The Vanna—Volga method is a mathematical tool used in finance. It is a technique for pricing first-generation exotic options in foreign exchange market FX derivatives. It consists of adjusting the Black—Scholes theoretical value BSTV by the cost of a portfolio which hedges three main risks associated to the volatility of the option: The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate:.
The rationale behind the above formulation of the Vanna-Volga price is that one can extract the smile cost of an exotic option by measuring the smile cost of a portfolio designed to hedge its Vanna and Volga risks. The reason why one chooses the strategies BF and RR to do this is because they are liquid FX instruments and they carry mainly Volga, and respectively Vanna risks.
The above approach ignores the small but non-zero fraction of Volga carried by the RR and the small fraction of Vanna carried by the BF. It further neglects the cost of hedging the Vega risk.
This has led to a more general formulation of the Vanna-Volga method in which one considers that within the Black—Scholes assumptions the exotic option's Vega, Vanna and Volga can be replicated by the weighted sum of three instruments:. Given this replication, the Vanna—Volga method adjusts the BS price of an exotic option by the smile cost of the above weighted sum note that the ATM smile cost is zero by construction:.
The resulting correction, however, typically turns out to be too large. The Vega contribution turns out to be several orders of magnitude smaller than the Vanna and Volga terms in all practical situations, hence one neglects it. Since the Vanna-Volga method is a simple rule-of-thumb and not a rigorous model, there is no guarantee that this will be a priori the case. The attenuation factors are of a different from for the Vanna or the Volga of an instrument.
This is because for barrier values close to the spot they behave differently: Hence the attenuation factors take the form:. Both of these quantities offer the desirable property that they vanish close to a barrier. For example, for a single barrier option we have. Similarly, for options with two barriers the survival probability is given through the undiscounted value of a double-no-touch option.
The first exit time FET is the minimum between: From Wikipedia, the free encyclopedia. Energy derivative Freight derivative Inflation derivative Property derivative Weather derivative. Retrieved from " https: Mathematical finance Derivatives finance Portfolio theories.