Module 8: Crypto Options
What Drives Option Prices
Chapter 5 of 14 · about 5 minutes
5.1 Same price, different premium
Mint checked the price of a 30-day $100,000 BTC call on Monday and on Thursday. Bitcoin was at almost exactly $100,000 both days. But Monday's premium was about $5,100, and Thursday's was about $7,400.
"Bitcoin didn't move," said Mint. "Why did the option get so much more expensive?"
"Because the market started expecting bitcoin to move more," said Hash. "An option's price isn't just about where the price is. It's about how much the market thinks the price might move. That expectation is the most important number in options."
5.2 The five inputs
Option prices depend mainly on five factors:
| Factor | Effect on call price when the factor rises | Effect on put price when the factor rises |
|---|---|---|
| Underlying price | Up | Down |
| Strike price | Down | Up |
| Time to expiry | Up | Up |
| Volatility | Up | Up |
| Interest rates | Slightly up | Slightly down |
The first two determine intrinsic value and moneyness (Chapter 4). The next two determine how much time value the option has. Interest rates matter much less in crypto than the others.
5.3 Underlying price
As BTC rises, calls gain value and puts lose value.
| BTC price | 30-day $100,000 call (55% IV) |
|---|---|
| $100,000 | ~$6,284 |
| $101,000 | ~$6,828 |
A $1,000 rise in BTC increased the call's value by about $544. That's roughly half the move, which makes sense for an at-the-money option that has about a 50/50 chance of finishing in the money. Chapter 7 measures this sensitivity with delta.
5.4 Time to expiry
More time means more chance for the price to move in the option's favour, so longer options cost more (Chapter 4).
| Expiry | $100,000 call premium |
|---|---|
| 7 days | ~$3,038 |
| 30 days | ~$6,284 |
| 90 days | ~$10,862 |
Notice that 90 days doesn't cost three times as much as 30 days. Option value grows roughly with the square root of time. Doubling the time doesn't double the price; it increases it by about 41%. This is why longer options are cheaper per day of protection.
5.5 Volatility
Volatility is the factor that explains Mint's puzzle.
An option's value comes from the chance of large moves. If the underlying is expected to move a lot, there's a bigger chance the option ends up deep in the money. And because the buyer's loss is capped at the premium, more movement mostly adds upside for the buyer. So higher expected volatility makes both calls and puts more expensive.
| Expected volatility | 30-day $100,000 call premium |
|---|---|
| 45% | ~$5,143 |
| 55% | ~$6,284 |
| 65% | ~$7,424 |
On Monday, the market expected about 45% volatility. By Thursday, ahead of a major announcement, it expected about 65%. The option's price rose by over 40% without bitcoin moving at all.
"So I can make or lose money on an option just from the market's mood changing," said Mint.
"Exactly," said Hash. "Options let you trade expected volatility itself. That's what makes them different from every instrument we've studied so far."
Chapter 6 explains implied volatility in detail.
5.6 Interest rates and the forward price
Interest rates have a small effect on option prices, through the cost of holding the underlying versus holding cash. In crypto, the more relevant factor is the futures basis (Module 7, Chapter 2): options are usually priced relative to the forward price for their expiry, rather than spot. If 30-day futures trade above spot, calls are slightly more expensive and puts slightly cheaper than they'd otherwise be.
5.7 Pricing models
Traders and exchanges use mathematical models to calculate fair option prices from these inputs. The best-known is the Black-Scholes model, developed in the early 1970s, which won a Nobel Prize in economics. The examples in this module are calculated using a version of it.
Black-Scholes makes simplifying assumptions, such as constant volatility and smooth price movements, that don't fully hold in crypto, where prices jump and volatility changes constantly. Professional traders adjust for this, notably through the volatility smile and skew (Chapter 9). But the model remains the common language of options markets.
You don't need to calculate it by hand. Options exchanges and calculators do it for you. What matters is understanding how each input pushes the price.
5.8 Prices in coins
On many crypto options venues, especially those using coin-margined contracts (Module 7, Chapter 3), option prices are quoted in the underlying coin rather than dollars.
| Value | |
|---|---|
| $100,000 call premium in dollars | $6,284 |
| Same premium quoted in BTC | 6,284 ÷ 100,000 ≈ 0.0628 BTC |
This can confuse new traders: the dollar value of an option quoted in BTC changes with both the option's value and BTC's price. Always check the quote currency.
5.9 Try it yourself
Use a free online option pricing calculator. Enter a BTC price, strike, expiry, and volatility. Then change one input at a time and record the effect on the call and put prices: raise the underlying by 5%, double the time, and raise volatility by 10 points. Which change had the biggest effect on the at-the-money option?
Key takeaways from this chapter
- Option prices depend on the underlying price, strike, time to expiry, volatility, and interest rates.
- Rising underlying prices increase call values and decrease put values.
- More time increases both call and put prices, roughly with the square root of time.
- Higher expected volatility increases both call and put prices, sometimes dramatically.
- Options can gain or lose value from changes in expected volatility alone, without the price moving.
- Crypto options are priced relative to the forward price, affected by the futures basis.
- Black-Scholes is the standard pricing language, despite simplifying assumptions, and prices may be quoted in coins.