Girish Mahajan (Editor)

Snell envelope

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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.

Contents

Definition

Given a filtered probability space ( Ω , F , ( F t ) t ∈ [ 0 , T ] , P ) and an absolutely continuous probability measure Q ≪ P then an adapted process U = ( U t ) t ∈ [ 0 , T ] is the Snell envelope with respect to Q of the process X = ( X t ) t ∈ [ 0 , T ] if

  1. U is a Q -supermartingale
  2. U dominates X , i.e. U t ≥ X t Q -almost surely for all times t ∈ [ 0 , T ]
  3. If V = ( V t ) t ∈ [ 0 , T ] is a Q -supermartingale which dominates X , then V dominates U .

Construction

Given a (discrete) filtered probability space ( Ω , F , ( F n ) n = 0 N , P ) and an absolutely continuous probability measure Q ≪ P then the Snell envelope ( U n ) n = 0 N with respect to Q of the process ( X n ) n = 0 N is given by the recursive scheme

U N := X N , U n := X n ∨ E Q [ U n + 1 ∣ F n ] for n = N − 1 , . . . , 0

where ∨ is the join.

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.

    • References

    Snell envelope Wikipedia
    (Text) CC BY-SA


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