Superhedging price

If the set of equivalent martingale measures is denoted by EMM then the superhedging price of a portfolio X is ${\displaystyle \rho (-X)}$ where ${\displaystyle \rho }$ is defined by

${\displaystyle \rho (X)=\sup _{Q\in \mathrm {EMM} }\mathbb {E} ^{Q}[-X]}$.

${\displaystyle \rho }$ defined as above is a coherent risk measure.[2]

The acceptance set for the superhedging price is the negative of the set of values of a self-financing portfolio at the terminal time. That is

${\displaystyle A=\{-V_{T}:(V_{t})_{t=0}^{T}{\text{ is the price of a self-financing portfolio at each time}}\}}$.

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 ${\displaystyle \inf _{Q\in \mathrm {EMM} }\mathbb {E} ^{Q}[X]}$. It is obvious to see that this is the negative of the superhedging price of the negative of the initial claim (${\displaystyle -\rho (X)}$). In a complete market then the supremum and infimum are equal to each other and a unique hedging price exists.[3] The upper and lower bounds created by the subhedging and superhedging prices respectively are the no-arbitrage bounds, an example of good-deal bounds.[4][5]

The dynamic superhedging price has conditional risk measures of the form:

${\displaystyle \rho _{t}(X)=\operatorname {ess\,sup} _{Q\in EMM}\mathbb {E} ^{Q}[-X|{\mathcal {F}}_{t}]}$

where ${\displaystyle \operatorname {ess\,sup} }$ denotes the essential supremum. It is a widely shown result that this is time consistent.[6]