The superhedging price is a

coherent risk measure In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent ris ...

. The superhedging price of a

portfolio Portfolio may refer to: Objects * Portfolio (briefcase), a type of briefcase Collections * Portfolio (finance), a collection of assets held by an institution or a private individual * Artist's portfolio, a sample of an artist's work or a c ...

(A) is equivalent to the smallest amount necessary to be paid for an admissible portfolio (B) at the current time so that at some specified future time the value of B is at least as great as A. In a

complete market In economics, a complete market (aka Arrow-Debreu market or complete system of markets) is a market with two conditions: # Negligible transaction costs and therefore also perfect information, # there is a price for every asset in every possible st ...

the superhedging price is equivalent to the price for hedging the initial portfolio.

Mathematical definition

If the set of equivalent martingale measures is denoted by EMM then the superhedging price of a portfolio ''X'' is

\rho(-X)

where

\rho

is defined by :

\rho(X) = \sup_ \mathbb^Q X /math>. \rho defined as above is a coherent risk measure.

Acceptance set

The

acceptance set In financial mathematics, acceptance set is a set of acceptable future net worth which is acceptable to the regulator. It is related to risk measures. Mathematical Definition Given a probability space (\Omega,\mathcal,\mathbb), and letting L^p = L ...

for the superhedging price is the negative of the set of values of a

self-financing portfolio In financial mathematics, a self-financing portfolio is a portfolio having the feature that, if there is no exogenous infusion or withdrawal of money, the purchase of a new asset must be financed by the sale of an old one. Mathematical definitio ...

at the terminal time. That is :

A = \

Subhedging price

The subhedging price is the greatest value that can be paid so that in any possible situation at the specified future time you have a second portfolio worth less or equal to the initial one. Mathematically it can be written as

\inf_ \mathbb^Q /math>.  It is obvious to see that this is the negative of the superhedging price of the negative of the initial claim (-\rho(X)).  In a complete market then the supremum and

infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...

are equal to each other and a unique hedging price exists. The upper and lower bounds created by the subhedging and superhedging prices respectively are the

no-arbitrage bounds In financial mathematics, no-arbitrage bounds are mathematical relationships specifying limits on financial portfolio prices. These price bounds are a specific example of good–deal bounds, and are in fact the greatest extremes for good–deal bo ...

, an example of good-deal bounds.

Dynamic superhedging price

The dynamic superhedging price has

conditional risk measure In financial mathematics, a conditional risk measure is a random variable of the financial risk (particularly the downside risk) as if measured at some point in the future. A risk measure can be thought of as a conditional risk measure on the triv ...

s of the form: :

\rho_t(X) = \operatorname_ \mathbb^Q \mathcal_t /math>
where \operatorname denotes the

essential supremum In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...

. It is a widely shown result that this is

time consistent Time consistency in the context of finance is the property of not having mutually contradictory evaluations of risk at different points in time. This property implies that if investment A is considered riskier than B at some future time, then A wi ...

References

{{Reflist Financial risk modeling