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 $(\Omega ,ℱ,({ℱ}_t{)}_{t\in [0,T]},\mathbb{P})$ 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. $U_t\ge X_t$ $\mathbb{Q}$-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 $(\Omega ,ℱ,({ℱ}_n{)}_{n=0}^N,\mathbb{P})$ 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

$U_N:=X_N,$

$U_n:=X_n\vee {𝔼}^{\mathbb{Q}}[U_{n+1}\mid {ℱ}_n]$ for $n=N-1,\mathrm{...},0$

where $\vee$ 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.