Calibrating FX Volatility Smiles to Market Strangles

# Calibration of parameters of implied vol smile


# Calibration of parameters of implied vol smile












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

- market strangle $\sigma_{25-d-ms}$

- 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:

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

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

Quote volatility curve by five delta

## Answer by BrownianBread (score 1)

https://quant.stackexchange.com/a/42568

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