The Greeks, Part 2: Theta and Vega

8.1 Losing money while nothing happens

Mint bought a 30-day at-the-money BTC call for $6,284. Over the next ten days, bitcoin drifted sideways and ended exactly where it started. Implied volatility didn't change.

The call was now worth about $5,100.

"The price didn't move and volatility didn't move," said Mint. "Why is my option down over $1,000?"

"Time moved," said Hash. "Every day, your option loses a little value, and that loss speeds up as expiry gets closer. That's theta. And its partner, vega, explains why you can lose money even when the price moves your way."

8.2 Theta: the cost of time

Theta measures how much an option's value changes as one day passes, with everything else unchanged. For option buyers, theta is negative: they lose value each day. For option sellers, theta is positive: they gain as time passes.

For Mint's 30-day at-the-money call, theta is about −$105 per day.

Theta isn't constant. It accelerates as expiry approaches, especially for at-the-money options (Chapter 4):

At-the-money call (BTC at $100,000, 55% IV)	Premium	Theta per day
90 days to expiry	~$10,862	~−$60
30 days to expiry	~$6,284	~−$105
7 days to expiry	~$3,038	~−$217

A 7-day at-the-money option loses about 7% of its value every day if nothing happens. A 90-day option loses about 0.6% a day.

"So short-dated options are like ice cubes melting in the sun," said Mint.

"And long-dated ones melt more slowly," said Hash. "That's why buyers who need time for their idea to work often choose longer expiries, and sellers who want to collect decay often choose shorter ones."

8.3 The theta and gamma trade-off

Theta and gamma (Chapter 7) are two sides of the same coin.

	Option buyer	Option seller
Gamma	Positive: benefits from big moves	Negative: hurt by big moves
Theta	Negative: pays time decay	Positive: earns time decay

Buyers pay theta every day for the chance that a large move will make gamma pay off. Sellers collect theta every day in exchange for accepting the risk of a large move. Whether buying or selling works out depends on whether the market moves more or less than implied volatility suggested (Chapter 6).

8.4 Vega: sensitivity to volatility

Vega measures how much an option's value changes when implied volatility changes by one percentage point.

For Mint's 30-day at-the-money call, vega is about $114. If IV rises from 55% to 56%, the call gains about $114. If IV falls to 54%, it loses about $114.

Both calls and puts have positive vega for buyers: higher IV makes all options more expensive (Chapter 5).

At-the-money call	Vega (per 1 point of IV)
7 days	~$55
30 days	~$114
90 days	~$196

Longer-dated options have more vega, because a change in expected volatility affects them over a longer period. At-the-money options have more vega than far in-the-money or out-of-the-money options.

Mint's volatility crush in Chapter 6 was a vega loss: IV fell 25 points on a short-dated option, overwhelming the gain from delta.

8.5 Putting the Greeks together

The Greeks let you estimate how an option's value will change when several things happen at once.

Example. Mint holds the 30-day at-the-money call. Over one day, BTC rises $1,000, and implied volatility falls 2 points.

Effect	Greek	Estimated change
BTC rises $1,000	Delta ~0.53, plus a little from gamma	+~$544
One day passes	Theta ~−$105	−~$105
IV falls 2 points	Vega ~$114	−~$228
Total		+~$211

Bitcoin rose $1,000, but the option gained only about $211, because time and falling volatility took most of the gain.

"So when I buy an option, I'm betting on price, time, and volatility all at once," said Mint.

"Exactly," said Hash. "Which is why options feel confusing until you think in Greeks."

Chapter 8 · The Greeks, together

Starting from zero, a day's option profit and loss adds about +$544 from delta and gamma as the price moves, subtracts about $105 from theta as time passes, and subtracts about $228 from vega as volatility falls, leaving a net of roughly +$211.

8.6 Position Greeks

Traders add up the Greeks across all their options to see their overall exposure.

Position	Delta	Gamma	Theta	Vega
Long calls or puts	Directional	Positive	Negative	Positive
Short calls or puts	Directional (opposite)	Negative	Positive	Negative

A long option position wants: a large move, quickly, with rising volatility.

A short option position wants: little movement, time passing, and falling volatility.

You expect	Consider being	Because
A big move, direction unknown	Long options	Positive gamma and vega
A quiet period	Short options	Positive theta
Volatility to rise	Long options	Positive vega
Volatility to fall	Short options	Negative vega

8.7 Rho

A fifth Greek, rho, measures sensitivity to interest rates. In crypto, its effect is usually small compared with the others, though the futures basis plays a related role in pricing (Chapter 5). Most crypto traders pay little attention to rho.

8.8 Node's insight

Node, the stock investor, thought it through. "Selling options gives you theta every day, which sounds like easy income. But you're short gamma and short vega, so a sudden crash with volatility spiking hits you twice."

"That's the most important sentence about options selling," said Hash. "Chapter 12 shows the income strategies. Chapter 14 shows how to survive them."

8.9 Try it yourself

On an options chain with Greeks, find the theta and vega of an at-the-money BTC call for three different expiries. Calculate how much each would lose over a week if nothing changed. Then estimate how each would change if implied volatility rose 5 points.

Finally, use the method in section 8.5 to estimate the change in an option's value if BTC moves 2%, a day passes, and IV changes by 3 points.

Key takeaways from this chapter

1. Theta measures daily time decay. Buyers pay it; sellers earn it.
2. Theta accelerates near expiry, especially for at-the-money options.
3. Buyers pay theta for positive gamma; sellers earn theta while accepting negative gamma.
4. Vega measures sensitivity to implied volatility and is larger for longer-dated, at-the-money options.
5. The Greeks combine to explain why options can disappoint even when the price moves your way.
6. Long options want big, fast moves and rising volatility; short options want calm markets and falling volatility.
7. Option sellers are exposed to sudden crashes with rising volatility, which hurts them through both gamma and vega.