Volatility smile

One expiry, many IVs

If Black-Scholes described the market perfectly, every option of one expiry would trade at the same implied volatility, whatever its strike. They don't. Plot each strike's IV for one expiry and you get a curve, usually lowest near the money and rising toward both wings: the smile.

Why the wings trade higher

• Fat tails. Large moves happen more often than the model's lognormal distribution says, so an option far from the money is worth more than at-the-money IV would price it.
• Demand for protection. Holders, funds and miners buy out-of-the-money puts as insurance against a crash, and sellers of those puts want extra for the jump risk. In crypto, that makes the put side the higher one in sell-offs and bear phases.
• Supply on the call side. Holders who sell out-of-the-money calls for yield add supply there, which tends to keep call IV lower.
• Rallies. In a strong rally, upside calls can be chased hard enough that the call wing rises above the put wing, and not only briefly: BTC's 25-delta risk reversal stayed positive for long stretches of the first half of 2021, late 2023 to early 2024, and late 2024.

A worked example

A 30-day expiry with the forward at $100,000: the $80,000 put trades at 60% IV, the $90,000 put at 54%, the money at 50%, the $110,000 call at 48% and the $120,000 call at 50%.

• At 54%, the $90,000 put costs $2,165. At a flat 50% it would be $1,828: the smile adds $337, 15.6% of the price (18% more than the flat price).
• At 60%, the $80,000 put costs $703, more than twice the $329 a flat 50% would give. The further out of the money, the more of the price is the smile.
• A short strangle on the $90,000 put and $110,000 call collects $2,165 + $2,085 = $4,250, against $4,103 at a flat 50%: its wings are priced off the smile, not off at-the-money IV. The same is true of what a long strangle pays.

Smile and skew

The two words describe the same curve. The smile is its curvature: both wings above the middle. The skew is its tilt: one side higher than the other. Crypto curves usually show both: a smile, tilted toward the puts or the calls depending on the regime. Skew is measured with 25-delta options rather than fixed strikes, and the smiles of all expiries together make the volatility surface.

How BOSS draws it

For the selected expiry, BOSS takes the venue's own mark IV at each strike from the out-of-the-money side: the put below the forward, the call above it. The out-of-the-money option is usually the more liquid quote, and by put-call parity a call and a put at one strike should carry nearly the same IV anyway. The chart keeps strikes within three standard deviations of the forward (ATM IV × √time), so the far wings of a sub-day expiry, quoted off near-zero premiums, don't flatten the meaningful part of the curve.