Financial Engineering

6. Exotic Options

For consistency, we switch to the following notation for a derivative VV: Vσ,r,…(t)(St,τ,K) V^{(\mathrm{t})}_{\sigma,r,\ldots}(S_t,\tau, K) where σ,r,…\sigma,r,\ldots are the option paramters, (t)(\mathrm{t}) further denotes the option type, and (St,τ,K)(S_t,\tau, K) are the underlying asset price, time to maturity τ=T−t\tau = T-t, and strike price, respectively. If it is clear from the context, we may omit KK and some of these parameters.

6.1 Gaussian Shift Theorem

The Gaussian Shift Theorem allows the simplification of many calculations in the Black-Scholes model setting and circumvents the need of changing probability measures.

6.1.1 One-dimensional GST

Theorem 6.1 (Gaussian Shift Theorem): Let Z∼N⁡(0,1)Z \sim \lawN(0,1), c∈Rc \in \R and f(Z)f(Z) be a measurable function of ZZ with finite expectation. Then E[ecZf(Z)]=e12c2E[f(Z+c)] \E\bk{e^{cZ} f(Z)} = e^{\frac{1}{2}c^2} \E[f(Z+c)]
Proof: We have the identity ecyϕ(y)=12πe−y22+cy=12πe−(y−c)22+12c2=e12c2ϕ(y−c) e^{cy} \phi(y) = \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2} + cy} = \frac{1}{\sqrt{2\pi}} e^{-\frac{(y-c)^2}{2} + \frac{1}{2}c^2} = e^{\frac{1}{2} c^2} \phi(y-c) hence E[ecZf(Z)]=∫−∞∞ecyf(y)ϕ(y) dy=∫−∞∞e12c2f(y)ϕ(y−c) dy=e12c2∫−∞∞f(z+c)ϕ(z) dy=e12c2E[f(Z+c)].\begin{align*} \E\bk{e^{cZ} f(Z)} &= \int_{-\infty}^{\infty} e^{cy} f(y) \phi(y) \dd y \\ &= \int_{-\infty}^{\infty} e^{\frac{1}{2} c^2} f(y) \phi(y-c) \dd y \\ &= e^{\frac{1}{2} c^2} \int_{-\infty}^{\infty} f(z+c) \phi(z) \dd y \\ &= e^{\frac{1}{2} c^2} \E[f(Z+c)]. \end{align*}

6.1.2 Multivariate GST

We use the notation N⁡Cor⁡(0,R)\lawN_{\Cor}(\vb{0}, \vb{R}) for when we use the correlation matrix R\vb{R} to define the multivariate Gaussian distribution.

Recap (Multivariate Gaussian): Let Z∼N⁡Cor⁡(0,R)\rb{Z} \sim \lawN_{\Cor}(\vb{0}, \vb{R}) be a multivariate Gaussian random vector with mean 0∈Rn\vb{0} \in \R^n and correlation matrix R∈Rn×n\vb{R} \in \R^{n \times n}, i.e. Cor⁡[Zi,Zj]=Rij\Cor[Z_i, Z_j] = \vb{R}_{ij}. The joint pdf of ZZ is ϕR(z)=e−12z⊤R−1z(2π)ndet⁡(R) \rb{\phi}_{\vb{R}}(\vb{z}) = \frac{e^{-\frac{1}{2} \vb{z}^\top \vb{R}^{-1} \vb{z}}}{\sqrt{(2\pi)^n \det(\vb{R})}} and the joint cdf is ΦR(z)=∫−∞zϕR(t) dt \rb{\Phi}_{\vb{R}}(\vb{z}) = \int_{-\infty}^{\vb{z}} \rb{\phi}_{\vb{R}}(\vb{t}) \dd \vb{t}
Theorem 6.2 (Multivariate Gaussian Shift Theorem): Let Z∼N⁡Cor⁡(0,R)\rb{Z} \sim \lawN_{\Cor}(\vb{0}, \vb{R}), c∈Rn\vb{c} \in \R^n and f(Z)f(\rb{Z}) be a measurable function of Z\rb{Z} with finite expectation. Then E[ec⊤Zf(Z)]=e12c⊤RcE[f(Z+c)] \E\bk{e^{\vb{c}^\top \rb{Z}} f(\rb{Z})} = e^{\frac{1}{2} \vb{c}^\top \vb{R} \vb{c}} \E[f(\rb{Z}+\vb{c})]

6.2 Simple Exotic Options

We will now consider and price some simple exotic options.

6.2.1 First-order Binaries

Definition 6.3 (First-order binary option): The derivative FB(±){FB}^{(\pm)} on a single underlying StS_t that at time TT pays FB(±)(ST,0,K)=f(ST)1{±ST>±K} {FB}^{(\pm)} (S_T, 0, K) = f(S_T) \ind{\pm S_T > \pm K} with payoff f(ST)f(S_T) is a first-order binary option with FB(+){FB}^{(+)} being an up-type and FB(−){FB}^{(-)} being a down-type first-order binary option.
Note:
  • When f(S)=Sf(S) = S, the binaries are known as asset binaries A(±)A^{(\pm)}
  • When f(S)=1f(S) = 1, the binaries are known as bond binaries B(±)B^{(\pm)}

We recall that under the risk-neutral measure Q\Q we have ST=Ste(r−12σ2)τ+στZS_T = S_t e^{(r - \frac{1}{2} \sigma^2) \tau + \sigma \sqrt{\tau} Z} with Z∼N⁡(0,1)Z \sim \lawN(0,1) and τ=T−t\tau = T - t. Then the condition ST≷KS_T \gtrless K is equivalent to Z≷−dKZ \gtrless -d_K where dK=ln⁡(StK)+(r−12σ2)τστ d_K = \frac{\ln{\frac{S_t}{K}} + (r - \frac{1}{2} \sigma^2) \tau}{\sigma \sqrt{\tau}}

Example (Price of a bond binary): The price of a bond binary is given by B(±)(St,τ,K)=e−rτEQ[1{±ST>±K}]=e−rτEQ[1{±Z>∓dK}]=e−rτΦ(±dK)\begin{align*} B^{(\pm)} (S_t, \tau, K) &= e^{-r\tau} \E_{\Q}[\ind{\pm S_T > \pm K}] = e^{-r\tau} \E_{\Q}[\ind{\pm Z > \mp d_K}] = e^{-r\tau} \Phi(\pm d_K) \end{align*}
Example (Price of an asset binary): Pricing an asset binary is slightly more involved but using the Gaussian Shift Theorem we avoid needing a measure change and the calculations become much simpler: A(±)(St,τ,K)=e−rτEQ[ST1{±ST>±K}]=e−rτEQ[Ste(r−12σ2)τ+στZ1{±Z>∓dK}]=e−rτSte(r−12σ2)τEQ[eστZ1{±Z>∓dK}]=Ste−12σ2τe12σ2τQ(±Z±στ>∓dK)=StQ(±Z>∓dK∓στ)=StΦ(±(dK+στ))\begin{align*} A^{(\pm)}(S_t, \tau, K) &= e^{-r\tau} \E_{\Q}[S_T \ind{\pm S_T > \pm K}] \\ &= e^{-r\tau} \E_{\Q}[S_t e^{(r - \frac{1}{2} \sigma^2) \tau + \sigma \sqrt{\tau} Z} \ind{\pm Z > \mp d_K}] \\ &= e^{-r\tau} S_t e^{(r - \frac{1}{2} \sigma^2) \tau} \E_{\Q}[e^{\sigma \sqrt{\tau} Z} \ind{\pm Z > \mp d_K}] \\ &= S_t e^{- \frac{1}{2} \sigma^2 \tau} e^{\frac{1}{2} \sigma^2 \tau} \Q(\pm Z \pm \sigma \sqrt{\tau} > \mp d_K) \\ &= S_t \Q(\pm Z > \mp d_K \mp \sigma \sqrt{\tau}) \\ &= S_t \Phi(\pm(d_K + \sigma \sqrt{\tau})) \end{align*}

6.2.2 Gap and Q-Options

Definition 6.4 (Gap option): Gap call options GC{GC} and gap put options GP{GP} are calls and puts with a strike price KK different from the exercise price ξ\xi, i.e. GC{GC} has the payoff function GCξ(ST,0,K)=(ST−K)1{ST>ξ} {GC}_{\xi}(S_T, 0, K) = (S_T - K) \ind{S_T > \xi} where K>ξK > \xi and GP{GP} has the payoff function GPξ(ST,0,K)=(K−ST)1{ST<ξ} {GP}_{\xi}(S_T, 0, K) = (K - S_T) \ind{S_T < \xi} where ξ<K\xi < K.
Example (Price of a gap call): We have GCξ(ST,0,K)=(ST−K)1{ST>ξ}=ST1{ST>ξ}−K1{ST>ξ}=A(+)(ST,0,ξ)−K⋅B(+)(ST,0,ξ)\begin{align*} {GC}_{\xi}(S_T, 0, K) &= (S_T - K) \ind{S_T > \xi} \\ &= S_T \ind{S_T > \xi} - K \ind{S_T > \xi} \\ &= A^{(+)}(S_T, 0, \xi) - K \cdot B^{(+)}(S_T, 0, \xi) \end{align*} hence GCξ(St,τ,K)=A(+)(St,τ,ξ)−K⋅B+(St,τ,ξ)=StΦ(dξ+στ)−Ke−rτΦ(dξ){GC}_{\xi}(S_t, \tau, K) = A^{(+)}(S_t, \tau, \xi) - K \cdot B^{+}(S_t, \tau, \xi) = S_t \Phi(d_{\xi} + \sigma \sqrt{\tau}) - K e^{-r\tau} \Phi(d_{\xi}).
Definition 6.5 (First-order Q-option): The first-order Q-option is the derivative Qξ(±)Q^{(\pm)}_{\xi} with strike KK different from the exercise price ξ\xi and payoff function Qξ(±)(ST,0,K)=(ST−K)1{±ST>±ξ} Q^{(\pm)}_{\xi}(S_T, 0, K) = (S_T - K) \ind{\pm S_T > \pm \xi}
Note: We note that gap calls and puts are special cases of first-order Q-options, i.e. Qξ(+)=GCξQ^{(+)}_{\xi} = {GC}_{\xi} and Qξ(−)=GPξQ^{(-)}_{\xi} = {GP}_{\xi}.

6.2.3 Log Contracts

Definition 6.6 (Log contract): A log contract LC{LC} with strike K>0K > 0 is an option on the log of the underlying asset price StS_t. It has the payoff function LC(ST,0,K)=ln⁡(STK) {LC}(S_T, 0, K) = \ln{\frac{S_T}{K}}
Example (Price of a log contract): We have LC(St,τ,K)=e−rτEQ[ln⁡(STK)]=e−rτEQ[ln⁡(StKe(r−12σ2)τ+στZ)]=e−rτ(EQ[ln⁡(StK)]+EQ[(r−12σ2)τ+στZ])=e−rτ(ln⁡(StK)+(r−12σ2)τ)\begin{align*} {LC}(S_t, \tau, K) & = e^{-r\tau} \E_{\Q}\bk{\ln{\frac{S_T}{K}}} \\ & = e^{-r\tau} \E_{\Q}\bk{\ln{\frac{S_t}{K} e^{(r - \frac{1}{2} \sigma^2) \tau + \sigma \sqrt{\tau} Z}}} \\ & = e^{-r\tau} \pa{\E_{\Q}\bk{\ln{\frac{S_t}{K}}} + \E_{\Q}\bk{\pa{r - \frac{1}{2} \sigma^2} \tau + \sigma \sqrt{\tau} Z}} \\ &= e^{-r\tau} \pa{\ln{\frac{S_t}{K}} + \pa{r - \frac{1}{2} \sigma^2} \tau} \end{align*}
Proposition 6.7 (Log contracts can be statically hedged): Let LCt{LC}_t be a log contract with strike K>0K > 0, FtF_t be a forward with the same strike KK and CK~,tC_{\tilde{K},t} and PK~,tP_{\tilde{K},t} be call and puts with strike K~\tilde{K}. The portfolio πt=∫0KK~−2PK~,t dK~+∫K∞K~−2CK~,t dK~−K−1Ft \pi_t = \int_{0}^{K} \tilde{K}^{-2} P_{\tilde{K},t} \dd \tilde{K} + \int_{K}^{\infty} \tilde{K}^{-2} C_{\tilde{K},t} \dd \tilde{K} - K^{-1} F_t consisting of a continuum of calls and puts at different strikes K~\tilde{K} and one forward statically hedges LC{LC}, i.e. πt=−LC(St,τ,K)\pi_t = -{LC}(S_t, \tau, K).
Proof: We compute the payoff of the portfolio πt\pi_t at expiry πT=∫0KK~−2PK~,T dK~+∫K∞K~−2CK~,T dK~−K−1FT=∫0K(K~−ST)+K~2 dK~+∫K∞(ST−K~)+K~2 dK~−1K=∫0KK~−STK~21{ST<K~} dK~+∫K∞ST−K~K~21{ST>K~} dK~−1K=[(−ln⁡(STK~)+1K~)1{ST<K~}]0K+[(ln⁡(STK~)−1K~)1{ST>K~}]K∞−1K=(−ln⁡(STK)+1K)1{ST<K}−(ln⁡(STK)−1K)1{ST>K}−1K=−ln⁡(STK)+1K−1K=−LC(ST,τ,K)\begin{align*} \pi_T &= \int_{0}^{K} \tilde{K}^{-2} P_{\tilde{K},T} \dd \tilde{K} + \int_{K}^{\infty} \tilde{K}^{-2} C_{\tilde{K},T} \dd \tilde{K} - K^{-1} F_T \\ & = \int_{0}^{K} \frac{(\tilde{K} - S_T)^{+}}{\tilde{K}^{2}} \dd \tilde{K} + \int_{K}^{\infty} \frac{(S_T - \tilde{K})^{+}}{\tilde{K}^{2}} \dd \tilde{K} - \frac{1}{K} \\ & = \int_{0}^{K} \frac{\tilde{K} - S_T}{\tilde{K}^{2}} \ind{S_T < \tilde{K}} \dd \tilde{K} + \int_{K}^{\infty} \frac{S_T - \tilde{K}}{\tilde{K}^{2}} \ind{S_T > \tilde{K}} \dd \tilde{K} - \frac{1}{K} \\ & = \bk{ \pa{- \ln{\frac{S_T}{\tilde{K}}} + \frac{1}{\tilde{K}}} \ind{S_T < \tilde{K}}}_0^K + \bk{ \pa{\ln{\frac{S_T}{\tilde{K}}} - \frac{1}{\tilde{K}}} \ind{S_T > \tilde{K}}}_K^{\infty} - \frac{1}{K} \\ & = \pa{- \ln{\frac{S_T}{K}} + \frac{1}{K}} \ind{S_T < K} - \pa{\ln{\frac{S_T}{K}} - \frac{1}{K}} \ind{S_T > K} - \frac{1}{K} \\ & = - \ln{\frac{S_T}{K}} + \frac{1}{K} - \frac{1}{K} \\ & = - {LC}(S_T, \tau, K) \end{align*}

6.3 Dual Expiry Options

For consistency, we define the notation τi=Ti−t\tau_i = T_i - t and τj,i=Tj−Ti\tau_{j,i} = T_j - T_i.

6.3.1 Forward Start Calls and Puts

Definition 6.8 (Forward start option): A forward start option F(C/S)(St,τ2)F^{(C/S)}(S_{t}, \tau_2) gives the holder at time T1T_1 an ATM option with K=ST1K = S_{T_1} expiring at T2>T1T_2 > T_1, i.e. a forward start call option is defined through F(C)(ST1,τ2,1)=C(ST1,τ2,1,ST1)F(C)(ST2,0)=(ST1−ST2)+\begin{align*} & F^{(C)}(S_{T_1}, \tau_{2,1}) = C(S_{T_1}, \tau_{2,1}, S_{T_1}) \\ & F^{(C)}(S_{T_2},0) = (S_{T_1} - S_{T_2})^+ \end{align*} A forward start put option is defined similarly.
Example (Price of a forwards start call): Consider a forwards start call at time T1T_1 F(C)(ST1,τ2,1)=C(ST1,τ2,1,ST1) F^{(C)}(S_{T_1}, \tau_{2,1}) = C(S_{T_1}, \tau_{2,1}, S_{T_1}) We note that for an ATM option at time T1T_1 with K=ST1K = S_{T_1} we have dATM,1=ln⁡(ST1ST1)+(r+σ22)τ2,1στ2=rστ2,1+σ2τ2,1 d_{\mathrm{ATM},1} = \frac{\ln{\frac{S_{T_1}}{S_{T_1}}} + \pa{r + \frac{\sigma^2}{2}} \tau_{2,1}}{\sigma \sqrt{\tau_2}} = \frac{r}{\sigma} \sqrt{\tau_{2,1}} + \frac{\sigma}{2} \sqrt{\tau_{2,1}} and dATM,2=dATM,1(τ2)−στ2,1=rστ2,1−σ2τ2,1d_{\mathrm{ATM},2} = d_{\mathrm{ATM},1}(\tau_2) - \sigma \sqrt{\tau_{2,1}} = \frac{r}{\sigma} \sqrt{\tau_{2,1}} - \frac{\sigma}{2} \sqrt{\tau_{2,1}}. We discount the expected value of F(C)(ST1,τ2)F^{(C)}(S_{T_1}, \tau_{2}) at time t<T1t < T_1 F(C)(St,τ2)=e−rτ1EQ[F(C)(ST1,τ2,1)]=e−rτ1EQ[C(BS)(ST1,τ2,1,ST1)]=e−rτ1EQ[ST1Φ(dATM,1)−ST1e−rτ2,1Φ(dATM,2)]=e−rτ1(Φ(dATM,1)−e−rτ2,1Φ(dATM,2))EQ[Ste(r−12σ2)τ1+στ1Z]=e−rτ1(Φ(dATM,1)−e−rτ2,1Φ(dATM,2))Ste(r−12σ2)τ1EQ[eστ1Z]=St(Φ(dATM,1)−e−rτ2,1Φ(dATM,2))e−12σ2τ1e12σ2τ1=St(Φ(dATM,1)−e−rτ2,1Φ(dATM,2))\begin{align*} F^{(C)}(S_{t}, \tau_2) &= e^{-r \tau_1} \E_{\Q}[F^{(C)}(S_{T_1},\tau_{2,1})] \\ & = e^{-r \tau_1} \E_{\Q}[C^{(\mathrm{BS})}(S_{T_1}, \tau_{2,1}, S_{T_1})] \\ & = e^{-r \tau_1} \E_{\Q}[S_{T_1} \Phi(d_{\mathrm{ATM},1}) - S_{T_1}e^{-r\tau_{2,1}} \Phi(d_{\mathrm{ATM},2})] \\ & = e^{-r \tau_1} (\Phi(d_{\mathrm{ATM},1}) - e^{-r\tau_{2,1}} \Phi(d_{\mathrm{ATM},2})) \E_{\Q}[S_te^{(r - \frac{1}{2}\sigma^2)\tau_1 + \sigma \sqrt{\tau_1} Z}] \\ & = e^{-r \tau_1} (\Phi(d_{\mathrm{ATM},1}) - e^{-r\tau_{2,1}} \Phi(d_{\mathrm{ATM},2})) S_t e^{(r - \frac{1}{2}\sigma^2)\tau_1} \E_{\Q}[e^{\sigma \sqrt{\tau_1} Z}] \\ & = S_t (\Phi(d_{\mathrm{ATM},1}) - e^{-r\tau_{2,1}} \Phi(d_{\mathrm{ATM},2})) e^{- \frac{1}{2}\sigma^2\tau_1} e^{\frac{1}{2}\sigma^2 \tau_1} \\ & = S_t (\Phi(d_{\mathrm{ATM},1}) - e^{-r\tau_{2,1}} \Phi(d_{\mathrm{ATM},2})) \\ \end{align*}

6.3.2 Second-order Binaries

Definition 6.9 (Second-order binary option): A second-order binary option SB(±1,±2)(St,τ2,K1,K2){SB}^{(\pm_1,\pm_2)}(S_t,\tau_2,K_1,K_2) gives the holder at time T1T_1 a first-order binary FB(±2)(St,τ2,K2){FB}^{(\pm_2)}(S_t, \tau_2, K_2) expiring at T2>T1T_2 > T_1, i.e. a second-order binary is defined through SB(±1,±2)(ST1,τ2,1,K1,K2)=FB(±2)(ST1,τ2,1,K2)1{±1ST1>±1K1}SB(±1,±2)(ST2,0,K1,K2)=f(ST2)1{±1ST1>±1K1}1{±2ST2>±2K2}\begin{align*} & {SB}^{(\pm_1,\pm_2)}(S_{T_1},\tau_{2,1},K_1,K_2) = {FB}^{(\pm_2)}(S_{T_1}, \tau_{2,1}, K_2) \ind{\pm_1 S_{T_1} > \pm_1 K_1} \\ & {SB}^{(\pm_1,\pm_2)}(S_{T_2},0,K_1,K_2) = f(S_{T_2}) \ind{\pm_1 S_{T_1} > \pm_1 K_1} \ind{\pm_2 S_{T_2} > \pm_2 K_2} \end{align*}
Example (Price of a second-order down-up asset binary): For second-order assets we have the payoff function f(S)=Sf(S) = S. We write STi=Ste(r−12σ2)τi+στiZiS_{T_i} = S_t e^{(r - \frac{1}{2}\sigma^2)\tau_i + \sigma \sqrt{\tau_i} Z_i} and di=ln⁡(StKi)+(r−12σ2)τiστid~i=ln⁡(StKi)+(r+12σ2)τiστi=di+στi\begin{align*} &d_i = \frac{\ln{\frac{S_{t}}{K_i}}+(r-\frac{1}{2} \sigma^2) \tau_i}{\sigma \sqrt{\tau_i}} \\ &\tilde{d}_i = \frac{\ln{\frac{S_{t}}{K_i}}+(r+\frac{1}{2} \sigma^2) \tau_i}{\sigma \sqrt{\tau_i}} = d_i + \sigma \sqrt{\tau_i} \end{align*} Hence [Z1Z2]∼N⁡Cor⁡(0,R)R=[1ρρ1]ρ=−12τ1τ2\begin{align*} & \begin{bmatrix} Z_1 \\ Z_2 \end{bmatrix} \sim \lawN_{\Cor}\of{\vb{0},\vb{R}} \\ & \vb{R} = \begin{bmatrix} 1 & \rho \\ \rho & 1 \end{bmatrix} \\ & \rho = -\frac{1}{2} \sqrt{\frac{\tau_1}{\tau_2}} \end{align*} For time t<T1t < T_1 we have A(−,+)(ST1,τ2,K1,K2)=e−rτ2EQ[ST21{ST1<K1}1{ST2>K2}]=e−rτ2EQ[ST1e(r−12σ2)τ2,1+στ2Z21{ST1<K1}1{ST2>K2}]=e−rτ2EQ[Ste(r−12σ2)τ2+στ1Z1+στ2Z21{Z1<−d1}1{Z2>−d2}]=Ste(r−12σ2)τ2e−rτ2EQ[eστ1Z1+στ2Z21{Z1<−d1}1{Z2>−d2}]=Ste−12σ2τ2EQ[exp⁡([στ1στ2]⊤Z)1{Z1<−d1}1{Z2>−d2}]=Ste−12σ2τ2exp⁡(12c⊤Rc)EQ[1{Z1+στ1<−d1}1{Z2+στ2>−d2}]=Ste−12σ2τ2e12σ2τ2EQ[1{Z1<−(d1+στ1)}1{Z2>−(d2+στ2)}]=StEQ[1{Z1<−d1~}1{Z2<d2~}]=StΦR([−d1~d2~])\begin{align*} A^{(-,+)}(S_{T_1},\tau_2,K_1,K_2) &= e^{-r \tau_{2}} E_{\Q}\bk{S_{T_2} \ind{S_{T_1} < K_1} \ind{S_{T_2} > K_2}} \\ &= e^{-r \tau_{2}} E_{\Q}\bk{S_{T_1} e^{(r - \frac{1}{2}\sigma^2)\tau_{2,1} + \sigma \sqrt{\tau_2} Z_2} \ind{S_{T_1} < K_1} \ind{S_{T_2} > K_2}} \\ &= e^{-r \tau_{2}} E_{\Q}\bk{S_{t} e^{(r - \frac{1}{2}\sigma^2)\tau_{2} + \sigma \sqrt{\tau_1} Z_1 + \sigma \sqrt{\tau_2} Z_2} \ind{Z_1 < -d_1} \ind{Z_2 > -d_2}} \\ &= S_t e^{(r - \frac{1}{2}\sigma^2)\tau_2} e^{-r \tau_{2}} E_{\Q}\bk{e^{\sigma \sqrt{\tau_1} Z_1 + \sigma \sqrt{\tau_2} Z_2} \ind{Z_1 < -d_1} \ind{Z_2 > -d_2}} \\ &= S_t e^{-\frac{1}{2}\sigma^2 \tau_2} E_{\Q}\bk{\exp{\begin{bmatrix} \sigma \sqrt{\tau_1} \\ \sigma \sqrt{\tau_2} \end{bmatrix}^{\top} \rb{Z}} \ind{Z_1 < -d_1} \ind{Z_2 > -d_2}} \\ &= S_t e^{-\frac{1}{2}\sigma^2 \tau_2} \exp{\frac{1}{2} \vb{c}^{\top} \vb{R} \vb{c}} E_{\Q}\bk{\ind{Z_1 + \sigma \sqrt{\tau_1} < -d_1} \ind{Z_2 + \sigma \sqrt{\tau_2} > -d_2}} \\ &= S_t e^{-\frac{1}{2}\sigma^2 \tau_2} e^{\frac{1}{2}\sigma^2 \tau_2} E_{\Q}\bk{\ind{Z_1 < -(d_1 + \sigma \sqrt{\tau_1})} \ind{Z_2 > -(d_2 + \sigma \sqrt{\tau_2})}} \\ &= S_t E_{\Q}\bk{\ind{Z_1 < -\tilde{d_1}} \ind{Z_2 < \tilde{d_2}}} \\ &= S_t \rb{\Phi}_{\vb{R}}\of{\begin{bmatrix} -\tilde{d_1} \\ \tilde{d_2} \end{bmatrix}} \end{align*} where we have made use of the multivariate Gaussian Shift Theorem.

6.3.3 Second Order Q-Options

Definition 6.10 (Second Order Q-Options): A second-order Q-option Qξ1,ξ2(±1,±2)(St,τ2,K)Q^{(\pm_1,\pm_2)}_{\xi_1, \xi_2}(S_{t},\tau_{2},K) of exercise prices ξ1\xi_1 and ξ2\xi_2 and strike price KK gives the holder at time T1T_1 a first-order Q-option Qξ2(±2)(St,τ2,K2)Q^{(\pm_2)}_{\xi_2}(S_{t},\tau_{2},K_2) expiring at T2>T1T_2 > T_1, i.e. a second-order Q-option is defined through Qξ1,ξ2(±1,±2)(ST1,τ2,1,K)=Qξ2(±2)(ST1,τ2,1,K)1{±1ST1>±1ξ1}Qξ1,ξ2(±1,±2)(ST2,0,K)=±2(ST2−K)1{±1ST1>±1ξ1}1{±2ST2>±2ξ2}\begin{align*} & Q^{(\pm_1,\pm_2)}_{\xi_1, \xi_2}(S_{T_1},\tau_{2,1},K) = Q^{(\pm_2)}_{\xi_2}(S_{T_1},\tau_{2,1},K) \ind{\pm_1 S_{T_1} > \pm_1 \xi_1} \\ & Q^{(\pm_1,\pm_2)}_{\xi_1, \xi_2}(S_{T_2},0,K) = \pm_2(S_{T_2} - K) \ind{\pm_1 S_{T_1} > \pm_1 \xi_1} \ind{\pm_2 S_{T_2} > \pm_2 \xi_2} \end{align*}
Example (Price of a second-order Q-option): Since second order Q-options are portfolios of second-order assets and second-order bond binaries, their prices are given by a static replication as Qξ1,ξ2(±1,±2)(St,τ2,K)=±2A(±1,±2)(St,τ2,ξ1,ξ2)−±2K⋅B(±1,±2)(St,τ2,ξ1,ξ2) Q^{(\pm_1,\pm_2)}_{\xi_1, \xi_2}(S_{t},\tau_2,K) = \pm_2 A^{(\pm_1,\pm_2)}(S_t, \tau_2, \xi_1, \xi_2) - \pm_2 K \cdot B^{(\pm_1,\pm_2)}(S_t, \tau_2, \xi_1, \xi_2)