From three quoted numbers to the two wing volatilities, the strikes they sit at, and what the skew does to the premium.
The 25-delta risk reversal is the implied volatility of the 25-delta call minus that of the 25-delta put, and the 25-delta butterfly is the average of the two minus at-the-money volatility. Turned around, call vol = ATM + butterfly + risk reversal / 2 and put vol = ATM + butterfly − risk reversal / 2. With an illustrative 20.0 per cent ATM, a −4.0 risk reversal and a 0.6 butterfly, the 3-month 25-delta put is at 22.6 per cent and the call at 18.6 per cent, struck at 94.19 and 108.01 on a spot of 100.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The quotes describe a 3-month equity index option smile. The levels are illustrative, chosen to show the usual equity shape: puts richer than calls. Deltas are Black-Scholes spot deltas with no dividends.
| Input | Value |
|---|---|
| Spot | 100.00 |
| Risk-free rate, continuous | 4.0% |
| Time to expiry, years | 0.25 |
| At-the-money volatility | 20.0% |
| 25-delta risk reversal (call minus put) | −4.0 |
| 25-delta butterfly | 0.6 |
Step 1, the wing vols: call = 20.0 + 0.6 + (−4.0 / 2) = 18.6%; put = 20.0 + 0.6 − (−4.0 / 2) = 22.6%
Check: RR = 18.6 − 22.6 = −4.0; BF = (18.6 + 22.6) / 2 − 20.0 = 0.6
Step 2, the d1 of a 25-delta option: call delta N(d1) = 0.25, so d1 = N−1(0.25) = −0.6745; put delta N(d1) − 1 = −0.25, so d1 = N−1(0.75) = 0.6745
Step 3, the strikes: K = S × e(r + vol²/2) T − d1 × vol × root T, each with its own vol: call 108.01, put 94.19
In Excel, for the call: =S*EXP((r+vc^2/2)*T-NORM.S.INV(0.25)*vc*SQRT(T)), and for the put replace 0.25 with 0.75 and vc with vp.
Step 3 is where the smile and the strike meet. A 25-delta option is defined by its delta, and its delta depends on its own volatility, so the strike has to be solved with the wing vol, not the ATM vol. Because the put carries more volatility, it sits further from spot than it would on a flat smile.
| Option | Strike | Vol | Premium on smile | Premium at flat 20% |
|---|---|---|---|---|
| 25-delta call | 108.01 | 18.6% | 1.3251 | 1.5516 |
| 25-delta put | 94.19 | 22.6% | 1.7858 | 1.3827 |
| Risk reversal, long call short put | n/a | −4.0 | −0.4607 | 0.1689 |
On the smile, buying the call and selling the put receives 0.4607 per option, 4,607 on 10,000. On a flat smile the same strikes would cost 0.1689. The skew makes the put 0.4031 dearer and the call 0.2265 cheaper than the flat-vol price, which is the market charging for crash protection. The 25-delta strangle, long both wings, costs 3.1109 on the smile.
The risk reversal shifts the two wings in opposite directions; the butterfly lifts both together.
| RR | BF | Call vol | Put vol | Call strike | Put strike | RR premium |
|---|---|---|---|---|---|---|
| −2.0 | 0.6 | 19.6% | 21.6% | 108.43 | 94.46 | −0.3094 |
| −4.0 | 0.6 | 18.6% | 22.6% | 108.01 | 94.19 | −0.4607 |
| −6.0 | 0.6 | 17.6% | 23.6% | 107.60 | 93.93 | −0.6128 |
| −4.0 | 0.0 | 18.0% | 22.0% | 107.76 | 94.35 | −0.4515 |
| −4.0 | 1.2 | 19.2% | 23.2% | 108.26 | 94.03 | −0.4702 |
Each extra point of negative risk reversal adds about 0.08 to the credit. That is less than the vegas of the two options would suggest at fixed strikes, because the strikes are fixed by delta, not by price: as the put's vol rises its 25-delta strike moves further out, and as the call's vol falls its strike moves closer to spot, which offsets part of the change. The butterfly barely changes the risk reversal premium; it changes the strangle, from 3.0199 at zero to 3.2019 at 1.2.
In FX markets the quoted butterfly is often a "market" or "broker" strangle, defined by a different pricing rule, and it is not the same number as the smile butterfly used here. Confirm which convention a quote follows before backing out wing vols from it.
Recover the wings with ATM + BF ± RR / 2, then solve each strike with its own vol. Here 20.0, −4.0 and 0.6 give a 22.6 per cent put at 94.19 and an 18.6 per cent call at 108.01, and the skew turns a 0.1689 cost into a 0.4607 credit. The book's volatility workbook, which shows vega across strikes and maturities and solves implied volatility by iteration, is in the free workbooks for this book. For the delta inversion behind the strikes, see how to calculate delta, and for what each vol point is worth, how to calculate vega.
Because downside puts trade at higher implied volatility than upside calls, and the convention is call minus put. In the illustrative case the 25-delta put is 22.6 per cent and the call 18.6 per cent, a risk reversal of -4.0. Some desks quote put minus call and would show 4.0, so always check the sign before comparing quotes.
The curvature of the smile: how much both wings sit above at-the-money volatility on average. A 0.6 butterfly on a 20.0 per cent ATM puts the average of the two 25-delta vols at 20.6 per cent. Raising it to 1.2 lifts both wings by the same 0.6 and makes the 3-month 25-delta strangle cost 3.2019 instead of 3.1109.
Invert the delta. For a call, N(d1) = 0.25 gives d1 = -0.6745, and K = S x e^((r + vol squared / 2) T - d1 x vol x root T). Using the 25-delta call's own 18.6 per cent vol gives 108.01; using the 20.0 per cent ATM vol by mistake gives 108.59, a 23-delta option.
This article is one calculation from Quantitative Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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