Crypto options and the Greeks

Calls, puts, intrinsic value and time value

A call pays the amount by which the underlying finishes above the strike. A put pays the amount by which it finishes below. At expiry nothing else matters:

Before expiry an option is worth its intrinsic value, what it would pay if it expired now, plus time value, the price of the chance that it pays more later. Here is one illustrative market used throughout this guide: bitcoin at 60,000, 30 days to expiry, implied volatility 50% for every strike, priced with the Black model shown below. Prices are in dollars and rounded.

Notice that the in the money 55,000 call has exactly the time value of the 55,000 put. That is put call parity at work: a call minus a put at the same strike is a forward, C − P = F − K, so the two can never drift apart without an arbitrage.

How crypto options are listed and settled

Deribit is the leading crypto options exchange, and Coinbase completed its acquisition of it on August 14, 2025. Its conventions are the ones most crypto option data is quoted in.

The inverse convention deserves a second look. A 3,428 dollar premium appears on the screen as 0.0571 BTC, and a call that finishes 5,000 dollars in the money at a settlement of 65,000 pays 5,000 ÷ 65,000, about 0.0769 BTC. That is harmless for a buyer, but it compounds risk for a seller of inverse puts who holds BTC as collateral: in a crash the option liability grows while the collateral paying for it shrinks.

What sets the price

Crypto options are priced off the forward, so the cleanest model is Black’s version of Black Scholes. With discounting ignored, which barely matters over a month:

Every input except σ can be read off the screen, which is why the market quotes options in implied volatility rather than dollars. For an at the money option there is a handy shortcut:

from math import log, sqrt, exp, pi, erf

N   = lambda x: 0.5 * (1 + erf(x / sqrt(2)))      # standard normal CDF
pdf = lambda x: exp(-x * x / 2) / sqrt(2 * pi)

def black(F, K, T, vol, kind='call'):
    d1 = (log(F / K) + vol * vol * T / 2) / (vol * sqrt(T))
    d2 = d1 - vol * sqrt(T)
    if kind == 'call':
        price, delta = F * N(d1) - K * N(d2), N(d1)
    else:
        price, delta = K * N(-d2) - F * N(-d1), N(d1) - 1
    gamma = pdf(d1) / (F * vol * sqrt(T))
    vega  = F * pdf(d1) * sqrt(T) / 100              # per 1 vol point
    theta = -F * pdf(d1) * vol / (2 * sqrt(T)) / 365  # per day
    return price, delta, gamma, theta, vega

black(60_000, 60_000, 30 / 365, 0.50)
# (3428.26, 0.5286, 0.0000463, -57.04, 68.45)

Implied volatility versus realized volatility

Implied volatility is the σ that makes the model match the market price. It is a price, set by supply and demand for options, not a measurement. Realized volatility is what the underlying actually did, measured from its returns:

To turn an annual figure into a daily move, divide by √365. Deribit gives the same rule for its DVOL index, a 30 day forward looking implied volatility index for bitcoin. At 50%, the market is pricing a one standard deviation daily move of about 2.62%, or 1,570 dollars at 60,000, and a 30 day move of about 14.3%.

Sellers of options carry the risk of sudden jumps, so they charge for it, and implied volatility often sits above what is later realized. That gap is the volatility risk premium. It is compensation for a real risk, and it is paid back, with interest, in the weeks when realized volatility blows through implied.

The Greeks, with numbers

The Greeks are the partial derivatives of the option price. Here they are for the 60,000 at the money call in the example market.

Vega is easy to check: lift implied volatility from 50% to 60% and the call goes from 3,428 to 4,112, a gain of 684 dollars, ten points at about 68 each. The signs follow from which side of the trade you are on:

Every long option is long gamma and vega and pays theta. Every short option is the mirror image. There is no position that collects theta without being short gamma, and that is the whole story of option selling.

Gamma and theta are the same trade

Over a short step, the change in an option’s value is well described by a second order expansion:

Move bitcoin up 1,000 dollars over one day in the example and the expansion predicts +494.66 dollars for the at the money call; the full model gives +493.87. Now hedge away the delta. What is left is the gamma term fighting the theta term (hedging with a perp, as in the perps guide, is the usual way to do it), and they break even when:

This is what implied volatility means in practice. A delta hedged option buyer makes money if bitcoin moves more than about 1,570 dollars a day on average and loses if it moves less. The seller has the opposite trade. Buying an option is a bet that realized volatility will exceed implied, not simply a bet on direction.

Smile, skew and term structure

If the model were complete, one volatility would price every strike. It never does. Plot implied volatility against strike and you get a smile, with the wings priced higher because large moves happen more often than the normal distribution allows. The asymmetry between the two wings is the skew, usually quoted as a 25 delta risk reversal:

Equity index options almost always have put skew, because investors pay for crash protection. Crypto skew swings: puts trade richer in fearful markets, and calls can trade over puts in strong rallies when traders chase upside. Term structure is volatility across expiries. Calm markets usually show longer expiries priced higher; stress or a known event inverts the curve as near term options get bid. The guide to reading derivatives data covers how to read these as context.

Common structures and their payoffs

All priced in the example market: bitcoin at 60,000, 30 days, 50% implied volatility, fees ignored. P&L is at expiry, per 1 BTC. Premiums use unrounded model prices, so they can differ by 1 from the rounded table above.

Spreads and collars trade away part of the payoff to cut the premium. The straddle shows the gamma and theta story in one line: it needs an 11.4% move in a month just to break even, which is exactly the average absolute move the model expects (about 0.8 times the 14.3% one standard deviation move). At fair implied volatility the straddle breaks even on average; it profits only if realized volatility beats implied.

Why selling options is not free yield

Selling the 70,000 call collects 668 dollars, 1.11% of the notional in a month, about 13.5% a year if it could be repeated at the same price. The model’s risk neutral probability that it finishes in the money, N(d₂), is about 12.6%. Most months it expires worthless and the income looks steady. Then bitcoin settles at 75,000, the call pays out 5,000 dollars and the seller is down 4,332 net of the premium, six and a half months of premium in one expiry. Like every yield, it has a payer and a reason; where yield comes from applies the same test across DeFi.

Onchain options and option vaults

Onchain options have moved toward the designs that work in traditional markets. Venues such as Derive and Aevo run order books and request for quote systems on top of margin accounts; Aevo, for example, matches orders off chain for speed and settles onchain. That keeps custody and settlement verifiable while borrowing the market structure of a centralized exchange, and it adds the usual onchain risks: contract bugs, oracle settlement and bridge or chain dependencies.

Option vaults automate the two income strategies above. Depositors’ assets are sold as covered calls or cash secured puts on a schedule, often weekly, typically to professional market makers, often through an auction. The headline yield is the premium, and every risk from the previous section comes with it:

• Short gamma by construction. A sharp rally caps the call vault; a crash makes the put vault pay out at the strike.
• Auction pricing. The premium is what a few bidders offered, which can be below fair value when demand is thin.
• Lockups. Funds are usually committed until the round expires, so you cannot exit mid move.
• Contract and settlement risk. The vault, the option contracts and the settlement oracle all have to work.

For primary sources, read Deribit’s inverse options specification and its DVOL methodology.

Common questions

Where to go next