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bengourion
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Questions about Vanna-Volga from Castagno and Mercurio

September 6th, 2018, 9:36 am

Working on a topic related to the vana-Volga method, i came back to the paper of Castagno and Mercurio and i have some concerns about the paper:

here is the link to the paper as i will quote some parts to expose my question

https://www.researchgate.net/profile/Fa ... lities.pdf

The Vanna-Volga is based on the idea that the price of an option is a function of the stock value S (the FX in the paper but it doesn't matter whether the underlying is a fx or not here) and the implied volatility v so that to build a locally hedged portfolio you not only need cancel Delta but also the Vega, the Vanna and the Volga components. The market products used to proceed are then the underlying S and 3 quoted options. 
Very important, in page 3 of the paper just before the remark 2.1, it is clearly mentionned that the dynamics of the state variables S and v are assumed to be continuous ("under the assumptions of no jumps").
Then to pursue the objective, the authors computes the dynamic of the portfolio V assuming S and v are the only two state variables using the Ito's Lemma. They obtain a dynamic exhibiting a theta, delta, vega and gamma components as well as Volga and Vanna components. So far so good. 
Then they say that to identify the hedging ratios, (the delta ratio is easily identified separately), they have to cancel the vega, the vanna and the volga components. 
My main concern is that i consider this approach to be theroretically inconsistent. Why ?
As your dynamic are continuous, the only components to be stochastic here are the ones depending on dSt and dvt . The other components, especially the vanna and the volga ones are depending on the quadratic variations <dSt> <dStdvt> and <dvt> and so are deterministic quantities. In the paper, applying the Itô lemma, the authors write the gamma, the volga and the Vanna as depending on (dSt)2 (dvt)2 and (dStdvt) but that is a mistake : the Itô lemma makes quadratic variations to appear and not simple square or simple multiplications.  So in fact, at time t,  the vanna and volga are NOT stochastic at all and so it is theoretically pointless to cancel them: they don't bear any risk (beware that the current time, the vanna and the volga are only depending on known quantities and so are known quantities (F-t adapted) and so don't bear randomness). So in fact, you only need to identify the hedge ratios by cancelling the delta and the Vega components. One direct consequence is that using 3 options to locally hedge the risks (aka the randomness of the brownians) is also useless, you only need one.
Apart of this, in fact, if the dynamic of V is studied at discrete times, the idea of Vanna-Volga becomes relevant as the quadratic variations are replaced by the squares and products above because you doesn't need anymore the Itô's lemma but the Taylor theorem. So in this paper, i feel the problem is ill-posed at the beginning because of this wrong application of the Itô's Lemma... 

It doesn't necessarily invalidate their end results but it is annoying...

Your opinion ?