The price of a vanilla option with payoff \(\varphi\) is given by \(e^{-rT} \mathbb{E} \left[\varphi \left(S_T\right)\right]\) for a monodimensional problem and by \(e^{-rT} \mathbb{E} \left[\varphi \left(\sum_{i=1}^d \frac 1 d S_T^i\right)\right]\) for dimension \(d>1\).
The price of a path dependent option with functional payoff \(\varphi\) is given by \(e^{-rT} \mathbb{E} \left[\varphi \left(\left(S_t\right)_{t\in [0,T]}\right)\right]\) for a monodimensional problem and by \(e^{-rT} \mathbb{E} \left[\varphi \left(\sum_{i=1}^d \frac 1 d \left(S_t^i\right)_{t\in [0,T]}\right)\right]\) for dimension \(d>1\).
Here a list of payoff :
Vanilla
Call option on stock price \((S_t)_{t\in [0,T]}\), with strike \(K\) :
\[\varphi(S_T) = (S_T - K)^+\]
Put option on stock price \((S_t)_{t\in [0,T]}\), with strike \(K\) :
\[\varphi(S_T) = (K-S_T)^+\]
Barrier Call
Barrier up-and-out call option on \((S_t)_{t\in [0,T]}\) with strike \(K\), barrier \(B>K\) and current price \(S_0<B\) :
\[\varphi(x) = \left( x(T) - K \right)^+ \mathbb{1} _{\left\{ \max_{t\in [0,T]} x(t)\leq B\right\}}\]
Barrier up-and-in call option on \((S_t)_{t\in [0,T]}\) with strike \(K\), barrier \(B\) and current price \(S_0<B\) :
\[\varphi(x) = \left( x(T) - K \right)^+ \mathbb{1} _{\left\{ \max_{t\in [0,T]} x(t)\geq B\right\}}\]
Barrier down-and-out call option on \((S_t)_{t\in [0,T]}\) with strike \(K\), barrier \(B\) and current price \(S_0>B\) :
\[\varphi(x) = \left( x(T) - K \right)^+ \mathbb{1} _{\left\{ \min_{t\in [0,T]} x(t)\geq B\right\}}\]
Barrier down-and-in call option on \((S_t)_{t\in [0,T]}\) with strike \(K\), barrier \(B>K\) and current price \(S_0>B\) :
\[\varphi(x) = \left( x(T) - K \right)^+ \mathbb{1} _{\left\{ \min_{t\in [0,T]} x(t)\leq B\right\}}\]
Barrier Put
Barrier up-and-out put option on \((S_t)_{t\in [0,T]}\) with strike \(K\), barrier \(B\) and current price \(S_0<B\) :
\[\varphi(x) = \left( K - x(T) \right)^+ \mathbb{1} _{\left\{ \max_{t\in [0,T]} x(t)\leq B\right\}}\]
Barrier up-and-in put option on \((S_t)_{t\in [0,T]}\) with strike \(K\), barrier \(B<K\) and current price \(S_0<B\) :
\[\varphi(x) = \left( K -x(T) \right)^+ \mathbb{1} _{\left\{ \max_{t\in [0,T]} x(t)\geq B \right\}}\]
Barrier down-and-out put option on \((S_t)_{t\in [0,T]}\) with strike \(K\), barrier \(B<K\) and current price \(S_0>B\) :
\[\varphi(x) = \left( K -x(T) \right)^+ \mathbb{1} _{\left\{ \min_{t\in [0,T]} x(t)\geq B\right\}}\]
Barrier down-and-in put option on \((S_t)_{t\in [0,T]}\) with strike \(K\), barrier \(B\) and current price \(S_0>B\) :
\[\varphi(x) = \left( K -x(T) \right)^+ \mathbb{1} _{\left\{ \min_{t\in [0,T]} x(t)\leq B\right\}}\]
Lookback
Lookback Call with floating strike option :
\[\varphi(x) = \left( x(T) - \lambda \min_{t \in [0,T]} x(t)\right)^+ \]
Lookback Put with floating strike option :
\[\varphi(x) = \left( \lambda \max_{t \in [0,T]} x(t) - x(T) \right)^+ \]
Nested Put-on-Call
A Put-on-Call option with expiration dates \(T_1 < T_2\) and strikes \(K_1\) and \(K_2\) on a stock price \(S_t\) is an option where the holder has at time \(T_1\) the right to sell, using the strike \(K_1\), a new Call with strike \(K_2\) and maturity \(T_2\). The payoff of such an option writes
\[\left( K_1 -\mathbb{E} \left[\left(S_{T_2} - K_2 \right) ^+ \middle\vert S_{T_1} \right] \right)^+\]