Snell envelope

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The Snell envelope, used in

.

Definition

Given a

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

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

Construction

Given a (discrete)

${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n=0}^{N},\mathbb {P} )}$ and an ${\displaystyle \mathbb {Q} \ll \mathbb {P} }$ then the Snell envelope ${\displaystyle (U_{n})_{n=0}^{N}}$ with respect to ${\displaystyle \mathbb {Q} }$ of the process ${\displaystyle (X_{n})_{n=0}^{N}}$ is given by the recursive scheme

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

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

where ${\displaystyle \lor }$ is the join (in this case equal to the maximum of the two random variables).^[1]

Application

• If ${\displaystyle X}$ is a discounted
  American option
  payoff with Snell envelope ${\displaystyle U}$ then ${\displaystyle U_{t}}$ is the minimal capital requirement to hedge ${\displaystyle X}$ from time ${\displaystyle t}$ to the expiration date.^[1]

References