rallies → your call's delta climbs from 0.50 toward 0.70, so you participate more in further upside. Stock drops → delta falls toward 0.30, so you bleed less on further downside. Long gamma decelerates losses and accelerates gains. That is what you are paying theta for.

Short gamma (you sold the option) is the mirror image. Stock rallies against your short call → your delta gets more short, so each additional dollar of upside hurts more. Stock drops against your short put → your delta gets more long, so each additional dollar of downside hurts more. Volatility tails wag the dog, and the mechanism is short gamma. The position looks like steady income at low realised vol and bleeds at compounding speed when vol arrives.

2.6 Theta - The Cost of Carrying Time

Theta is negative for long options (you lose value each calendar day) and positive for short options (you collect each calendar day). Three things to know about theta:

Theta is highest at-the-money. Same logic as gamma - the

ATM option has the most extrinsic value to lose, so the daily melt is biggest there in absolute dollars. ITM and OTM options have less extrinsic to lose, so their theta is smaller.

Theta accelerates as expiry approaches. A 90-day ATM call

loses theta at maybe $0.013 per day; a 30-day ATM call loses $0.04 per day; a 7-day ATM call loses $0.10 per day; a 1-day ATM call can lose $0.20 per day or more. The decay is non- linear - roughly two-thirds of a 90-day ATM call's premium is gone by the time DTE hits 30.

Theta and gamma are joined at the hip. A position cannot

be long gamma without paying theta and cannot be short gamma without collecting theta. The arbitrage-free price of carrying gamma is theta. *That trade-off is what option-selling income strategies are.*

The 30-45 DTE window is the conventional sweet spot for theta- collectors. Earlier than that, daily decay is small relative to premium received; later than that, gamma starts to bite and the position swings around violently with small spot moves. Weeks 27 and 28 used 30-45 DTE for exactly this reason - now you can see on the chart why.

2.7 Vega - The Greek That Hides In Plain Sight

Vega is the change in premium per one percentage point move in implied vol. It is positive for both long calls and long puts (buyers benefit when vol expands), negative for both short calls and short puts (sellers benefit when vol contracts).

Three properties:

Vega peaks at-the-money - same shape as gamma.

Vega is largest for long-dated options. This is the

cleanest counter-example to "all Greeks behave like gamma": gamma peaks near expiry, vega peaks far from expiry. A 1-year ATM option might have vega ~$0.40; a 30-day ATM option ~$0.11; a 1-day ATM option ~$0.02. Long-duration options are *vol instruments* almost more than they are direction instruments.

Vol crush is a vega event. When you buy an option ahead of

earnings and IV ranks at the 90th percentile, you are buying 30-50 vega-points of "earnings premium" that the market will evaporate the moment the print drops. The stock can move exactly the way you predicted and you can still lose money, because the directional gain (delta + gamma) does not cover the vega loss. Beginners learn this once.

Vega-aware position management is what separates *income strategies from vol strategies*. A covered call sold at 15% IV in a calm market has small vega; the same covered call sold at 40% IV during a market panic has large negative vega and can print you a fast profit if vol mean-reverts even with the stock unchanged. Side25 develops this further.

2.8 Rho - The Greek You Mostly Ignore

Rho measures sensitivity to the risk-free rate. For a 30-day ATM call on a $100 stock, rho is about $0.04 per 1% rate move - so even a Fed-meeting-day swing of 25 bp moves the option's price by about a penny. For most retail option positions on most days of most months, rho is a rounding error.

When does rho actually matter?

LEAPS - long-dated equity options (1-3 year expiries) have

rho large enough to register. A 2-year ATM LEAPS call on a $100 stock has rho near $1.40 per 1% rate move - that is meaningful.

Regime transitions. When the Fed is moving 75 bp at a

meeting (think Q1 2022 or Q3 2025), even short-dated rho compounds across positions in a way you should at least be aware of.

Outside those, rho lives at the back of the dashboard. We mention it for completeness; we do not optimise around it.

2.9 How the Greeks Drive the Strategies You Already Know

The four conservative strategies from Weeks 25-28 each have a characteristic Greek signature. Knowing it lets you predict how each will breathe.

Long calls (Week 25/26): $\Delta > 0$, $\Gamma > 0$, $\Theta < 0$, $\nu > 0$. You are paying theta and rho-bleed every day for the right to participate in upside and benefit from a vol expansion. This is a directional + vol bet wearing a leverage costume.

Long puts (Week 25): $\Delta < 0$, $\Gamma > 0$, $\Theta < 0$, $\nu > 0$. Same Greek shape as a long call but with negative delta - you are buying insurance against a drawdown, and the vega component is precisely the reason puts get more expensive during a sell-off (vol expands, vega works in your favour, but you already paid for the put before the move).

Covered calls (Week 27): stock + short call. Net Greek: $\Delta \approx 100 - 50 = +50$ per share-pair, $\Gamma < 0$ (small), $\Theta > 0$ (you collect daily), $\nu < 0$ (small). You have muted upside, retained most of the downside, and traded both for a steady theta drip. The income SHOULD count as a tax-shifting tool, not as alpha.

Cash-secured puts (Week 28): short put + cash. Net Greek: $\Delta \approx +50$ per contract (similar directional exposure to a covered call), $\Gamma < 0$, $\Theta > 0$, $\nu < 0$. The trade is equivalent to a covered call by put-call parity - same Greek skeleton, same risk profile, same theta engine. The brokerage-margin treatment differs, the strategic logic does not.

Iron condors and verticals (Week 30, preview): combinations designed to be delta-neutral and *short gamma + short vega + long theta*. The trade is "I think realised vol will be lower than implied vol." The Greeks let you measure exactly how much vol-edge you need to break even.

The interactive at the end of this lesson lets you set spot, strike, DTE, vol, and rate, and watch all five Greeks live as sliders move - including a chart that sweeps the chosen Greek across spot. *Drag the DTE slider and watch gamma climb at the strike; that is the entire risk profile of selling weekly options in one animation.*

[interactive: interactive/week29_greeks_lab.html]

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3. Common Misconceptions

"Delta is the probability of finishing in the money." It is

close, not equal. The risk-neutral probability of finishing ITM is $N(d_2)$, not $N(d_1) = \Delta$. The two diverge when vol or time is large. Use delta as a quick proxy; do not bet money on it being exact.

"Theta is the same every day." No. Theta on a 90-day option

is small; on a 7-day option it is large. The headline number you read at trade entry is today's theta, not the path-average. The whole point of the theta-decay chart is that it accelerates.

"If gamma is positive, I always make money on a move." Long

gamma helps you on any move, but you still pay theta to hold it. If realised volatility is below implied (i.e. the move is smaller than what the option priced in), theta wins and you lose. Long gamma is profitable only when realised vol exceeds implied vol over the holding period. *Every long-gamma trade is implicitly a long-vol trade.*

"Vega and gamma are the same Greek." They peak at the same

strike (ATM) and have similar bell shapes versus spot, but they do opposite things versus time: gamma rises as expiry approaches, vega falls. A 1-day ATM option is mostly gamma; a 1-year ATM option is mostly vega. Confusing them costs money when DTE matters.

"Short options have unlimited risk because of theta." Theta

is positive for short options - it is your friend, not your enemy. The unlimited risk on a naked short call comes from delta (the underlying can run away from you) compounded by gamma (your delta gets shorter as it does). Theta is the payment you collect for accepting that risk, not the source of it.

"Higher delta is always better for a call buyer." Higher

delta means more directional participation, but it also means more capital outlay (deep-ITM calls cost almost as much as the stock) and lower vega (you give up the IV-expansion bet). The 0.30-0.40 delta range is the conventional "speculative long" sweet spot precisely because it balances directional exposure against capital efficiency and convex vega payoff.

"Vega only matters around earnings." It matters every day

any time IV is far from its long-run average. Owning long options at 30% IV when the post-FOMC reality is 18% IV is a guaranteed slow bleed independent of what the stock does. Persistent IV crush is the reason most retail directional options trades lose money even when the directional view was right.

"Rho is irrelevant." Almost true for short-dated options;

not true for LEAPS. A 2-year ATM call on a $200 stock has meaningful rho. If you trade LEAPS without checking rho, you are leaving exposure unmonitored.

**"I can ignore the Greeks because my broker shows me a P&L

chart."** A P&L chart at expiration is one slice of a five- dimensional surface. Greeks tell you the path between today and expiration. The chart tells you what happens only if you hold to the bell. Most positions are closed before that.

"The Greeks are too mathematical for retail traders." The

formulas are mathematical; the concepts are arithmetic. If you can read a speedometer (delta), accept that the speedometer itself shifts under you (gamma), pay rent on parking (theta), and own a thermostat for fear (vega), you understand the Greeks. You do not need to derive Black-Scholes to use them.

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4. Q&A Section

**Q: What is the easiest way to estimate delta if I don't have a calculator?** A: For ATM options, delta is approximately 0.50 for calls and -0.50 for puts. For each $1 of moneyness (ITM or OTM), shift delta by about $0.05 / (\sigma \sqrt{T})$ - so for a 30-day, 20% vol option, that is roughly $0.05 / 0.057 \approx 0.88$ per $1 of moneyness, capped at $\pm 1$. In practice, just read it off your broker.

Q: Why does my long call lose money on a flat day? A: Theta. Even with spot, vol, and rate unchanged, one calendar day passing burns extrinsic value. For a 30-day ATM call that is roughly $0.04 per share. If you bought 5 contracts, that is $20 of theta across the day - on a quiet Tuesday, that is the entire P&L.

Q: When should I worry about gamma the most? A: Two situations. First, when you are short options inside the last two weeks before expiration with the stock close to your strike - gamma there is large enough to flip you from "winning the trade" to "underwater on the trade" with one news headline. Second, when you are running multiple short-option positions in the same underlying - gammas add, and a portfolio that is short 30 gamma is qualitatively different from one short 5 gamma.

Q: Are gamma and vega the same thing? A: No, but they share the at-the-money peak. Their time profiles are opposite: gamma rises into expiry, vega falls. A short-dated ATM option is mostly a gamma instrument; a long-dated ATM option is mostly a vega instrument. This is why "selling weeklies" and "selling LEAPS-puts" are very different strategies even though both are short premium.

Q: What is "vol crush" and which Greek explains it? A: Vega. Implied volatility tends to be elevated before known-event dates (earnings, FOMC, biotech FDA decisions) and to collapse the moment the event passes. If you own long options across the event, you collect the directional move (delta and gamma) but you also eat the IV collapse (a vega loss). Net P&L is whatever wins. Beginners assume "stock moved my way → I made money"; the Greek decomposition makes the loss diagnosable.

Q: How big is rho really? Should I ever worry about it? A: For a 30-day ATM call, rho is about $0.04 per 1% rate move per share. So a 25 bp Fed move is about a penny. Don't worry about it on short-dated trades. For LEAPS (1-3 year expiries) rho can be $1-$3 per 1% per share - large enough that a Fed pivot moves the LEAPS premium materially even with the stock unchanged. So: LEAPS yes, weeklies no.

**Q: If I sell a covered call, what does my net Greek profile look like?** A: Stock + short call. Per-100-shares-plus-one-short-call: delta $\approx +50$ (you have ceded half your upside delta to the short call), gamma slightly negative, theta positive (you are collecting), vega slightly negative. The position has muted upside, almost full downside, and a positive theta drip. The daily theta is income, not alpha.

Q: What changes if I sell a cash-secured put instead? A: The Greek skeleton is identical to a covered call by put-call parity - same delta, same gamma, same theta, same vega. The broker margin treatment differs (one ties up shares, the other ties up cash), but the risk profile is the same trade in two costumes.

Q: Can a short option position ever be long gamma? A: Not on a single short option. Combinations can engineer any Greek profile - a long calendar (sell front-month, buy back- month at the same strike) is short gamma but long vega; a long straddle (buy a call and a put at the same strike) is long both. Once you start composing positions, the Greeks add and you can build essentially any signature you want.

Q: How fast do the Greeks change in real markets? A: Delta changes continuously as spot moves (that is gamma). Gamma drifts up as expiry approaches and explodes in the last week. Theta accelerates non-linearly - the chart in §2.6 shows the curve. Vega changes any time IV moves, which on equity options is daily and on event-driven names hourly. Rho changes with the rate curve, which is slow on most days. The five Greeks together are a time-varying dashboard; nothing about them is set at trade entry.

Q: Where do dividends fit in? A: We deliberately used the no-dividend version of Black-Scholes to keep this lesson clean. For dividend-paying stocks, a known discrete dividend reduces call values and increases put values (forward price drops by the PV of dividends). The Greeks shift correspondingly, and continuous-dividend yield $q$ enters the formulas as $S \to S e^{-qT}$ throughout. The shape of every Greek profile is unchanged; the levels move modestly. For most retail option work, ignoring dividends is a small error.

Q: What is the single biggest practical use of the Greeks? A: Position sizing. "Three short puts" is uninformative; "+150 deltas, +$8 theta per day, -$35 vega per vol-point, -12 gamma" describes the trade. With those numbers you can ask: *at what spot move do I lose all my theta? at what vol expansion does my vega loss exceed two weeks of theta collected? what does my delta become if the stock drops 5%?* Those are the questions a Greek-aware trader answers before the trade is open; a beginner answers them after the loss is realised.