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Info on Risk Reversals for context

In the FX vanilla options market buying risk reversal involves selling a lower strike put and buying a higher strike call. The price of such a structure is a volatility spread. Thus, if we assume the existence of a delta-volatility smile function, which has the appropriate abilities to convert deltas to strikes and vice versa then one can easily use that Smile to price the following Risk Reversals:

a) -25d put with vol 8.9% and a 25d call with vol 10.15% = RR vol spread of 1.25%.

b) 1.035 strike put with vol 8.9% and a 1.100 strike call with vol 10.15% = RR vol spread of 1.25%.

The corresponding strikes in a) are determined using the given vols, and the premiums associated with each option are derived also from those given vols, using Black-76.

Question regarding Strangle

Buying a strangle is similar to an RR except it involves buying both the lower strike put and the higher strike call. With the same assumption of the existence of a delta-volatility smile function it is again possible to calculate the mid-market values of such an option combination:

a) -25d put with vol 8.9% and a 25d call with vol 10.15% gives a total premium amount.

b) 1.035 strike put with vol 8.9% and a 1.100 strike call with vol 10.15% also gives a total premium amount.

It is my understanding that FX strangles are quoted with a convention that specifies a single volatility, which is used to define the premium on both options. For b) this would then involve reverse solving for a single vol value that returns the same total premium as calculated by the Smile. Is this correct?

For a) this would seem less trivial, because the definition of the strike for the option is tied to the volatility value. So if the single quoted volatility value is used then the strikes used in a) would not be the same strikes as those implied from the Smile, and thus the overall total premium calculable from the Smile (the real market vol) would be different. So this seems like another iterative procedure to return the correct price for the strangle if it is quoted in delta terms. Is this correct?

(edit for comments) Numerical example

Suppose that a Delta-Vol Smile exists and has been calibrated by some market quotations, and it is as follows:

enter image description here

The request is for a "-20delta, 20delta strangle".

If this is directly input and calculated with the smile then we get the following (on 1mm EUR EURUSD):

  • Put: expiry: 92 cal days, strike: 1.022178, vol: 10.57422, premium: \$ 6489.09
  • Call: expiry: 92 cal days, strike: 1.110987, vol: 9.22305, premium: \$ 5398.60
  • Total premium is \$ 11887.69

Now if we apply the formula for a single (vol averaged) vol quotation:

$$ \sigma_{strangle} = \frac{\sigma_{call} v_{call} + \sigma_{put} v_{put}} {v_{call} + v_{put}} = 9.898635$$

When this volatility is then agreed the strikes and premiums on each option are calculated with this agreed price, hence the new information:

  • Put: expiry: 92 cal days, strike: 1.024921, vol: 9.898635, premium: \$ 6064.53
  • Call: expiry: 92 cal days, strike: 1.114344, vol: 9.898635, premium: \$ 5784.88
  • Total premium is \$ 11849.41

But actually calculating the premiums of the above struck options with the real volatilities from the smile gives the respective premiums \$6948.41 and \$4931.11 which totals \$ 11,879.52.

One can observe that the two total premiums are close but not exact, therefore this procedure and approximating the single vol price of the strangle has resulted in small mid-market error. Is this just ignored in practice or, does some iteration occur to derive an exact result?

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  • $\begingroup$ Where do you get/see a strangle IV quote? Usually, the entire vol surface is defined by ATM delta neutral straddles (ATM DNS), risk reversal and butterflies for various deltas. $\endgroup$
    – AKdemy
    Commented Apr 28 at 21:01
  • $\begingroup$ Yes I understand that the surface is usually calibrated by Straddle/RR/Fly. That alludes to the assumption that the surface exists. If one were asked to quote a strangle the convention (as I am led to understand) is to quote a single vol, as above, with the pricing considerations and confusions as highlighted. $\endgroup$
    – Attack68
    Commented Apr 28 at 21:06
  • $\begingroup$ I think reading page 15 onwards in the CPQF Working Paper Series No. 20 , FX Volatility Smile Construction by Dimitri Reiswich and Uwe Wystup should help. $\endgroup$
    – AKdemy
    Commented May 1 at 0:59
  • $\begingroup$ Thanks, it would appear he acknowledges the same issue. On page 19: "However, the smile strangle and quoted strangle volatilities differ significantly for the skewed JPYUSD smile." And on page 20: " a significant risk reversal will lead to a failure of the [Maltz] formula". Then on page 23 he suggests that a root search algorithm is indeed required for strangle calibration, and that in extreme cases it may not even be solvable. $\endgroup$
    – Attack68
    Commented May 3 at 5:23

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You don't need any information from your private volatility surface to price the broker straddle / fly. The volatility of the put leg and the call leg is just $\sigma=\sigma_{atm}+\sigma_{fly}$ and you then back out the $\delta$-strikes using this volatility. You then price each leg using this volatility and strike via Black-76. This premium does put a constraint on the shape of your private volatility surface.

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  • $\begingroup$ Thanks, the question was specifically regarding the quoting convention of the strangle. If you have a volatility surface/smile available, i.e. vols available for various deltas are you suggesting to price this strangle one needs to first use the smile to calculate the $\sigma_{atm}$ and the $\sigma_{fly}$ (for the specific deltas) and to then imply the vol for the target delta strangle? Seems quite convoluted? $\endgroup$
    – Attack68
    Commented Apr 28 at 21:02
  • $\begingroup$ Its simpler for vol traders and brokers this way as the thing you are trading is fully specified by two numbers and a forward. What you don't want is an hour of haggling over strikes and eventually abandoning the trade entirely in a busy market. $\endgroup$
    – river_rat
    Commented Apr 30 at 7:31

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