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Vanna-Volga Pricing and the Cost of Volatility Skew Hedges

Article Quant Q&A · Author: Wyngarden83
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Summary

The document explains why Vanna-Volga option pricing uses the difference between market and flat-volatility theoretical prices for risk-reversal and butterfly instruments. These instruments serve as calibration pillars: their observed prices are treated as a flat-volatility value plus an adjustment associated with hedging vega, vanna, and volga exposures.

Under this decomposition, the hedge cost is the adjustment beyond the flat-volatility price, rather than the full market price of the risk-reversal or butterfly. The same idea is used to estimate a price adjustment for another vanilla option from the pillar instruments’ hedge costs. The document provides a conceptual explanation but no derivation, numerical example, or discussion of assumptions and limitations beyond the decomposition itself; it points readers toward further technical treatment.

Key ideas

  • Vanna-Volga pricing decomposes pillar option prices into flat-volatility prices and hedge adjustments.
  • The market-minus-flat-volatility difference represents the cost attributed to vega, vanna, and volga hedging.
  • Risk-reversal and butterfly instruments provide the inputs for estimating adjustments to other vanilla option prices.
  • Using the full market price would include the flat-volatility component as well as the hedge adjustment.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Risk-Reversal cost in the Vanna-Volga pricing


# Risk-Reversal cost in the Vanna-Volga pricing












To hedge the Vanna in the Vanna-Volga pricing, the methodology does not use the market price of the risk-reversal but uses the difference between the market price (with skew/smile) and the theoretical price (no skew/smile) (same process for Volga).

If the goal is to set up a replicating portfolio for the Vanna and volga exposure to adjust the option's price, why not using the market prices of the RR and BF directly instead of the difference in prices? Given the market prices are what a dealer would spend to hedge the vanna/volga of the option.

## Answer by Frido (score 2)

https://quant.stackexchange.com/a/81157

(Too long for a comment)

Because it's the difference between the market price and the flat vola price that gives the cost of the vega-vanna-volga hedge.

The assumption of VV pricing is that you are given the market prices of three options (or alternatively RR and BF). It is then assumed that the market prices of these "pillar options" can be decomposed into a flat vola price and an adjustment due to the cost of hedging vega, vanna and volga.

The VV method then says that the VV (market) price of any vanilla option can be expressed as the sum of a flat vola price (the same flat vola that is used for the RR and BF) and an adustment due to the cost of hedging vega vanna volga. This cost, in turn, can be expressed in terms of the cost of the vega,vanna,volga hedge of the RR and BF.

More details in this paper by Castagna and Mercurio, if you've not already read it.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.