Snell envelope

From Wikipedia, the free encyclopedia

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

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

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.^[1]

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.^[1]

References

↑ 1.0 1.1 1.2 Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 280–282. ISBN 9783110183467.

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.