Financial Engineering 2025-04-03

6.Exotic Options

For consistency, we switch to the following notation for a derivative $V$ $V^{(\mathrm{t})}_{\sigma,r,\ldots}(S_t,\tau, K)$ where $\sigma,r,\ldots$ are the option paramters, $(\mathrm{t})$ further denotes the option type, and $(S_t,\tau, K)$ are the underlying asset price, time to maturity $\tau = T-t$ and strike price, respectively. If it is clear from the context, we may omit $K$ 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 \sim \lawN(0,1)

c \in \R

and

f(Z)

be a measurable function of

Z

with finite expectation. Then

\E\bk{e^{cZ} f(Z)} = e^{\frac{1}{2}c^2} \E[f(Z+c)]

Proof: We have the identity

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

\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

\lawN_{\Cor}(\vb{0}, \vb{R})

for when we use the correlation matrix

\vb{R}

to define the multivariate Gaussian distribution.

Recap (Multivariate Gaussian): Let

\rb{Z} \sim \lawN_{\Cor}(\vb{0}, \vb{R})

be a multivariate Gaussian random vector with mean

\vb{0} \in \R^n

and correlation matrix

\vb{R} \in \R^{n \times n}

i.e.

\Cor[Z_i, Z_j] = \vb{R}_{ij}.

The joint pdf of

Z

\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

\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

\rb{Z} \sim \lawN_{\Cor}(\vb{0}, \vb{R}),

\vb{c} \in \R^n

and

f(\rb{Z})

be a measurable function of

\rb{Z}

with finite expectation. Then

\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}^{(\pm)}

on a single underlying

S_t

that at time

T

pays

{FB}^{(\pm)} (S_T, 0, K) = f(S_T) \ind{\pm S_T > \pm K}

with payoff

f(S_T)

is a first-order binary option with

{FB}^{(+)}

being an up-type and

{FB}^{(-)}

being a down-type first-order binary option.

Note:

When $f(S) = S$ the binaries are known as asset binaries $A^{(\pm)}$
When $f(S) = 1$ the binaries are known as bond binaries $B^{(\pm)}$

We recall that under the risk-neutral measure $\Q$ we have $S_T = S_t e^{(r - \frac{1}{2} \sigma^2) \tau + \sigma \sqrt{\tau} Z}$ with $Z \sim \lawN(0,1)$ and $\tau = T - t$ Then the condition $S_T \gtrless K$ is equivalent to $Z \gtrless -d_K$ where $d_K = \frac{\ln{\frac{S_t}{K}} + (r - \frac{1}{2} \sigma^2) \tau}{\sigma \sqrt{\tau}}$

Example (Price of a bond binary): The price of a bond binary is given by

\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:

\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}

and gap put options

{GP}

are calls and puts with a strike price

K

different from the exercise price

\xi

i.e.

{GC}

has the payoff function

{GC}_{\xi}(S_T, 0, K) = (S_T - K) \ind{S_T > \xi}

where

K > \xi

and

{GP}

has the payoff function

{GP}_{\xi}(S_T, 0, K) = (K - S_T) \ind{S_T < \xi}

where

\xi < K

Example (Price of a gap call): We have

\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

{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

Q^{(\pm)}_{\xi}

with strike

K

different from the exercise price

\xi

and payoff function

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.

Q^{(+)}_{\xi} = {GC}_{\xi}

and

Q^{(-)}_{\xi} = {GP}_{\xi}

6.2.3 Log Contracts

Definition 6.6 (Log contract): A log contract

{LC}

with strike

K > 0

is an option on the log of the underlying asset price

S_t

It has the payoff function

{LC}(S_T, 0, K) = \ln{\frac{S_T}{K}}

Example (Price of a log contract): We have

\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

{LC}_t

be a log contract with strike

K > 0

F_t

be a forward with the same strike

K

and

C_{\tilde{K},t}

and

P_{\tilde{K},t}

be call and puts with strike

\tilde{K}

The portfolio

\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

\tilde{K}

and one forward statically hedges

{LC}

i.e.

\pi_t = -{LC}(S_t, \tau, K)

Proof: We compute the payoff of the portfolio

\pi_t

at expiry

\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 $\tau_i = T_i - t$ and $\tau_{j,i} = T_j - T_i$

6.3.1 Forward Start Calls and Puts

Definition 6.8 (Forward start option): A forward start option

F^{(C/S)}(S_{t}, \tau_2)

gives the holder at time

T_1

an ATM option with

K = S_{T_1}

expiring at

T_2 > T_1

i.e. a forward start call option is defined through

\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

T_1

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

T_1

with

K = S_{T_1}

we have

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

d_{\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)}(S_{T_1}, \tau_{2})

at time

t < T_1

\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}^{(\pm_1,\pm_2)}(S_t,\tau_2,K_1,K_2)

gives the holder at time

T_1

a first-order binary

{FB}^{(\pm_2)}(S_t, \tau_2, K_2)

expiring at

T_2 > T_1

i.e. a second-order binary is defined through

\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) = S

We write

S_{T_i} = S_t e^{(r - \frac{1}{2}\sigma^2)\tau_i + \sigma \sqrt{\tau_i} Z_i}

and

\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

\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 < T_1

we have

\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

Q^{(\pm_1,\pm_2)}_{\xi_1, \xi_2}(S_{t},\tau_{2},K)

of exercise prices

\xi_1

and

\xi_2

and strike price

K

gives the holder at time

T_1

a first-order Q-option

Q^{(\pm_2)}_{\xi_2}(S_{t},\tau_{2},K_2)

expiring at

T_2 > T_1

i.e. a second-order Q-option is defined through

\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

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)