Finance: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 [math]\displaystyle{ (\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,T]},\mathbb{P}) }[/math] and an absolutely continuous probability measure [math]\displaystyle{ \mathbb{Q} \ll \mathbb{P} }[/math] then an adapted process [math]\displaystyle{ U = (U_t)_{t \in [0,T]} }[/math] is the Snell envelope with respect to [math]\displaystyle{ \mathbb{Q} }[/math] of the process [math]\displaystyle{ X = (X_t)_{t \in [0,T]} }[/math] if
- [math]\displaystyle{ U }[/math] is a [math]\displaystyle{ \mathbb{Q} }[/math]-supermartingale
- [math]\displaystyle{ U }[/math] dominates [math]\displaystyle{ X }[/math], i.e. [math]\displaystyle{ U_t \geq X_t }[/math] [math]\displaystyle{ \mathbb{Q} }[/math]-almost surely for all times [math]\displaystyle{ t \in [0,T] }[/math]
- If [math]\displaystyle{ V = (V_t)_{t \in [0,T]} }[/math] is a [math]\displaystyle{ \mathbb{Q} }[/math]-supermartingale which dominates [math]\displaystyle{ X }[/math], then [math]\displaystyle{ V }[/math] dominates [math]\displaystyle{ U }[/math].[1]
Construction
Given a (discrete) filtered probability space [math]\displaystyle{ (\Omega,\mathcal{F},(\mathcal{F}_n)_{n = 0}^N,\mathbb{P}) }[/math] and an absolutely continuous probability measure [math]\displaystyle{ \mathbb{Q} \ll \mathbb{P} }[/math] then the Snell envelope [math]\displaystyle{ (U_n)_{n = 0}^N }[/math] with respect to [math]\displaystyle{ \mathbb{Q} }[/math] of the process [math]\displaystyle{ (X_n)_{n = 0}^N }[/math] is given by the recursive scheme
- [math]\displaystyle{ U_N := X_N, }[/math]
- [math]\displaystyle{ U_n := X_n \lor \mathbb{E}^{\mathbb{Q}}[U_{n+1} \mid \mathcal{F}_n] }[/math] for [math]\displaystyle{ n = N-1,...,0 }[/math]
where [math]\displaystyle{ \lor }[/math] is the join (in this case equal to the maximum of the two random variables).[1]
Application
- If [math]\displaystyle{ X }[/math] is a discounted American option payoff with Snell envelope [math]\displaystyle{ U }[/math] then [math]\displaystyle{ U_t }[/math] is the minimal capital requirement to hedge [math]\displaystyle{ X }[/math] from time [math]\displaystyle{ t }[/math] 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.
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