Pricing two-asset alternating barrier options with icicles and their variations

Abstract

This paper introduces a new class of barrier options and its variations. We call the new class of options as two-asset alternating barrier options, since we consider alternating barrier levels for two underlying assets. The alternating barrier levels are placed in the sub-periods of the option’s lifetime; each being applied to one of the two underlying assets. We also consider vertical branches of the barrier, which are termed as icicles. The alternating barrier with icicles can be often seen as an embedded form in various equity-linked financial products. To price such new options, we obtain the joint distribution of two underlying asset prices at an intermediate time point and the maturity, along with their partial maximums under the Black–Scholes model. This joint distribution plays a critical role in the derivation of the pricing formulas for alternating barrier options and their variants. As in ordinary barrier options, we consider eight types of alternating barrier options and derive their explicit option pricing formulas. To our knowledge, the pricing formulas for these options have never been obtained explicitly in the literature even under the Black–Scholes model. We also examine an autocallable equity-linked investment product to derive its explicit pricing formula. Our results are illustrated with numerical examples, showing the effect of different barrier levels and different values of correlation coefficient between two underlying asset prices.

Appendices

Appendix A: Proof of Theorem 3.3

Let us first assume $\rho >0$. Using (13),

$$\begin{aligned} Pr(X_1(t_1)&\le x_{11}, X_1(t_2) \le x_{12}, X_2(t_1) \le x_{21}, X_2(t_2) \le x_{22}, M_2(t_1)>m)\\&=P(Z(t_1) +\rho \frac{\sigma _1}{\sigma _2}X_2(t_1) \le x_{11}, Z(t_2) +\rho \frac{\sigma _1}{\sigma _2}X_2 (t_2) \le x_{12},\\&\qquad X_2(t_1) \le x_{21}, X_2(t_2) \le x_{22}, M_2 (t_1)>m)\\&= E[Pr( \rho \frac{\sigma _1}{\sigma _2}X_2(t_1) \le x_{11}-Z(t_1), X_2(t_1) \le x_{21},\\&\qquad \rho \frac{\sigma _1}{\sigma _2}X_2(t_2) \le x_{12}-Z(t_2),\\&\qquad X_2(t_2) \le x_{22}, M_2 (t_1)>m|Z(t_1), Z(t_2))]\\&=E[Pr(X_2(t_1) \le c_1, X_2(t_2) \le c_2, M_2(t_1)>m|Z(t_1), Z(t_2))], \end{aligned}$$

where $c_1(Z(t_1))=\min \{ \frac{x_{11}-Z(t_1)}{\rho \frac{\sigma _1}{\sigma _2}}, x_{21}\}$ and $c_2(Z(t_2))=\min \{ \frac{x_{12}-Z(t_2)}{\rho \frac{\sigma _1}{\sigma _2}}, x_{22}\}$.

Then using Lemma 3.2, the above formula will be

$$\begin{aligned}&E\bigg [e^{\frac{2 \mu _2}{\sigma _2^2}m}Pr(X_2(t_1) +2m\le c_1(Z(t_1)), \\&\qquad \quad X_2(t_2)+2m \le c_2(Z(t_2))|Z(t_1), Z(t_2))\bigg ]\\ {}&\qquad =e^{\frac{2 \mu _2}{\sigma _2^2}m}Pr(X_2(t_1)+2m \le \frac{x_{11}-Z(t_1)}{\rho \frac{\sigma _1}{\sigma _2}}, X_2(t_1) +2m \le x_{21},\\ {}&\qquad \qquad X_2(t_2)+2m \le \frac{x_{12}-Z(t_2)}{\rho \frac{\sigma _1}{\sigma _2}}, X_2(t_2) +2m \le x_{22} )\\ {}&\qquad =e^{\frac{2 \mu _2}{\sigma _2^2}m}Pr(X_1(t_1) \le x_{11} -2m \rho \frac{\sigma _1}{\sigma _2}, X_1(t_2) \le x_{12} -2m \rho \frac{\sigma _1}{\sigma _2},\\ {}&\qquad \qquad X_2(t_1) \le x_{21}-2m, X_2(t_2) \le x_{22} -2m). \end{aligned}$$

Now, let us assume $\rho <0$. Define $d_1(Z(t_1))=\frac{x_{11}-Z(t_1)}{\rho \frac{\sigma _1}{\sigma _2}}$ and $d_2(Z(t_2))=\frac{x_{12}-Z(t_2)}{\rho \frac{\sigma _1}{\sigma _2}}$ and suppose $d_1(Z(t_1))<x_{21}$ and $d_2(Z(t_2))<x_{22}$ when $Z(t_1)$ and $Z(t_2)$ are given. Then

$$\begin{aligned}&Pr(X_1(t_1) \le x_{11}, X_1(t_2) \le x_{12}, X_2(t_1) \le x_{21}, X_2(t_2) \le x_{22}, M_2(t_1)>m)\\ {}&\quad =E[Pr(d_1(Z(t_1))<X_2(t_1) \le x_{21}, d_2(Z(t_2)) <X_2(t_2) \le x_{22}, \\&\qquad \quad M_2(t_1)>m |Z(t_1), Z(t_2))]\\ {}&\quad =E[Pr(X_2(t_1) \le x_{21}, X_2(t_2) \le x_{22}, M_2(t_1)>m|Z(t_1), Z(t_2))\\ {}&\qquad -Pr(X_2(t_1) \le d_1(Z(t_1)), X_2(t_2) \le x_{22}, M_2(t_1)>m|Z(t_1), Z(t_2))\\ {}&\qquad -Pr(X_2(t_1) \le x_{21}, X_2(t_2) \le d_2(Z(t_2)), M_2(t_1)>m|Z(t_1), Z(t_2))\\ {}&\qquad +Pr(X_2(t_1) \le d_1(Z(t_1)), X_2(t_2) \le d_2(Z(t_2)), M_2(t_1)>m|Z(t_1), Z(t_2))]. \end{aligned}$$

Using Lemma 3.2 for each probability, it turns to

$$\begin{aligned} \begin{aligned}&e^{\frac{2 \mu _2}{\sigma _2^2}m}E[Pr(d_1(Z(t_1)) \le X_2(t_1)+2m \le x_{21}, \\&\quad \qquad d_2(Z(t_2)) \le X_2(t_2)+2m \le x_{22}|Z(t_1), Z(t_2))]\\ {}&=e^{\frac{2 \mu _2}{\sigma _2^2}m}E\left[ Pr\left( \frac{x_{11}-Z(t_1)}{\rho \frac{\sigma _1}{\sigma _2}} \le X_2(t_1)+2m \le x_{21},\right. \right. \\ {}&\qquad \left. \left. \frac{x_{12}-Z(t_2)}{\rho \frac{\sigma _1}{\sigma _2}} \le X_2(t_2)+2m \le x_{22} |Z(t_1), Z(t_2)\right) \right] \\ {}&=e^{\frac{2 \mu _2}{\sigma _2^2}m}Pr(X_1(t_1) \le x_{11}-2m \rho \frac{\sigma _1}{\sigma _2}, X_1(t_2) \le x_{12}-2m \rho \frac{\sigma _1}{\sigma _2},\\ {}&\qquad X_2(t_1) \le x_{21}-2m, X_2(t_2) \le x_{22}-2m). \end{aligned} \end{aligned}$$

(21)

Similarly as seen in the proof of Proposition 3.1, for any element in the set $\{ d_1(Z(t_1))<x_{21}, d_2(Z(t_2))<x_{22}\}^c$,

$$\begin{aligned}&Pr(d_1(Z(t_1)) \le X_2(t_1) \le x_{21}, d_2(Z(t_2)) \le X_2(t_2) \le x_{22}, \\&\qquad M_2(t_1)>m|Z(t_1), Z(t_2))=0 \end{aligned}$$

and (21) is also 0. Combining all of the above, we obtain the desired result.

Appendix B: Proof of Theorem 3.6

For $x_{21} \le m$, $x_{22} \le m$ and $M_2 (t_1, t_2) = \max \{ X_2 (\tau ): t_1 \le \tau \le t_2 \}$,

$$\begin{aligned}&Pr(X_1 (t_1) \le x_{11}, X_1 (t_2) \le x_{12}, X_2 (t_1) \le x_{21}, X_2 (t_2) \le x_{22}, M_2 (t_1, t_2)>m)\nonumber \\&=E[I(X_1 (t_1) \le x_{11}, X_2 (t_1) \le x_{21})\nonumber \\&\qquad \times Pr(X_1(t_2)-X_1 (t_1) \le x_{12} -X_1 (t_1),\nonumber \\&\qquad X_2(t_2) -X_2(t_1) \le x_{22} -X_2 (t_1), \nonumber \\&\qquad M_2 (t_1, t_2)-X_2 (t_1) > m -X_2 (t_1))| X_1(t_1), X_2(t_1))], \end{aligned}$$

(22)

and by Proposition 3.1, the probability inside the expectation in (22) is

$$\begin{aligned}&e^{R_2(m-X_2 (t_1))}Pr(X_1(t_2) -X_1(t_1) \le x_{12}-X_1 (t_1)-2(m-X_2 (t_1))\rho \frac{\sigma _1}{\sigma _2},\\&\quad X_2(t_2)-X_2(t_1) \le x_{22} -X_2 (t_1) -2(m-X_2 (t_1))|X_1 (t_1), X_2(t_1)) \end{aligned}$$

using the notation of $R_2 =\frac{2 \mu _2}{\sigma _2^2}$. Thus, (22) is

$$\begin{aligned}&e^{R_2m}E[e^{-R_2 X_2(t_1)} I(X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}) \\&\qquad \times Pr(X_1(t_2) -X_1(t_1) \le x_{12}-X_1 (t_1)-2(m-X_2 (t_1))\rho \frac{\sigma _1}{\sigma _2},\\&\qquad X_2(t_2)-X_2(t_1) \le x_{22} -X_2 (t_1) -2(m-X_2 (t_1))|X_1 (t_1), X_2(t_1))]\\&\quad =e^{R_2 m} E[I(X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}) \\&\qquad \times Pr(X_1(t_2) -X_1(t_1) \le x_{12}-X_1 (t_1)-2(m-X_2 (t_1))\rho \frac{\sigma _1}{\sigma _2},\\&\qquad X_2(t_2)-X_2(t_1) \le x_{22} -X_2 (t_1) -2(m-X_2 (t_1))|X_1 (t_1), X_2(t_1)); (0, -R_2)']. \end{aligned}$$

Here, we used the factorization formula, (9), and the fact that $E(e^{-R_2 X_2(t_1)})=E(e^{(0, -R_2)' \mathbf{X}(t_1)})=1$. Also, note that the distribution of $(X_1(t_1), X_2(t_1))$ under the Esscher measure of $(0, -R_2)'$ is the same as the distribution of $(X_1^*(t_1), X_2^*(t_1))$ under the original measure by Lemma 3.5. So the above formula turns to

$$\begin{aligned}&e^{R_2 m} E[Pr(X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}, \\ {}&\qquad X_1(t_2) -X_1(t_1) \le x_{12} -X_1 (t_1)-2(m-X_2 (t_1))\rho \frac{\sigma _1}{\sigma _2},\\ {}&\qquad X_2(t_2)-X_2(t_1) \le x_{22} -X_2 (t_1) -2(m-X_2 (t_1))|X_1 (t_1), X_2(t_1)); (0, -R_2)']\\ {}&\quad =e^{R_2 m} E[Pr(X_1^*(t_1) \le x_{11}, X_2^*(t_1) \le x_{21}, \\ {}&\qquad X_1(t_2) -X_1(t_1) \le x_{12}-X_1^*(t_1)-2(m-X_2^*(t_1))\rho \frac{\sigma _1}{\sigma _2},\\ {}&\qquad X_2(t_2)-X_2(t_1) \le x_{22}-X_2^*(t_1)-2(m-X_2^*(t_1))|X_1^*(t_1), X_2^*(t_1))]\\ {}&\quad =e^{R_2 m} Pr(X_1^*(t_1) \le x_{11}, X_2^*(t_1) \le x_{21}, \\ {}&\qquad X_1(t_2) \le x_{12} -2m\rho \frac{\sigma _1}{\sigma _2}, X_2(t_2) \le x_{22}-2m). \end{aligned}$$

This produces the desired result.

Appendix C: Proof of Lemma 3.8

Using $Z^*$ defined after (13),

$$\begin{aligned} \begin{aligned}&Pr(X_1(t_1) \le x_{11}, M_1 (t_1)> m_1, X_2 (t_1) \le x_{21},\\ {}&\qquad X_1 (t_2) -2 \rho \frac{\sigma _1}{\sigma _2}X_2 (t_2) \le x_{12} -2m_2 \rho \frac{\sigma _1}{\sigma _2}, -X_2(t_2) \le x_{22}-2m_2) \\ {}&\quad =Pr\left( X_1(t_1) \le x_{11}, M_1(t_1)> m_1, Z^*(t_1)+\rho \frac{\sigma _2}{\sigma _1}X_1(t_1) \le x_{21}, \right. \\ {}&\qquad X_1(t_2)-2 \rho \frac{\sigma _1}{\sigma _2}(Z^*(t_2)+\rho \frac{\sigma _2}{\sigma _1}X_1(t_2)) \le x_{12}-2m_2 \rho \frac{\sigma _1}{\sigma _2},\\ {}&\qquad \left. -Z^*(t_2) - \rho \frac{\sigma _2}{\sigma _1}X_1(t_2) \le x_{22}-2m_2\right) \\ {}&\quad =E[Pr(X_1(t_1) \le x_{11}, M_1(t_1) > m_1, \rho \frac{\sigma _2}{\sigma _1}X_1(t_1) \le x_{21} -Z^*(t_1),\\ {}&\qquad (1-2 \rho ^2) X_1(t_2) \le x_{12}-2m_2 \rho \frac{\sigma _1}{\sigma _2}+ 2 \rho \frac{\sigma _1}{\sigma _2}Z^*(t_2),\\ {}&\qquad -\rho \frac{\sigma _2}{\sigma _1}X_1(t_2) \le x_{22} -2m_2+Z^*(t_2) | Z^*(t_1), Z^*(t_2))]\\ {}&\quad =:E((A)). \end{aligned} \end{aligned}$$

(23)

We will look at the expectation above, E((A)), in 5 different cases, depending on the value of $\rho$. We will use $c_1(Z^*(t_1))$, $c_2$, $d_1(Z^*(t_2))$, and $d_2(Z^*(t_2))$ defined as follows.

$$\begin{aligned} c_1(Z^*(t_1))&=\frac{x_{21}-Z^*(t_1)}{\rho \frac{\sigma _2}{\sigma _1}}, \qquad \qquad c_2=x_{11}\\ d_1(Z^*(t_2))&=\frac{x_{22}-2m_2+Z^*(t_2)}{-\rho \frac{\sigma _2}{\sigma _1}}, ~d_2(Z^*(t_2))=\frac{x_{12}-2m_2\rho \frac{\sigma _1}{\sigma _2}+2\rho \frac{\sigma _1}{\sigma _2}Z^*(t_2)}{1-2\rho ^2}\\ \end{aligned}$$

• (i) $0< \rho <\frac{1}{\sqrt{2}}$: Assume $d_1 (Z^*(t_2))<d_2(Z^*(t_2))$, since $(A)=0$ when $d_1(Z^*(t_2))>d_2(Z^*(t_2))$.

  $$\begin{aligned} (A)&=Pr(X_1(t_1) \le (c_1(Z^*(t_1)) \wedge c_2), d_1(Z^*(t_2)) \le X_1(t_2) \le d_2(Z^*(t_2)),\\&\qquad M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&=Pr(X_1(t_1) \le (c_1(Z^*(t_1)) \wedge c_2), X_1(t_2) \le d_2(Z^*(t_2)), \\&\qquad M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&\qquad -Pr(X_1(t_1) \le (c_1(Z^*(t_1)) \wedge c_2), X_1(t_2) \le d_1(Z^*(t_2)), \\&\qquad M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2)). \end{aligned}$$

  By Lemma 3.2,

  $$\begin{aligned} (A)&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} [Pr(X_1(t_1) +2m_1 \le (c_1(Z^*(t_1)) \wedge c_2), \\ {}&\qquad X_1(t_2) +2m_1 \le d_2(Z^*(t_2))|Z^*(t_1), Z^*(t_2))\\ {}&\quad -Pr(X_1(t_1) +2m_1 \le (c_1(Z^*(t_1)) \wedge c_2), \\ {}&\qquad X_1(t_2) +2m_1 \le d_1(Z^*(t_2))|Z^*(t_1), Z^*(t_2))]\\ {}&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(X_1(t_1) +2m_1 \le (c_1(Z^*(t_1)) \wedge c_2),\\ {}&\qquad d_1(Z^*(t_2)) \le X_1(t_2) +2m_1 \le d_2(Z^*(t_2))|Z^*(t_1), Z^*(t_2)). \end{aligned}$$

  Then the expectation in (23) is

  $$\begin{aligned} \begin{aligned} E((A))&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr\left. \bigg (X_1(t_1) +2m_1 \le x_{11}, \right. \\ {}&\quad X_1(t_1) +2m_1 \le \frac{x_{21}-X_2(t_1)+\rho \frac{\sigma _2}{\sigma _1}X_1(t_1)}{\rho \frac{\sigma _2}{\sigma _1}},\\ {}&\quad X_1(t_2) +2m_1 \le \frac{x_{12}-2m_2\rho \frac{\sigma _1}{\sigma _2}+2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2)-2 \rho ^2 X_1(t_2)}{1-2\rho ^2},\\ {}&\quad \left. X_1(t_2) +2m_1 \ge \frac{x_{22}-2m_2+X_2(t_2)-\rho \frac{\sigma _2}{\sigma _1}X_1(t_2)}{-\rho \frac{\sigma _2}{\sigma _1}}\right) \\ {}&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(X_1(t_1) +2m_1 \le x_{11}, X_2(t_1)+2m_1 \rho \frac{\sigma _2}{\sigma _1}\le x_{21},\\ {}&\quad X_1(t_2) -2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2) +2m_1(1-2\rho ^2)+2 m_2 \rho \frac{\sigma _1}{\sigma _2}\le x_{12},\\ {}&\quad -X_2(t_2)+2m_2-2m_1 \rho \frac{\sigma _2}{\sigma _1}\le x_{22})\\ {}&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(X_1(t_1) \le x_{11}-2m_1, X_2(t_1) \le x_{21}-2m_1 \rho \frac{\sigma _2}{\sigma _1},\\ {}&\quad X_1(t_2) -2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2) \le x_{12}-2\left( m_2 \rho \frac{\sigma _1}{\sigma _2}-2m_1 \rho ^2+m_1\right) ,\\ {}&\quad -X_2(t_2) \le x_{22}-2\left( m_2-m_1 \rho \frac{\sigma _2}{\sigma _1}\right) ). \end{aligned} \end{aligned}$$

  (24)

• (ii) $\frac{1}{\sqrt{2}}<\rho \le 1$:

  $$\begin{aligned} (A)&=Pr(X_1(t_1) \le (c_1(Z^*(t_1)) \wedge c_2), \\ {}&\quad X_1(t_2) \ge (d_1(Z^*(t_2)) \vee d_2(Z^*(t_2))), M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&=Pr(X_1(t_1) \le (c_1(Z^*(t_1)) \wedge c_2), M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&\quad -Pr(X_1(t_1) \le (c_1(Z^*(t_1)) \wedge c_2), \\&\quad X_1(t_2) \le (d_1(Z^*(t_2)) \vee d_2(Z^*(t_2))), M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} [Pr(X_1(t_1) +2m_1 \le (c_1(Z^*(t_1)) \wedge c_2)|Z^*(t_1), Z^*(t_2))\\ {}&\quad -Pr(X_1(t_1) +2m_1 \le (c_1(Z^*(t_1)) \wedge c_2), \\ {}&\qquad X_1(t_2) +2m_1 \le (d_1(Z^*(t_2)) \vee d_2(Z^*(t_2)))|Z^*(t_1), Z^*(t_2))]\\&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(X_1(t_1) +2m_1 \le (c_1(Z^*(t_1)) \wedge c_2),\\ {}&\qquad X_1(t_2) +2m_1 \ge (d_1(Z^*(t_2)) \vee d_2(Z^*(t_2)))|Z^*(t_1), Z^*(t_2)). \end{aligned}$$

  Then the expectation in (23) is

  $$\begin{aligned} E((A))&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(X_1(t_1) +2m_1 \le x_{11}, \\&\quad X_1(t_1) +2m_1 \le \frac{x_{21}-X_2(t_1)+\rho \frac{\sigma _2}{\sigma _1}X_1(t_1)}{\rho \frac{\sigma _2}{\sigma _1}},\\ {}&\quad X_1(t_2) +2m_1 \ge \frac{x_{12}-2m_2\rho \frac{\sigma _1}{\sigma _2}+2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2)-2 \rho ^2 X_1(t_2)}{1-2\rho ^2},\\ {}&\quad X_1(t_2) +2m_1 \ge \frac{x_{22}-2m_2+X_2(t_2)-\rho \frac{\sigma _2}{\sigma _1}X_1(t_2)}{-\rho \frac{\sigma _2}{\sigma _1}})\\ {}&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(X_1(t_1) \le x_{11}-2m_1, X_2(t_1) \le x_{21}-2m_1 \rho \frac{\sigma _2}{\sigma _1},\\ {}&\qquad X_1(t_2) -2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2) \le x_{12}-2(m_2 \rho \frac{\sigma _1}{\sigma _2}-2m_1 \rho ^2+m_1),\\ {}&\qquad -X_2(t_2) \le x_{22}-2\left( m_2-m_1 \rho \frac{\sigma _2}{\sigma _1}\right) ). \end{aligned}$$

• (iii) $-\frac{1}{\sqrt{2}}<\rho <0$: Assume $c_1(Z^*(t_1)) <c_2$, since $(A)=0$ when $c_1(Z^*(t_1))>c_2$.

  $$\begin{aligned} (A)&=Pr( c_1(Z^*(t_1)) \le X_1(t_1) \le c_2,\\&\qquad X_1(t_2) \le (d_1(Z^*(t_2)) \wedge d_2(Z^*(t_2))), \\&\qquad M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&=Pr(X_1(t_1) \le c_2, X_1(t_2) \le (d_1(Z^*(t_2)) \wedge d_2(Z^*(t_2))), \\&\qquad M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&\qquad -Pr(X_1(t_1) \le c_1(Z^*(t_1)), X_1(t_2) \le (d_1(Z^*(t_2)) \wedge d_2(Z^*(t_2))), \\&\qquad M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} [Pr(X_1(t_1) +2m_1 \le c_2,\\&\qquad X_1(t_2) +2m_1 \le (d_1(Z^*(t_2)) \wedge d_2(Z^*(t_2))) |Z^*(t_1), Z^*(t_2))\\&\qquad -Pr(X_1(t_1) +2m_1 \le c_1(Z^*(t_1)), \\&\qquad X_1(t_2) +2m_1 \le (d_1(Z^*(t_2)) \wedge d_2(Z^*(t_2)))|Z^*(t_1), Z^*(t_2))]\\&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(c_1(Z^*(t_1)) \le X_1(t_1) +2m_1 \le c_2, \\&\qquad X_1(t_2) +2m_1 \le (d_1(Z^*(t_2)) \wedge d_2(Z^*(t_2)))|Z^*(t_1), Z^*(t_2)). \end{aligned}$$

  Then the expectation in (23) is

  $$\begin{aligned} E((A))&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr\left( \frac{x_{21}-X_2(t_1)+\rho \frac{\sigma _2}{\sigma _1}X_1(t_1)}{\rho \frac{\sigma _2}{\sigma _1}} \le X_1(t_1) +2m_1 \le x_{11},\right. \\&\quad X_1(t_2) +2m_1 \le \frac{x_{12}-2m_2\rho \frac{\sigma _1}{\sigma _2}+2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2)-2 \rho ^2 X_1(t_2)}{1-2\rho ^2},\\&\quad \left. X_1(t_2) +2m_1 \le \frac{x_{22}-2m_2+X_2(t_2)-\rho \frac{\sigma _2}{\sigma _1}X_1(t_2)}{-\rho \frac{\sigma _2}{\sigma _1}}\right) \\&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(X_1(t_1) \le x_{11}-2m_1, X_2(t_1) \le x_{21}-2m_1 \rho \frac{\sigma _2}{\sigma _1},\\&\quad X_1(t_2) -2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2) \le x_{12}-2(m_2 \rho \frac{\sigma _1}{\sigma _2}-2m_1 \rho ^2+m_1),\\&\quad -X_2(t_2) \le x_{22}-2\left( m_2-m_1 \rho \frac{\sigma _2}{\sigma _1}\right) . \end{aligned}$$

• (iv) $-1 \le \rho <-\frac{1}{\sqrt{2}}$: Assume $d_1(Z^*(t_2)) >d_2(Z^*(t_2))$, since $(A)=0$ when $d_1(Z^*(t_2))<d_2(Z^*(t_2))$.

  $$\begin{aligned} (A)&=Pr( c_1(Z^*(t_1)) \le X_1(t_1) \le c_2, d_2(Z^*(t_2)) \le X_1(t_2) \le d_1(Z^*(t_2)), \\&\quad M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&=Pr(X_1(t_1) \le c_2, X_1(t_2) \le d_1(Z^*(t_2)), M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&\quad -Pr(X_1(t_1) \le c_1(Z^*(t_1)), X_1(t_2) \le d_1(Z^*(t_2)), \\&\qquad \, M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&\quad -Pr(X_1(t_1) \le c_2, X_1(t_2) \le d_2(Z^*(t_2)), M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&\quad +Pr(X_1(t_1) \le c_1(Z^*(t_1)), X_1(t_2) \le d_2(Z^*(t_2)), \\&\quad M_1(t_1)>m_1|Z^*(t_1), Z^*(t_2))\\&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} [Pr(X_1(t_1) +2m_1 \le c_2,\\&\quad X_1(t_2) +2m_1 \le d_1(Z^*(t_2)) |Z^*(t_1), Z^*(t_2))\\&\qquad -Pr(X_1(t_1) +2m_1 \le c_1(Z^*(t_1)), \\&\quad X_1(t_2) +2m_1 \le d_1(Z^*(t_2))|Z^*(t_1), Z^*(t_2))\\&\qquad -Pr(X_1(t_1) +2m_1 \le c_2, X_1(t_2) +2m_1 \le d_2(Z^*(t_2))|Z^*(t_1), Z^*(t_2))\\&\qquad +Pr(X_1(t_1) +2m_1 \le c_1(Z^*(t_1)), \\&\quad X_1(t_2) +2m_1 \le d_2(Z^*(t_2))|Z^*(t_1), Z^*(t_2))]\\&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(c_1(Z^*(t_1)) \le X_1(t_1) +2m_1 \le c_2, \\&\quad d_2(Z^*(t_2)) \le X_1(t_2) +2m_1 \le d_1(Z^*(t_2))|Z^*(t_1), Z^*(t_2)). \end{aligned}$$

  Then the expectation in (23) is

  $$\begin{aligned} E((A))&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr\left( X_1(t_1) +2m_1 \le x_{11}, \right. \\ X_1(t_1) +2m_1&\ge \frac{x_{21}-X_2(t_1)+\rho \frac{\sigma _2}{\sigma _1}X_1(t_1)}{\rho \frac{\sigma _2}{\sigma _1}}, \\&\quad X_1(t_2) +2m_1 \ge \frac{x_{12}-2m_2\rho \frac{\sigma _1}{\sigma _2}+2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2)-2 \rho ^2 X_1(t_2)}{1-2\rho ^2},\\&\quad \left. X_1(t_2) +2m_1 \le \frac{x_{22}-2m_2+X_2(t_2)-\rho \frac{\sigma _2}{\sigma _1}X_1(t_2)}{-\rho \frac{\sigma _2}{\sigma _1}}\right) \\&=e^{\frac{2\mu _1}{\sigma _1^2}m_1} Pr(X_1(t_1) \le x_{11}-2m_1, X_2(t_1) \le x_{21}-2m_1 \rho \frac{\sigma _2}{\sigma _1},\\&\quad X_1(t_2) -2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2) \le x_{12}-2(m_2 \rho \frac{\sigma _1}{\sigma _2}-2m_1 \rho ^2+m_1),\\&\quad -X_2(t_2) \le x_{22}-2\left( m_2-m_1 \rho \frac{\sigma _2}{\sigma _1}\right) ). \end{aligned}$$

• (v) $\rho =\pm \frac{1}{\sqrt{2}}$: Define

  $$\begin{aligned} G=\left\{ (1-2 \rho ^2) X_1(t_2) \le x_{12}-2m_2 \rho \frac{\sigma _1}{\sigma _2}+ 2 \rho \frac{\sigma _1}{\sigma _2}Z^*(t_2), 1-2\rho ^2=0 \right\} \end{aligned}$$

  Then (A) in (23) is written as the product of the indicator function of G and

  $$\begin{aligned}&Pr(X_1(t_1) \le x_{11}, M_1(t_1) > m_1, \rho \frac{\sigma _2}{\sigma _1}X_1(t_1) \le x_{21} -Z^*(t_1),\\&\quad -\rho \frac{\sigma _2}{\sigma _1}X_1(t_2) \le x_{22} -2m_2+Z^*(t_2) | Z^*(t_1), Z^*(t_2)). \end{aligned}$$

  Using Lemma 3.2 and putting I(G) back into the conditional probability obtained from applying Lemma 3.2, E((A)) can again be expressed as (24).

Since the case of $\rho =0$ is trivial, this completes the proof.

Appendix D: Proof of Theorem 3.9

For $x_{11}<m_1$, $x_{21}<m_2$, and $x_{22} <m_2$,

$$\begin{aligned}&Pr(X_1(t_1) \le x_{11}, X_1(t_2) \le x_{12}, M_1(t_1)>m_1, X_2(t_1) \le x_{21}, \\ {}&\qquad X_2(t_2) \le x_{22}, M_2(t_1, t_2)>m_2)\\ {}&\quad = E[I(X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}, M_1(t_1)>m_1)\\ {}&\qquad \times Pr(X_1(t_2)-X_1(t_1) \le x_{12}-X_1(t_1),\\ {}&\qquad X_2(t_2)-X_2(t_1) \le x_{22}-X_2(t_1), \\ {}&\qquad M_2(t_1, t_2)-X_2(t_1) >m_2 -X_2(t_1)| X_1(t_1), X_2(t_1), M_1(t_1))].\end{aligned}$$

Here, applying Proposition 3.1, the conditional probability above can be written as

$$\begin{aligned}&Pr(X_1(t_2)-X_1(t_1) \le x_{12}-X_1(t_1), X_2(t_2)-X_2(t_1) \le x_{22}-X_2(t_1), \\ {}&\qquad M_2(t_1, t_2)-X_2(t_1) >m_2 -X_2(t_1)| X_1(t_1), X_2(t_1), M_1(t_1))\\ {}&\quad =e^{\frac{2 \mu _2}{\sigma _2^2}(m_2 -X_2(t_1))} \\&\qquad \times Pr(X_1(t_2)-X_1(t_1) \le x_{12}-X_1(t_1)-2(m_2-X_2(t_1))\rho \frac{\sigma _1}{\sigma _2},\\ {}&\qquad X_2(t_2)-X_2(t_1) \le x_{22}-X_2(t_1)-2(m_2-X_2(t_1)) | X_1(t_1), X_2(t_1), M_1(t_1)). \end{aligned}$$

Thus, with $R_2 = \frac{2 \mu _2}{\sigma _2^2}$,

$$\begin{aligned} Pr(X_1(t_1)&\le x_{11}, X_1(t_2) \le x_{12}, M_1(t_1)>m_1, X_2(t_1) \le x_{21}, \\ {}&\quad X_2(t_2) \le x_{22}, M_2(t_1, t_2)>m_2)\\ {}&=e^{R_2 m_2} E[ e^{-R_2 X_2 (t_1)} I( X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}, M_1(t_1)>m_1) \\ {}&\quad \times Pr(X_1(t_2)-X_1(t_1) \le x_{12}-X_1(t_1)-2(m_2 -X_2(t_1))\rho \frac{\sigma _1}{\sigma _2}, \\ {}&\qquad X_2(t_2)-X_2(t_1) \le x_{22}-X_2(t_1)-2(m_2 -X_2(t_1))|X_1(t_1), \\ {}&\qquad X_2(t_1), M_1(t_1))]\\ {}&=e^{R_2 m_2} E[ I( X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}, M_1(t_1)>m_1) \\ {}&\quad \times Pr(X_1(t_2)-X_1(t_1) \le x_{12}-X_1(t_1)-2(m_2 -X_2(t_1))\rho \frac{\sigma _1}{\sigma _2}, \\ {}&\quad X_2(t_2)-X_2(t_1) \le x_{22}-X_2(t_1)-2(m_2 -X_2(t_1))|X_1(t_1), \\ {}&\quad X_2(t_1), M_1(t_1));(0, -R_2)']\\ {}&=e^{R_2 m_2} Pr( X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}, M_1(t_1)>m_1,\\ {}&\quad X_1(t_2)-X_1(t_1)-2\rho \frac{\sigma _1}{\sigma _2}(X_2(t_2)-X_2(t_1)) \le x_{12}\\ {}&\quad -X_1(t_1)-2(m_2 -X_2(t_1))\rho \frac{\sigma _1}{\sigma _2}, \\ {}&\quad -(X_2(t_2)-X_2(t_1)) \le x_{22}-X_2(t_1)-2(m_2 -X_2(t_1));(0, -R_2)')\\ {}&=e^{R_2 m_2} Pr( X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}, M_1(t_1)>m_1, \\ {}&\quad X_1(t_2) -2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2) \le x_{12}-2m_2\rho \frac{\sigma _1}{\sigma _2}, \\ {}&\quad -X_2(t_2) \le x_{22}-2m_2;(0, -R_2)'). \end{aligned}$$

For the third equality, we used the fact that $X_i(t_2)-X_i(t_1)$ under the original measure has the same distribution as $X_i^*(t_2)-X_i^*(t_1)$ under the Esscher measure of $(0, -R_2)'$, $i=1,2$. Note that $X_1$ has the drift $\mu _1 -2 \rho \frac{\sigma _1}{\sigma _2}\mu _2$ under the Esscher measure of $(0, -R_2)'$. By Lemma 3.8,

$$\begin{aligned}&e^{R_2 m_2} Pr( X_1(t_1) \le x_{11}, X_2(t_1) \le x_{21}, M_1(t_1)>m_1, \\ {}&\qquad X_1(t_2) -2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2) \le x_{12}-2m_2\rho \frac{\sigma _1}{\sigma _2}, \\ {}&\qquad -X_2(t_2) \le x_{22}-2m_2;(0, -R_2)')\\ {}&\quad =e^{R_2 m_2+\frac{2(\mu _1-2\rho \frac{\sigma _1}{\sigma _2}\mu _2)}{\sigma _1^2}m_1} Pr(X_1(t_1) \le x_{11}-2m_1,\\ {}&\qquad \quad X_2(t_1) \le x_{21}-2m_1 \rho \frac{\sigma _2}{\sigma _1}, \\ {}&\qquad X_1(t_2) -2\rho \frac{\sigma _1}{\sigma _2}X_2(t_2) \le x_{12}-2\left( m_2 \rho \frac{\sigma _1}{\sigma _2}-2m_1 \rho ^2+m_1\right) ,\\ {}&\qquad -X_2(t_2) \le x_{22}-2\left( m_2-m_1 \rho \frac{\sigma _2}{\sigma _1}\right) ; (0, -R_2)').\end{aligned}$$

Applying Lemma 3.5, we obtain the desired result.

Appendix E: (III) in the ELS price from Sect. 5

Recall that we put $Y_i(t)=-X_i(t)$ for $0 \le t \le T$, $y_{ij}=-x_{ij}$, $m_{y_i}=-m_i$ for $i,j=1, 2$, and $M_{y_i}(s,t)=\max \{ Y_i(\tau ): s \le \tau \le t\}.$ Define an index set, $I=\{(1,1), (1,2), (2,1), (2,2)\}$ and assume that $(i_1, j_1)$, $(i_2, j_2)$, and $(i_3, j_3)$ are distinct elements in I. Then

$$\begin{aligned} (III)&=Pr(M_{y_1}(0,t_1) \le m_{y_1}, M_{y_2}({t_1},{t_2}) \le m_{y_2};\mathbf {h}^*)\\&\quad -Pr((Y_1(t_1) \ge y_{11}, Y_2(t_1) \ge y_{21}, Y_1(t_2) \ge y_{12}, \\&\qquad Y_2(t_2) \ge y_{22})^{c},M_{y_1}(0,t_1) \le m_{y_1}, M_{y_2}({t_1},{t_2}) \le m_{y_2};\mathbf {h}^*)\\ {}&=Pr(M_{y_1}(0,t_1) \le m_{y_1}, M_{y_2}({t_1},{t_2}) \le m_{y_2};\mathbf {h}^*)\\ {}&\quad -\sum _{i=1}^2 \sum _{j=1}^2 Pr(Y_i(t_j) \le y_{ij}, M_{y_1}(0,t_1) \le m_{y_1}, M_{y_2}({t_1},{t_2}) \le m_{y_2}; \mathbf {h}^*)\\ {}&\quad +\sum _{(i_1, j_1), (i_2, j_2) \in I} Pr(Y_{i_1}(t_{j_1}) \le y_{i_1 j_1}, Y_{i_2}(t_{j_2}) \le y_{i_2 j_2}, M_{y_1}(0,t_1) \le m_{y_1}, \\ {}&\quad M_{y_2}({t_1},{t_2}) \le m_{y_2};\mathbf {h}^*)\\ {}&\quad -\sum _{(i_1, j_1), (i_2, j_2), (i_3, j_3) \in I} Pr(Y_{i_1}(t_{j_1}) \le y_{i_1 j_1}, Y_{i_2}(t_{j_2}) \le y_{i_2 j_2}, Y_{i_3}(t_{j_3}) \le y_{i_3 j_3}, \\ {}&\qquad \qquad \qquad \quad M_{y_1}(0,t_1) \le m_{y_1}, M_{y_2}({t_1},{t_2}) \le m_{y_2};\mathbf {h}^*)\\ {}&\quad +Pr(Y_1(t_1) \le y_{11}, Y_2(t_1) \le y_{21}, Y_1(t_2) \le y_{12}, Y_2(t_2) \le y_{22},\\&\quad M_{y_1}(0,t_1) \le m_{y_1}, M_{y_2}({t_1},{t_2}) \le m_{y_2};\mathbf {h}^*)\\ {}&=(i)-(ii)+(iii)-(iv)+(v). \end{aligned}$$

Probabilities, (i) through (v), can be computed using $PA_u$ function defined in Section 4 as follows.

$$\begin{aligned} (i)&=PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -m_1, -m_2, \infty , -m_2, -m_1, -m_2),\\ (ii)&=PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -x_{11}, -m_2, \infty , -m_2, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -m_1, -x_{21}, \infty , -m_2, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -m_1, -m_2, -x_{12}, -m_2, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -m_1, -m_2, \infty , -x_{22}, -m_1, -m_2), \\ (iii)&=PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -x_{11}, -x_{21}, \infty , -m_2, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -x_{11}, -m_2, -x_{12}, -m_2, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -x_{11}, -m_2, \infty , -x_{22}, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -m_1, -x_{21}, -x_{12}, -m_2, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -m_1, -x_{21}, \infty , -x_{22}, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -m_1, -m_2, -x_{12}, -x_{22}, -m_1, -m_2), \\ (iv)&=PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -x_{11}, -x_{21}, -x_{12}, -m_2, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -x_{11}, -x_{21}, \infty , -x_{22}, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -x_{11}, -m_2, -x_{12}, -x_{22}, -m_1, -m_2)\\&\quad +PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -m_1, -x_{21}, -x_{12}, -x_{22}, -m_1, -m_2),\\ (v)&=PA_u(-r+\sigma _1^2/2, -r+\sigma _2^2/2, -x_{11}, -x_{21}, -x_{12}, -x_{22}, -m_1, -m_2). \end{aligned}$$

Appendix F: Partial differential equation for alternating barrier options

Let us express the partial differential equation that governs the alternating barrier option. As we took it as an example in formula (19), let us consider the up and out put option case. PDEs for other types of options can be similarly expressed.

When $\{W_1 (t); t \ge 0\}$ and $\{W_2(t); t\ge 0\}$ are standard Brownian motions with $d<W_1, W_2>_t=\rho dt$, underlying asset price processes, $S_1$ and $S_2$ can be written as follows.

$$\begin{aligned} \frac{dS_1(t)}{S_1(t)}&=(\mu _1 + 0.5 \sigma _1^2)dt+\sigma _1 dW_1(t),\\ \frac{dS_2(t)}{S_2(t)}&=(\mu _2 + 0.5 \sigma _2^2)dt+\sigma _1 dW_2(t).\\ \end{aligned}$$

Denote $V(S_1(t), S_2(t), t)$ is the price of the alternating up-and-out put option at time $t \le T$. And define functions of $B_1(t)$ and $B_2(t)$ as

$$\begin{aligned} B_1(t)=\left\{ \begin{array}{ll} B_1, &{} t \in (0, t_1)\\ B_1 \wedge L_{11}, &{} t=t_1\\ \infty , &{} t \in (t_1, T)\\ L_{12}, &{} t=T \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} B_2(t)=\left\{ \begin{array}{ll} \infty , &{} t \in (0, t_1)\\ L_{21}, &{} t=t_1\\ B_2, &{} t \in (t_1, T)\\ B_2 \wedge L_{22}, &{} t=T \end{array}\right. , \end{aligned}$$

respectively.

Then the Black–Scholes PDE of the alternating barrier up-and-out put option is given by

$$\begin{aligned} rV&=r \frac{\partial V}{\partial S_1} S_1 + r \frac{\partial V}{\partial S_2} S_2 + \frac{1}{2} \frac{\partial ^2 V}{\partial S_1^2}S_1^2 \sigma _1^2+ \frac{1}{2} \frac{\partial ^2 V}{\partial S_2^2}S_2^2 \sigma _2^2 \nonumber \\&\quad +\frac{1}{2} \frac{\partial ^2 V}{\partial S_1 \partial S_2 }S_1 S_2 \sigma _1 \sigma _2 \rho +\frac{\partial V}{\partial t}, \quad S_1(t)< B_1(t), \quad S_2(t) < B_2(t), \end{aligned}$$

(25)

subject to the boundary condition $V(S_1(T), S_2(T),T)=(K-S_1(T))^+$. Here, $B_1$, $B_2$, $L_{11}$, $L_{12}$ are horizontal barriers or icicles as used in Sect. 4, K is the strike price, T is the expiration, and r is the interest rate. The knock-out region for this option can be also seen in Fig. 3.

Fig. 3

Knock-out region for the alternating up-and-out barrier option

By Feynman–Kac theorem, the solution of the PDE (25) will be the expectation of the discounted payoff under the risk neutral measure. The explicit solution should be the same as the option price given in formula (19).