The valuation of forward-start rainbow options

Abstract

This paper studies the valuation and hedging problems of forward-start rainbow options (FSROs). By combining the characteristics of both multiple assets and forward-start feature, this new type of derivative has many potential applications, for instance, to incorporate the reset provision in rainbow options for investors or hedgers or design more effective executive compensation plans. The main contribution of this paper is a novel martingale pricing technique for options whose payoffs are associated with multiple assets and time points. Equipped with this technique, the analytic pricing formula and the formulae of the delta and gamma of the FSRO are first derived. We conduct numerical experiments to verify these formulae and examine the characteristics of the FSRO’s price and Greek letters. To demonstrate the importance and general applicability of the proposed technique, we also apply it to deriving the pricing formula for the discrete-sampling lookback rainbow options.

Appendices

Appendix 1: Proof of Theorem 2.1

The martingale pricing method relies heavily on the Cameron–Martin–Girsanov theorem, which is the key to evaluating the conditional expectation of the price processes with the multivariate standard normal cumulative distribution function (CDF). Therefore, the proof starts with the review of the general $n$-dimensional Cameron-Martin-Girsanov theorem based on the description in Baxter and Rennie (2000), with a slight change in notations.

Cameron–Martin–Girsanov Theorem

Let $\vec {W}^Q(z)=(W_1^Q(z),\ldots ,W_{n}^Q(z))$ be a vector of $n$-dimensional independent $Q$-Brownian motions up to time $T$ adapted with the natural filtration $\mathcal {F}$. Suppose that $\vec {\nu }(z)=(\nu _1(z),\ldots ,\nu _{n}(z))$ is an $\mathcal {F}$-previsible $n$-vector process that satisfies the condition $E^Q[\exp (\frac{1}{2}\int _0^{T}\left| \vec {\nu }^2(\tau )\right| d\tau )]<\infty$. Then there is a new measure $R$ such that

• 1. $Q$ is equivalent to $R$ up to time $T$,

• 2. $\frac{dR}{dQ}=\exp \left( \sum \limits _{j=1}^{n}\int _0^{T}\nu _{j}(\tau )dW_j^Q(\tau )-\frac{1}{2}\int _0^{T}\left| \vec {\nu }(\tau )\right| ^2d\tau \right)$, and

• 3. $W_j^{R}(z)=W_j^Q(z)-\int _0^{z}\nu _{j}(\tau )d\tau ,$ and $\vec {W}^{R}(z)=(W_1^{R}(z),\ldots ,W_{n}^{R}(z))$ is a vector of $n$-dimensional $R$-Brownian motion up to time $T$.

Based upon the general $n$-dimensional Cameron–Martin–Girsanov theorem, the most important finding in this paper is stated in Theorem 2.1.

Theorem 2.1

Let $W_j^Q(z)$and $W_i^Q(z)$ be two correlated $Q$-Brownian motions, and $\rho _{ji}$ is their correlation coefficient. By defining the Radon–Nikodym derivative as $dR_{is}/dQ=\exp (\sigma _iW_i^Q(s) -\sigma _i^2 s/2)$, we can obtain that

$$\begin{aligned} W_j^{R_{is}}(z)=W_j^Q(z)-\rho _{ji}\sigma _i\min (s,z) \end{aligned}$$

is an $R_{is}$-Brownian motion for any time points $s$ and $z$.

Before proving Theorem 2.1, we state a useful identity equation. Suppose $W_j^{P}(z)$ and $W_i^{P}(z)$ are correlated Brownian motions with a correlation coefficient $\rho _{ji}$ under any probability measure $P$. Once we define $\widetilde{W}_{j}^{P}(z)\equiv (W_j^{P}(z)-\rho _{ji}W_i^{P}(z))/\sqrt{1-\rho _{ji}^2}$, then $\widetilde{W}_{j}^{P}(z)$ is an independent $P$-Brownian motion with respect to $W_i^{P}(z)$. Rearranging the above equation, we can obtain

$$\begin{aligned} W_j^{P}(z)=\rho _{ji}W_i^{P}(z)+\sqrt{1-\rho _{ji}^2}\widetilde{W}_{j}^{P}(z), \end{aligned}$$

(6.1)

an equation that is helpful in the following proof.

Proof

Suppose $W_i^Q(z)$ and $\widetilde{W}_{j}^Q(z)$ are two independent $Q$-Brownian motions satisfying Eq. (6.1), that is, $W_j^Q(z)=\rho _{ji}W_i^Q(z)+\sqrt{1-\rho _{ji}^2}\widetilde{W}_{j}^Q(z)$. By considering $\vec {\nu }(z)=(\nu _{i}(z),\nu _{j}(z))=(\sigma _i\cdot I(z \le s),0)$ in the Cameron–Martin–Girsanov theorem, the Radon–Nikodym derivative can be derived as

$$\begin{aligned} \frac{dR_{is}^{{}}}{dQ}&= \exp (\sum \limits _{u=i,j}\int _0^{T}\nu _{u}(\tau )dW_{u}^Q(\tau )-\frac{1}{2}\int _0^{T}\left| \vec {\nu } (\tau )\right| ^2d\tau ) \\&= \exp (\sigma _iW_i^Q(s)-\frac{1}{2}\sigma _i^2s), \end{aligned}$$

and for any time point $z$,

$$\begin{aligned} W_i^{R_{is}}(z)&= W_i^Q(z)-\int _0^{z}\nu _{i}(\tau )d\tau =W_i^Q(z)-\int _0^{z}\sigma _i\cdot I(\tau \le s) d\tau \\&= W_i^Q(z)-\sigma _i\min (s,z), \\ \widetilde{W}_{j}^{R_{is}}(z)&= \widetilde{W}_{j}^Q(z)-\int _0^{z}\nu _{j}(\tau )d\tau =\widetilde{W}_{j}^Q(z). \end{aligned}$$

Finally, we have the following equations and the proof is complete:

$$\begin{aligned} W_j^{R_{is}}(z)&= \rho _{ji}W_i^{R_{is}}(z)+\sqrt{1-\rho _{ji}^2}\widetilde{W}_{j}^{R_{is}}(z)\qquad \qquad (\hbox {by } (6.1)) \\&= \rho _{ji}(W_i^Q(z)-\sigma _i\min (s,z))+\sqrt{1-\rho _{ji}^2}\widetilde{W}_{j}^Q(z) \\&= \rho _{ji}W_i^Q(z)+\sqrt{1-\rho _{ji}^2}\widetilde{W}_{j}^Q(z)-\rho _{ji}\sigma _i\min (s,z) \\&= W_j^Q(z)-\rho _{ji}\sigma _i\min (s,z).\qquad \qquad (\hbox {by } (6.1)). \end{aligned}$$

$\square$

After proving Theorem 2.1, we would like to illustrate the specialty of Theorem 2.1 regarding the time dimension by evaluating $E^Q [S_a(s) \cdot I(S_b(u) \ge S_c (v))]$ (given $u \le s \le v$ for example), which is a problem arising exclusively when pricing FSROs or other path-dependent rainbow options. To evaluate this expectation, we first rewrite it with $S_a (s)=S_a (0)\exp ((r-\sigma _a^2/2)s+\sigma _a W_a^Q(s))$, i.e.,

$$\begin{aligned}&E^Q\left[ S_a(s) \cdot I(S_b(u) \ge S_c(v))\right] \nonumber \\&\quad = S_a(0) e^{rs} E^Q\left[ \exp (\sigma _a W_a^Q(s)-\sigma _a^2 s/2) \cdot I(S_b (u)\ge S_c (v))\right] . \end{aligned}$$

(6.2)

By defining the Radon–Nikodym derivative as $dR_{as}/dQ=\exp (\sigma _a W_a^Q (s) -\sigma _a^2 s/2)$ and applying Theorem 2.1, we can obtain

$$\begin{aligned} W_b^{R_{as}} (u)&= W_b^Q (u)-\rho _{ba} \sigma _a \min (s,u) = W_b^Q (u)-\rho _{ba} \sigma _a u, \end{aligned}$$

(6.3)

$$\begin{aligned} W_c^{R_{as}} (v)&= W_c^Q (v)-\rho _{ca} \sigma _a \min (s,v) = W_c^Q (v)-\rho _{ca} \sigma _a s. \end{aligned}$$

(6.4)

As a result, Eq. (6.2) can be expressed as

$$\begin{aligned}&S_a(0) e^{rs} E^{R_{as}} [I(S_b(u) \ge S_c(v))] \\&\quad = S_a(0) e^{rs} Pr^{R_{as}} (S_b (u) \ge S_c (v)) \\&\quad =S_a(0) e^{rs} Pr^{R_{as}} (\ln S_b (u) \ge \ln S_c (v)) \\&\quad =S_a(0) e^{rs} Pr^{R_{as}} (\ln S_b (0)+(r-\sigma _b^2/2)u+\sigma _b (W_b^{R_{as}}(u)+\rho _{ba} \sigma _a u) \\&\quad \ge \ln S_c(0)+(r-\sigma _c^2/2)v + \sigma _c (W_c^{R_{as}} (v)+\rho _{ca} \sigma _a s)) \end{aligned}$$

The last equation is derived by replacing $W_b^Q (u)$ and $W_c^Q (v)$ according to Eqs. (6.3) and (6.4), respectively. The straightforward algebraic calculation leads to

$$\begin{aligned}&S_a (0) e^{rs} Pr^{R_{as}} \left( \frac{\sigma _c W_c^{R_{as}} (v) - \sigma _b W_b^{R_{as}} (u)}{\sqrt{\text{ var }(\sigma _c W_c^{R_{as}} (v) - \sigma _b W_b^{R_{as}} (u))}} \right. \\&\quad \le \left( \frac{\ln ( S_b(0) /S_c(0)) + r(u-v) + \sigma _c^2 v/2 - \sigma _b^2 u/2 + \rho _{ba} \sigma _b \sigma _a u - \rho _{ca} \sigma _c \sigma _a s}{\sqrt{\text{ var }(\sigma _c W_c^{R_{as}} (v) - \sigma _b W_b^{R_{as}} (u))}}\right) \\&\quad \equiv S_a (0) e^{rs} Pr^{R_{as}} (Z_{bu,cv}^{R_{as}} \le d_{bu,cv}^{R_{as}}) \\&\quad = S_a (0) e^{rs} N(d_{bu,cv}^{R_{as}})\!, \end{aligned}$$

where $\text{ var }(\sigma _c W_c^{R_{as}} (v)-\sigma _b W_b^{R_{as}} (u))$ equals $\sigma _c^2 v - 2\rho _{cb} \sigma _c \sigma _b \min (v,u) + \sigma _b^2 u$ by definition. In addition, since $Z_{bu,cv}^{R_{as}} \equiv \frac{\sigma _c W_c^{R_{as}} (v) - \sigma _b W_b^{R_{as}} (u)}{\sqrt{\text{ var }(\sigma _c W_c^{R_{as}} (v) - \sigma _b W_b^{R_{as}} (u))}}$ follows the standard normal distribution, the last equation in the above can be derived with defining $N(\cdot )$ as the cumulative distribution function of the standard normal distribution.

It should be noted that when we transform $W_b^Q(u)$ to $W_b^{R_{as}} (u)$ in Eq. (6.3), the adjustment term is $\rho _{ba} \sigma _a u$, which is proportional to $u$ as expected. On the other hand, when we transform $W_c^Q (v)$ to $W_c^{R_{as}} (v)$ in Eq. (6.4), although both $W_c^Q (v)$ and $W_c^{R_{as}} (v)$ are Brownian motions evolving up to the time point $v$, the adjustment term is $\rho _{ca} \sigma _a s$ rather than $\rho _{ca} \sigma _a v$ according to Theorem 2.1. In fact, if we apply, without any modification, the commonly used change-of-probability-measure techniques for option pricing, e.g., those in Ouwehand and West (2006) and Liao and Wang (2003), this special characteristic associated with the time dimension cannot be captured such that we cannot derive the correct formula corresponding to $E^Q [S_a(s) \cdot I(S_b(u) \ge S_c(v))]$.

Appendix 2: Derivations of the pricing formula

Theorem 2.1 makes it straightforward to derive the pricing formula for the FSRPO. Recall that in Eq. (2.5) the price today is

$$\begin{aligned}&P(0)=e^{-rT}E^Q\left[ P(T)|\mathcal {F}_0\right] \nonumber \\&\quad = \sum _{m=1}^{n}Ke^{-rT}E^Q\left[ I\left( \left\{ K\ge S_i(t)\right\} _{1\le i\le n},\right. \right. \nonumber \\&\quad \left. \left. \left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},K\ge S_m(T)\right) |\mathcal {F}_0\right] \nonumber \\&\qquad -\,\sum _{m=1}^{n}e^{-rT}E^Q\left[ S_m(T)\cdot I\left( \left\{ K\ge S_i(t)\right\} _{1\le i\le n},\right. \right. \nonumber \\&\qquad \left. \left. \left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},K\ge S_m(T)\right) |\mathcal {F}_0\right] \nonumber \\&\qquad +\,\sum _{1\le m,M\le n}e^{-rT}E^Q\left[ S_{M}(t)\cdot I\left( \left\{ S_{M}(t)\ge S_{it}\right\} _{1\le i\le n},\right. \right. \nonumber \\&\qquad \left. \left. \left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},S_{M}(t)\ge S_m(T)\right) |\mathcal {F}_0\right] \nonumber \\&\qquad -\,\sum _{1\le m,M\le n}e^{-rT}E^Q\left[ S_m(T)\cdot I\left( \left\{ S_{M}(t)\ge S_{it}\right\} _{1\le i\le n},\right. \right. \nonumber \\&\qquad \left. \left. \left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},S_{M}(t)\ge S_m(T)\right) |\mathcal {F}_0\right] , \end{aligned}$$

(7.1)

where $S_{it}\equiv \left\{ \begin{array}{ll} K &{} i=M \\ S_i(t) &{} i\ne M \end{array} \right. .$

We evaluate the option value in Eq. (7.1) term by term in the following paragraphs.

1.1 The first summation in Eq. (7.1)

The expectation inside the first summation in Eq. (7.1) can be evaluated directly in the probability measure $Q$:

$$\begin{aligned}&E^Q\left[ I\left( \left\{ K\ge S_i(t)\right\} _{1\le i\le n},\left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},K\ge S_m(T)\right) |\mathcal {F }_0\right] \\&\quad =Pr^Q\left( \left\{ K\ge S_i(t)\right\} _{1\le i\le n},\left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},K\ge S_m(T)|\mathcal {F}_0\right) \\&\quad \!=\!Pr^Q\left( \left\{ d_{K,it}^Q\!\ge \! Z_{K,it}^Q\right\} _{1\le i\le n},\left\{ d_{jT,mT}^Q\!\ge \! Z_{jT,mT}^Q\right\} _{1\le j\ne m\le n},d_{K,mT}^Q\ge Z_{K,mT}^Q|\mathcal {F}_0\right) \\&\quad =N_{2n}\left( \left\{ d_{K,it}^Q\right\} _{1\le i\le n},\left\{ d_{jT,mT}^Q\right\} _{1\le j\ne m\le n},d_{K,mT}^Q;R^{K,m}\right) \!, \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} d_{K,it}^Q=\frac{\ln (K/S_i(0))-(r-\sigma _i^2/2)t}{\sigma _i\sqrt{t}}, &{} Z_{K,it}^Q=\frac{W_i^Q(t)}{\sqrt{t}}, \\ d_{jT,mT}^Q=\frac{\ln (S_{j}(0)/S_m(0))+(\sigma _m^2-\sigma _j^2)T/2}{\sigma _{jm}\sqrt{T}}, &{} Z_{jT,mT}^Q=\frac{\sigma _mW_m^Q(T)-\sigma _jW_j^Q(T)}{\sigma _{jm}\sqrt{T}}, \\ d_{K,mT}^Q=\frac{\ln (K/S_m(0))-(r-\sigma _m^2/2)T}{\sigma _m\sqrt{T}}, &{} Z_{K,mT}^Q=\frac{W_m^Q(T)}{\sqrt{T}}\!. \end{array} \end{aligned}$$

In the above equations, $\sigma _{ab}$ is defined as $\sqrt{\sigma _{a}^2-2\rho _{ab}\sigma _{a}\sigma _{b}+\sigma _{b}^2}$, and the notation $d_{as,b\tau }^{P}$ is referred to as the parameter of the multivariate standard normal CDF under the event $\{S_{a}(s)\ge S_{b}(\tau )\}$ with respect to any probability measure $P$. The notation $d_{K,b\tau }^{P}$ is defined similarly but corresponding to the event $\{K \ge S_{b}(\tau )\}$.

For the correlation matrix $R^{K,m}$, we first concatenate $\{Z_{K,it}^Q\}_{1\le i\le n}$, $\{Z_{jT,mT}^Q\}_{1 \le j\ne m\le n}$, and $Z_{K,mT}^Q$ to form $\{Z_{p}^{K,m}\}_{1\le p \le 2n}$, i.e., $\{Z_{p}^{K,m}\}_{1 \le p\le 2n} \equiv \{\{Z_{\!K,it}^Q\}_{1\le i\le n},\!\{Z_{\!jT,mT}^Q\}_{1\le j\ne m\le n},Z_{K,mT}^Q\}$. Similarly, $\{Z_{q}^{K,m}\}_{1\le q\le 2n} \!\equiv \! \{\{Z_{\!K,kt}^Q\}_{1\le k\le n}$, $\{Z_{lT,mT}^Q\}_{1\le l\ne m\le n},Z_{K,mT}^Q\}$. In addition, we also define $\rho _{p,q}^{K,m}\equiv \text {corr}(Z_{p}^{K,m},Z_{q}^{K,m})$, and $R^{K,m} \equiv (\rho _{p,q}^{K,m})_{1\le p,q\le 2n}$ is a $2n\times 2n$ correlation coefficient matrix as follows:

$$\begin{aligned} R^{K,m}=\left( \begin{array}{c} (I)_{n\times n} \\ (II)_{(n-1)\times n}^{^{\prime }} \\ (III)_{1\times n}^{^{\prime }} \end{array} \begin{array}{c} (II)_{n\times (n-1)} \\ (IV)_{(n-1)\times (n-1)} \\ (V)_{1\times (n-1)}^{^{\prime }} \end{array} \begin{array}{c} (III)_{n\times 1} \\ (V)_{(n-1)\times 1} \\ 1 \end{array} \right) _{2n\times 2n}\!, \end{aligned}$$

where

$$\begin{aligned} (I)_{n\times n}=\left( \text {corr}(Z_{K,it}^Q,Z_{K,kt}^Q)\right) _{\begin{array}{c} 1\le i\le n \\ 1\le k\le n \end{array}}&= \left( \text {corr}\left( \frac{W_i^Q(t)}{\sqrt{t}},\frac{W_{k}^Q(t)}{\sqrt{t}}\right) \right) _{\begin{array}{c} 1\le i\le n \\ 1\le k\le n \end{array}} \\&= \left( \rho _{ik}\right) _{\begin{array}{c} 1\le i\le n \\ 1\le k\le n \end{array}}\!, \end{aligned}$$

and other parts listed below can be computed similarly:

$$\begin{aligned} (II)_{n\times (n-1)}&= \left( \text {corr}(Z_{K,it}^Q,Z_{lT,mT}^Q)\right) _{\begin{array}{c} 1\le i\le n \\ 1\le l\ne m\le n \end{array}} \!=\!\left( \frac{\rho _{im}\sigma _m\!-\!\rho _{il}\sigma _{l}}{\sigma _{lm}}\sqrt{\frac{t}{T}}\right) _{\begin{array}{c} 1\le i\le n \\ 1\le l\ne m\le n \end{array}}, \\ (III)_{n\times 1}&= \left( \text {corr}(Z_{K,it}^Q,Z_{K,mT}^Q)\right) _{1\le i\le n}=\left( \rho _{im}\sqrt{\frac{t}{T}}\right) _{1\le i\le n}, \\ (IV)_{(n-1)\times (n-1)}&= \left( \text {corr}(Z_{jT,mT}^Q,Z_{lT,mT}^Q)\right) _{\begin{array}{c} 1\le l\ne m\le n \\ 1\le l\ne m\le n \end{array}} \\&= \left( \frac{\sigma _m^2-\rho _{lm}\sigma _m\sigma _{l}-\rho _{jm}\sigma _j\sigma _m+\rho _{jl}\sigma _j\sigma _{l}}{\sigma _{jm}\sigma _{lm}}\right) _{\begin{array}{c} 1\le j\ne m\le n \\ 1\le l\ne m\le n \end{array}}, \\ (V)_{(n-1)\times 1}&= \left( \text {corr}(Z_{jT,mT}^Q,Z_{K,mT}^Q)\right) _{1\le l\ne m\le n} =\left( \frac{\sigma _m-\rho _{jm}\sigma _j}{\sigma _{jm}}\right) _{1\le j\ne m\le n}\!. \end{aligned}$$

1.2 The second summation in Eq. (7.1)

Under the risk-neutral measure $Q$, the underlying price of asset $m$ at time $T$ is

$$\begin{aligned} S_m(T)=S_m(0)\exp \left( (r-\sigma _m^2/2)T+\sigma _mW_m^Q(T)\right) . \end{aligned}$$

It is convenient to introduce the probability measure $R_{mT}$ by setting the corresponding Radon–Nikodym derivative to

$$\begin{aligned} \frac{dR_{mT}}{dQ}=\exp \left( \sigma _mW_m^Q(T)-\sigma _m^2T/2\right) . \end{aligned}$$

With Theorem 2.1, $W_{j}^{R_{mT}}(z)=W_{j}^Q(z )-\rho _{jm}\sigma _m\min (z ,T)=W_{j}^Q(z )-\rho _{jm}\sigma _m z$ is a standard Brownian motion under the probability measure $R_{mT}$, where $j$ is the index for the underlying asset. Then we can rewrite the expectation inside the second summation in Eq. (7.1) evaluated with respect to the $R_{mT}$ measure as follows:

$$\begin{aligned}&E^Q\left[ S_m(T)\cdot I\left( \left\{ K\ge S_i(t)\right\} _{1\le i\le n},\left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},K\ge S_m(T)\right) |\mathcal {F}_0\right] \\&\quad =S_m(0)e^{rT}\,\,E^{R_{mT}}\left[ I\left( \left\{ K\ge S_i(t)\right\} _{1\le i\le n},\left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},\right. \right. \\&\quad \left. \left. K\ge S_m(T)\right) |\mathcal {F}_0\right] \\&\quad =S_m(0)e^{rT}\,\,Pr^{R_{mT}}\left( \left\{ d_{K,it}^{R_{mT}}\ge Z_{K,it}^{R_{mT}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{mT}}\ge Z_{jT,mT}^{R_{mT}}\right\} _{1\le j\ne m\le n},\right. \\&\quad \left. d_{K,mT}^{R_{mT}}\ge Z_{K,mT}^{R_{mT}}|\mathcal {F}_0\right) \\&\quad =S_m(0)e^{rT}\,\,N_{2n}\left( \left\{ d_{K,it}^{R_{mT}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{mT}}\right\} _{1\le j\ne m\le n},d_{K,mT}^{R_{mT}};\,\,R^{K,m}\right) , \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} d_{K,it}^{R_{mT}}=\frac{\ln (K/S_i(0))-(r-(\sigma _{im}^2-\sigma _m^2)/2)t}{\sigma _i\sqrt{t}}, &{} Z_{K,it}^{R_{mT}}=\frac{W_i^{R_{mT}}(t)}{\sqrt{t}}, \\ d_{jT,mT}^{R_{mT}}=\frac{\ln (S_{j}(0)/S_m(0))-\sigma _{jm}^2T/2}{\sigma _{jm}\sqrt{T}}, &{} Z_{jT,mT}^{R_{mT}}=\frac{\sigma _mW_m^{R_{mT}}(T)-\sigma _jW_j^{R_{mT}}(T)}{\sigma _{jm}\sqrt{T}}, \\ d_{K,mT}^{R_{mT}}=\frac{\ln (K/S_m(0))-(r+\sigma _m^2/2)T}{\sigma _m\sqrt{T}}, &{} Z_{K,mT}^{R_{mT}}=\frac{W_m^{R_{mT}}(T)}{\sqrt{T}}. \end{array} \end{aligned}$$

It is worth noting that $Z_{\bullet ,\bullet }^{R_{mT}}$ are essentially the same as $Z_{\bullet ,\bullet }^Q$ in the first summation of Eq. (7.1) except under a different probability measure. Hence for the expectation inside the second summation, the correlation matrix of the multivariate standard normal CDF is the same as $R^{K,m}$ in the first summation in Eq. (7.1).

1.3 The third summation in Eq. (7.1)

Following the same technique, we compute the expectation inside the third summation in Eq. (7.1). The underlying price of asset $M$ at time $t$ under the risk-neutral measure $Q$ is

$$\begin{aligned} S_{M}(t)=S_{M}(0)\exp \left( (r-\sigma _{M}^2/2)t+\sigma _{M}W_{M}^Q(t)\right) . \end{aligned}$$

Let $R_{Mt}$ be the equivalent probability measure defined by

$$\begin{aligned} \frac{dR_{Mt}}{dQ}=\exp \left( \sigma _{M}W_{M}^Q(t)-\sigma _{M}^2t/2\right) . \end{aligned}$$

According to Theorem 2.1, $W_{j}^{R_{Mt}}(z )=W_{j}^Q(z)-\rho _{jM}\sigma _{M}\min (z ,t)$ is a standard Brownian motion under the probability measure $R_{Mt}$, where $j$ is the index for the underlying asset. Note that if the examined time point $z$ is before $t$, $W_{j}^{R_{Mt}}(z )=W_{j}^Q(z)-\rho _{jM}\sigma _{M} z$; if the examined time point $z$ is after $t$, $W_{j}^{R_{Mt}}(z )=W_{j}^Q(z)-\rho _{jM}\sigma _{M} t$ even though both $W_{j}^{R_{Mt}}(z)$ and $W_{j}^{Q}(z)$ evolve up to $z$. The expectation inside the third summation in Eq. (7.1) evaluated with respect to the $R_{Mt}$ measure is as follows:

$$\begin{aligned}&E^Q\left[ S_{M}(t)\cdot I\left( \left\{ S_{M}(t)\ge S_{it}\right\} _{1\le i\le n},\left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},\right. \right. \nonumber \\&\quad \left. \left. S_{M}(t)\ge S_m(T)\right) |\mathcal {F}_0\right] \\&\quad =S_{M}(0)e^{rt}\,\,Pr^{R_{Mt}}\left( \left\{ d_{it}^{R_{Mt}}\ge Z_{it}^{R_{Mt}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{Mt}}\ge Z_{jT,mT}^{R_{Mt}}\right\} _{1\le j\ne m\le n},\right. \nonumber \\&\quad \left. d_{Mt,mT}^{R_{Mt}}\ge Z_{Mt,mT}^{R_{Mt}}|\mathcal {F}_0\right) \\&\quad =S_{M}(0)e^{rt}\,\,N_{2n}\left( \left\{ d_{it}^{R_{Mt}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{Mt}}\right\} _{1\le j\ne m\le n},d_{Mt,mT}^{R_{Mt}};\,\,R^{M,m}\right) , \end{aligned}$$

where

$$\begin{aligned}&d_{it}^{R_{Mt}} =\left\{ \begin{array}{ll} d_{Mt,K}^{R_{Mt}}=\frac{\ln (S_{M}(0)/K)+(r+\sigma _{M}^2/2)t}{\sigma _{M} \sqrt{t}} &{} i=M \\ d_{Mt,it}^{R_{Mt}}=\frac{\ln (S_{M}(0)/S_i(0))+\sigma _{iM}^2t/2}{\sigma _{iM}\sqrt{t}} &{} i\ne M \end{array}\right. , \\&Z_{it}^{R_{Mt}} =\left\{ \begin{array}{ll} Z_{Mt,K}^{R_{Mt}}=-\frac{W_{M}^{R_{Mt}}(t)}{\sqrt{t}} &{} i=M \\ Z_{Mt,it}^{R_{Mt}}=\frac{\sigma _{i}W_i^{R_{Mt}}(t)-\sigma _{M}W_{M}^{R_{Mt}}(t)}{\sigma _{iM}\sqrt{t}}, &{} i\ne M \end{array}\right. ,\\&\begin{array}{ll} d_{jT,mT}^{R_{Mt}}=\frac{\ln (S_{j}(0)/S_m(0))+(\sigma _{Mt,mT}^2-\sigma _{Mt,jT}^2)/2}{\sigma _{jm}\sqrt{T}}, &{} Z_{jT,mT}^{R_{Mt}}=\frac{\sigma _mW_m^{R_{Mt}}(T)-\sigma _jW_j^{R_{Mt}}(T)}{\sigma _{jm}\sqrt{T}}, \\ d_{Mt,mT}^{R_{Mt}}=\frac{\ln (S_{M}(0)/S_m(0))+r(t-T)+\sigma _{Mt,mT}^2/2}{\sigma _{Mt,mT}}, &{} Z_{Mt,mT}^{R_{Mt}}=\frac{\sigma _mW_m^{R_{Mt}}(T)-\sigma _{M}W_{M}^{R_{Mt}}(t)}{\sigma _{Mt,mT}}, \end{array} \end{aligned}$$

and the notation $\sigma _{as,b\tau }$ is defined as $\sqrt{\sigma _{a}^2s-2\rho _{ab}\sigma _{a}\sigma _{b}\min (s,\tau )+\sigma _{b}^2\tau }$.

Next, we define $\{Z_{p}^{M,m}\}_{1 \le p\le 2n}\equiv \{\{Z_{it}^{R_{Mt}}\}_{1\le i\le n},\{Z_{jT,mT}^{R_{Mt}} \}_{1\le j\ne m\le n},Z_{Mt,mT}^{R_{Mt}}\}$, $\{Z_{q}^{K,m}\}_{1\le q\le 2n} \equiv \{\{Z_{kt}^{R_{Mt}}\}_{1\le i\le n},\{Z_{lT,mT}^{R_{Mt}}\}_{1\le j\ne m\le n},Z_{Mt,mT}^{R_{Mt}}\}$, $\rho _{p,q}^{M,m} \equiv \text {corr}(Z_{p}^{M,m},Z_{q}^{M,m})$, and $R^{M,m} \equiv (\rho _{p,q}^{M,m})_{1\le p,q\le 2n}$ as a $2n\times 2n$ correlation coefficient matrix. For simplicity, we abuse the notation slightly and set $\sigma _i=0$ when $i=M$. Consequently, as $i=M$, $\sigma _{iM}=\sqrt{\sigma _{i}^2-2\rho _{iM}\sigma _i\sigma _{M}+\sigma _{M}^2}=\sqrt{\sigma _{M}^2}=\sigma _{M}$. Then the correlation coefficient matrix $R^{M,m}$ is presented as follows:

$$\begin{aligned} R^{M,m}=\left( \begin{array}{ccc} (VI)_{n\times n} &{} (VII)_{n\times (n-1)}&{}(VIII)_{n\times 1} \\ (VII)_{(n-1)\times n}^{^{\prime }}&{} (IX)_{(n-1)\times (n-1)} &{} (X)_{(n-1)\times 1}\\ (VIII)_{1\times n}^{^{\prime }}&{} (X)_{1\times (n-1)}^{^{\prime }} &{} 1\\ \end{array}\right) _{2n\times 2n}, \end{aligned}$$

where

$$\begin{aligned} (VI)_{n\times n}&= \left( \text {corr}(Z_{it}^{R_{Mt}},Z_{kt}^{R_{Mt}})\right) _{\begin{array}{c} 1\le i\le n \\ 1\le k\le n \end{array}} \\&= \left( \frac{\rho _{ik}\sigma _i\sigma _{k}-\rho _{kM}\sigma _{k}\sigma _{M}-\rho _{iM}\sigma _i\sigma _{M}+\sigma _{M}^2}{\sigma _{iM}\,\, \sigma _{kM}}\right) _{\begin{array}{c} 1\le i\le n \\ 1\le k\le n \end{array}},\\ (VII)_{n\times (n-1)}&= \left( \text {corr}(Z_{it}^{R_{Mt}},Z_{lT,mT}^{R_{Mt}})\right) _{\begin{array}{c} 1\le i\le n \\ 1\le l\ne m\le n \end{array}} \\&= \left( \frac{\rho _{im}\sigma _i\sigma _m-\rho _{il}\sigma _i\sigma _{l}-\rho _{mM}\sigma _m\sigma _{M}+\rho _{lM}\sigma _{l}\sigma _{M}}{\sigma _{iM}\,\,\sigma _{lm}}\sqrt{\frac{t}{T}}\right) _{\begin{array}{c} 1\le i\le n \\ 1\le l\ne m\le n \end{array}}, \\ (VIII)_{n\times 1}&= \left( \text {corr}(Z_{it}^{R_{Mt}},Z_{Mt,mT}^{R_{Mt}})\right) _{1\le i\le n} \\&= \left( \frac{\rho _{im}\sigma _i\sigma _m-\rho _{iM}\sigma _i\sigma _{M}-\rho _{mM}\sigma _m\sigma _{M}+\sigma _{M}^2}{\sigma _{iM}\,\,\sigma _{Mt,mT}}\sqrt{t}\right) _{1\le i\le n}, \\ (IX)_{(n-1)\times (n-1)}&= \left( \text {corr}(Z_{jT,mT}^{R_{Mt}},Z_{lT,mT}^{R_{Mt}})\right) _{\begin{array}{c} 1\le l\ne m\le n \\ 1\le l\ne m\le n \end{array}} \\&= \left( \frac{\sigma _m^2-\rho _{jm}\sigma _j\sigma _m-\rho _{lm}\sigma _{l}\sigma _m+\rho _{jl}\sigma _j\sigma _{l}}{\sigma _{jm}\,\,\sigma _{lm}}\right) _{\begin{array}{c} 1\le j\ne m\le n \\ 1\le l\ne m\le n \end{array}}, \\ (X)_{(n-1)\times 1}&= \left( \text {corr}(Z_{jT,mT}^{R_{Mt}},Z_{Mt,mT}^{R_{Mt}})\right) _{1\le l\ne m\le n} \\&= \left( \frac{\sigma _m^2T-\rho _{mM}\sigma _m\sigma _{M}t-\rho _{jm}\sigma _j\sigma _mT+\rho _{jM}\sigma _j\sigma _{M}t}{\sigma _{jm} \sqrt{T}\,\,\sigma _{Mt,mT}}\right) _{1\le j\ne m\le n}. \end{aligned}$$

1.4 The fourth summation in Eq. (7.1)

The probability measure $R_{mT}$ used for the expectation inside the fourth summation in Eq. (7.1) is the same as that in the second summation of Eq. (7.1). In addition, comparing with the derivation details of the third summation, it can be easily shown that $Z_{\bullet ,\bullet }^{R_{mT}}$ are in essence the same as $Z_{\bullet ,\bullet }^{R_{Mt}}$ except under a different probability measure. Hence the correlation matrix for the multivariate standard normal CDF inside the fourth summation will be $R^{M,m}$. Consequently, the expectation inside the fourth summation in Eq. (7.1) is evaluated as follows:

$$\begin{aligned}&E^Q\left[ S_m(T)\cdot I\left( \left\{ S_{M}(t)\ge S_{it}\right\} _{1\le i\le n},\left\{ S_{j}(T)\ge S_m(T)\right\} _{1\le j\ne m\le n},\right. \right. \\&\quad \left. \left. S_{M}(t)\ge S_m(T)\right) |\mathcal {F}_0\right] \\&\quad =S_m(0)e^{rT}\,\,Pr^{R_{mT}}\left( \left\{ d_{it}^{R_{mT}}\ge Z_{it}^{R_{mT}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{mT}}\ge Z_{jT,mT}^{R_{mT}}\right\} _{1\le j\ne m\le n},\right. \\&\quad \left. d_{Mt,mT}^{R_{mT}}\ge Z_{Mt,mT}^{R_{mT}}|\mathcal {F}_0\right) \\&\quad =S_m(0)e^{rT}\,\,N_{2n}\left( \left\{ d_{it}^{R_{mT}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{mT}}\right\} _{1\le j\ne m\le n},d_{Mt,mT}^{R_{mT}};\,\,R^{M,m}\right) , \end{aligned}$$

where

$$\begin{aligned}&d_{it}^{R_{mT}} =\left\{ \begin{array}{ll} d_{Mt,K}^{R_{mT}}=\frac{\ln (S_{M}(0)/K)+(r-(\sigma _{Mm}^2-\sigma _m^2)/2)t}{\sigma _{M}\sqrt{t}} &{} i=M \\ d_{Mt,it}^{R_{mT}}=\frac{\ln (S_{M}(0)/S_i(0))+(\sigma _{im}^2-\sigma _{Mm}^2)t/2}{\sigma _{iM}\sqrt{t}} &{} i\ne M \end{array}\right. , \\&Z_{it}^{R_{mT}} =\left\{ \begin{array}{ll} Z_{Mt,K}^{R_{mT}}=-\frac{W_{M}^{R_{mT}}(t)}{\sqrt{t}} &{} i=M \\ Z_{Mt,it}^{R_{mT}}=\frac{\sigma _{i}W_i^{R_{mT}}(t)-\sigma _{M}W_{M}^{R_{mT}}(t)}{\sigma _{iM}\sqrt{t}} &{} i\ne M \end{array} \right. ,\\&\begin{array}{ll} d_{jT,mT}^{R_{mT}}=\frac{\ln (S_{j}(0)/S_m(0))-\sigma _{jm}^2T/2}{\sigma _{jm}\sqrt{T}}, &{} Z_{jT,mT}^{R_{mT}}=\frac{\sigma _mW_m^{R_{mT}}(T)-\sigma _jW_j^{R_{mT}}(T)}{\sigma _{jm}\sqrt{T}},\\ d_{Mt,mT}^{R_{mT}}=\frac{\ln (S_{M}(0)/S_m(0))+r(t-T)-\sigma _{Mt,mT}^2/2}{\sigma _{Mt,mT}}, &{} Z_{Mt,mT}^{R_{mT}}=\frac{\sigma _mW_m^{R_{mT}}(T)-\sigma _{M}W_{M}^{R_{mT}}(t)}{\sigma _{Mt,mT}}. \end{array} \end{aligned}$$

As a result, the pricing formula of the forward-start put option can be summarized as follows.

$$\begin{aligned}&\sum _{m=1}^{n}Ke^{-rT}N_{2n}\left( \left\{ d_{K,it}^Q\right\} _{1\le i\le n},\left\{ d_{jT,mT}^Q\right\} _{1\le j\ne m\le n},d_{K,mT}^Q;R^{K,m}\right) \\&\quad -\,\sum _{m=1}^{n}S_m(0)\,\,N_{2n}\left( \left\{ d_{K,it}^{R_{mT}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{mT}}\right\} _{1\le j\ne m\le n},d_{K,mT}^{R_{mT}};\,\, R^{K,m}\right) \\&\quad +\,\sum _{1\le m,M\le n}S_{M}(0)e^{-r(T-t)}\,\,N_{2n}\left( \{d_{it}^{R_{Mt}} \}_{1\le i\le n},\left\{ d_{jT,mT}^{R_{Mt}}\right\} _{1\le j\ne m\le n},d_{Mt,mT}^{R_{Mt}};\,\,R^{M,m}\right) \\&\quad -\,\sum _{1\le m,M\le n}S_m(0)\,\,N_{2n}\left( \left\{ d_{it}^{R_{mT}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{mT}}\right\} _{1\le j\ne m\le n},d_{Mt,mT}^{R_{mT}}; \,\,R^{M,m}\right) \!. \end{aligned}$$

Appendix 3: Delta and gamma

The delta of an option can be directly derived by differentiating the pricing formula with respect to the initial price of the underlying asset. However, here we employ an alternative method that uses the linear homogeneity of the option pricing formula to derive the formula for delta. For any linearly homogeneous function, that is, $f(\lambda x,\lambda y)=\lambda f(x,y)$, the Euler’s rule implies

$$\begin{aligned} x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=f(x,y). \end{aligned}$$

To simplify the expression, we use $S_i$ as shorthand for $S_i(0)$ hereafter. Since the general pricing formula (2.6) is linearly homogeneous, that is,

$$\begin{aligned} P(\lambda S_1,\lambda S_2,\ldots ,\lambda S_n,\lambda K)=\lambda P(S_1,S_2,\ldots ,S_n,K), \end{aligned}$$

by the Euler’s rule the equation can be written as

$$\begin{aligned} P(S_1,S_2,\ldots ,S_n,K)=\sum _{a=1}^{n}S_{a}\frac{\partial P}{\partial S_{a}}+K\frac{\partial P}{\partial K}=S_1\frac{\partial P}{\partial S_1} +\cdots +S_n\frac{\partial P}{\partial S_n}+K\frac{\partial P}{\partial K}. \end{aligned}$$

To obtain the delta with respect to the $a$-th asset, $\frac{\partial P}{\partial S_{a}}$, we need to collect all the terms that contain $S_{a}$. Recall that the pricing formula of $P(0)$ in Eq. (2.6) is

$$\begin{aligned}&\sum _{m=1}^{n}Ke^{-rT}N_{2n}\left( \left\{ d_{K,it}^Q\right\} _{1\le i\le n},\left\{ d_{jT,mT}^Q\right\} _{1\le j\ne m\le n},d_{K,mT}^Q;R^{K,m}\right) \nonumber \\&\quad -\,\sum _{m=1}^{n}S_mN_{2n}\left( \left\{ d_{K,it}^{R_{mT}}\right\} _{1\le i\le n},\{d_{jT,mT}^{R_{mT}}\}_{1\le j\ne m\le n},d_{K,mT}^{R_{mT}};\,\, R^{K,m}\right) \nonumber \\&\quad +\,\sum _{1\le m,M\le n}S_{M}e^{-r(T-t)}\,\,N_{2n}\left( \left\{ d_{it}^{R_{Mt}} \right\} _{1\le i\le n},\{d_{jT,mT}^{R_{Mt}}\}_{1\le j\ne m\le n},d_{Mt,mT}^{R_{Mt}};\,\,R^{M,m}\right) \nonumber \\&\quad -\,\sum _{1\le m,M\le n}S_mN_{2n}\left( \left\{ d_{it}^{R_{mT}}\right\} _{1\le i\le n},\left\{ d_{jT,mT}^{R_{mT}}\right\} _{1\le j\ne m\le n},d_{Mt,mT}^{R_{mT}}; \,\,R^{M,m}\right) \!. \end{aligned}$$

(8.1)

Inside the above formula, $S_{a}$ appears in the terms of the second and fourth summations when $m$ is equal to $a$ and in the third summation when $M$ is equal to $a$. The term that contains $S_{a}$ in the second summation of the formula (8.1) is

$$\begin{aligned} -S_{a}N_{2n}\left( \left\{ d_{K,it}^{R_{aT}}\right\} _{1\le i\le n},\left\{ d_{jT,aT}^{R_{aT}}\right\} _{1\le j\ne a\le n},d_{K,aT}^{R_{aT}};R^{K,a}\right) \!. \end{aligned}$$

Similarly, the terms with $S_{a}$ in the third summation of the formula (8.1) can be grouped as follows.

$$\begin{aligned} \sum _{a^{\prime }=1}^{n}S_{a}e^{-r(T-t)}N_{2n}\left( \left\{ d_{it}^{R_{at}}\right\} _{1\le i\le n},\left\{ d_{jT,a^{\prime }T}^{R_{at}}\right\} _{1\le j\ne a^{\prime }\le n},d_{at,a^{\prime }T}^{R_{at}};R^{a,a^{\prime }}\right) \!, \end{aligned}$$

where $a^{\prime }$ is merely an index of the underlying assets. Finally, we consider the terms that contain $S_{a}$ in the fourth summation of the formula (8.1) and the corresponding sum of these terms is

$$\begin{aligned} -\sum _{a^{\prime }=1}^{n}S_{a}N_{2n}\left( \left\{ d_{it}^{R_{aT}}\right\} _{1\le i\le n},\left\{ d_{jT,aT}^{R_{aT}}\right\} _{1\le j\ne a\le n},d_{a^{\prime }t,aT}^{R_{aT}};R^{a^{\prime },a}\right) \!. \end{aligned}$$

After rearranging the terms in the formula (8.1) with respect to each $S_{a}$ and the strike price $K$, we can obtain the formula for delta with respect to the $a$-th asset as follows:

$$\begin{aligned}&\frac{\partial P}{\partial S_{a}}=-N_{2n}\left( \left\{ d_{K,it}^{R_{aT}}\right\} _{1\le i\le n},\left\{ d_{jT,aT}^{R_{aT}}\right\} _{1\le j\ne a\le n},d_{K,aT}^{R_{aT}}; \,\,R^{K,a}\right) \\&\quad +\,\sum _{a^{\prime }=1}^{n}e^{-r(T-t)}N_{2n}\left( \left\{ d_{it}^{R_{at}}\right\} _{1\le i\le n},\left\{ d_{jT,a^{\prime }T}^{R_{at}}\right\} _{1\le j\ne a^{\prime }\le n},d_{at,a^{\prime }T}^{R_{at}};R^{a,a^{\prime }}\right) \\&\quad -\,\sum _{a^{\prime }=1}^{n}N_{2n}\left( \left\{ d_{it}^{R_{aT}}\right\} _{1\le i\le n},\left\{ d_{jT,aT}^{R_{aT}}\right\} _{1\le j\ne a\le n},d_{a^{\prime }t,aT}^{R_{aT}};R^{a^{\prime },a}\right) \!. \end{aligned}$$

Based on the formula for delta, we can proceed to derive the formulae for gamma with respect to the $a$-th asset and cross gamma with respect to the $a$-th and $b$-th assets, $\frac{\partial ^2 P}{\partial S_{a}^2}$ and $\frac{\partial ^2 P}{\partial S_{a}\partial S_{b}}$. Consider the derivation of $\frac{\partial ^2 P}{\partial S_{a}^2}$ first:

$$\begin{aligned} \frac{\partial ^2 P}{\partial S_{a}^2}&= \frac{\partial }{\partial S_{a}} \left( \frac{\partial P}{\partial S_{a}}\right) \\ \qquad&= -\frac{\partial }{\partial S_{a}}N_{2n}\left( \left\{ d_{K,it}^{R_{aT}}\right\} _{1\le i\le n},\left\{ d_{jT,aT}^{R_{aT}}\right\} _{1\le j\ne a\le n},d_{K,aT}^{R_{aT}};\,\,R^{K,a}\right) \\&+\,\sum _{a^{\prime }=1}^{n}e^{-r\left( T-t\right) }\,\,\frac{\partial }{\partial S_{a} }N_{2n}\left( \left\{ d_{it}^{R_{at}}\right\} _{1\le i\le n},\left\{ d_{jT,a^{\prime }T}^{R_{at}}\right\} _{1\le j\ne a^{\prime }\!\le \! n},d_{at,a^{\prime }T}^{R_{at}}; \,\,R^{a,a^{\prime }}\right) \\&-\,\sum _{a^{\prime }=1}^{n}\frac{\partial }{\partial S_{a}} N_{2n}\left( \left\{ d_{it}^{R_{aT}}\right\} _{1\le i\le n},\left\{ d_{jT,aT}^{R_{aT}}\right\} _{1\le j\ne a\le n},d_{a^{\prime }t,aT}^{R_{aT}};\,\,R^{a^{\prime },a}\right) \!. \end{aligned}$$

We obtain the derivative of the first term in the above equation by the chain rule.

$$\begin{aligned}&\frac{\partial }{\partial S_{a}}N_{2n}\left( \left\{ d_{K,it}^{R_{aT}}\right\} _{1\le i\le n},\left\{ d_{jT,aT}^{R_{aT}}\right\} _{1\le j\ne a\le n},d_{K,aT}^{R_{aT}};R^{K,a}\right) \\&\quad =\sum _{1\le i \le n}\frac{\partial N_{2n}}{\partial d_{K,it}^{R_{aT}}}\frac{\partial d_{K,it}^{R_{aT}}}{\partial S_{a}}+\sum _{1\le j\ne a\le n}\frac{\partial N_{2n}}{\partial d_{jT,aT}^{R_{aT}}}\frac{\partial d_{jT,aT}^{R_{aT}}}{ \partial S_{a}}+\frac{\partial N_{2n}}{\partial d_{K,aT}^{R_{aT}}}\frac{ \partial d_{K,aT}^{R_{aT}}}{\partial S_{a}}. \end{aligned}$$

The key element in deriving the formula for gamma is the calculation of the derivatives of the multivariate standard normal CDFs. The method adopted in this paper is as follows. Let $\{d_{u}\}_{1\le u\le 2n}\equiv \{\{d_{K,it}^{R_{aT}}\}_{1\le i\le n},\{d_{jT,aT}^{R_{aT}}\}_{1\le j\ne a\le n},d_{K,aT}^{R_{aT}}\}$, and $\rho _{u,v}^{K,a}$ be the correlation coefficient element of $R^{K,a}$. Taking $\partial N_{2n}/\partial d_{K,aT}^{R_{aT}}$ as an example, we rewrite the above multivariate standard normal CDF according to the results in Curnow and Dunnett (1962).

$$\begin{aligned}&N_{2n}\left( \left\{ d_{K,it}^{R_{aT}}\right\} _{1\le i\le n},\left\{ d_{jT,aT}^{R_{aT}}\right\} _{1\le j\ne a\le n},d_{K,aT}^{R_{aT}};R^{K,a}\right) \\&\quad =N_{2n}\left( \left\{ d_{u}\right\} _{1\le u\le 2n};R^{K,a}\right) \\&\quad =\int _{-\infty }^{d_{2n}}N_{2n-1}\left( \left\{ \widehat{d}_{u}\left( x\right) \right\} _{1\le u<2n};\left( \widehat{\rho }_{u,v}^{K,a}\right) _{\begin{array}{c} 1\le u<2n \\ 1\le v<2n \end{array}}\right) \phi \left( x\right) dx. \end{aligned}$$

where

$$\begin{aligned} d_{2n} \equiv d_{K,aT}^{R_{aT}},\,\, \widehat{d}_{u}(x)\!=\!\frac{d_{u}-\rho _{u,2n}^{K,a}x}{\sqrt{1-\left( \rho _{u,2n}^{K,a}\right) ^2}},\text { and }\widehat{\rho }_{u,v}^{K,a}\!=\!\frac{\rho _{u,v}^{K,a}-\rho _{u,2n}^{K,a}\rho _{v,2n}^{K,a}}{\sqrt{1-\left( \rho _{u,2n}^{K,a}\right) ^2}\sqrt{1-\left( \rho _{v,2n}^{K,a}\right) ^2}}, \end{aligned}$$

and $\phi (x)$ is the standard normal probability density function. Hence, $\partial N_{2n}/\partial d_{2n}=\partial N_{2n}/\partial d_{K,aT}^{R_{aT}}$ can be calculated as

$$\begin{aligned} \left. \frac{\partial N_{2n}}{\partial d_{2n}}=N_{2n-1}\left( \left\{ \widehat{d} _{u}(x)\right\} _{1\le u<2n};\left( \widehat{\rho }_{u,v}^{K,a}\right) _{\begin{array}{c} 1\le u<2n \\ 1\le v<2n \end{array}}\right) \phi (x)\right| _{x=d_{2n}=d_{K,aT}^{R_{aT}}}. \end{aligned}$$

After similarly calculating other terms, we obtain the formula for the FSRPO’s gamma. This paper only briefly describes the method to derive the formulae of gamma and cross gamma, and the detailed formulae are omitted for brevity but available upon request.

Appendix 4: Formulae of discrete-sampling lookback options

This appendix demonstrates the generality of our main theorem by applying it to discrete-sampling lookback rainbow options. In order to describe the formulae more clearly, the notation is slightly different from that in Sect. 2. Let $S_i(t_j)$ denote the price of the $i$-th stock at time $t_{j}$ for $i=1,2,\ldots ,m$, $j=1,2,\ldots ,n$, and $0=t_0<t_1<\cdots <t_n=T$. The return dynamics of $S_i$ for $i=1,2,\ldots ,m$ follow the same stochastic differential equations as Eq. (2.2). In addition, $M$ denotes the historical highest price up to now. Except for above differences, the rest of the notation, such as $r$, $\sigma _i$, $K$, $T$, and $\rho _{ij}$, is the same.

Consider the discrete-sampling lookback rainbow option that is a call option on the maximum among all the prices of every risky asset at every monitoring time during the option’s life. Let $C_{\max }(T)$ be the payoff of this option at maturity $T$ with the historical highest price $M$ until $t_0$ and constant strike price $K$:

$$\begin{aligned} C_{\max }(T)=\left( \underset{k,l}{\max }(M,S_{k}(t_{l}))-K\right) ^+=\left( \underset{k,l}{\max } (M,S_{k}(t_{l}))-K\right) \cdot I_{A}, \end{aligned}$$

where

$$\begin{aligned} A_{ij}&= \left\{ S_i(t_j)=\underset{k,l}{\max }(M,S_{k}(t_{l}))\ge K\right\} , \\ A_{M}&= \left\{ M=\underset{k,l}{\max }(M,S_k(t_l))\ge K\right\} , \\ A&= \underset{i,j}{\cup }\left\{ S_i(t_j)=\underset{k,l}{\max } (M,S_{k}(t_{l}))\ge K\right\} \cup \left\{ M=\underset{k,l}{\max }(M,S_k(t_l))\ge K\right\} \\&= \left( \underset{i,j}{\cup }A_{ij}\right) \cup A_{M}. \end{aligned}$$

By the risk-neutral valuation, the option value today can be expressed as follows:

$$\begin{aligned} C_{\max }(0)&= e^{-rT}E^Q[C_{\max }(T)] \nonumber \\&= e^{-rT}E^Q[(\underset{k,l}{\max }(M,S_{k}(t_{l}))-K)\cdot I_{A}] \nonumber \\&= e^{-rT}E^Q[(\underset{k,l}{\max }(M,S_{k}(t_{l}))\cdot I_{A}]-Ke^{-rT}E^Q[I_{A}]\nonumber \\&= e^{-rT}\underset{i,j}{\sum }E^Q[(S_i(t_j)\cdot I_{A_{ij}})]+Me^{-rT}E^Q[I_{A_{M}}]-Ke^{-rT}E^Q[I_{A}].\nonumber \\ \end{aligned}$$

(9.1)

As $M\ge K$, $I_{A}=1$, the equation (9.1) can be expressed as

$$\begin{aligned} C_{\max }(0)&= e^{-rT}\underset{i,j}{\sum }E^Q[(S_i(t_j)\cdot I_{A_{ij}})]+Me^{-rT}E^Q\left[ I_{A_{M}}\right] -Ke^{-rT} \nonumber \\&= \left\{ \sum \limits _{i,j} S_i(0)e^{-r(T-t_{j})}N_{mn}\left[ \left( D1_{kl}^{ij}\right) _{mn};\left( F1_{kl,pq}^{ij}\right) _{mn \times mn}\right] \right\} \nonumber \\&\quad +\,Me^{-rT}\,\,N_{mn}\left[ \left( d1_{ij}\right) _{mn};(f1_{ij,kl})_{mn\times mn}\right] -Ke^{-rT}, \end{aligned}$$

(9.2)

where

$$\begin{aligned} d1_{ij}&= -\frac{\ln (S_i(0)/M)+(r-\frac{1}{2}\sigma _i^2)t_{j}}{ \sigma _i\sqrt{t_{j}}}, \\ \ f1_{ij,kl}&= \frac{\rho _{ik}\min (t_{j},t_{l})}{\sqrt{t_{j}\cdot t_{l}}},\\ D1_{kl}^{ij}&= \left\{ \begin{array}{cl} \frac{\ln (S_i(0)/M)+(r+\frac{1}{2}\sigma _i^2)t_{j}}{\sigma _i\sqrt{ t_{j}}} &{} \text {for }k=i\text { and }l=j \\ \frac{\ln (S_i(0)/S_{k}(0))+r(t_{j}-t_{l})+\frac{1}{2}\Sigma _{ijkl}^2}{ \Sigma _{ijkl}} &{} \text {otherwise} \end{array} \right. ,\\ F1_{kl,pq}^{ij}&= \left\{ \begin{array}{cl} 1 &{} \text {for }kl=pq=ij \\ \frac{\sigma _it_{j}-\rho _{ip}\sigma _{p}\min (t_{j},t_{q})}{\sqrt{t_{j}} \Sigma _{ijpq}} &{} \text {for }kl=ij\text {, }pq\ne ij \\ \frac{\sigma _it_{j}-\rho _{ik}\sigma _{k}\min (t_{j},t_{l})}{\sqrt{t_{j}} \Sigma _{ijkl}} &{} \text {for }kl\ne ij\text {, }pq=ij \\ \frac{H}{\Sigma _{ijkl}\Sigma _{ijpq}} &{} \text {for }kl\ne ij\text {, }pq\ne ij \end{array} \right. , \end{aligned}$$

and

$$\begin{aligned} \Sigma _{ijkl}=\sqrt{\sigma _i^2t_{j}-2\rho _{ik}\sigma _i\sigma _{k}\min (t_{j},t_{l})+\sigma _{k}^2t_{l}}, \end{aligned}$$

and

$$\begin{aligned} H=\sigma _i^2t_{j}-\rho _{ik}\sigma _i\sigma _{k}\min (t_{j},t_{l})-\rho _{ip}\sigma _i\sigma _{p}\min (t_{j},t_{q})+\rho _{kp}\sigma _{k}\sigma _{p}\min (t_{l},t_{q}). \end{aligned}$$

In the other case of $M<K$, $I_{A_{M}}=0$, the formula is similar to the case of $M\ge K,$ and Eq. (9.1) can be expressed as

$$\begin{aligned} C_{\max }(0)&= e^{-rT}\underset{i,j}{\sum }E^Q\left[ (S_i(t_j)\cdot I_{A_{ij}})\right] -Ke^{-rT}E^Q[I_{A}] \nonumber \\&= \left\{ \sum \limits _{i,j} S_i(0)e^{-r(T-t_{j})}N_{mn}\left[ (D2_{kl}^{ij})_{mn};(F2_{kl,pq}^{ij})_{mn \times mn}\right] \right\} \nonumber \\&\quad -\,Ke^{-rT}\left( 1-N_{mn}\left[ \left( d2_{ij}\right) _{mn};(f2_{ij,kl})_{mn\times mn}\right] \right) , \end{aligned}$$

(9.3)

where

$$\begin{aligned} d2_{ij}&= -\frac{\ln (S_i(0)/K)+(r-\frac{1}{2}\sigma _i^2)t_{j}}{ \sigma _i\sqrt{t_{j}}}, \\ \ f2_{ij,kl}&= f1_{ij,kl}, \\ D2_{kl}^{ij}&= \left\{ \begin{array}{ll} \frac{\ln (S_i(0)/K)+(r+\frac{1}{2}\sigma _i^2)t_{j}}{\sigma _i\sqrt{ t_{j}}} &{}\, \text {for }k=i\text { and }l=j \\ D1_{kl}^{ij} &{} \,\text {otherwise} \end{array} \right. \!\!\!, \\ F2_{kl,pq}^{ij}&= F1_{kl,pq}^{ij}. \end{aligned}$$

This appendix shows that the pricing technique of Theorem 2.1 can be employed to price different types of discrete-sampling path-dependent rainbow options. Therefore, the technique proposed in this paper can broach many possible directions in the design of new contracts with both the path dependent and multiple asset features.