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I am looking for a reference on how to approximate implied volatility in a stochastic model vis-a-vis vanna, volga, vega, and other model parameters, in particular the derivation of such equations and the underlying intuition.

Any good articles/books out there?

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I find Dimitri Reiswich's Ph.D. thesis quite useful when it comes to FX smile construction and market conventions. Section 3.3 is on vanna/volga method.

Also have a look at Uwe Wystup's book, especially Section 3.1 "The Trader's Rule of Thump".

References

Reiswich, Dimitri (2010) "The Foreign Exchange Volatility Surface", Ph.D. Dissertation, Frankfurt School of Finance & Management

Wystup, Uwe (2006) "FX Options and Structured Products", John Wiley & Sons

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perhaps this is what you're looking for? - lookup Mercurio Vanna VOlga "Consistent Pricing of FX Options " this paper derives a formula for the vol smile in terms of atm vol and 25 delta call and put vols

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Here's more references to the vanna-volga method:

  • Castagna, Mercurio (2007), The vanna-volga method for implied volatilities, Risk [ Download ]

  • Perederiy (2018), Vanna-Volga Method for Normal Volatilities, arXiv. Note: As the title indicates, this paper is about the normal volatility

  • Wikipedia Note: also see the references there

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