Intoroduction

以下のSDEをもつものをSABR modelと呼ぶ。

\[\begin{eqnarray} S(t) & = & S(0) + \int_{0}^{t} \alpha(s) S(s)^{\beta} dW_{1}(s), \\ \alpha(t) & = & \alpha(0) + \int_{0}^{t} \nu\alpha(s) dW_{2}(s), \\ \langle W_{1}(\cdot), W_{2}(\cdot) \rangle(t) & = & \rho t \end{eqnarray}\]

微分形で書くと以下のようになる。　

\[\begin{eqnarray} d S(t) & = & \alpha(t)S(t)^{\beta} dW_{1}(t), \quad S(0) = S_{0}, \\ d\alpha(t) & = & \nu \alpha(t) dW_{2}(t), \quad \alpha(0) = \alpha_{0}, \\ dW_{1}(t)dW_{2}(t) & = & \rho dt, \end{eqnarray}\]

measure変換した結果マルチンゲールとなる商品について考えることが殆どであるため、drift項は考慮しない。 SABRの名はStochastic Alpha Beta Rhoに由来する。

Pricing

において、以下の2つが示されている。

• sigular purtabation theoryによるSABR modelのimplied volatility function
• implied volatilityによるcall option（及びput option）の評価

implied volatility

Haganは、SABR modelのimplied volatilityを以下のように近似して求めた。

\[\begin{eqnarray} \sigma_{B}(K, S; T) & \approx & \frac{ \alpha }{ (SK)^{(1-\beta)/2} \left( 1 + \frac{(1 - \beta)^{2}}{24} \log^{2}\frac{S}{K} + \frac{(1 - \beta)^{4}}{1920} \log^{4}\frac{S}{K} \right) } \left( \frac{z}{x(z)} \right) \left[ 1 + \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{(SK)^{1-\beta}} + \frac{1}{4} \frac{\rho\beta\nu\alpha}{(SK)^{(1-\beta)/2}} + \frac{2 - 3\rho^{2}}{24}\nu^{2} \right) T \right], \\ z & := & \frac{\nu}{\alpha} (SK)^{(1-\beta)/2} \log\left( \frac{S}{K} \right), \\ x(z) & := & \log \left( \frac{ \sqrt{1 - 2\rho z + z^{2}} + z - \rho }{ 1 - \rho } \right) \end{eqnarray}\]

更にATM($S = K$)の場合には以下のようにかける。

\[\sigma_{ATM}(S; T) := \sigma_{B}(S, S; T) \approx \frac{\alpha}{S^{(1-\beta)}} \left[ 1 + \left( \frac{(1-\beta)^{2}}{24} \frac{\alpha^{2}}{S^{2 - 2\beta}} + \frac{1}{4} \frac{\rho \beta \alpha \nu}{S^{1-\beta}} + \frac{2 - 3\rho^{2}}{24} \nu^{2} \right) T \right]\]

European Option

European call optionの価格は以下であたえられる。

\[\begin{eqnarray} V_{\mathrm{SABR}}(S, K, r, T) & := & V_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \nonumber \\ & = & S \Phi(d_{1}) - Ke^{-rT} \Phi(d_{2}) \nonumber \\ d_{1} & = & \frac{ \ln\left(\frac{S}{K} \right) + (r + \frac{1}{2}\sigma_{B}(K, S; T)^{2}))T }{ \sigma_{B}(K, S; T) \sqrt{T} }, \nonumber \\ d_{2} & := & \frac{ \ln\left(\frac{S}{K} \right) + (r - \frac{1}{2}\sigma_{B}(K, S; T)^{2})T }{ \sigma_{B}(K, S; T) \sqrt{T} } \nonumber \end{eqnarray}\]

European put optionはput-call parityにより求める。

\[\begin{eqnarray} V_{\mathrm{SABRput}}(S, K, r, T) & := & V_{\mathrm{SABRcall}}(S, K, r, T) - (S - e^{-rT}K) \nonumber \end{eqnarray}\]

swaptionの場合は、$r=1$として計算する。

Greeks

簡単のため、Black Scholes call option formulaのGreeksを以下のようにかく。

\[\begin{eqnarray} \Delta_{\mathrm{BScall}}(S, K, r, T, \sigma) & := & \frac{\partial}{\partial S} V_{\mathrm{BScall}}(S, K, r, T, \sigma) \nonumber \\ \Gamma_{\mathrm{BScall}}(S, K, r, T, \sigma) & = & \frac{\partial^{2}}{\partial S^{2}} V_{\mathrm{BScall}}(S, K, r, T, \sigma) \nonumber \\ \Theta_{\mathrm{BScall}}(S, K, r, T, \sigma) & = & \frac{\partial}{\partial T} V_{\mathrm{BScall}}(S, K, r, T, \sigma) \nonumber \\ \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma) & = & \frac{\partial}{\partial \sigma} V_{\mathrm{BScall}}(S, K, r, T, \sigma) \nonumber \\ \mathrm{Volga}_{\mathrm{BScall}}(S, K, r, T, \sigma) & = & \frac{\partial^{2}}{\partial \sigma^{2}} V_{\mathrm{BScall}}(S, K, r, T, \sigma) \nonumber \\ \mathrm{Vanna}_{\mathrm{BScall}}(S, K, r, T, \sigma) & = & \frac{\partial^{2}}{\partial \sigma \partial S} V_{\mathrm{BScall}}(S, K, r, T, \sigma) \end{eqnarray}\]

以下で見るように、SABR modelの微分はBSのGreeksとimplied volatilityの微分で表現できる。

合成関数の微分より、

\[\begin{eqnarray} \frac{\partial}{\partial S} V_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) & = & \left. \frac{\partial}{\partial S} V_{\mathrm{BScall}}(S, K, r, T, \sigma) \right|_{\sigma = \sigma_{B}(K, S; T)} + \left. \frac{\partial}{\partial \sigma} V_{\mathrm{BScall}}(S, K, r, T, \sigma) \right|_{\sigma = \sigma_{B}(K, S; T)} \frac{\partial}{\partial S} \sigma_{B}(K, S; T) \nonumber \\ & = & \Delta_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) + \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial S} \sigma_{B}(K, S; T) \end{eqnarray}\]

Delta

\[\begin{eqnarray} \frac{\partial}{\partial S} V_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) & = & \Delta_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) + \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial S} \sigma_{B}(K, S; T) \end{eqnarray}\]

Gamma

\[\begin{eqnarray} \frac{\partial^{2}}{\partial S^{2}} V_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) & = & \Gamma_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \nonumber \\ & & + \mathrm{Vanna}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial S} \sigma_{B}(K, S; T) \nonumber \\ & & + \left( \mathrm{Vanna}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) + \mathrm{Volga}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial S} \sigma_{B}(K, S; T) \right) \frac{\partial}{\partial S} \sigma_{B}(K, S; T) \nonumber \\ & & + \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial^{2}}{\partial S^{2}} \sigma_{B}(K, S; T) \nonumber \\ & = & \Gamma_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \nonumber \\ & & + 2\mathrm{Vanna}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial S} \sigma_{B}(K, S; T) \nonumber \\ & & + \mathrm{Volga}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \left( \frac{\partial}{\partial S} \sigma_{B}(K, S; T) \right)^{2} \nonumber \\ & & + \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial^{2}}{\partial S^{2}} \sigma_{B}(K, S; T) \end{eqnarray}\]

Theta

\[\begin{eqnarray} \frac{\partial}{\partial T} V_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) & = & \Theta_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) + \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial T} \sigma_{B}(K, S; T) \end{eqnarray}\]

Vega

\[\begin{eqnarray} \frac{\partial}{\partial \sigma} V_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) & = & \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial \sigma} \sigma_{B}(K, S; T) \end{eqnarray}\]

Volga

Distribution of underlying

call option価格の1階微分及び2階微分はそれぞれ原資産の分布と密度を与える。

\[\mathrm{E} \left[ (S(T) - K)^{+} \right] = V_{\mathrm{SABRcall}}(S, K, 1, T)\]

より、

\[\begin{eqnarray} \Phi_{\mathrm{SABR}}(s) & = & 1 + \left. \frac{\partial}{\partial K} V_{\mathrm{SABRcall}}(S, K, 1, T) \right|_{K=s} \\ \phi_{\mathrm{SABR}}(s) & = & \left. \frac{\partial^{2}}{\partial K^{2}} V_{\mathrm{SABRcall}}(S, K, 1, T) \right|_{K=s} \end{eqnarray}\]

を計算すれば良い。 以下では、BS modelでの分布と密度関数を$\Phi_{\mathrm{BS}}(\cdot)$, $\phi_{\mathrm{BS}}(\cdot)$とおく。

\[\begin{eqnarray} \Phi_{\mathrm{BS}}(s; S, T, \sigma) & = & 1 + \left. \frac{\partial}{\partial K} V_{\mathrm{BScall}}(S, K, 1, T, \sigma) \right|_{K=s} \nonumber \\ \phi_{\mathrm{BS}}(s; S, T, \sigma) & = & \left. \frac{\partial^{2}}{\partial K^{2}} V_{\mathrm{BScall}}(S, K, 1, T, \sigma) \right|_{K=s} \end{eqnarray}\]

Distribution

SABR modelの分布を考える。

\[\begin{eqnarray} \frac{\partial}{\partial K} V_{\mathrm{SABRcall}}(S, K, 1, T) & = & \frac{\partial}{\partial K} V_{\mathrm{BScall}}(S, K, 1, T, \sigma_{B}(K, S; T)) + \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \end{eqnarray}\]

より、

\[\begin{eqnarray} \Phi_{\mathrm{SABR}}(s; S, T) & = & 1 + \left. \frac{\partial}{\partial K} V_{\mathrm{SABRcall}}(S, K, 1, T) \right|_{K=s} \nonumber \\ & = & \Phi_{\mathrm{BS}}(s; S, T, \sigma_{B}(K, S; T)) + \mathrm{Vega}_{\mathrm{BScall}}(S, s, r, T, \sigma_{B}(s, S; T)) \left. \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \right|_{K=s} \end{eqnarray}\]

また、密度関数は

\[\begin{eqnarray} \frac{\partial^{2}}{\partial K^{2}} V_{\mathrm{SABRcall}}(S, K, 1, T) & = & \frac{\partial^{2}}{\partial K^{2}} V_{\mathrm{BScall}}(S, K, 1, T, \sigma_{B}(K, S; T)) \\ & & + \frac{\partial}{\partial K} \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \\ & & + \left( \frac{\partial}{\partial K} \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) + \mathrm{Volga}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \right) \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \\ & & + \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial^{2}}{\partial K^{2}} \sigma_{B}(K, S; T) \nonumber \\ & = & \frac{\partial^{2}}{\partial K^{2}} V_{\mathrm{BScall}}(S, K, 1, T, \sigma_{B}(K, S; T)) \\ & & + 2 \frac{\partial}{\partial K} \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \\ & & + \mathrm{Volga}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \left( \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \right)^{2} \\ & & + \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \frac{\partial^{2}}{\partial K^{2}} \sigma_{B}(K, S; T) \end{eqnarray}\]

以上より、

\[\begin{eqnarray} \phi_{\mathrm{SABR}}(s; S, T) & = & \phi_{\mathrm{BS}}(s; S, T, \sigma_{B}(s, S; T)) \\ & & + 2 \left. \frac{\partial}{\partial K} \mathrm{Vega}_{\mathrm{BScall}}(S, K, r, T, \sigma_{B}(K, S; T)) \right|_{K=s} \left. \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \right|_{K=s} \\ & & + \mathrm{Volga}_{\mathrm{BScall}}(S, s, r, T, \sigma_{B}(s, S; T)) \left( \left. \frac{\partial}{\partial K} \sigma_{B}(K, S; T) \right|_{K=s} \right)^{2} \\ & & + \mathrm{Vega}_{\mathrm{BScall}}(S, s, r, T, \sigma_{B}(s, S; T)) \left. \frac{\partial^{2}}{\partial K^{2}} \sigma_{B}(K, S; T) \right|_{K=s} \end{eqnarray}\]

derivative of implied volatility

分布やGreeksの計算でimplied volatilityの微分が必要となる。 以下では、implied volatilityの微分を考える。

with respect to underlying

\[\begin{eqnarray} A_{1}(K, S; T) & := & (SK)^{(1-\beta)/2} \left( 1 + \frac{(1 - \beta)^{2}}{24} \log^{2}\frac{S}{K} + \frac{(1 - \beta)^{4}}{1920} \log^{4}\frac{S}{K} \right) \nonumber \\ A_{2}(K, S; T) & := & z \nonumber \\ & = & \frac{\nu}{\alpha} (SK)^{(1-\beta)/2} \log\left( \frac{S}{K} \right), \nonumber \\ A_{3}(K, S; T) & := & x(z) \nonumber \\ & = & \log \left( \frac{ \sqrt{1 - 2\rho z + z^{2}} + z - \rho }{ 1 - \rho } \right) \nonumber \\ A_{4}(K, S; T) & := & 1 + \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{(SK)^{1-\beta}} + \frac{1}{4} \frac{\rho\beta\nu\alpha}{(SK)^{(1-\beta)/2}} + \frac{2 - 3\rho^{2}}{24}\nu^{2} \right) T \nonumber \\ \sigma_{B}(K, S; T) & \approx & \frac{ \alpha }{ A_{1}(K, S; T) } \frac{A_{2}(K, S; T)}{A_{3}(K, S; T)} A_{4}(K, S; T) \end{eqnarray}\]

とおく。 $A_{1}$の微分を考える。 まず、$A_{11} := (SK)^{(1-\beta)/2}$とおく。

\[\begin{eqnarray} \frac{\partial}{\partial S} A_{11} & = & \frac{1 - \beta}{2} K^{(1 - \beta)/2} S^{(-1 - \beta)/2} \nonumber \\ \frac{\partial^{2}}{\partial S^{2}} A_{11} & = & \frac{1 - \beta}{2} K^{(1 - \beta)/2} \frac{-1 - \beta}{2} S^{(-3 - \beta)/2} \nonumber \\ & = & -\frac{1 - \beta^{2}}{4} K^{(1 - \beta)/2} S^{(-3 - \beta)/2} \nonumber \end{eqnarray}\]

また、

\[\begin{eqnarray} A_{12} & := & \left( 1 + \frac{(1 - \beta)^{2}}{24} \log^{2}\frac{S}{K} + \frac{(1 - \beta)^{4}}{1920} \log^{4}\frac{S}{K} \right) \\ \frac{\partial}{\partial S} A_{12} & = & \frac{(1 - \beta)^{2}}{12S} \log\left( \frac{S}{K} \right) \frac{1}{S} + \frac{(1 - \beta)^{4}}{480S} \log^{3}\left( \frac{S}{K} \right) \frac{1}{S} \nonumber \\ & = & \frac{(1 - \beta)^{2}}{12S} \log \left( \frac{S}{K} \right) + \frac{(1 - \beta)^{4}}{480S} \log^{3}\left( \frac{S}{K} \right) \nonumber \\ \frac{\partial^{2}}{\partial S^{2}} A_{12} & = & -\frac{(1 - \beta)^{2}}{12S^{2}} \log \left( \frac{S}{K} \right) + \frac{(1 - \beta)^{2}}{12S} \frac{1}{S} - \frac{(1 - \beta)^{4}}{480S^{2}} \log^{3}\left( \frac{S}{K} \right) + \frac{(1 - \beta)^{4}}{480S} 3 \log^{2}\left( \frac{S}{K} \right) \frac{1}{S} \nonumber \\ & = & \frac{(1 - \beta)^{2}}{12S^{2}} \left( -\log \left( \frac{S}{K} \right) + 1 \right) + \frac{(1 - \beta)^{4}}{480S^{2}} \left( -\log^{3}\left( \frac{S}{K} \right) + 3\log^{2}\left( \frac{S}{K} \right) \right) \nonumber \end{eqnarray}\]

以上より、

\[\begin{eqnarray} \frac{\partial}{\partial S} A_{1}(K, S; T) & = & A_{11} A_{12}^{\prime} + A_{11}^{\prime} A_{12} \nonumber \\ \frac{\partial^{2}}{\partial S^{2}} A_{1}(K, S; T) & = & A_{11} A_{12}^{\prime\prime} + 2 A_{11}^{\prime} A_{12}^{\prime} + A_{11}^{\prime\prime} A_{12} \end{eqnarray}\]

となる。 $A_{2}$の微分を考える。

\[\begin{eqnarray} \frac{\partial }{\partial S} A_{2}(K, S; T) & = & \frac{\nu}{\alpha} \left( \frac{1 - \beta}{2} K^{(1-\beta)/2} S^{(-1-\beta)/2} \log\left( \frac{S}{K} \right) + (SK)^{(1-\beta)/2} \frac{1}{S} \right) \nonumber \\ & = & \frac{\nu}{\alpha} \left( \frac{1 - \beta}{2} K^{(1-\beta)/2} S^{(-1-\beta)/2} \log\left( \frac{S}{K} \right) + K^{(1-\beta)/2} S^{(-1-\beta)/2} \right) \nonumber \\ & = & \frac{\nu}{\alpha} K^{(1-\beta)/2} \left( \frac{1 - \beta}{2} S^{(-1-\beta)/2} \log\left( \frac{S}{K} \right) + S^{(-1-\beta)/2} \right) \nonumber \\ & = & \frac{\nu}{\alpha} K^{(1-\beta)/2} S^{(-1-\beta)/2} \left( \frac{1 - \beta}{2} \log\left( \frac{S}{K} \right) + 1 \right) \nonumber \\ \frac{\partial^{2} }{\partial S^{2}} A_{2}(K, S; T) & = & \frac{\nu}{\alpha} K^{(1-\beta)/2} \left( \frac{-1-\beta}{2} S^{(-3-\beta)/2} \left( \frac{1 - \beta}{2} \log\left( \frac{S}{K} \right) + 1 \right) + S^{(-1-\beta)/2} \frac{1 - \beta}{2S} \right) \nonumber \\ & = & \frac{\nu}{\alpha} K^{(1-\beta)/2} \left( S^{(-3-\beta)/2} \left( - \frac{1 - \beta^{2}}{4} \log\left( \frac{S}{K} \right) + \frac{-1-\beta}{2} \right) + S^{(-3-\beta)/2} \frac{1 - \beta}{2} \right) \nonumber \\ & = & \frac{\nu}{\alpha} K^{(1-\beta)/2} S^{(-3-\beta)/2} \left( - \frac{1-\beta^{2}}{4} \log\left( \frac{S}{K} \right) - \beta \right) \end{eqnarray}\]

次に、$A_{3}$の部分を考える。 簡単のため、$A_{3}(K) := A_{3}(K, S; T)$, $A_{2}(K) := A(K, S; T)$と書く。 $A_{2}(K)$の$K$での微分を$A_{2}^{\prime}(K)$とかく。 $A_{2}(K) = z$に注意すると、

となる。 次に、$A_{4}(K, S; T)$の微分を考える。

\[\begin{eqnarray} \frac{\partial}{\partial S} A_{4}(K, S; T) & = & \frac{\partial}{\partial S} \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{K^{1-\beta}} S^{-(1-\beta)} + \frac{1}{4} \frac{\rho\beta\nu\alpha}{K^{(1-\beta)/2}} S^{-(1-\beta)/2} \right) T \nonumber \\ & = & \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{K^{1-\beta}} (\beta - 1) S^{\beta - 2} + \frac{1}{4} \frac{\rho\beta\nu\alpha}{K^{(1-\beta)/2}} \frac{(\beta - 1)}{2} S^{(\beta-3)/2} \right) T \nonumber \\ & = & \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{K^{1-\beta}} S^{\beta - 2} + \frac{1}{8} \frac{\rho\beta\nu\alpha}{K^{(1-\beta)/2}} S^{(\beta-3)/2} \right) T (\beta - 1) \nonumber \\ \frac{\partial^{2}}{\partial S^{2}} A_{4}(K, S; T) & = & \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{K^{1-\beta}} (\beta - 2) S^{\beta - 3} + \frac{1}{8} \frac{\rho\beta\nu\alpha}{K^{(1-\beta)/2}} \frac{\beta - 3}{2} S^{(\beta-5)/2} \right) T (\beta - 1) \nonumber \\ & = & \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{K^{1-\beta}} (\beta - 2) S^{\beta - 3} + \frac{1}{16} \frac{\rho\beta\nu\alpha}{K^{(1-\beta)/2}} (\beta - 3) S^{(\beta-5)/2} \right) T (\beta - 1) \end{eqnarray}\]

ここで、以下に注意する。

\[\begin{eqnarray} \left. \frac{\partial}{\partial \bar{K}} A_{4}(\bar{K}, S; T) \right|_{\bar{K}=K} & = & \left. \frac{\partial}{\partial \bar{S}} A_{4}(S, \bar{S}; T) \right|_{\bar{S}=K} \nonumber \end{eqnarray}\]

with respect to strike

strikeによる微分を考える。 まず、以下のようにおく。

\[\begin{eqnarray} A_{1}(K, S; T) & := & (SK)^{(1-\beta)/2} \left( 1 + \frac{(1 - \beta)^{2}}{24} \log^{2}\frac{S}{K} + \frac{(1 - \beta)^{4}}{1920} \log^{4}\frac{S}{K} \right) \nonumber \\ A_{2}(K, S; T) & := & z \nonumber \\ & = & \frac{\nu}{\alpha} (SK)^{(1-\beta)/2} \log\left( \frac{S}{K} \right), \nonumber \\ A_{3}(K, S; T) & := & x(z) \nonumber \\ & = & \log \left( \frac{ \sqrt{1 - 2\rho z + z^{2}} + z - \rho }{ 1 - \rho } \right) \nonumber \\ A_{4}(K, S; T) & := & 1 + \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{(SK)^{1-\beta}} + \frac{1}{4} \frac{\rho\beta\nu\alpha}{(SK)^{(1-\beta)/2}} + \frac{2 - 3\rho^{2}}{24}\nu^{2} \right) T \nonumber \\ \sigma_{B}(K, S; T) & \approx & \frac{ \alpha }{ A_{1}(K, S; T) } \frac{A_{2}(K, S; T)}{A_{3}(K, S; T)} A_{4}(K, S; T) \end{eqnarray}\]

$A_{1}$の微分を考える。 まず、$B_{11} := (SK)^{(1-\beta)/2}$とおく。

\[\begin{eqnarray} \frac{\partial}{\partial K} B_{11} & = & \frac{1 - \beta}{2} S^{(1 - \beta)/2}K^{(-1 - \beta)/2} \nonumber \\ \frac{\partial^{2}}{\partial K^{2}} B_{11} & = & \frac{1 - \beta}{2} S^{(1 - \beta)/2} \frac{-1 - \beta}{2} K^{(-3 - \beta)/2} \nonumber \\ & = & -\frac{1 - \beta^{2}}{4} S^{(1 - \beta)/2} K^{(-3 - \beta)/2} \nonumber \end{eqnarray}\]

また、

\[\begin{eqnarray} B_{12} & := & \left( 1 + \frac{(1 - \beta)^{2}}{24} \log^{2}\frac{S}{K} + \frac{(1 - \beta)^{4}}{1920} \log^{4}\frac{S}{K} \right) \\ \frac{\partial}{\partial K} B_{12} & = & \frac{(1 - \beta)^{2}}{12} \log\left( \frac{S}{K} \right) \frac{K}{S} (- \frac{S}{K^{2}}) + \frac{(1 - \beta)^{4}}{480} \log^{3}\left( \frac{S}{K} \right) \frac{K}{S} (- \frac{S}{K^{2}}) \nonumber \\ & = & -\frac{(1 - \beta)^{2}}{12K} \log \left( \frac{S}{K} \right) - \frac{(1 - \beta)^{4}}{480K} \log^{3}\left( \frac{S}{K} \right) \nonumber \\ \frac{\partial^{2}}{\partial K^{2}} B_{12} & = & \frac{(1 - \beta)^{2}}{12K^{2}} \log \left( \frac{S}{K} \right) -\frac{(1 - \beta)^{2}}{12K} (-\frac{1}{K}) + \frac{(1 - \beta)^{4}}{480K^{2}} \log^{3}\left( \frac{S}{K} \right) - \frac{(1 - \beta)^{4}}{480K} 3 \log^{2}\left( \frac{S}{K} \right) (-\frac{1}{K}) \nonumber \\ & = & \frac{(1 - \beta)^{2}}{12K^{2}} \left( \log \left( \frac{S}{K} \right) + 1 \right) + \frac{(1 - \beta)^{4}}{480K^{2}} \left( \log^{3}\left( \frac{S}{K} \right) + 3\log^{2}\left( \frac{S}{K} \right) \right) \nonumber \end{eqnarray}\]

以上より、

\[\begin{eqnarray} \frac{\partial}{\partial K} A_{1}(K, S; T) & = & A_{11} A_{12}^{\prime} + A_{11}^{\prime} A_{12} \nonumber \\ \frac{\partial^{2}}{\partial K^{2}} A_{1}(K, S; T) & = & A_{11} A_{12}^{\prime\prime} + 2 A_{11}^{\prime} A_{12}^{\prime} + A_{11}^{\prime\prime} A_{12} \end{eqnarray}\]

となる。 $A_{2}$の微分を考える。

\[\begin{eqnarray} \frac{\partial }{\partial K} A_{2}(K, S; T) & = & \frac{\nu}{\alpha} \left( \frac{1 - \beta}{2} S^{(1-\beta)/2} K^{(-1-\beta)/2} \log\left( \frac{S}{K} \right) + (SK)^{(1-\beta)/2} \frac{K}{S} (-\frac{S}{K^{2}}) \right) \nonumber \\ & = & \frac{\nu}{\alpha} \left( \frac{1 - \beta}{2} S^{(1-\beta)/2} K^{(-1-\beta)/2} \log\left( \frac{S}{K} \right) - S^{(1-\beta)/2} K^{(-1-\beta)/2} \right) \nonumber \\ & = & \frac{\nu}{\alpha} S^{(1-\beta)/2} \left( \frac{1 - \beta}{2} K^{(-1-\beta)/2} \log\left( \frac{S}{K} \right) - K^{(-1-\beta)/2} \right) \nonumber \\ & = & \frac{\nu}{\alpha} S^{(1-\beta)/2} K^{(-1-\beta)/2} \left( \frac{1 - \beta}{2} \log\left( \frac{S}{K} \right) - 1 \right) \nonumber \\ \frac{\partial^{2} }{\partial K^{2}} A_{2}(K, S; T) & = & \frac{\nu}{\alpha} S^{(1-\beta)/2} \left( - \frac{1-\beta^{2}}{4} K^{(-3-\beta)/2} \log\left( \frac{S}{K} \right) + \frac{1-\beta}{2} K^{(-1-\beta)/2} (-\frac{1}{K}) - \frac{-1-\beta}{2} K^{(-3-\beta)/2} \right) \nonumber \\ & = & \frac{\nu}{\alpha} S^{(1-\beta)/2} K^{(-3-\beta)/2} \left( - \frac{1-\beta^{2}}{4} \log\left( \frac{S}{K} \right) - \frac{1-\beta}{2} - \frac{-1-\beta}{2} \right) \nonumber \\ & = & \frac{\nu}{\alpha} S^{(1-\beta)/2} K^{(-3-\beta)/2} \left( - \frac{1-\beta^{2}}{4} \log\left( \frac{S}{K} \right) + \beta \right) \end{eqnarray}\]

次に、$A_{3}$の部分を考える。 簡単のため、$A_{3}(K) := A_{3}(K, S; T)$, $A_{2}(K) := A(K, S; T)$と書く。 $A_{2}(K)$の$K$での微分を$A_{2}^{\prime}(K)$とかく。 $A_{2}(K) = z$に注意すると、

\[\begin{eqnarray} A_{31} & := & \sqrt{1 - 2\rho z + z^{2}} + z - \rho \nonumber \\ \frac{\partial }{\partial K} A_{31} & = & \frac{1}{2} \frac{ (-2\rho A_{2}^{\prime}(K) + 2zA_{2}^{\prime}(K)) }{ \sqrt{1 - 2\rho z + z^{2}} } + A_{2}^{\prime}(K) \nonumber \\ & = & \frac{ (-\rho A_{2}^{\prime}(K) + zA_{2}^{\prime}(K)) }{ \sqrt{1 - 2\rho z + z^{2}} } + A_{2}^{\prime}(K) \nonumber \\ \frac{\partial^{2} }{\partial K^{2}} A_{31} & = & \frac{ (-\rho A_{2}^{\prime\prime}(K) + A_{2}^{\prime}(K)^{2} + zA_{2}^{\prime\prime}(K)) \sqrt{1 - 2\rho z + z^{2}} - (-\rho A_{2}^{\prime}(K) + zA_{2}^{\prime}(K)) \frac{1}{2} (1 - 2\rho z + z^{2})^{-1/2} (-2 \rho A_{2}^{\prime}(K) + 2 z A_{2}^{\prime}(K)) }{ 1 - 2\rho z + z^{2} } + A_{2}^{\prime\prime}(K) \nonumber \\ & = & \frac{ (-\rho A_{2}^{\prime\prime}(K) + A_{2}^{\prime}(K)^{2} + zA_{2}^{\prime\prime}(K)) (1 - 2\rho z + z^{2}) - (-\rho A_{2}^{\prime}(K) + zA_{2}^{\prime}(K))^{2} }{ (1 - 2\rho z + z^{2})^{3/2} } + A_{2}^{\prime\prime}(K) \end{eqnarray}\]

以上より、

\[\begin{eqnarray} \frac{\partial }{\partial K} A_{3}(K, S; T) & = & \frac{ 1 - \rho }{ A_{31} } \frac{\partial}{\partial K} \left( \frac{ A_{31} }{ 1 - \rho } \right) \nonumber \\ & = & \frac{ 1 }{ A_{31} } \frac{\partial}{\partial K} A_{31} \nonumber \\ \frac{\partial^{2} }{\partial K^{2}} A_{3}(K, S; T) & = & -A_{31}^{-2} \frac{\partial}{\partial K} A_{31} \frac{\partial}{\partial K} A_{31} + A_{31}^{-1} \frac{\partial^{2}}{\partial K^{2}} A_{31} \nonumber \\ & = & -A_{31}^{-2} \left( \frac{\partial}{\partial K} A_{31} \right)^{2} + A_{31}^{-1} \frac{\partial^{2}}{\partial K^{2}} A_{31} \end{eqnarray}\]

となる。 次に、$A_{4}(K, S; T)$の微分を考える。

\[\begin{eqnarray} \frac{\partial}{\partial K} A_{4}(K, S; T) & = & \frac{\partial}{\partial K} \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{S^{1-\beta}} K^{-(1-\beta)} + \frac{1}{4} \frac{\rho\beta\nu\alpha}{S^{(1-\beta)/2}} K^{-(1-\beta)/2} \right) T \nonumber \\ & = & \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{S^{1-\beta}} (\beta - 1) K^{\beta - 2} + \frac{1}{4} \frac{\rho\beta\nu\alpha}{S^{(1-\beta)/2}} \frac{(\beta - 1)}{2} K^{(\beta-3)/2} \right) T \nonumber \\ & = & \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{S^{1-\beta}} K^{\beta - 2} + \frac{1}{8} \frac{\rho\beta\nu\alpha}{S^{(1-\beta)/2}} K^{(\beta-3)/2} \right) T (\beta - 1) \nonumber \\ \frac{\partial^{2}}{\partial K^{2}} A_{4}(K, S; T) & = & \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{S^{1-\beta}} (\beta - 2) K^{\beta - 3} + \frac{1}{8} \frac{\rho\beta\nu\alpha}{S^{(1-\beta)/2}} \frac{\beta - 3}{2} K^{(\beta-5)/2} \right) T (\beta - 1) \nonumber \\ & = & \left( \frac{(1 - \beta)^{2}}{24} \frac{\alpha^{2}}{S^{1-\beta}} (\beta - 2) K^{\beta - 3} + \frac{1}{16} \frac{\rho\beta\nu\alpha}{S^{(1-\beta)/2}} (\beta - 3) K^{(\beta-5)/2} \right) T (\beta - 1) \end{eqnarray}\]

以上より、以下を計算すれば良い。 簡単のため、$A_{i}(K) := A_{i}(K, S; T)$とおく。

\[\begin{eqnarray} \frac{\partial}{\partial K} \alpha A_{1}(K)^{-1} & = & -\alpha A_{1}(K)^{-2} A_{1}^{\prime}(K) \nonumber \\ \frac{\partial^{2}}{\partial K^{2}} \alpha A_{1}(K)^{-1} & = & -\alpha (-2 A_{1}(K)^{-3} A_{1}^{\prime}(K) A_{1}^{\prime}(K) + A_{1}(K)^{-2} A_{1}^{\prime\prime}(K) ) \nonumber \\ & = & \alpha ( 2 A_{1}(K)^{-3} A_{1}^{\prime}(K)^{2} - A_{1}(K)^{-2} A_{1}^{\prime\prime}(K) ) \nonumber \\ \frac{\partial}{\partial K} \frac{A_{2}(K)}{A_{3}(K)} & = & A_{2}^{\prime}(K) A_{3}(K)^{-1} - A_{2}(K) A_{3}^{\prime}(K) A_{3}(K)^{-2} \nonumber \\ \frac{\partial^{2}}{\partial K^{2}} \frac{A_{2}(K)}{A_{3}(K)} & = & A_{2}^{\prime\prime}(K) A_{3}(K)^{-1} + A_{2}^{\prime}(K) (- A_{3}(K)^{-2} A_{3}^{\prime}(K) ) - A_{2}^{\prime}(K) A_{3}^{\prime}(K) A_{3}(K)^{-2} - A_{2}(K) A_{3}^{\prime\prime}(K) A_{3}(K)^{-2} + 2 A_{2}(K) A_{3}^{\prime}(K) A_{3}(K)^{-3} A_{3}^{\prime}(K) \nonumber \\ & = & A_{2}^{\prime\prime}(K) A_{3}(K)^{-1} -2 A_{2}^{\prime}(K) A_{3}^{\prime}(K) A_{3}(K)^{-2} - A_{2}(K) A_{3}^{\prime\prime}(K) A_{3}(K)^{-2} + 2 A_{2}(K) A_{3}^{\prime}(K)^{2} A_{3}(K)^{-3} \end{eqnarray}\]

以上より、

\[\begin{eqnarray} \frac{\partial}{\partial K} \sigma_{B}(K, S; T) & \approx & \alpha \frac{\partial}{\partial K} A_{1}(K)^{-1} A_{2}(K)A_{3}(K)^{-1} A_{4}(K) + \alpha A_{1}(K)^{-1} \frac{\partial}{\partial K} A_{2}(K)A_{3}(K)^{-1} A_{4}(K) + \alpha A_{1}(K)^{-1} A_{2}(K)A_{3}(K)^{-1} \frac{\partial}{\partial K} A_{4}(K) \\ \frac{\partial^{2}}{\partial K^{2}} \sigma_{B}(K, S; T) & \approx & \alpha \frac{\partial^{2}}{\partial K^{2}} A_{1}(K)^{-1} A_{2}(K)A_{3}(K)^{-1} A_{4}(K) + \alpha A_{1}(K)^{-1} \frac{\partial^{2}}{\partial K^{2}} A_{2}(K)A_{3}(K)^{-1} A_{4}(K) + \alpha A_{1}(K)^{-1} A_{2}(K)A_{3}(K)^{-1} \frac{\partial^{2}}{\partial K^{2}} A_{4}(K) \nonumber \\ & & + 2 \alpha \frac{\partial}{\partial K} A_{1}(K)^{-1} \frac{\partial}{\partial K} A_{2}(K)A_{3}(K)^{-1} A_{4}(K) + 2 \alpha \frac{\partial}{\partial K} A_{1}(K)^{-1} A_{2}(K)A_{3}(K)^{-1} \frac{\partial}{\partial K} A_{4}(K) + 2 \alpha A_{1}(K)^{-1} \frac{\partial}{\partial K} A_{2}(K)A_{3}(K)^{-1} \frac{\partial}{\partial K} A_{4}(K) \end{eqnarray}\]

を計算すれば良い。

Reference