MTM Hedging Performance of Vanna-Volga

Question

I was wondering how well Vanna-Volga (VV) Implied Vols "perform". So I experimented with the following option parameters $$S_0=100,\ K=92,\ r=0.03,\ q=0.01,\ T=2$$ and VV parameters $$K_1,K_2,K_3=94,\ 105,\ 118,\quad \sigma_1,\sigma_2,\sigma_3=0.18,\ 0.12,\ 0.14$$ As per The Vanna-Volga method for implied volatilities, I computed the weights $$x_1,\ x_2,\ x_3=0.6887,\ 0.4208,\ -0.1026$$ Now we can define two portfolios $$P_1 = C(K) = \text{Call with Strike }K\text{ and Notional 1}$$ $$P_2=\sum_{i=1}^3x_iC(K_i)= \text{3 Calls with strikes }K_1,K_2,K_3\text{ and Notionals }x_1,x_2,x_3$$

In short, we can now price $P_1^{FLAT}$ and $P_2^{FLAT}$ using a Flat Vol of $\sigma_2=0.12$ for all strikes. Then we price $P_2^{MTM}$ using the actual "MTM" vols $\sigma_1,\sigma_2,\sigma_3$ for $K_1,K_2,K_3$. After computing the MTM-adjusted price $$P_1^{MTM}=P_1^{FLAT}+(P_2^{MTM}-P_2^{FLAT} )$$ for portfolio 1, we can then back out implied vol $\sigma_{imp}=0.1678$ that matches $P_1^{MTM}$.

However, when looking at the risk breakdown below we can see that Vega, Volga & Vanna match for the FLAT risk (green numbers), but not when we mark-to-market as there is a noticeable difference (red numbers). Different settings can even give opposite signs for Vanna or Volga.

Is the risk mismatch a known drawback? And more importantly, is there a way to overcome this?

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Just as a sanity check, I plotted the VV implied vols and they look fine.

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Answer 1

This mismatch is known, at least to the extent that, as someome familiar with the Vanna-Volga model, I wouldn't have expected it to have the property that you are testing for.

There various ways of managing risks of vanilla options. Three very classical ways are to compute hedges for each vanilla in a portfolio according to the Black-Scholes model with either (1) the at-the-money (ATM) volatility or (2) the implied volatility for the option in question, or (3) to compute volatility greeks assuming the ATM volatility moves while the risk-reversal (RR) and butterfly (BF) quotes remain fixed.

The Vanna-Volga model is closely related to approach (1). In the Vanna-Volga approach each vanilla option has a base value according to its Black scholes price with the at-the-money volatility. It also has a price adjustment for the "value" of the option's Vega, Vanna, and Volga, again under the Black-Scholes model with at-the-money volatility. The "value" of the Vega, Vanna, and Volga are implied from the RR and BF vol quotes. [1]

You are asking essentially if it is known that hedge approach (1) and (2) give different results, in that you are comparing the vega, vanna, and volga of a hedge portfolio using flat vols (approach 1) vs smile vols (approach 2). Yes these are known to differ.

Probably what you want is to use one of the approaches (1-3) on top of the Vanna-Volga model as a curve model. Each of hedge approaches (1-3) is agnostic to the underlying pricing model. I think generally approaches (2) and (3) are comparable in effectiveness and both a bit better than approach (1). If you will try to manage smile risk, approach (3) has the advantage of letting you find sensitivities to each hedge instrument (ATM, RR, BF) to do that.

[1] https://www.researchgate.net/publication/247606459_Variations_on_the_Vanna-Volga_Adjustment