Superhedging price

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_{Q \in \mathrm{EMM}} \mathbb{E}^Q[-X]. \rho defined as above is a coherent risk measure.

Acceptance set

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

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_{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 (-\rho(X)). In a complete market then the supremum and infimum 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, an example of good-deal bounds.

Dynamic superhedging price

The dynamic superhedging price has conditional risk measures of the form: :\rho_t(X) = \operatorname{ess,sup}_{Q \in EMM} \mathbb{E}^Q[-X | \mathcal{F}_t] where \operatorname{ess,sup} denotes the essential supremum. It is a widely shown result that this is time consistent.

References

References

1. "Dynamic Replication".
2. (October 8, 2008). "Convex and Coherent Risk Measures".
3. Lei (Nick) Guo. (August 23, 2006). "Pricing and hedging in incomplete markets".
4. John R. Birge. (2008). "Financial Engineering". Elsevier.
5. (2011). "Convex risk measures for good deal bounds".
6. Penner, Irina. (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability".