Appendix A
Proof of proposition 1: Equation (5)
In a risk-neutral world, the price of the Type-1 Quanto reset put option is computed as follows:
$$\begin{aligned} RP_{1} \left( {S_{0} ,\;K,\;0} \right) & =\, e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {\bar{X}\left( {S_{{t_{0} }} - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{0} } \right.} \right] + E^{Q} \left[ {\bar{X}\left( {K - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ & =\, e^{{ - r_{d} T}} E^{Q} \left[ {\bar{X}\left( {S_{{t_{0} }} - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{0} } \right.} \right] + e^{{ - r_{d} T}} E^{Q} \left[ {\bar{X}\left( {K - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] \\ & =\, RP_{11} + RP_{12} \\ \end{aligned}.$$
(A.1)
We compute the first component \(RP_{11}\) of (A.1) and use the law of iterated conditional expectations. Rewrite \(RP_{11}\) as follows:18
$$\begin{aligned} RP_{11} & = e^{{ - r_{d} T}} E^{Q} \left[ {E^{Q} \left[ {\bar{X}\left( {S_{{t_{0} }} - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] \\ & = \bar{X}e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {E^{Q} \left[ {S_{{t_{0} }} \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] - E^{Q} \left[ {E^{Q} \left[ {S{}_{T} \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ \end{aligned}.$$
(A.2)
Using the independence property,19 the first component of (A.2) is computed as follows:20
$$\begin{aligned} E^{Q} \left[ {S_{{t_{0} }} \cdot 1_{{\left( {S_{{t_{0} }} > K} \right)}} E^{Q} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] = E^{Q} \left[ {S_{{t_{0} }} \cdot 1_{{\left( {S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]E^{Q} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] \hfill \\ = S_{0} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)t_{0} }} E^{{R_{1} }} \left[ {1_{{\left( {S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]E^{Q} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] = S_{0} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)t_{0} }} N\left( {d_{1} } \right)N\left( { - b_{2} } \right). \hfill \\ \end{aligned}$$
(A.3)
Similarly, the second component of (A.2) can be computed using change of probability measure based on Girsanov theorem and then we decompose the bivariate cumulative normal probability into the product of two univariate cumulative normal probabilities. This is given below:
$$\begin{aligned} E^{Q} \left[ {E^{Q} \left[ {S{}_{T} \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] = E^{Q} \left[ {S{}_{{t_{0} }}e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)\left( {T - t_{0} } \right)}} 1_{{\left( {S_{{t_{0} }} > K} \right)}} E^{{R_{1} }} \left[ {1_{{\left( {\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] \hfill \\ \quad = E^{Q} \left[ {S{}_{{t_{0} }}e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)\left( {T - t_{0} } \right)}} \cdot 1_{{\left( {S_{{t_{0} }} > K} \right)}} \left| {F_{{t_{0} }} } \right.} \right]E^{{R_{1} }} \left[ {1_{{\left( {\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] \hfill \\ \quad = S{}_{0}e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)T}} E^{{R_{1} }} \left[ {1_{{\left( {S_{{t_{0} }} > K} \right)}} \left| {F_{{t_{0} }} } \right.} \right]E^{{R_{1} }} \left[ {1_{{\left( {\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] = S{}_{0}e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)T}} N\left( {d_{1} } \right)N\left( { - b_{1} } \right). \hfill \\ \end{aligned}$$
(A.4)
Using (A.3), (A.4), and (A.2), we get
$$RP_{11}\, =\, \bar{X}e^{{ - r_{d} T}} \left\{ {S_{0} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)t_{0} }} N\left( {d_{1} } \right)N\left( { - b_{2} } \right) - S{}_{0}e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)T}} N\left( {d_{1} } \right)N\left( { - b_{1} } \right)} \right\} .$$
(A.5)
Next, we compute the second component \(RP_{12}\) of (A.1). Decompose it into two terms and use change of probability measure to obtain the following:
$$\begin{aligned} RP_{12} &= e^{{ - r_{d} T}} E^{Q} \left[ {\bar{X}\left( {K - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] \\ \quad \quad &= \bar{X}e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {K \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] - E^{Q} \left[ {S{}_{T} \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ \quad \quad& =\bar{X}e^{{ - r_{d} T}} \left\{ {KE^{Q} \left[ {1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] - S_{0} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)T}} E^{{R_{1} }} \left[ {1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ \quad \quad &= \bar{X}e^{{ - r_{d} T}} \left\{ {KN_{2} \left( { - d_{2} , - \tilde{b}_{2} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - S_{0} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)T}} N_{2} \left( { - d_{1} , - \tilde{b}_{1} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right)} \right\}. \\ \end{aligned}$$
(A.6)
Substituting (A.5) and (A.6) into (A.1), we obtain the pricing model for the Type-1 Quanto reset put option as given below:
$$\begin{aligned} RP_{1} \left( {S_{0} ,K,0} \right) = \bar{X}e^{{ - r_{d}^{T} }} \left\{ {S_{0} e^{{\left( {r_{f} - \rho \sigma_{s} \sigma_{X} } \right)}} N(d_{1} )N( - b_{2} ) - S_{0} e^{{\left( {r_{f} - \rho \sigma_{s} \sigma_{X} } \right)}} N(d_{1} )N( - b_{1} )} \right. \hfill \\ \left. { + \,KN_{2} \left( { - d_{2} , - \tilde{b}_{2} ,\sqrt {\frac{{t_{0} }}{T}} } \right) - S_{0} e^{{\left( {r_{f} - \rho \sigma_{s} \sigma_{X} } \right)}} N_{2} \left( { - d_{2} , - \tilde{b}_{2} ,\sqrt {\frac{{t_{0} }}{T}} } \right)} \right\} \hfill \\ \end{aligned}.$$
\(\quad \square\)
Appendix B
Proof of proposition 2: equation (9)
Based on the payoff structure as given in equation (8), the price of the Type-2 Quanto reset put option is computed as follows:
$$\begin{aligned} RP_{2} \left( {S_{0} ,\;K,\;0} \right) = & e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {X_{T} \left( {S_{{t_{0} }} - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{0} } \right.} \right] + E^{Q} \left[ {X_{T} \left( {K - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ = e^{{ - r_{d} T}} E^{Q} \left[ {X_{T} \left( {S_{{t_{0} }} - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{0} } \right.} \right] + e^{{ - r_{d} T}} E^{Q} \left[ {X_{T} \left( {K - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] \\ = RP_{21} + RP_{22} \\ \end{aligned}.$$
(B.1)
The first component \(RP_{21}\) of (B.1) is computed using the law of iterated conditional expections as follows:
$$\begin{aligned} RP_{21} &= e^{{ - r_{d} T}} E^{Q} \left[ {E^{Q} \left[ {X_{T} \left( {S_{{t_{0} }} - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {E^{Q} \left[ {X_{T} S_{{t_{0} }} \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] - E^{Q} \left[ {E^{Q} \left[ {X_{T} S{}_{T} \cdot 1_{{\left( {S_{{t_{0} }} > K,\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {X_{{t_{0} }} S_{{t_{0} }} E^{Q} \left[ {e^{{\left( {r_{d} - r_{f} - \tfrac{1}{2}\sigma_{X}^{2} } \right)\left( {T - t_{0} } \right) + \sigma_{X} w_{{T - t_{0} }}^{Q} }} \cdot 1_{{\left( {S_{{t_{0} }} > K} \right)}} 1_{{\left( {\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right. \\ &\left. { - E^{Q} \left[ {X_{{t_{0} }} S{}_{{t_{0} }}E^{Q} \left[ {e^{{\left[ {r_{d} - \tfrac{1}{2}\left( {\sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2} } \right)} \right]\left( {T - t_{0} } \right) + \sigma_{S} Z_{{T - t_{0} }}^{Q} + \sigma_{X} w_{{T - t_{0} }}^{Q} }} \cdot 1_{{\left( {S_{{t_{0} }} > K} \right)}} 1_{{\left( {\,S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {X_{{t_{0} }} S_{{t_{0} }} e^{{\left( {r_{d} - r_{f} } \right)\left( {T - t_{0} } \right)}} 1_{{\left( {S_{{t_{0} }} > K} \right)}} E^{{R_{2} }} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right. \\ &\left. { - E^{Q} \left[ {X_{{t_{0} }} S{}_{{t_{0} }}e^{{r_{d} \left( {T - t_{0} } \right)}} 1_{{\left( {S_{{t_{0} }} > K} \right)}} E^{{R_{3} }} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\}. \\ \end{aligned}$$
(B.2)
The above computation employs Girsanov theorem to change Q measure into two probability measures (\(R_{2}\) measure and \(R_{3}\) measure).21 Hence, equation (B.2) is rewritten as follows:22
$$\begin{aligned} &e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {X_{{t_{0} }} S_{{t_{0} }} e^{{\left( {r_{d} - r_{f} } \right)\left( {T - t_{0} } \right)}} 1_{{\left( {S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]E^{{R_{2} }} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] - E^{Q} \left[ {X_{{t_{0} }} S{}_{{t_{0} }}e^{{r_{d} \left( {T - t_{0} } \right)}} 1_{{\left( {S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]E^{{R_{3} }} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]} \right\} \hfill \\ &= e^{{ - r_{d} T}} \left\{ {X_{0} S_{0} e^{{r_{d} t_{0} }} e^{{\left( {r_{d} - r_{f} } \right)\left( {T - t_{0} } \right)}} E^{{R_{3} }} \left[ {1_{{\left( {S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]E^{{R_{2} }} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]} \right. \hfill \\ & \left. { - X_{0} S{}_{0}e^{{r_{d} t_{0} }} e^{{r_{d} \left( {T - t_{0} } \right)}} E^{{R_{3} }} \left[ {1_{{\left( {S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]E^{{R_{3} }} \left[ {1_{{\left( {S_{{t_{0} }} > S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]} \right\}, \hfill \\ \end{aligned}$$
(B.3)
where we again use Girsanov theorem to make change of measure. Using (B) in Appendix E and arranging the terms, we obtain \(RP_{21}\):
$$RP_{21} = X_{0} \left\{ {S_{0} e^{{ - r_{f} \left( {T - t_{0} } \right)}} N\left( {d_{3} } \right)N\left( { - b_{4} } \right) - S_{0} N\left( {d_{3} } \right)N\left( { - b_{3} } \right)} \right\}.$$
(B.4)
Next, we compute the second component \(RP_{22}\) and decompose it into two terms. Using Girsanov theorem to make change of measure and (B) in Appendix E, we obtain \(RP_{22}\):
$$\begin{aligned} RP_{22} = & e^{{ - r_{d} T}} E^{Q} \left[ {X_{T} \left( {K - S{}_{T}} \right) \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] \\ = e^{{ - r_{d} T}} \left\{ {KE^{Q} \left[ {X_{T} \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] - E^{Q} \left[ {X_{T} S_{T} \cdot 1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ = e^{{ - r_{d} T}} \left\{ {KX_{0} e^{{\left( {r_{d} - r_{f} } \right)T}} E^{{R_{2} }} \left[ {1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] - X_{0} S_{0} e^{{r_{d} T}} E^{{R_{3} }} \left[ {1_{{\left( {S_{{t_{0} }} \le K,\,S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ = e^{{ - r_{d} T}} X_{0} \left\{ {Ke^{{\left( {r_{d} - r_{f} } \right)T}} N\left( { - d_{4} ,\; - \tilde{b}_{4} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - S_{0} e^{{r_{d} T}} N\left( { - d_{3} ,\; - \tilde{b}_{3} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right)} \right\} \\ = X_{0} \left\{ {Ke^{{ - r_{f} T}} N\left( { - d_{4} ,\; - \tilde{b}_{4} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - S_{0} N\left( { - d_{3} ,\; - \tilde{b}_{3} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right)} \right\}. \\ \end{aligned}$$
(B.5)
Substituting (B.4) and (B.5) into (B.1) to obtain the pricing model for the Type-2 Quanto reset put option:
$$\begin{aligned} RP_{2} \left( {S_{0} ,\;K,\;0} \right) = & X_{0} \left\{ {S_{0} e^{{ - r_{f} \left( {T - t_{0} } \right)}} N\left( {d_{3} } \right)N\left( { - b_{4} } \right) - S_{0} N\left( {d_{3} } \right)N\left( { - b_{3} } \right)} \right. \\ \left. { +\, Ke^{{ - r_{f} T}} N\left( { - d_{4} ,\; - \tilde{b}_{4} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - S_{0} N\left( { - d_{3} ,\; - \tilde{b}_{3} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right)} \right\}. \\ \end{aligned}$$
\(\quad \square\)
Appendix C
Proof of Proposition 3: equation (13)
Based on the payoff structure as given in equation (12), the price of the Type-3 Quanto reset put option is computed below:
$$\begin{aligned} RP_{3} \left( {X_{0} S_{0} ,\;K,\;0} \right) &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {\left( {X_{{t_{0} }} S_{{t_{0} }} - X_{T} S{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K,\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{0} } \right.} \right] + E^{Q} \left[ {\left( {K - X_{T} S{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} E^{Q} \left[ {\left( {X_{{t_{0} }} S_{{t_{0} }} - X_{T} S{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K,\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{0} } \right.} \right] + e^{{ - r_{d} T}} E^{Q} \left[ {\left( {K - X_{T} S{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] \\ &= RP_{31} + RP_{32} . \\ \end{aligned}$$
(C.1)
The first component \(RP_{31}\) of (C.1) is computed below:
$$\begin{aligned} RP_{31} &= e^{{ - r_{d} T}} E^{Q} \left[ {E^{Q} \left[ {\left( {X_{{t_{0} }} S_{{t_{0} }} - X_{T} S{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K,\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {E^{Q} \left[ {X_{{t_{0} }} S_{{t_{0} }} \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K,\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right. \\ &\left. { -\, E^{Q} \left[ {E^{Q} \left[ {X_{T} S{}_{T} \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K,\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {X_{{t_{0} }} S_{{t_{0} }} E^{Q} \left[ {1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K} \right)}} 1_{{\left( {\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right. \\ &\left. { -\, E^{Q} \left[ {X_{{t_{0} }} S{}_{{t_{0} }}E^{Q} \left[ {e^{{\left[ {r_{d} - \tfrac{1}{2}\left( {\sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2} } \right)} \right]\left( {T - t_{0} } \right) + \sigma_{S} Z_{{T - t_{0} }}^{Q} + \sigma_{X} w_{{T - t_{0} }}^{Q} }} \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K} \right)}} 1_{{\left( {\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {X_{{t_{0} }} S_{{t_{0} }} 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K} \right)}} E^{Q} \left[ {1_{{\left( {\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right. &\\ \left. { - E^{Q} \left[ {X_{{t_{0} }} S{}_{{t_{0} }}e^{{r_{d} \left( {T - t_{0} } \right)}} 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K} \right)}} E^{{R_{4} }} \left[ {1_{{\left( {\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {X_{0} S_{0} E^{Q} \left[ {e^{{\left[ {r_{d} - \tfrac{1}{2}\left( {\sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2} } \right)} \right]t_{0} + \sigma_{S} Z_{{t_{0} }}^{Q} + \sigma_{X} w_{{t_{0} }}^{Q} }} \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]} \right.E^{Q} \left[ {1_{{\left( {\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] \\ &\left. { - X_{0} S{}_{0}e^{{r_{d} \left( {T - t_{0} } \right)}} E^{Q} \left[ {e^{{\left[ {r_{d} - \tfrac{1}{2}\left( {\sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2} } \right)} \right]t_{0} + \sigma_{S} Z_{{t_{0} }}^{Q} + \sigma_{X} w_{{t_{0} }}^{Q} }} \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]E^{{R_{4} }} \left[ {1_{{\left( {\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {X_{0} S_{0} e^{{r_{d} t_{0} }} E^{{R_{4} }} \left[ {1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]} \right.E^{Q} \left[ {1_{{\left( {\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] \\ \left. { - X_{0} S{}_{0}e^{{r_{d} T}} E^{{R_{4} }} \left[ {1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} > K} \right)}} \left| {F_{0} } \right.} \right]E^{{R_{4} }} \left[ {1_{{\left( {\,X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]} \right\}. \\ \end{aligned}$$
(C.2)
The above derivation uses Girsanov theorem to make change of measure from Q measure to \(R_{4}\) measure23, and then employs (C) in Appendix E. Arranging the terms, we get
$$RP_{21} = \left\{ {X_{0} S_{0} e^{{ - r_{f} \left( {T - t_{0} } \right)}} N\left( {d_{5} } \right)N\left( { - b_{6} } \right) - X_{0} S_{0} N\left( {d_{5} } \right)N\left( { - b_{5} } \right)} \right\}.$$
(C.3)
Next, the second component \(RP_{32}\) is decomposed into two terms. Then we employ Girsanov theorem to make change of measure and use (C) in Appendix E to derive the following:24
$$\begin{aligned} RP_{32} &= e^{{ - r_{d} T}} E^{Q} \left[ {\left( {K - X_{T} S{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] \\ &= e^{{ - r_{d} T}} \left\{ {KE^{Q} \left[ {1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] - E^{Q} \left[ {X_{T} S_{T} \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {KE^{Q} \left[ {1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] - X_{0} S_{0} E^{Q} \left[ {e^{{\left[ {r_{d} - \tfrac{1}{2}\left( {\sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2} } \right)} \right]T + \sigma_{S} Z_{T}^{Q} + \sigma_{X} w_{T}^{Q} }} \cdot 1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {KE^{Q} \left[ {1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right] - X_{0} S_{0} e^{{r_{d} T}} E^{{R_{4} }} \left[ {1_{{\left( {X_{{t_{0} }} S_{{t_{0} }} \le K,\,X_{T} S_{T} < K} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {KN\left( { - d_{6} ,\; - \tilde{b}_{6} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - X_{0} S_{0} e^{{r_{d} T}} N\left( { - d_{5} ,\; - \tilde{b}_{5} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right)} \right\} \\ &= Ke^{{ - r_{d} T}} N\left( { - d_{6} ,\; - \tilde{b}_{6} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - X_{0} S_{0} N\left( { - d_{5} ,\; - \tilde{b}_{5} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right). \\ \end{aligned}$$
(C.4)
Substituting (C.3) and (C.4) into (C.1), we obtain the pricing model for the Type-3 Quanto reset put options as given below:
$$\begin{aligned} RP_{3} \left( {X_{0} S_{0} ,\;K,\;0} \right) &= X_{0} S_{0} e^{{ - r_{f} \left( {T - t_{0} } \right)}} N\left( {d_{5} } \right)N\left( { - b_{6} } \right) - X_{0} S_{0} N\left( {d_{5} } \right)N\left( { - b_{5} } \right) \\ & \quad+\, Ke^{{ - r_{d} T}} N\left( { - d_{6} ,\; - \tilde{b}_{6} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - X_{0} S_{0} N\left( { - d_{5} ,\; - \tilde{b}_{5} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) \\ \end{aligned}.$$
\(\quad \square\)
Appendix D
Proof of Proposition 4: equation (17)
Based on the payoff structure as given in equation (16), the price of the Type-4 Quanto reset put option is computed as follows:
$$\begin{aligned} RP_{4} \left( {X_{0} ,\;L,\;0} \right) &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {S_{T} \left( {X_{{t_{0} }} - X{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} > L,\,X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{0} } \right.} \right] + E^{Q} \left[ {S_{T} \left( {L - X{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} \le L,\,X_{T} < L} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} E^{Q} \left[ {S_{T} \left( {X_{{t_{0} }} - X{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} > L,\,X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{0} } \right.} \right] + e^{{ - r_{d} T}} E^{Q} \left[ {S_{T} \left( {L - X{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} \le L,\,X_{T} < L} \right)}} \left| {F_{0} } \right.} \right] \\ &= RP_{41} + RP_{42} . \\ \end{aligned}$$
(D.1)
The first component \(RP_{41}\) of (D.1) is derived using the law of iterated conditional expections:
$$\begin{aligned} RP_{41} &= e^{{ - r_{d} T}} E^{Q} \left[ {E^{Q} \left[ {S_{T} \left( {X_{{t_{0} }} - X{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} > L,\,X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {E^{Q} \left[ {S_{T} X_{{t_{0} }} \cdot 1_{{\left( {X_{{t_{0} }} > L,\,X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right] - E^{Q} \left[ {E^{Q} \left[ {S_{T} X{}_{T} \cdot 1_{{\left( {X_{{t_{0} }} > L,\,X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {S_{{t_{0} }} X_{{t_{0} }} E^{Q} \left[ {e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} - \tfrac{1}{2}\sigma_{X}^{2} } \right)\left( {T - t_{0} } \right) + \sigma_{X} w_{{T - t_{0} }}^{Q} }} \cdot 1_{{\left( {X_{{t_{0} }} > L} \right)}} 1_{{\left( {\,X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right. \\ &\quad \left. { -\, E^{Q} \left[ {S_{{t_{0} }} X{}_{{t_{0} }}E^{Q} \left[ {e^{{\left[ {r_{d} - \tfrac{1}{2}\left( {\sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2} } \right)} \right]\left( {T - t_{0} } \right) + \sigma_{S} Z_{{T - t_{0} }}^{Q} + \sigma_{X} w_{{T - t_{0} }}^{Q} }} \cdot 1_{{\left( {X_{{t_{0} }} > L} \right)}} 1_{{\left( {\,X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {E^{Q} \left[ {S_{{t_{0} }} X_{{t_{0} }} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)\left( {T - t_{0} } \right)}} 1_{{\left( {X_{{t_{0} }} > L} \right)}} E^{{R_{5} }} \left[ {1_{{\left( {X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right. \\ &\quad \left. { -\, E^{Q} \left[ {S_{{t_{0} }} X{}_{{t_{0} }}e^{{r_{d} \left( {T - t_{0} } \right)}} 1_{{\left( {X_{{t_{0} }} > L} \right)}} E^{{R_{6} }} \left[ {1_{{\left( {X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]\left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {S_{0} X_{0} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)\left( {T - t_{0} } \right)}} E^{Q} \left[ {e^{{\left[ {r_{d} - \tfrac{1}{2}\left( {\sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2} } \right)} \right]t_{0} + \sigma_{S} Z_{{t_{0} }}^{Q} + \sigma_{X} w_{{t_{0} }}^{Q} }} \cdot 1_{{\left( {X_{{t_{0} }} > L} \right)}} \left| {F_{0} } \right.} \right]} \right.E^{{R_{5} }} \left[ {1_{{\left( {X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] \\ & \quad \left. { -\, S_{0} X{}_{0}e^{{r_{d} \left( {T - t_{0} } \right)}} E^{Q} \left[ {e^{{\left[ {r_{d} - \tfrac{1}{2}\left( {\sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2} } \right)} \right]t_{0} + \sigma_{S} Z_{{t_{0} }}^{Q} + \sigma_{X} w_{{t_{0} }}^{Q} }} \cdot 1_{{\left( {X_{{t_{0} }} > L} \right)}} \left| {F_{0} } \right.} \right]E^{{R_{6} }} \left[ {1_{{\left( {X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {S_{0} X_{0} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)\left( {T - t_{0} } \right)}} e^{{r_{d} t_{0} }} E^{{R_{6} }} \left[ {1_{{\left( {X_{{t_{0} }} > L} \right)}} \left| {F_{0} } \right.} \right]} \right.E^{{R_{5} }} \left[ {1_{{\left( {X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right] \\ & \quad \left. { -\, S_{0} X{}_{0}e^{{r_{d} T}} E^{{R_{6} }} \left[ {1_{{\left( {X_{{t_{0} }} > L} \right)}} \left| {F_{0} } \right.} \right]E^{{R_{6} }} \left[ {1_{{\left( {X_{{t_{0} }} > X_{T} } \right)}} \left| {F_{{t_{0} }} } \right.} \right]} \right\}. \\ \end{aligned}$$
(D.2)
The above derivation employs Girsanov theorem to make change of measure from \(Q\) measure into \(R_{5}\) measure and \(R_{6}\) measure25 and (D) in Appendix E. Arranging the terms, we get26
$$RP_{41} = S_{0} \left\{ {X_{0} e^{{ - \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} } \right)\left( {T - t_{0} } \right)}} N\left( {d_{7} } \right)N\left( { - b_{8} } \right) - X{}_{0}N\left( {d_{7} } \right)N\left( { - b_{7} } \right)} \right\}.$$
(D.3)
Next, the second component \(RP_{42}\) is decomposed into two terms. Then we employ Girsanov theorem to make change of measure and use (D) in Appendix E to derive the following:27
$$\begin{aligned} RP_{42} &= e^{{ - r_{d} T}} E^{Q} \left[ {S_{T} \left( {L - X{}_{T}} \right) \cdot 1_{{\left( {X_{{t_{0} }} \le L,\,X_{T} < L} \right)}} \left| {F_{0} } \right.} \right] \\ &= e^{{ - r_{d} T}} \left\{ {LE^{Q} \left[ {S_{T} \cdot 1_{{\left( {X_{{t_{0} }} \le L,\,X_{T} < L} \right)}} \left| {F_{0} } \right.} \right] - E^{Q} \left[ {S_{T} X_{T} \cdot 1_{{\left( {X_{{t_{0} }} \le L,\,X_{T} < L} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} \left\{ {LS_{0} e^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)T}} E^{{R_{5} }} \left[ {1_{{\left( {X_{{t_{0} }} \le L,\,X_{T} < L} \right)}} \left| {F_{0} } \right.} \right] - X_{0} S_{0} e^{{r_{d} T}} E^{{R_{6} }} \left[ {1_{{\left( {X_{{t_{0} }} \le L,\,X_{T} < L} \right)}} \left| {F_{0} } \right.} \right]} \right\} \\ &= e^{{ - r_{d} T}} S_{0} \left\{ {Le^{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} } \right)T}} N\left( { - d_{8} ,\; - \tilde{b}_{8} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - X_{0} e^{{r_{d} T}} N\left( { - d_{7} ,\; - \tilde{b}_{7} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right)} \right\} \\ &= S_{0} \left\{ {Le^{{ - \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} } \right)T}} N\left( { - d_{8} ,\; - \tilde{b}_{8} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - X_{0} N\left( { - d_{7} ,\; - \tilde{b}_{7} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right)} \right\}. \\ \end{aligned}$$
(D.4)
Substituting (D.3) and (D.4) into (D.1), we obtain the pricing model for the Type-4 Quanto reset put option as given below:
$$\begin{aligned} RP_{4} \left( {X_{0} ,\;L,\;0} \right) &= S_{0} \left\{ {X_{0} e^{{ - \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} } \right)\left( {T - t_{0} } \right)}} N\left( {d_{7} } \right)N\left( { - b_{8} } \right) - X{}_{0}N\left( {d_{7} } \right)N\left( { - b_{7} } \right)} \right. \\ &\quad \left. { +\, Le^{{ - \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} } \right)T}} N\left( { - d_{8} ,\; - \tilde{b}_{8} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right) - X_{0} N\left( { - d_{7} ,\; - \tilde{b}_{7} ,\;\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right)} \right\}. \\ \end{aligned}$$
\(\quad \square\)
Appendix E
This Appendix summarizes the necessary mathematical results for the previous proofs. The following conditional expectations are computed using Girsanov theorem:28
$$\begin{aligned} E^{{R_{1} }} \left\{ {1_{{\left[ {S_{{t_{0} }} > K} \right]}} \left| {F_{{t_{0} }} } \right.} \right\} &= \Pr^{{R_{1} }} \left\{ {S_{{t_{0} }} > K\left| {F_{{t_{0} }} } \right.} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad &= \Pr^{Q} \left\{ {\tfrac{{ - z_{{t_{0} }}^{{R_{1} }} }}{{\sqrt {t_{0} } }} < \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }}} \right\} &= N\left( {\tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }}} \right) &= N\left( {d_{1} } \right), \hfill \\ E^{{R_{1} }} \left\{ {1_{{\left[ {S_{{t_{0} }} > S_{T} } \right]}} \left| {F_{{t_{0} }} } \right.} \right\} &= \Pr^{{R_{1} }} \left\{ {S_{{t_{0} }} > S_{T} \left| {F_{{t_{0} }} } \right.} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad = \Pr^{{R_{1} }} \left\{ {\tfrac{{z_{{T - t_{0} }}^{{R_{1} }} }}{{\sqrt {T - t_{0} } }} < - \tfrac{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right\} &= N\left( { - \tfrac{{\left( {r_{f} - \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right) = N\left( { - b_{1} } \right), \hfill \\ E^{Q} \left\{ {1_{{\left[ {S_{{t_{0} }} < K,S_{T} < K} \right]}} } \right\} &= \Pr^{Q} \left\{ {S_{{t_{0} }} < K,S_{T} < K} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad &= \Pr^{Q} \left\{ {\tfrac{{z_{{t_{0} }}^{Q} }}{{\sqrt {t_{0} } }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \rho \sigma_{S} \sigma_{X} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }},\tfrac{{z_{T}^{Q} }}{\sqrt T } < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \rho \sigma_{S} \sigma_{X} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\,T}}{{\sigma_{S} \sqrt T }}} \right\} = N_{2} \left( { - d_{2} , - \tilde{b}_{2} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right), \hfill \\ E^{{R_{1} }} \left\{ {1_{{\left[ {S_{{t_{0} }} < K,S_{T} < K} \right]}} } \right\} &= \Pr^{{R_{1} }} \left\{ {S_{{t_{0} }} < K,S_{T} < K} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad &= \Pr^{{R_{1} }} \left\{ {\tfrac{{z_{{t_{0} }}^{{R_{1} }} }}{{\sqrt {t_{0} } }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }},\tfrac{{z_{T}^{{R_{1} }} }}{\sqrt T } < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,T}}{{\sigma_{S} \sqrt T }}} \right\} = N_{2} \left( { - d_{1} , - \tilde{b}_{1} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right), \hfill \\ \end{aligned}$$
(A)
where we use Givsanov theorem to transform the risk-neutral probability measure \(Q\) into the risk-neutral probability measure \(R_{1}\) and Brownian motion under \(R_{1}\) measure transformed from \(Q\) measure is \(dz_{S}^{Q} = dz_{S}^{{R_{1} }} + \sigma_{S} dt\). And let \(d_{1} \left( {S_{0} ,K,t_{0} } \right) = \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }}\), \(b_{1} = d_{1} \left( {S_{0} ,S_{0} ,T - t_{0} } \right)\), \(\tilde{b}_{1} = d_{1} \left( {S_{0} ,K,T} \right)\), \(d_{2} = d_{1} - \sigma_{S} \sqrt {t_{0} }\), \(b_{2} = b_{1} - \sigma_{S} \sqrt {T - t_{0} }\), \(\tilde{b}_{2} = \tilde{b}_{1} - \sigma_{S} \sqrt T\). \(N\left( \cdot \right)\) and \(N_{2} \left( { \cdot , \cdot } \right)\) represent, respectively, the univariate and bivariate cumulative normal probabilities.
$$\begin{aligned} E^{{R_{2} }} \left\{ {1_{{\left[ {S_{{t_{0} }} > S_{T} } \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{2} }} \left\{ {S_{{t_{0} }} > S_{T} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{2} }} \left\{ {\tfrac{{z_{{T - t_{0} }}^{{R_{2} }} }}{{\sqrt {T - t_{0} } }} < - \tfrac{{\left( {r_{f} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\,\left( {T - t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right\} = N\left( { - \tfrac{{\left( {r_{f} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\left( {T - \,t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right) = N\left( { - b_{4} } \right), \hfill \\ E^{{R_{3} }} \left\{ {1_{{\left[ {S_{{t_{0} }} > K} \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{3} }} \left\{ {S_{{t_{0} }} > K\left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{3} }} \left\{ {\tfrac{{ - z_{{t_{0} }}^{{R_{3} }} }}{{\sqrt {t_{0} } }} < \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }}} \right\} = N\left( {\tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }}} \right) = N\left( {d_{3} } \right), \hfill \\ E^{{R_{3} }} \left\{ {1_{{\left[ {S_{{t_{0} }} > S_{T} } \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{3} }} \left\{ {S_{{t_{0} }} > S_{T} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{3} }} \left\{ {\tfrac{{z_{{T - t_{0} }}^{{R_{3} }} }}{{\sqrt {T - t_{0} } }} < - \tfrac{{\left( {r_{f} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right\} = N\left( { - \tfrac{{\left( {r_{f} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right) = N\left( { - b_{3} } \right), \hfill \\ E^{{R_{2} }} \left\{ {1_{{\left[ {S_{{t_{0} }} < K,S_{T} < K} \right]}} } \right\} = \Pr^{{R_{2} }} \left\{ {S_{{t_{0} }} < K,S_{T} < K} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad = \Pr^{{R_{2} }} \left\{ {\tfrac{{z_{{t_{0} }}^{{R_{2} }} }}{{\sqrt {t_{0} } }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }},\tfrac{{z_{T}^{{R_{2} }} }}{\sqrt T } < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\,T}}{{\sigma_{S} \sqrt T }}} \right\} = N_{2} \left( { - d_{4} , - \tilde{b}_{4} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right), \hfill \\ E^{{R_{3} }} \left\{ {1_{{\left[ {S_{{t_{0} }} < K,S_{T} < K} \right]}} } \right\} = \Pr^{{R_{3} }} \left\{ {S_{{t_{0} }} < K,S_{T} < K} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad = \Pr^{{R_{3} }} \left\{ {\tfrac{{z_{{t_{0} }}^{{R_{3} }} }}{{\sqrt {t_{0} } }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }},\tfrac{{z_{T}^{{R_{3} }} }}{\sqrt T } < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,T}}{{\sigma_{S} \sqrt T }}} \right\} = N_{2} \left( { - d_{3} , - \tilde{b}_{3} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right), \hfill \\ \end{aligned}$$
(B)
where the Brownian motions under \(R_{2}\) and \(R_{3}\) measures transformed from \(Q\) measure are, respectively, given by \(dz_{S}^{Q} = dz_{S}^{{R_{2} }} + \rho \sigma_{X} dt\), \(dw_{X}^{Q} = dw_{X}^{{R_{2} }} + \sigma_{X} dt\), \(dz_{S}^{Q} = dz_{S}^{{R_{3} }} + \left( {\sigma_{S} + \rho \sigma_{X} } \right)dt\), and \(dw_{X}^{Q} = dw_{X}^{{R_{3} }} + \left( {\rho \sigma_{S} + \sigma_{X} } \right)dt\). And let \(d_{3} \left( {S_{0} ,K,t_{0} } \right) = \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{f} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }},\)
\(b_{3} = d_{3} \left( {S_{0} ,S_{0} ,T - t_{0} } \right)\), \(\tilde{b}_{3} = d_{3} \left( {S_{0} ,K,T} \right)\), \(d_{4} = d_{3} - \sigma_{S} \sqrt {t_{0} }\), \(b_{4} = b_{3} - \sigma_{S} \sqrt {T - t_{0} }\), \(\tilde{b}_{4} = \tilde{b}_{3} - \sigma_{S} \sqrt T\).
$$\begin{aligned} E^{Q} \left\{ {1_{{\left[ {X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{Q} \left\{ {X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{Q} \left\{ {\tfrac{{\sigma_{S} z_{{T - t_{0} }}^{Q} + \sigma_{X} w_{{T - t_{0} }}^{Q} }}{{\sigma_{SX} \sqrt {T - t_{0} } }} < - \tfrac{{\left( {r_{d} - \tfrac{1}{2}\sigma_{SX}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{SX} \sqrt {T - t_{0} } }}} \right\} = N\left( { - \tfrac{{\left( {r_{d} - \tfrac{1}{2}\sigma_{SX}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{SX} \sqrt {T - t_{0} } }}} \right) = N\left( { - b_{6} } \right), \hfill \\ E^{{R_{4} }} \left\{ {1_{{\left[ {X_{{t_{0} }} S_{{t_{0} }} > K} \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{4} }} \left\{ {X_{{t_{0} }} S_{{t_{0} }} > K\left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{4} }} \left\{ { - \tfrac{{\sigma_{S} z_{{t_{0} }}^{{R_{4} }} + \sigma_{X} w_{{t_{0} }}^{{R_{4} }} }}{{\sigma_{SX} \sqrt {t_{0} } }} < \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{d} + \tfrac{1}{2}\sigma_{SX}^{2} } \right)\,t_{0} }}{{\sigma_{SX} \sqrt {t_{0} } }}} \right\} = N\left( {\tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{d} + \tfrac{1}{2}\sigma_{SX}^{2} } \right)\,t_{0} }}{{\sigma_{SX} \sqrt {t_{0} } }}} \right) = N\left( {d_{5} } \right), \hfill \\ E^{{R_{4} }} \left\{ {1_{{\left[ {X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} } \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{4} }} \left\{ {X_{{t_{0} }} S_{{t_{0} }} > X_{T} S_{T} \left| {F_{{t_{0} }} } \right.} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad = \Pr^{{R_{4} }} \left\{ {\tfrac{{\sigma_{S} z_{{T - t_{0} }}^{Q} + \sigma_{X} w_{{T - t_{0} }}^{Q} }}{{\sigma_{SX} \sqrt {T - t_{0} } }} < - \tfrac{{\left( {r_{d} + \tfrac{1}{2}\sigma_{SX}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{SX} \sqrt {T - t_{0} } }}} \right\} = N\left( { - \tfrac{{\left( {r_{d} + \tfrac{1}{2}\sigma_{SX}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{SX} \sqrt {T - t_{0} } }}} \right) = N\left( { - b_{5} } \right), \hfill \\ E^{Q} \left\{ {1_{{\left[ {X_{{t_{0} }} S_{{t_{0} }} < K,X_{T} S_{T} < K} \right]}} } \right\} = \Pr^{Q} \left\{ {X_{{t_{0} }} S_{{t_{0} }} < K,X_{T} S_{T} < K} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad = \Pr^{Q} \left\{ {\tfrac{{\sigma_{S} z_{{T - t_{0} }}^{Q} + \sigma_{X} w_{{T - t_{0} }}^{Q} }}{{\sigma_{SX} \sqrt {T - t_{0} } }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{d} - \tfrac{1}{2}\sigma_{SX}^{2} } \right)\,t_{0} }}{{\sigma_{SX} \sqrt {t_{0} } }},\tfrac{{\sigma_{S} z_{T}^{Q} + \sigma_{X} w_{T}^{Q} }}{{\sigma_{SX} \sqrt T }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{d} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\,T}}{{\sigma_{S} \sqrt T }}} \right\} = N_{2} \left( { - d_{6} , - \tilde{b}_{6} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right), \hfill \\ E^{{R_{4} }} \left\{ {1_{{\left[ {X_{{t_{0} }} S_{{t_{0} }} < K,X_{T} S_{T} < K} \right]}} } \right\} = \Pr^{{R_{4} }} \left\{ {X_{{t_{0} }} S_{{t_{0} }} < K,X_{T} S_{T} < K} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad = \Pr^{{R_{4} }} \left\{ {\tfrac{{\sigma_{S} z_{{t_{0} }}^{{R_{4} }} + \sigma_{X} w_{{t_{0} }}^{{R_{4} }} }}{{\sigma_{SX} \sqrt {t_{0} } }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{d} + \tfrac{1}{2}\sigma_{SX}^{2} } \right)\,t_{0} }}{{\sigma_{SX} \sqrt {t_{0} } }},\tfrac{{\sigma_{S} z_{T}^{{R_{4} }} + \sigma_{X} w_{T}^{{R_{4} }} }}{{\sigma_{SX} \sqrt T }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{d} + \tfrac{1}{2}\sigma_{SX}^{2} } \right)\,T}}{{\sigma_{SX} \sqrt T }}} \right\} = N_{2} \left( { - d_{5} , - \tilde{b}_{5} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right), \hfill \\ \end{aligned}$$
(C)
where the Brownian motions under \(R_{4}\) measure transformed from \(Q\) measure are given by \(dz_{S}^{Q} = dz_{S}^{{R_{4} }} + \left( {\sigma_{S} + \rho \sigma_{X} } \right)dt\), and \(dw_{X}^{Q} = dw_{X}^{{R_{4} }} + \left( {\rho \sigma_{S} + \sigma_{X} } \right)dt\). And let \(d_{5} \left( {X_{0} S_{0} ,K,t_{0} } \right) = \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle K$}}} \right) + \left( {r_{d} + \tfrac{1}{2}\sigma_{SX}^{2} } \right)\,t_{0} }}{{\sigma_{SX} \sqrt {t_{0} } }}\), \(b_{5} = d_{5} \left( {X_{0} S_{0} ,X_{0} S_{0} ,T - t_{0} } \right)\), \(\tilde{b}_{5} = d_{5} \left( {X_{0} S_{0} ,K,T} \right)\), \(d_{6} = d_{5} - \sigma_{SX} \sqrt {t_{0} }\), \(b_{6} = b_{5} - \sigma_{SX} \sqrt {T - t_{0} }\), \(\tilde{b}_{6} = \tilde{b}_{5} - \sigma_{SX} \sqrt T\), \(\sigma_{SX}^{2} = \sigma_{S}^{2} + 2\rho \sigma_{S} \sigma_{X} + \sigma_{X}^{2}\).
$$\begin{aligned} E^{{R_{5} }} \left\{ {1_{{\left[ {X_{{t_{0} }} > X_{T} } \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{5} }} \left\{ {X_{{t_{0} }} > X_{T} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{5} }} \left\{ {\tfrac{{w_{{T - t_{0} }}^{{R_{5} }} }}{{\sqrt {T - t_{0} } }} < - \tfrac{{\left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\,\left( {T - t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right\} = N\left( { - \tfrac{{\left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} - \tfrac{1}{2}\sigma_{S}^{2} } \right)\left( {T - \,t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right) = N\left( { - b_{8} } \right), \hfill \\ E^{{R_{6} }} \left\{ {1_{{\left[ {X_{{t_{0} }} > L} \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{6} }} \left\{ {X_{{t_{0} }} > L\left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{6} }} \left\{ {\tfrac{{ - w_{{t_{0} }}^{{R_{6} }} }}{{\sqrt {t_{0} } }} < \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle L$}}} \right) + \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }}} \right\} = N\left( {\tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle L$}}} \right) + \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\,t_{0} }}{{\sigma_{S} \sqrt {t_{0} } }}} \right) = N\left( {d_{7} } \right), \\ E^{{R_{6} }} \left\{ {1_{{\left[ {X_{{t_{0} }} > X_{T} } \right]}} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{6} }} \left\{ {X_{{t_{0} }} > X_{T} \left| {F_{{t_{0} }} } \right.} \right\} = \Pr^{{R_{6} }} \left\{ {\tfrac{{w_{{T - t_{0} }}^{{R_{6} }} }}{{\sqrt {T - t_{0} } }} < - \tfrac{{\left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right\} = N\left( { - \tfrac{{\left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{S}^{2} } \right)\left( {T - t_{0} } \right)}}{{\sigma_{S} \sqrt {T - t_{0} } }}} \right) = N\left( { - b_{7} } \right), \\ E^{{R_{5} }} \left\{ {1_{{\left[ {X_{{t_{0} }} < L,X_{T} < L} \right]}} } \right\} = \Pr^{{R_{5} }} \left\{ {X_{{t_{0} }} < L,X_{T} < L} \right\} \quad = \Pr^{{R_{5} }} \left\{ {\tfrac{{w_{{t_{0} }}^{{R_{5} }} }}{{\sqrt {t_{0} } }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle L$}}} \right) + \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} - \tfrac{1}{2}\sigma_{X}^{2} } \right)\,t_{0} }}{{\sigma_{X} \sqrt {t_{0} } }},\tfrac{{w_{T}^{{R_{5} }} }}{\sqrt T } < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {S_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle L$}}} \right) + \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} - \tfrac{1}{2}\sigma_{X}^{2} } \right)\,T}}{{\sigma_{X} \sqrt T }}} \right\} = N_{2} \left( { - d_{8} , - \tilde{b}_{8} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right), \\ E^{{R_{6} }} \left\{ {1_{{\left[ {X_{{t_{0} }} < L,X_{T} < L} \right]}} } \right\} = \Pr^{{R_{6} }} \left\{ {X_{{t_{0} }} < L,X_{T} < L} \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad = \Pr^{{R_{6} }} \left\{ {\tfrac{{w_{{t_{0} }}^{{R_{6} }} }}{{\sqrt {t_{0} } }} < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle L$}}} \right) + \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{X}^{2} } \right)\,t_{0} }}{{\sigma_{X} \sqrt {t_{0} } }},\tfrac{{w_{T}^{{R_{6} }} }}{\sqrt T } < - \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle L$}}} \right) + \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{X}^{2} } \right)\,T}}{{\sigma_{X} \sqrt T }}} \right\} = N_{2} \left( { - d_{7} , - \tilde{b}_{7} ,\sqrt {{\raise0.5ex\hbox{$\scriptstyle {t_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle T$}}} } \right), \hfill \\ \end{aligned}$$
(D)
where the Brownian motions under \(R_{5}\) and \(R_{6}\) measures transformed from \(Q\) measure are given by \(dz_{S}^{Q} = dz_{S}^{{R_{5} }} + \sigma_{S} dt\), \(dw_{X}^{Q} = dw_{X}^{{R_{5} }} + \rho \sigma_{S} dt\), \(dz_{S}^{Q} = dz_{S}^{{R_{6} }} + \left( {\sigma_{S} + \rho \sigma_{X} } \right)dt\), and \(dw_{X}^{Q} = dw_{X}^{{R_{6} }} + \left( {\rho \sigma_{S} + \sigma_{X} } \right)dt\). And let \(d_{7} \left( {X_{0} ,L,t_{0} } \right) = \tfrac{{\ln \left( {{\raise0.5ex\hbox{$\scriptstyle {X_{0} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle L$}}} \right) + \left( {r_{d} - r_{f} + \rho \sigma_{S} \sigma_{X} + \tfrac{1}{2}\sigma_{X}^{2} } \right)\,t_{0} }}{{\sigma_{X} \sqrt {t_{0} } }},\)
\(b_{7} = d_{7} \left( {X_{0} ,X_{0} ,T - t_{0} } \right)\), \(\tilde{b}_{7} = d_{7} \left( {X_{0} ,L,T} \right)\), \(d_{8} = d_{7} - \sigma_{X} \sqrt {t_{0} }\), \(b_{8} = b_{7} - \sigma_{X} \sqrt {T - t_{0} }\), \(\tilde{b}_{8} = \tilde{b}_{7} - \sigma_{X} \sqrt T\).
