Snell envelope

The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating a stochastic process. The Snell envelope is named after James Laurie Snell.

Definition

Given a filtered probability space $(\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in [0,T]},\mathbb {P} )$ and an absolutely continuous probability measure $\mathbb {Q} \ll \mathbb {P}$ then an adapted process $U=(U_{t})_{t\in [0,T]}$ is the Snell envelope with respect to $\mathbb {Q}$ of the process $X=(X_{t})_{t\in [0,T]}$ if

1. $U$ is a $\mathbb {Q}$-supermartingale
2. $U$ dominates $X$, i.e. $U_{t}\geq X_{t}$ $\mathbb {Q}$-almost surely for all times $t\in [0,T]$
3. If $V=(V_{t})_{t\in [0,T]}$ is a $\mathbb {Q}$-supermartingale which dominates $X$, then $V$ dominates $U$.

Construction

Given a (discrete) filtered probability space $(\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n=0}^{N},\mathbb {P} )$ and an absolutely continuous probability measure $\mathbb {Q} \ll \mathbb {P}$ then the Snell envelope $(U_{n})_{n=0}^{N}$ with respect to $\mathbb {Q}$ of the process $(X_{n})_{n=0}^{N}$ is given by the recursive scheme

$U_{N}:=X_{N},$

$U_{n}:=X_{n}\lor \mathbb {E} ^{\mathbb {Q} }[U_{n+1}\mid {\mathcal {F}}_{n}]$ for $n=N-1,...,0$

where $\lor$ is the join (in this case equal to the maximum of the two random variables).

Application

• If $X$ is a discounted American option payoff with Snell envelope $U$ then $U_{t}$ is the minimal capital requirement to hedge $X$ from time $t$ to the expiration date.