Snell envelope explained

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

(\Omega, l{F},( l{F} _{t)} _{t} , P ) and an absolutely continuous probability measure Q \ll P then an adapted process U = (U _{t)} _{t} is the Snell envelope with respect to Q of the process X = (X _{t)} _{t} if U is a Q -supermartingaleU dominates X , i.e. U _{t} \geq X _{t} Q -almost surely for all times t \in [ 0 ,T]

1. If

V = (V _{t)} _{t} is a Q -supermartingale which dominates X , then V dominates U .^[1]

Construction

(\Omega, l{F},( l{F} _{n)} ^{N, P )} ^{n =0} and an absolutely continuous probability measure Q \ll P then the Snell envelope (U _{n)} ^{N} ^{n =0} with respect to Q of the process (X _{n)} ^{N} ^{n =0} is given by the recursive schemeU _{N} : = X _{N,} U _{n} : = X _{n} \lor E ^{Q} [U _{n+1} \mid l{F} _{n]} for n = N -1 ,..., 0 where \lor 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.

Notes and References

1. Book: Hans. Föllmer. Alexander. Schied. Stochastic finance: an introduction in discrete time. Walter de Gruyter. 2004. 2. 9783110183467. 280–282.