Here is the book Foreign Exchange Option Pricing: A Practitioner’s Guide, p.56 by Clark (2015).

The context is a little bit long. I summery my understanding as follow:

We first assume the form of volatility such as $\sigma(K) = aK^2+bK+c$ (just a example), then what we can obtain from the market is

  1. market strangle $\sigma_{25-d-ms}$
  2. risk reversal $\sigma_{25-d-RR}$

here we don't know that ATM voal $\sigma_{ATM}$ and smile strangle $\sigma_{25-d-SS}.$

Then use the following equations and least squares optimiser to obtain the best value of $a,b,c.$

Is it right? What I confused is the original items we obtained from the market. Since sometime we interpolate the vol smile use the sample point 10/25-delta-risk reversal 10/25-delta-butterfly (smile strangle) and ATM vol such five points:

$$\Delta_{Q}\left( -1,K_{25-d-P},T,\sigma_X \left( K_{25-d-P} \right)\right) = -0.25,$$

$$\Delta_{Q}\left( +1,K_{25-d-C},T,\sigma_X \left( K_{25-d-C} \right)\right) = +0.25.$$

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The $\sigma_{ATM}$ is given by the market along with the market strangle and risk-reversal, you don't solve for it in the sense you don't know it from the market.

You are trying to find the parametrisation of $\sigma(K)$ such that you fit $\sigma(K_{ATM}) = \sigma_{ATM}$. To clarify the whole routine, you need to find the solutions to $a,b,c$ such that the parametric smile fits through the ATM point as well as respecting equations 3.19 and when you calculate the smile-strangle 3.20 and plug that into the the pricer for a strangle, you obtain the same price as plugging in the vols from the market strangle.

Note that the market strangle will imply a different set of strikes per delta than the smile strangle, that is OK since you are looking to match the prices of the market strangle $V_{MS}(\sigma_{25-d-MS}, \{K_{25-d-MS}\})$ with the the smile strangle $V_{MS}(\sigma_{25-d-SS}, \{K_{25-d-SS}\})$

One final comment, your example parametrisation will give you a lot of trouble, especially in the wings. Typically you want your vol as a function of log-moneyness to avoid blow-ups. A good start would be to try the SABR interpolation formula with $\beta=1$ to simplify the maths and avoid instabilities in the calibration of the skew. Another option is to try Gatheral's SVI model that you can find here: https://arxiv.org/abs/1204.0646

  • so actually the words 'Determine $K_{ATM}$ by using either (3.7) or (3.8) with $\sigma_{ATM}$' mean $K_{ATM}$ is calculated by given value $\sigma_{ATM}$ from the equation (3.8). And actually only the strikes $K_{25-d-C}$ and $K_{25-d-P}$ are dependent on the parameters $a,b,c,$ $K_{ATM}, K_{25-d-C-MS}, K_{25-d-P-MS}$ are independent on the parameter $a,b,c?$ – user6703592 2 days ago
  • Another question is that we usually use ATM vol, 10/25-d-RR, 10/25-d-butterfly, those five quoted points to interpolate/extrapolate the whole smile curves. Are they totally two different stories? – user6703592 2 days ago
  • Yes that’s right, only the smile-strangle implied strikes will change under your choice of interpolation scheme, i.e. $a,b,c$. You must be careful though, as there is no guarantee that the solution to the unknowns will ensure the ATM strike gives the ATM vol from the final surface, so you need to use/modify schemes to enforce this. You can add penalty functions etc or use a function of $ln(K/F)$ that immediately solves $c=\sigma_{ATM}$ in your example. – BrownianBread yesterday
  • For your second point about 10-deltas, you can repeat the procedure for these in the same optimisation, so your procedure above for implying the 25-delta is used twice for 10/25-delta. In your optimiser you can then do a least squares fit on residuals for both deltas’ strangles, risk reversals and ATM points. So you end up with one curve that fits through 5 points. You may want weights in your calibration to ensure fitting is accurate where it matters most. – BrownianBread yesterday
  • sorry, for the second question i mean in some book, I see we directly obtain those five values from market quoting and no above procedure. We obtain the whole smile curve by interpolating those five given points. Namely we quote the smile strange (butterfly) instead of the market strange? So is it the true story? – user6703592 22 hours ago

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