Calculation methods of quanto cms

Symbols

• $P_{d}(t, T)$
  • value of domestic currency $T$ maturity zero coupon bond at time $t$.
• $P_{f}(t, T)$
  • value of foreign currency $T$ maturity zero coupon bond at time $t$.
• $S(t)$
  • domestic forward swap rate at time $t$.
• $X(t)$
  • FX rate [DOM/FOR] at time $t$.
  • domestic currency per unit of foreign currency.
• $X_{T}(t)$
  • $T$ forward FX rate at time $t$.
• $T_{p}$
  • payment date of quanto CMS
• $\Phi$
  • standard gaussian c.d.f.
• $\phi$
  • standard gaussian p.d.f.
• $c(t, S(t); K, T)$
  • strike $K$, Maturity $T$のcall optionの$t$での価格
• $p(t, S(t); K, T)$
  • strike $K$, Maturity $T$のput optionの$t$での価格
• $\Psi^{A}(K)$
  • $S(T)$の分布
  • $\Psi^{A}(K) = 1 + \frac{\partial }{\partial K} c(0, S(0); K, T)$
• $\psi^{A}$
  • $S(T)$の密度関数
  • $\psi^{A}(K) = \frac{\partial^{2} }{\partial K^{2}} c(0, S(0); K, T)$

Simple case

volatility $\sigma_{X}$は$T$満期のFX rateのATM optionにキャリブレーションして求める。 $X_{T_{p}}$が$Q^{A,d}$の下でlog-normalであるとする。

\[\begin{equation} X_{T_{p}}(T) = X(0) e^{\sigma_{X}\sqrt{T}\xi_{1} + m_{X}T}, \end{equation}\]

ここで、$\xi_{1}$は標準正規乱数で、$\sigma_{X}$はvolatility、$m_{X}$は定数である。

$S(T)$と$X_{T_{p}}$の同時分布を表現する為に、copula methodを使う。 Chapter 17でcopula methodについて詳しく述べるが、ここでは次のように$S(T)$を定義する。 $\xi_{2}$が標準正規乱数とすると、

\[S(T) := (\Psi^{A})^{-1}(\Phi(\xi_{2})),\]

ここで、$\Phi(\cdot)$は標準正規分布のc.d.f.である。 $S(T)$と$X_{T_{p}}(T)$の相関は$\xi_{1}$と$\xi_{2}$の相関$\rho_{XS}$で表現される。

\[\begin{eqnarray*} X_{T_{p}}(T) & = & X(0) e^{\sigma_{X} \sqrt{T} \xi_{1} + m_{X}T}, \\ S(T) & = & (\Psi^{A})^{-1}(\Phi(\xi_{2})), \\ Corr(\xi_{1}, \xi_{2}) & = & \rho_{XS}. \end{eqnarray*}\] \[\chi(s) = X(0) e^{m_{X}T} \hat{\chi}(s),\]

ここで、

\[\tilde{\chi}(s) := \exp \left( \rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(s)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right).\]

である。 annuity mapping function $\alpha(\cdot)$は条件付き期待値で定義されているとする。

\[\begin{equation} V_{\mathrm{QuantoCMS}}(0) \approx P_{f}(0, T_{p}) \frac{\mathrm{E}^{A,d} \left[ g(S(T)) \alpha(S(T)) \tilde{\chi}(S(T)) \right] }{\mathrm{E}^{A,d} \left[ \alpha(S(T)) \tilde{\chi}(S(T)) \right] } \end{equation}\]

Calculation of numerator

\[\mathrm{E}^{A,d} \left[ g(S(T)) \alpha(S(T)) \tilde{\chi}(S(T)) \right] = g(S(0)) \alpha(S(0)) \tilde{\chi}(S(0)) + \int_{-\infty}^{S(0)} p(0, S(0); k, T) w(k) \ dk + \int_{S(0)}^{\infty} c(0, S(0); k, T) w(k) \ dk\]

ここで、

\[\begin{eqnarray*} w(s) & := & \frac{d^{2} }{d s^{2}} (g(s) \alpha(s) \tilde{\chi}(s)) \\ & = & g^{\prime\prime}(s)\alpha(s)\tilde{\chi}(s) + g(s)\alpha^{\prime\prime}(s)\tilde{\chi}(s) + g(s)\alpha(s)\tilde{\chi}^{\prime\prime}(s) + 2g^{\prime}(s)\alpha^{\prime}(s)\tilde{\chi}(s) + 2g^{\prime}(s)\alpha(s)\tilde{\chi}^{\prime}(s) + 2g(s)\alpha^{\prime}(s)\tilde{\chi}^{\prime}(s) \end{eqnarray*}\]

$g, \alpha, \tilde{\chi}$の微分はAppendixで計算している。

linear TSR model + cap

$\alpha$がlinear annuity mapping functionで、$g$がcapの場合を考える。

\[\begin{eqnarray} g_{\mathrm{cap}}^{\prime\prime}(s; K)\alpha(s)\tilde{\chi}(s) & = & \delta(s - K)\alpha(s)\tilde{\chi}(s), \\ g_{\mathrm{cap}}(s; K)\alpha^{\prime\prime}(s)\tilde{\chi}(s) & = & 0, \\ g_{\mathrm{cap}}(s; K)\alpha(s)\tilde{\chi}^{\prime\prime}(s) & = & (s - K)^{+} (\alpha_{1}s + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T} \left( h^{\prime\prime}(s)\tilde{\chi}(s) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(s)^{2} \tilde{\chi}(s) \right), \\ 2g_{\mathrm{cap}}^{\prime}(s; K)\alpha^{\prime}(s)\tilde{\chi}(s) & = & 2 1_{[K, \infty)}(s) \alpha_{1} \exp \left( \rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(s)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right), \\ 2g_{\mathrm{cap}}^{\prime}(s; K)\alpha(s)\tilde{\chi}^{\prime}(s) & = & 2 1_{[K, \infty)}(s) (\alpha_{1}s + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(s)\tilde{\chi}(s), \\ 2g_{\mathrm{cap}}(s; K)\alpha^{\prime}(s)\tilde{\chi}^{\prime}(s) & = & 2(s - K)^{+} \alpha_{1} \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(s)\tilde{\chi}(s), \end{eqnarray}\]

より、

\[\begin{eqnarray} \int_{-\infty}^{S(0)} p(0, S(0); k, T)w(k) \ dk & = & p(0, S(0); K, T) \alpha(K) \tilde{\chi}(K) 1_{(-\infty, S(0)]}(K) \nonumber \\ & & + 0 \nonumber \\ & & + \int_{-\infty}^{S(0)} p(0, S(0); k, T) (k - K)^{+} (\alpha_{1}k + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T} \left( h^{\prime\prime}(k)\tilde{\chi}(k) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(k)^{2} \tilde{\chi}(k) \right) \ dk \nonumber \\ & & + \int_{-\infty}^{S(0)} 2 \alpha_{1} p(0, S(0); k, T) 1_{[K, \infty)}(k) \exp \left( \rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(k)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right) \ dk \nonumber \\ & & + \int_{-\infty}^{S(0)} 2 (\alpha_{1}k + \alpha_{2}) p(0, S(0); k, T) 1_{[K, \infty)}(k) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(k)\tilde{\chi}(k) \ dk \nonumber \\ & & + \int_{-\infty}^{S(0)} 2(k - K)^{+} \alpha_{1} p(0, S(0); k, T) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(k)\tilde{\chi}(k) \ dk \nonumber \end{eqnarray}\]

である。 第一項は、積分範囲が行使価格$K$を含んでいれば1となる。

更に、$S(T)$が非負過程(BSなど)とすると、$k < 0$では、$p(0, S(0); k, T) = 0$である。 よって、その場合は

\[\begin{eqnarray} \int_{-\infty}^{S(0)} p(0, S(0); k, T)w(k) \ dk & = & p(0, S(0); K, T) \alpha(K) \tilde{\chi}(K)1_{(0, S(0)]}(K) \nonumber \\ & & + 0 \nonumber \\ & & + \int_{0}^{S(0)} p(0, S(0); k, T) (k - K)^{+} (\alpha_{1}k + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T} \left( h^{\prime\prime}(k)\tilde{\chi}(k) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(k)^{2} \tilde{\chi}(k) \right) \ dk \nonumber \\ & & + \int_{0}^{S(0)} 2 \alpha_{1} p(0, S(0); k, T) 1_{[K, \infty)}(k) \exp \left( \rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(k)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right) \ dk \nonumber \\ & & + \int_{0}^{S(0)} 2 (\alpha_{1}k + \alpha_{2}) p(0, S(0); k, T) 1_{[K, \infty)}(k) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(k)\tilde{\chi}(k) \ dk \nonumber \\ & & + \int_{0}^{S(0)} 2(k - K)^{+} \alpha_{1} p(0, S(0); k, T) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(k)\tilde{\chi}(k) \ dk \nonumber \end{eqnarray}\]

となる。 第三項も同様に計算すると、

\[\begin{eqnarray} \int_{-\infty}^{S(0)} c(0, S(0); k, T)w(k) \ dk & = & c(0, S(0); K, T) \alpha(K) \tilde{\chi}(K) 1_{[S(0), \infty)}(K) \nonumber \\ & & + 0 \nonumber \\ & & + \int_{S(0)}^{\infty} c(0, S(0); k, T) (k - K)^{+} (\alpha_{1}k + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T} \left( h^{\prime\prime}(k)\tilde{\chi}(k) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(k)^{2} \tilde{\chi}(k) \right) \ dk \nonumber \\ & & + \int_{S(0)}^{\infty} 2 \alpha_{1} c(0, S(0); k, T) 1_{[K, \infty)}(k) \exp \left( \rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(k)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right) \ dk \nonumber \\ & & + \int_{S(0)}^{\infty} 2 (\alpha_{1}k + \alpha_{2}) c(0, S(0); k, T) 1_{[K, \infty)}(k) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(k)\tilde{\chi}(k) \ dk \nonumber \\ & & + \int_{S(0)}^{\infty} 2(k - K)^{+} \alpha_{1} c(0, S(0); k, T) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(k)\tilde{\chi}(k) \ dk, \nonumber \end{eqnarray}\]

第一項は、積分範囲が行使価格$K$を含んでいれば1となる。

linear TSR model + floor

linear TSR modelにおいて、payoffがfloor(put)型のstochastic weightの計算をする。

\[\begin{eqnarray*} g_{\mathrm{floor}}^{\prime\prime}(s; K)\alpha(s)\tilde{\chi}(s) & = & \delta(s - K)\alpha(s)\tilde{\chi}(s), \\ g_{\mathrm{floor}}(s; K)\alpha^{\prime\prime}(s)\tilde{\chi}(s) & = & 0, \\ g_{\mathrm{floor}}(s; K)\alpha(s)\tilde{\chi}^{\prime\prime}(s) & = & (s - K)^{+} (\alpha_{1}s + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T} \left( h^{\prime\prime}(s)\tilde{\chi}(s) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(s)^{2} \tilde{\chi}(s) \right), \\ 2g_{\mathrm{floor}}^{\prime}(s; K)\alpha^{\prime}(s)\tilde{\chi}(s) & = & 2 1_{[K, \infty)}(s) \alpha_{1} \exp \left( \rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(s)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right), \\ 2g_{\mathrm{floor}}^{\prime}(s; K)\alpha(s)\tilde{\chi}^{\prime}(s) & = & 2 1_{[K, \infty)}(s) (\alpha_{1}s + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(s)\tilde{\chi}(s), \\ 2g_{\mathrm{floor}}(s; K)\alpha^{\prime}(s)\tilde{\chi}^{\prime}(s) & = & 2(s - K)^{+} \alpha_{1} \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(s)\tilde{\chi}(s), \end{eqnarray*}\]

linear TSR model + bull spread

\[\begin{eqnarray*} g_{\mathrm{bullspread}}^{\prime\prime}(s; K_{f}, K_{c})\alpha(s)\tilde{\chi}(s) & = & \left( \delta(s - K_{f}) + \delta(K_{c} - s) \right) \alpha(s)\tilde{\chi}(s), \\ g_{\mathrm{bullspread}}(s; K_{f}, K_{c})\alpha^{\prime\prime}(s)\tilde{\chi}(s) & = & 0, \\ g_{\mathrm{bullspread}}(s; K_{f}, K_{c})\alpha(s)\tilde{\chi}^{\prime\prime}(s) & = & g_{\mathrm{bullspread}}(s; K_{f}, K_{c}) (\alpha_{1}s + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T} \left( h^{\prime\prime}(s)\tilde{\chi}(s) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(s)^{2} \tilde{\chi}(s) \right), \\ 2g_{\mathrm{bullspread}}^{\prime}(s; K_{f}, K_{c})\alpha^{\prime}(s)\tilde{\chi}(s) & = & 2 1_{[K_{f}, K_{c}]}(s) \alpha_{1} \exp \left( \rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(s)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right), \\ 2g_{\mathrm{bullspread}}^{\prime}(s; K_{f}, K_{c})\alpha(s)\tilde{\chi}^{\prime}(s) & = & 2 1_{[K_{f}, K_{c}]}(s) (\alpha_{1}s + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(s)\tilde{\chi}(s), \\ 2g_{\mathrm{bullspread}}(s; K_{f}, K_{c})\alpha^{\prime}(s)\tilde{\chi}^{\prime}(s) & = & 2 g_{\mathrm{bullspread}}(s; K_{f}, K_{c}) \alpha_{1} \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(s)\tilde{\chi}(s), \end{eqnarray*}\]

Calculation of denominator

分母を計算する。

\[\mathrm{E}^{A,d} \left[ \alpha(S(T)) \tilde{\chi}(S(T)) \right] = \alpha(S(0)) \tilde{\chi}(S(0)) + \int_{-\infty}^{S(0)} p(0, S(0); k, T) w(k) \ dk + \int_{S(0)}^{\infty} c(0, S(0); k, T) w(k) \ dk\]

ここで、

\[\begin{eqnarray*} w(s) & := & \frac{d^{2} }{d s^{2}} (\alpha(s) \tilde{\chi}(s)) \\ & = & \alpha^{\prime\prime}(s)\tilde{\chi}(s) + \alpha(s)\tilde{\chi}^{\prime\prime}(s) + 2\alpha^{\prime}(s)\tilde{\chi}^{\prime}(s) \end{eqnarray*}\]

$\alpha, \tilde{\chi}$の微分はAppendixで計算している。 以下では、$\alpha$はTSR modelとして何を選択するのかに依存するので、$\alpha$で分けて計算する。

linear TSR model

linear TSR modelの場合を考える。

\[\begin{eqnarray} \alpha^{\prime\prime}(s)\tilde{\chi}(s) & = & 0 \\ \alpha(s)\tilde{\chi}^{\prime\prime}(s) & = & (\alpha_{1}s + \alpha_{2}) \rho_{XS}\sigma_{X}\sqrt{T} \left( h^{\prime\prime}(s)\tilde{\chi}(s) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(s)^{2} \tilde{\chi}(s) \right), \\ 2\alpha^{\prime}(s)\tilde{\chi}^{\prime}(s) & = & 2\alpha_{1} \rho_{XS}\sigma_{X}\sqrt{T}h^{\prime}(s)\tilde{\chi}(s), \end{eqnarray}\]

swap yield TSR model

TBD.

Appendix

ここでは、$g, \alpha, \tilde{\chi}$の微分を計算する。

payoff $g$の微分

各payoff $g$の微分を考える。

cap option

call optionのとき、$g_{\mathrm{cap}}$の微分を考える。

\[\begin{eqnarray} g_{\mathrm{cap}}(s;K) & := & (s - K)^{+} = \max(s - K, 0), \label{def_payoff_cap} \\ g_{\mathrm{cap}}^{\prime}(s; K) & = & 1_{[K, \infty)}(s), \label{derivative_payoff_cap_by_strike} \\ g_{\mathrm{cap}}^{\prime\prime}(s; K) & = & \delta(s - K), \label{derivative2_payoff_cap_by_strike} \end{eqnarray}\]

floor option

put(floor) optionのとき、$g_{\mathrm{floor}}$の微分を考える。

\[\begin{eqnarray} g_{\mathrm{floor}}(s; K) & := & (K - s)^{+} = \max(K - s, 0) \label{def_payoff_floor} \\ g_{\mathrm{floor}}^{\prime}(s; K) & = & -1_{(-\infty, K]}(s), \label{derivative_payoff_floor_by_strike} \\ g_{\mathrm{floor}}^{\prime\prime}(s; K) & = & -\delta(s - K), \label{derivative2_payoff_floor_by_strike} \end{eqnarray}\]

cap-floor (bull spread) option

cap floor optionのとき、$g_{\mathrm{capfloor}}$の微分を考える。

\[\begin{eqnarray} g_{\mathrm{capfloor}}(s; K_{f}, K_{c}) & := & \min(\max(s - K_{f}, 0), K_{c}), & = & \min((s - K_{f})^{+}, K_{c}), \label{def_payoff_cap_floor} \\ g_{\mathrm{capfloor}}^{\prime}(s; K_{f}, K_{c}) & = & 1_{[K_{f}, K_{c}]}(s), \label{derivaitve_payoff_cap_floor_by_strike} \\ g_{\mathrm{capfloor}}^{\prime\prime}(s; K_{f}, K_{c}) & = & \delta(s - K_{f}) + \delta(K_{c} - s), \label{derivaitve2_payoff_cap_floor_by_strike} \end{eqnarray}\]

annuity mapping funciton $\alpha$の微分

annuity mapping function $\alpha$の各modelごとの微分を考える

linear TSR model

linear TSR modelの場合、$\alpha$の微分を考える。

\[\begin{eqnarray} \alpha(s) & := & \alpha_{1}s + \alpha_{2} \\ \alpha^{\prime}(s) & = &\alpha_{1} \\ \alpha^{\prime\prime}(s) & = & 0 \end{eqnarray}\]

swap yield model

TBD.

$\chi$の微分

$\tilde{\chi}$の微分を考える。 まず、$h(s) := \Phi^{-1}(\Psi^{A}(s))$とおくと

\[\begin{eqnarray} h(s) & := & \Phi^{-1}(\Psi^{A}(s)), \label{def_h} \\ h^{\prime}(s) & = & \frac{1}{\phi(\Phi^{-1}(\Psi^{A}(s)))} \psi^{A}(s) = \frac{1}{\phi(h(s))} \psi^{A}(s), \label{derivaitve_h} \\ h^{\prime\prime}(s) & = & \frac{ (\psi^{A})^{\prime}(s) \phi(h(s)) - \psi^{A}(s) \phi^{\prime}(h(s)) h^{\prime}(s) }{ \phi(h(s))^{2} } \label{derivaitve2_h} \end{eqnarray}\] \[\begin{eqnarray} \tilde{\chi}(s) & = & \exp \left( \rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(s)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right), \label{forward_fx_diffusion} \\ \tilde{\chi}^{\prime}(s) & = & \rho_{XS}\sigma_{X}\sqrt{T}h'(s) \tilde{\chi}(s) \label{derivative_forward_fx_diffusion} \\ \tilde{\chi}^{\prime\prime}(s) & = & \rho_{XS}\sigma_{X}\sqrt{T} \left( h^{\prime\prime}(s)\tilde{\chi}(s) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(s)^{2} \tilde{\chi}(s) \right) \label{derivative2_forward_fx_diffusion} \end{eqnarray}\]

更に、$\Psi^{A}(s)$の計算をする為に、$S(T)$の分布を考える。

\[\begin{eqnarray} P(S(T) \le x) & = & P((\Psi^{A})^{-1}(\Phi(\xi_{2})) \le x) \nonumber \\ & = & P(\xi_{2} \le \Phi^{-1}((\Psi^{A}(x))) \nonumber \\ & = & \Phi(\Phi^{-1}(\Psi^{A}(x))) \nonumber \\ & = & \Psi^{A}(x) \nonumber \\ & = & 1 + \frac{1}{A(t)} \frac{\partial}{\partial K} V_{\mathrm{payer}}(t; S(t), K, A(t), T, \sigma) \end{eqnarray}\]

である。 payer’s swaptionのfirst derivative, second derivative, third derivativeより

\[\begin{eqnarray} V_{\mathrm{payer}}(t; S, K, 1, T, \sigma) & = & \mathrm{E}^{A} \left[ (S(T) - K)^{+} \right], \nonumber \\ \Psi^{A}(\tilde{K}) & = & 1 + \left. \frac{\partial}{\partial K} V_{\mathrm{payer}}(t; S, K, 1, T, \sigma) \right|_{K=\tilde{K}} \nonumber \\ & = & 1 - \Phi(d_{2}(\tilde{K})), \nonumber \\ \psi^{A}(\tilde{K}) & = & \left. \frac{\partial^{2}}{\partial K^{2}} V_{\mathrm{payer}}(t; S, K, 1, T, \sigma) \right|_{K=\tilde{K}} \nonumber \\ & = & -\phi(d_{2}(\tilde{K})) d_{2}^{\prime}(\tilde{K}), \nonumber \\ (\psi^{A})^{\prime}(\tilde{K}) & = & \left. \frac{\partial^{3}}{\partial K^{3}} V_{\mathrm{payer}}(t; S, K, 1, T, \sigma) \right|_{K=\tilde{K}} \nonumber \\ & = & -\left( \phi^{\prime}(d_{2}(\tilde{K})) (d^{\prime}(\tilde{K}))^{2} + \phi(d_{2}(\tilde{K})) d^{\prime\prime}(\tilde{K}) \right), \end{eqnarray}\]

Tips

replicationの積分範囲の決め方。 以下はQuantLibの実装より引用。

• 自分で指定する
• atm swaptionのvegaが1%以下になるようなvanilla swaptionのstrike
• 割り引かれていないpayer、reciever swaptionの価格が与えられたしきい値以下になるようなStrike
• ATM volatilityを持つBlack scholes process

Examples of simple case

ドルCMSを参照する$T$での円払いのQuanto CMSを考える。

• $S^{$}(T)$
  • ドルのswap rate
• $\beta_{$}(T)$
  • ドルのrisk neutral measureの下でのnumeraire
• $\beta_{\yen}(T)$
  • 円のrisk neutral measureの下でのnumeraire
• $T_{p}$
  • Quanto CMSのpayment date
• $T$
  • $S(T)$のfixing date
• $Q^{$}$
  • ドルのrisk neutral measure
• $Q^{\yen}$
  • 円のrisk neutral measure
• $X(t)$
  • 時刻$t$でのDOM=円、FOR=ドル、ドル円のtoday FX[DOM/FOR]
• $X_{T}(t)$
  • 時刻$t$での$T$のドル円のforward FX[DOM/FOR]
• $g(\cdot)$
  • payoff関数
• $A^{$}$
  • ドルのannuity measure
• $P_{$}(t, T)$
  • ドルのZero Coupon Bond
• $P_{\yen}(t, T)$
  • 円のZero Coupon Bond

\[\begin{eqnarray} V_{\mathrm{QuantoCMS}}(0) := \mathrm{E}_{t}^{\yen} \left[ \frac{\beta_{\yen}(t)}{\beta_{\yen}(T_{p})} g(S^{\$}(T)) \right] \end{eqnarray}\]

ここで、

\[\begin{eqnarray} \mathrm{E}_{t}^{\yen} \left[ \frac{dQ^{\$}}{dQ^{\yen}} \right] & = & \frac{ \frac{\beta_{\yen}(0)}{\beta_{\yen}(t)} }{ \frac{\beta_{\$}(0) X(0)}{\beta_{\$}(t) X(t)} } \nonumber \\ & = & \frac{ \beta_{\$}(t) X(t) }{ \beta_{\yen}(t) X(0) } \end{eqnarray}\]

となる。

\[\begin{eqnarray} V_{\mathrm{QuantoCMS}}(t) & = & \mathrm{E}_{t}^{\yen} \left[ \frac{\beta_{\yen}(t)}{\beta_{\yen}(T_{p})} g(S^{\$}(T)) \right] \nonumber \\ & = & \mathrm{E}_{t}^{\yen} \left[ \frac{\beta_{\yen}(t)}{\beta_{\yen}(T_{p})} \mathrm{E}_{T_{p}}^{\yen} \left[ \frac{dQ^{\$}}{dQ^{\yen}} \right] \frac{ \beta_{\yen}(T_{p}) X(0) }{ \beta_{\$}(T_{p}) X(T_{p}) } g(S^{\$}(T)) \right] \nonumber \\ & = & \mathrm{E}_{t}^{\yen} \left[ \frac{ X(0) \beta_{\yen}(t) }{ \beta_{\$}(T_{p}) X(T_{p}) } g(S^{\$}(T)) \right] \nonumber \\ & = & \mathrm{E}_{t}^{\$} \left[ \frac{ X(0) \beta_{\yen}(t) P^{\$}(T_{p}, T_{p}) }{ \beta_{\$}(T_{p}) X_{T_{p}}(T_{p}) P^{\yen}(T_{p}, T_{p}) } g(S^{\$}(T)) \right] \nonumber \\ & = & \mathrm{E}_{t}^{\$} \left[ \frac{ X(0) \beta_{\yen}(t) }{ \beta_{\$}(T) } \mathrm{E}_{T}^{\$} \left[ \beta_{\$}(T) \frac{ P^{\$}(T_{p}, T_{p}) }{ \beta_{\$}(T_{p}) X_{T_{p}}(T_{p}) } \right] g(S^{\$}(T)) \right] \nonumber \\ & = & \mathrm{E}_{t}^{\$} \left[ \frac{ X(0) \beta_{\yen}(t) }{ \beta_{\$}(T) } \beta_{\$}(T) \frac{ P^{\$}(T, T_{p}) }{ \beta_{\$}(T) X_{T_{p}}(T) } g(S^{\$}(T)) \right] \nonumber \\ & = & \mathrm{E}_{t}^{\$} \left[ \frac{ X(0) \beta_{\yen}(t) }{ \beta_{\$}(T) } \frac{ P^{\$}(T, T_{p}) }{ X_{T_{p}}(T) } g(S^{\$}(T)) \right] \end{eqnarray}\]

となる。 更にannuity measureの下では

\[\begin{eqnarray} V_{\mathrm{QuantoCMS}}(t) & = & \mathrm{E}_{t}^{A^{\$}} \left[ \frac{ X(0) \beta_{\yen}(t) A^{\$}(t) }{ \beta_{\$}(t) A^{\$}(T) } \frac{ P^{\$}(T, T_{p}) }{ X_{T_{p}}(T) } g(S^{\$}(T)) \right] \nonumber \\ & = & \frac{ X(0) \beta_{\yen}(t) A^{\$}(t) }{ \beta_{\$}(t) } \mathrm{E}_{t}^{A^{\$}} \left[ \frac{ P^{\$}(T, T_{p}) }{ A^{\$}(T) } \frac{ 1 }{ X_{T_{p}}(T) } g(S^{\$}(T)) \right] \end{eqnarray}\]

となる。 特に時刻$t=0$では

\[\begin{eqnarray} V_{\mathrm{QuantoCMS}}(0) & = & X(0) A^{\$}(0) \mathrm{E}^{A^{\$}} \left[ \frac{ P^{\$}(T, T_{p}) }{ A^{\$}(T) } \frac{ 1 }{ X_{T_{p}}(T) } g(S^{\$}(T)) \right] \end{eqnarray}\]

となる。 Andersen and Piterbargと同様の議論で

\[\begin{eqnarray} V_{\mathrm{QuantoCMS}}(0) & \approx & X(0) A^{\$}(0) \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \chi(S^{\$}(T)) g(S^{\$}(T)) \right] \label{dollar_yen_quanto_cms_approx_value_by_replication} \end{eqnarray}\]

と近似する。 ここで、

\[\begin{eqnarray} \chi(S^{\$}(T)) & := & \mathrm{E}^{A^{\$}} \left[ \left. \frac{1}{X_{T_{p}}(T)} \right| S^{\$}(T) = s \right] \end{eqnarray}\]

以下では具体的なモデルを考える。 $X_{T}(t)$が以下で与えられていると

\[\begin{equation} X_{T_{p}}(T) = X(0) e^{\sigma_{X}\sqrt{T}\xi_{1} + m_{X}T}, \end{equation}\]

その逆数は

\[\begin{equation} \frac{1}{X_{T_{p}}(T)} = \frac{1}{X(0)} e^{-\sigma_{X}\sqrt{T}\xi_{1} - m_{X}T}, \end{equation}\]

となる。 上記の下、$chi(\cdot)$を計算すると

\[\begin{eqnarray*} \chi(s) & = & \frac{1}{X(0)} e^{-m_{X}T} \mathrm{E}^{A,d} \left[ e^{-\sigma_{X}\sqrt{T}\xi_{1}} | \xi_{2} = \Psi^{-1}(\Phi^{A}(s)) \right] \nonumber \\ & = & \frac{1}{X(0)} e^{-m_{X}T} \hat{\chi}(s), \end{eqnarray*}\]

ここで、最後の等式は以下による。

\[\begin{eqnarray*} \mathrm{E}^{A,d} \left[ e^{-\sigma_{X}\sqrt{T}\xi_{1}} | \xi_{2} = x_{2} \right] & = & \frac{1}{\sqrt{2 \pi (1 - \rho^{2})}} \int_{-\infty}^{\infty} \exp \left( \frac{-2 \sigma_{X} \sqrt{T} x_{1}(1 - \rho^{2})}{2(1 - \rho^{2})} - \frac{ (x_{1} - x_{2}\rho)^{2} }{ 2(1 - \rho^{2}) } \right) \ dx_{1} \\ & = & \frac{1}{\sqrt{2 \pi (1 - \rho^{2})}} \int_{-\infty}^{\infty} \exp \left( \frac{ -2 \sigma_{X} \sqrt{T} x_{1}(1 - \rho^{2}) - x_{1}^{2} + 2x_{1}x_{2}\rho - x_{2}^{2}\rho^{2} }{ 2(1 - \rho^{2}) } \right) \ dx_{1} \\ & = & \frac{1}{\sqrt{2 \pi (1 - \rho^{2})}} \int_{-\infty}^{\infty} \exp \left( \frac{ -x_{1}^{2} - 2 x_{1}(\sigma_{X} \sqrt{T}(1 - \rho^{2}) + x_{2}\rho) -x_{2}^{2}\rho^{2} }{ 2(1 - \rho^{2}) } \right) \ dx_{1} \\ & = & \frac{1}{\sqrt{2 \pi (1 - \rho^{2})}} \int_{-\infty}^{\infty} \exp \left( \frac{ -(x_{1} - (-\sigma_{X} \sqrt{T}(1 - \rho^{2}) + x_{2}\rho))^{2} +(-\sigma_{X} \sqrt{T}(1 - \rho^{2}) + x_{2}\rho)^{2} -x_{2}^{2}\rho^{2} }{ 2(1 - \rho^{2}) } \right) \ dx_{1} \\ & = & \frac{1}{\sqrt{2 \pi (1 - \rho^{2})}} \int_{-\infty}^{\infty} \exp \left( \frac{ -(x_{1} - (-\sigma_{X} \sqrt{T}(1 - \rho^{2}) + x_{2}\rho))^{2} \sigma_{X}^{2} T(1 - \rho^{2})^{2} - 2\sigma_{X} \sqrt{T} (1 - \rho^{2})x_{2}\rho + x_{2}^{2}\rho^{2} -x_{2}^{2}\rho^{2} }{ 2(1 - \rho^{2}) } \right) \ dx_{1} \\ & = & \frac{1}{\sqrt{2 \pi (1 - \rho^{2})}} \int_{-\infty}^{\infty} \exp \left( \frac{1}{2} \sigma_{X}^{2} T(1 - \rho^{2}) - \sigma_{X} \sqrt{T} x_{2}\rho \right) \exp \left( \frac{ -(x_{1} - (\sigma_{X} \sqrt{T}(1 - \rho^{2}) + x_{2}\rho))^{2} }{ 2(1 - \rho^{2}) } \right) \ dx_{1} \\ & = & \exp \left( \frac{1}{2} \sigma_{X}^{2} T(1 - \rho^{2}) - \sigma_{X} \sqrt{T} x_{2}\rho \right). \end{eqnarray*}\]

ここで、

\[\begin{eqnarray} \frac{1}{X_{T_{p}}(0)} & = & \mathrm{E}^{T_{p},d} \left[ \frac{1}{X_{T_{p}}(T)} \right] \nonumber \\ & = & \frac{A^{\$}(0)}{P_{\$}(0,T_{p})} \mathrm{E}^{A^{\$}} \left[ \frac{P_{\$}(T, T_{p})}{A^{\$}(T)} \frac{1}{X_{T_{p}}(T)} \right] \nonumber \\ & = & \frac{A^{\$}(0)}{P_{\$}(0,T_{p})} \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \chi(S^{\$}(T)) \right] \nonumber \\ & = & \frac{A^{\$}(0)}{P_{\$}(0,T_{p})} \frac{1}{X(0)} e^{-m_{X}T} \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) \right]. \end{eqnarray}\]

よって、

\[\begin{eqnarray} e^{-m_{X}T} & = & \frac{P_{\$}(0,T_{p})}{A^{\$}(0)} \frac{X(0)}{X_{T_{p}}(0)} \frac{ 1 }{ \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) \right] } \label{dollar_yen_quanto_cms_forward_fx} \end{eqnarray}\]

\(\eqref{dollar_yen_quanto_cms_forward_fx}\)と\(\eqref{dollar_yen_quanto_cms_approx_value_by_replication}\)と

\[\begin{eqnarray} V_{\mathrm{QuantoCMS}}(0) & \approx & X(0) A^{\$}(0) \frac{1}{X(0)} e^{-m_{X}T} \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) g(S^{\$}(T)) \right] \nonumber \\ & = & A^{\$}(0) \frac{P_{\$}(0,T_{p})}{A^{\$}(0)} \frac{X(0)}{X_{T_{p}}(0)} \frac{ 1 }{ \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) \right] } \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) g(S^{\$}(T)) \right] \nonumber \\ & = & P_{\$}(0,T_{p}) \frac{X(0)}{X_{T_{p}}(0)} \frac{ \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) g(S^{\$}(T)) \right] }{ \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) \right] } \nonumber \\ & = & P_{\yen}(0,T_{p}) \frac{ \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) g(S^{\$}(T)) \right] }{ \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \hat{\chi}(S^{\$}(T)) \right] } \end{eqnarray}\]

また、$\chi(\cdot)$の微分を計算すると、

\[\begin{eqnarray} \tilde{\chi}(s) & = & \exp \left( -\rho_{XS}\sigma_{X}\sqrt{T}\Phi^{-1}(\Psi^{A}(s)) + \frac{\sigma_{X}^{2}T}{2}(1 - \rho_{XS}^{2}) \right), \label{dollar_yen_quanto_cms_forward_fx_diffusion} \\ \tilde{\chi}^{\prime}(s) & = & -\rho_{XS}\sigma_{X}\sqrt{T}h'(s) \tilde{\chi}(s) \label{dollar_yen_quanto_cms_derivative_forward_fx_diffusion} \\ \tilde{\chi}^{\prime\prime}(s) & = & \rho_{XS}\sigma_{X}\sqrt{T} \left( -h^{\prime\prime}(s)\tilde{\chi}(s) + \rho_{XS}\sigma_{X}\sqrt{T} h^{\prime}(s)^{2} \tilde{\chi}(s) \right) \label{dollar_yen_quanto_cms_derivative2_forward_fx_diffusion} \end{eqnarray}\]

となる。

Analysis

linear TSR modelで$\alpha_{0} = 0$かつ$\rho=0$の特別な場合を考える。

\[\begin{eqnarray} V_{\mathrm{QuantoCMS}}(0) & \approx & P_{\yen}(0,T_{p}) \frac{ \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \exp \left( \frac{1}{2} \sigma_{X}^{2}T \right) g(S^{\$}(T)) \right] }{ \mathrm{E}^{A^{\$}} \left[ \alpha(S^{\$}(T)) \exp \left( \frac{1}{2} \sigma_{X}^{2}T \right) \right] } \nonumber \\ & = & P_{\yen}(0,T_{p}) \frac{ \alpha_{0} \mathrm{E}^{A^{\$}} \left[ S^{\$}(T) g(S^{\$}(T)) \right] + \mathrm{E}^{A^{\$}} \left[ \alpha_{1} g(S^{\$}(T)) \right] }{ \alpha_{0}S^{\$}(0) + \alpha_{1} } \nonumber \\ & = & P_{\yen}(0,T_{p}) \mathrm{E}^{A^{\$}} \left[ g(S^{\$}(T)) \right] \end{eqnarray}\]

となる。 replication methodで計算される2階微分を考えると

\[\begin{eqnarray} \alpha^{\prime\prime}(S^{\$}(T)) g(S^{\$}(T)) & = & 0 \nonumber \\ 2\alpha^{\prime}(S^{\$}(T)) g^{\prime}(S^{\$}(T)) & = & \alpha_{0} g^{\prime}(S^{\$}(T)) \nonumber \\ \alpha(S^{\$}(T)) g^{\prime\prime}(S^{\$}(T)) & = & \alpha(S^{\$}(T)) g^{\prime\prime}(S^{\$}(T)) \end{eqnarray}\]

更にpayoff関数がstrike $K$のcall optionの場合は、

\[\begin{eqnarray} 2\alpha^{\prime}(S^{\$}(T)) g^{\prime}(S^{\$}(T)) & = & \alpha_{0} 1_{[K, \infty)}(S^{\$}(T)) \nonumber \\ \alpha(S^{\$}(T)) g^{\prime\prime}(S^{\$}(T)) & = & \alpha(S^{\$}(T)) \delta(S^{\$}(T) - K) \end{eqnarray}\]

となり、payoff関数が$K_{\mathrm{lower}}, K_{\mathrm{upper}}$をstrikeとするbull-spreadの場合は

\[\begin{eqnarray} 2\alpha^{\prime}(S^{\$}(T)) g^{\prime}(S^{\$}(T)) & = & \alpha_{0} 1_{[K_{\mathrm{lower}}, K_{\mathrm{upper}})}(S^{\$}(T)) \nonumber \\ \alpha(S^{\$}(T)) g^{\prime\prime}(S^{\$}(T)) & = &{ \alpha(S^{\$}(T))(\delta(S^{\$}(T) - K_{\mathrm{lower} - \delta{}) \end{eqnarray}\]

となる。