Vanna–Volga pricing
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. Description. 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 Vega formula_1, the Vanna and the Volga. The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate: formula_2. Similarly, the Volga is the sensitivity of the Vega with respect to a change  of the implied volatility formula_3: formula_4. If we consider a smile volatility term structure formula_5 with ATM strike formula_6, ATM volatility formula_7, 25-Delta call/put volatilities formula_8, and where formula_9 are the 25-Delta call/put strikes (obtained by solving the equations formula_10 and formula_11 where formula_12 denotes the Black–Scholes Delta sensitivity) then the hedging portfolio will be composed of the "at-the-money" (ATM), "risk-reversal" (RR) and "butterfly" (BF) strategies: formula_13 with formula_14 the Black–Scholes price of a call option (similarly for the put). The simplest formulation of the Vanna–Volga method suggests that the  Vanna–Volga price formula_15 of an exotic instrument formula_16 is given by formula_17 where by formula_18 denotes the Black–Scholes price of the exotic and the Greeks are calculated with ATM volatility and formula_19 These quantities represent a "smile cost", namely the difference between the price computed with/without including the smile effect. 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 weighting factors formula_20 and formula_21 represent respectively the amount of RR needed to replicate the option's Vanna, and the amount of BF needed to replicate the option's Volga. 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: formula_22 where the weightings are obtained by solving the system: formula_23 with formula_24, formula_25, formula_26 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): formula_27 where formula_28 and formula_29 The quantities formula_30 can be interpreted as the market prices attached to a unit amount of Vega , Vanna and Volga, respectively. The resulting correction, however, typically turns out to be too large. Market practitioners thus modify formula_15 to formula_32 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. The terms formula_33 and formula_34 are put in by-hand and represent factors that ensure the correct behaviour of the price of an exotic option near a barrier: as the knock-out barrier level formula_35 of an option is gradually moved toward the spot level formula_36, the BSTV price of a knock -out option must be a monotonically decreasing function, converging to zero exactly at formula_37. 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: the Vanna becomes large while, on the contrary, the Volga becomes small. Hence the attenuation factors take the form: formula_38 where formula_39 represents some measure of the barrier(s ) vicinity to the spot with the features formula_40 The coefficients formula_41 are found through calibration of the model to ensure that it reproduces the vanilla smile. Good candidates for formula_42 that ensure the appropriate behaviour close to the barriers are the "survival probability" and the "expected first exit time". Both of these quantities offer the desirable property that they vanish close to a barrier. Survival probability. The survival probability formula_43 refers to the probability that the spot does not touch one or more barrier levels formula_44. For example, for a single barrier option we have formula_45 is the value of  a "no-touch" option and formula_46 the discount factor between today and maturity. Similarly, for options with two barriers the survival probability is given through the undiscounted value of a double-no-touch option. First-exit time. The first exit time (FET) is the minimum between: (i) the time in the future when the spot is expected to exit a barrier zone before maturity, and (ii) maturity, if the spot has not hit any of the barrier levels up to maturity. That is, if we denote the FET by formula_47 then formula_48minformula_49 where formula_50 such that formula_51 or formula_52 where  formula_53 are the 'low' vs 'high' barrier levels and formula_54 the spot of today. The first-exit time is the solution of the following PDE formula_55 This equation is solved backwards in time starting from the terminal condition formula_56 where formula_57 is the time to maturity and boundary conditions formula_58. In case of a single barrier option we use the same PDE with either formula_59 or formula_60. The parameter formula_61 represents the risk-neutral drift of the underlying stochastic process.