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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Vanna–Volga pricing ： ウィキペディア英語版 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 $\backslash mathcal$, the Vanna and the Volga. The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate: $\backslash textrm\; =\; \backslash frac$. Similarly, the Volga is the sensitivity of the Vega with respect to a change of the implied volatility $\backslash sigma$: $\backslash textrm=\; \backslash frac$. If we consider a smile volatility term structure $\backslash sigma(K)$ with ATM strike $K\_0$, ATM volatility $\backslash sigma\_0$, and where $K\_$ are the 25-Delta call/put strikes (obtained by solving the equations $\backslash Delta\_(K\_c,\backslash sigma\_0)=1/4$ and $\backslash Delta\_(K\_p,\backslash sigma\_0)=-1/4$ where $\backslash Delta\_(K,\backslash sigma)$ 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: $$ \begin \textrm(K_0) &= \frac12 \left(\textrm(K_0,\sigma_0) + \textrm(K_0,\sigma_0)\right) \\ \textrm(K_c,K_p) &= \frac(K_c,\sigma(K_c))-\textrm(K_p,\sigma(K_p)) \\ \textrm(K_c,K_p) &= \frac12 \left(\textrm(K_c,\sigma(K_c)) + \textrm(K_p,\sigma(K_p))\right)- \textrm(K_0) \end with $\backslash textrm(K,\backslash sigma)$ the Black–Scholes price of a call option (similarly for the put). The simplest formulation of the Vanna–Volga method suggests that the Vanna–Volga price $X^$ of an exotic instrument $X$ is given by $$ X^ = X ^ + \underbrace}}}__ + \underbrace}}}__ where by $X^$ denotes the Black–Scholes price of the exotic and the Greeks are calculated with ATM volatility and $$ \begin RR_ &=\left(\textrm(K_c,\sigma(K_c))-\textrm(K_p,\sigma(K_p)) \right ) - \left(\textrm(K_c,\sigma_0)-\textrm(K_p,\sigma_0) \right ) \\ BF_ &= \frac12 \left( \textrm(K_c,\sigma(K_c))+\textrm(K_p,\sigma(K_p)) \right ) - \frac12 \left(\textrm(K_c,\sigma_0)+\textrm(K_p,\sigma_0) \right ) \end These quantities represent a ''smile cost'', namely the difference between the price computed with/without including the smile effect. The rationale behind the above formulation of the Vanna-Volga price is that one can extract the ''smile cost'' of an exotic option by measuring the ''smile cost'' of a portfolio designed to hedge its Vanna and Volga risks. The reason why one chooses the strategies BF and RR to do this is because they are liquid FX instruments and they carry mainly Volga, and respectively Vanna risks. The weighting factors $w\_$ and $w\_$ represent respectively the amount of RR needed to replicate the option's Vanna, and the amount of BF needed to replicate the option's Volga. The above approach ignores the small (but non-zero) fraction of Volga carried by the RR and the small fraction of Vanna carried by the BF. It further neglects the cost of hedging the Vega risk. This has led to a more general formulation of the Vanna-Volga method in which one considers that within the Black–Scholes assumptions the exotic option's Vega, Vanna and Volga can be replicated by the weighted sum of three instruments: $$ X_i = w_\, + w_\, + w_\, \, \, \, \, \, i\text where the weightings are obtained by solving the system: $$ \vec = \mathbb \vec with $$ \mathbb = \begin ATM_ & RR_ & BF_ \\ ATM_ & RR_ & BF_ \\ ATM_ & RR_ & BF_ \end , $$ \vec= \begin w_ \\ w_ \\ w_ \end , $\backslash vec=\; \backslash begin\; X\_\; \backslash \backslash$ X_ \\ X_ \end Given this replication, the Vanna–Volga method adjusts the BS price of an exotic option by the ''smile cost'' of the above weighted sum (note that the ATM smile cost is zero by construction): $$ \begin X^ &= X^ + w_ (^-^) + w_ (^-^) \\ &= X ^ + \vec^T(\mathbb^T)^\vec \\ & = X ^ + X_ \, \Omega_+ X_ \, \Omega_ + X_ \, \Omega_ \\ \end where $$ \vec = \begin 0 \\ ^ - ^\\ ^ - ^ \end and $$ \begin \Omega_ \\ \Omega_ \\ \Omega_ \end = (\mathbb^T)^\vec The quantities $\backslash Omega\_i$ 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 $X^$ to $$ \begin X^ &= X ^ + p_ X_ \Omega_ + p_ X_ \Omega_ \end The Vega contribution turns out to be several orders of magnitude smaller than the Vanna and Volga terms in all practical situations, hence one neglects it. The terms $p\_$ and $p\_$ are put in by-hand and represent factors that ensure the correct behaviour of the price of an exotic option near a barrier: as the knock-out barrier level $B$ of an option is gradually moved toward the spot level $S\_0$, the BSTV price of a knock-out option must be a monotonically decreasing function, converging to zero exactly at $B=S\_0$. Since the Vanna-Volga method is a simple rule-of-thumb and not a rigorous model, there is no guarantee that this will be a priori the case. The attenuation factors are of a different from for the Vanna or the Volga of an instrument. This is because for barrier values close to the spot they behave differently: the Vanna becomes large while, on the contrary, the Volga becomes small. Hence the attenuation factors take the form: $$ \begin p_ &= a \, \gamma \\ p_ &= b + c \gamma \end where $\backslash gamma\backslash in()$ represents some measure of the barrier(s) vicinity to the spot with the features $$ \begin \gamma=0 \ \ &\ \ S_0\to B \\ \gamma=1 \ \ &\ \ |S_0-B|\gg 0 \end The coefficients $a,b,c$ are found through calibration of the model to ensure that it reproduces the vanilla smile. Good candidates for $\backslash gamma$ that ensure the appropriate behaviour close to the barriers are the ''survival probability'' and the ''expected first exit time''. Both of these quantities offer the desirable property that they vanish close to a barrier. ==Survival probability== The survival probability $p\_\backslash in()$ refers to the probability that the spot does not touch one or more barrier levels $\backslash$. For example, for a single barrier option we have $$ p_ = \mathbb(1_}} ) = \mathrm(B) / \mathrm(t_}) where $\backslash mathrm(B)$ is the value of a ''no-touch'' option and $\backslash mathrm(t\_\})$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Vanna–Volga pricing」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.