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.

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 $\mathcal{V}$ , the Vanna and the Volga. The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate:

$\textrm{Vanna} = \frac{\partial \mathcal{V}}{\partial S}$ .

Similarly, the Volga is the sensitivity of the Vega with respect to a change of the implied volatility $\sigma$ :

$\textrm{Volga}= \frac{\partial \mathcal{V}}{\partial \sigma}$ .

If we consider a smile volatility term structure $\sigma(K)$ with ATM strike $K_0$ , ATM volatility $\sigma_0$ , and where $K_{c/p}$ are the 25-Delta call/put strikes (obtained by solving the equations $\Delta_{ call}(K_c,\sigma_0)=1/4$ and $\Delta_{ put}(K_p,\sigma_0)=-1/4$ where $\Delta_{call/put}(K,\sigma)$ 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:

$\begin{align} \textrm{ATM}(K_0) &= \frac12 \left(\textrm{Call}(K_0,\sigma_0) + \textrm{Put}(K_0,\sigma_0)\right) \\ \textrm{RR}(K_c,K_p) &= \frac{}{}\textrm{Call}(K_c,\sigma(K_c))-\textrm{Put}(K_p,\sigma(K_p)) \\ \textrm{BF}(K_c,K_p) &= \frac12 \left(\textrm{Call}(K_c,\sigma(K_c)) + \textrm{Put}(K_p,\sigma(K_p))\right)- \textrm{ATM}(K_0) \end{align}$

with $\textrm{Call}(K,\sigma)$ the Black–Scholes price of a call option (similarly for the put).

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Vanna–Volga_pricing

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