Snell envelope

Last updated March 19, 2019

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.

Mathematical finance, also known as quantitative finance, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.

Martingale (probability theory) model in probability theory

In probability theory, a martingale is a sequence of random variables for which, at a particular time, the conditional expectation of the next value in the sequence, given all prior values, is equal to the present value.

Stochastic process mathematical object usually defined as a collection of random variables

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

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

Probability measure Measure of total value one, generalizing probability distributions

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure is that a probability measure must assign value 1 to the entire probability space.

In the study of stochastic processes, an adapted process is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, X_n is known at time n. The concept of an adapted process is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process.

$U$ 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 (in this case equal to the maximum of the two random variables).^[1]

In a partially ordered set P, the join and meet of a subset S are respectively the supremum of S, denoted ⋁S, and infimum of S, denoted ⋀S. In general, the join and meet of a subset of a partially ordered set need not exist; when they do exist, they are elements of P.

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]

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References

1 2 3 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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[FollmerSchied-1] 1 2 3 Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 280–282. ISBN 9783110183467.