Risk Reversals & Butterflies by Delta
Objectives
- Construct the delta-strike risk reversal and butterfly from a quoted volatility smile and state what each one measures.
- Read the sign and magnitude of a risk reversal as directional risk pricing, and the butterfly as the market's price of tail convexity.
- Interpret SVQ's delta-strike RR = IV_call − IV_put and BF outputs as evidence about where the distribution's mass is leaning.
By now you know that a single implied volatility number is a summary, and like all summaries it throws information away.
When you quote one volatility for an asset, you are pretending the market believes returns are symmetric and well-behaved — a tidy bell curve. But the option market almost never believes that. It charges different volatilities for different strikes, and that pattern of volatilities across strikes is the smile. The whole craft of skew trading is learning to read the shape of that smile, because the shape is where the market hides its real opinion about which way things might break and how violently. Your job in this chapter is to learn the two numbers that compress the smile's shape into something you can trade and compare: the risk reversal and the butterfly, both measured by delta.
Let us build the intuition before any formula.
Imagine the smile as a curve. It has a level — roughly how high the whole thing sits, which is the at-the-money volatility. It has a tilt — whether the left side, the downside puts, costs more than the right side, the upside calls. And it has a curvature — how much the wings lift up above the middle, how pronounced the smile actually is. Level, tilt, curvature. That is genuinely most of what a smile is, and two of those three are exactly what the risk reversal and the butterfly capture. The risk reversal is the tilt. The butterfly is the curvature.
Now, why measure by delta instead of by strike? Because delta is the market's own native ruler. A strike of four hundred means nothing without context, but a twenty-five delta call is a recognizable, comparable point on every smile, on every asset, on every day — it is the option that is moderately out of the money in a way the market consistently treats the same. Quoting the smile at fixed deltas, typically the twenty-five delta and the ten delta wings against the at-the-money, lets you compare today's skew to last month's, and one asset's skew to another's, without the strikes drifting underneath you as spot and time move. This is why FX desks have quoted vol surfaces in delta terms for decades, and why SVQ does the same.
So here is the risk reversal, formalized. The twenty-five delta risk reversal is the implied volatility of the twenty-five delta call minus the implied volatility of the twenty-five delta put:
RR = IV_call − IV_put.
That is it. It is a difference of two volatilities taken at symmetric distances out on either wing. Read the sign carefully, because the sign is the whole message. If the risk reversal is negative, puts are more expensive than calls — the market is paying up for downside protection, which is the normal state for equity indices, where crashes are the feared event. If the risk reversal is positive, calls are bid over puts — the market fears, or craves, the upside more, which is common in some commodities and in assets gripped by a melt-up. The magnitude tells you how lopsided that fear is. A risk reversal of minus-six volatility points is a market leaning hard to the downside; a risk reversal hovering near zero is a market that genuinely sees the two tails as balanced.
The butterfly measures the other thing — the curvature. The twenty-five delta butterfly is the average of the two wing volatilities minus the at-the-money volatility:
BF = ½ (IV_call + IV_put) − IV_atm.
Think about what that difference captures. If the wings, on average, sit far above the middle, the smile is deeply curved — the market is paying a premium for both tails relative to the body, which is the market's price of convexity, its expectation of fat tails and large moves in either direction. A large butterfly says the market expects the unexpected. A butterfly near zero says the market thinks the distribution is close to lognormal, with thin, ordinary tails. Crucially, the butterfly is symmetric by construction — it does not care which way the move goes, only how violent. That is exactly why it pairs so cleanly with the risk reversal, which carries all the directional information. Together, tilt and curvature, they reconstruct the smile's shape from its flat midpoint.
SVQ's vol surface carries a delta-strike row for each single expiry. SVQ computes RR = IV_call − IV_put directly from the twenty-five delta call and put it has fitted on the smile, and BF = ½(IV_call + IV_put) − IV_atm against the at-the-money node. Suppose for a thirty-day expiry SVQ shows a twenty-five delta call vol of eighteen, a put vol of twenty-four, and an at-the-money of twenty. The risk reversal reads eighteen minus twenty-four, or minus six — a strong downside lean, puts firmly bid. The butterfly reads the average of eighteen and twenty-four, which is twenty-one, minus the at-the-money twenty, giving plus one — a modestly fat-tailed smile. Two numbers, and together they summarize the entire shape of that day's distribution: leaning down, with slightly heavy wings.
This is also where the institute's throughline reappears. A risk reversal is not a forecast — it is a price. When you see a sharply negative risk reversal, the honest question is not "is the market right that it will fall," but "is this lean genuine evidence of an impending move, or merely the standing cost of insurance that everyone pays regardless?" An extreme reading is only evidence when it is extreme relative to its own history. A risk reversal of minus-six is unremarkable in an index that always sits there and screaming in an asset that normally sits near zero. Learning to tell those two apart — the structural lean from the genuine signal — is the work of the rest of this course.