DYNAMIC NO-GOOD-DEAL BOUNDS AND PRICING MEASURES

The fundamental theorem of asset pricing provides a framework for pricing based on the marriage between the economic principle of no-arbitrage and the mathematical tools of martingales. A crucial observation is that, provided existence, there is no uniqueness of equivalent martingale measure guaranteed (with the exception of markets that are complete).
In the more realistic case of incompleteness, the problem of selecting one equivalent martingale measure out of the infinite many available has been largely treated.
More recently another idea was developed: instead of selecting a single measure, one can restrict the set of equivalent martingale measures characterizing those that are in some sense reasonable . In this talk we will consider the restriction to equivalent martingale measures that rule out deals that are too good to be true and we will focus on linear price systems in that are consistent with lower and upper bounds set on the Sharpe ratio.
The results show the existence of an equivalent martingale measure that both represents the fair prices and guarantees the non availability of deals that are too good to be true. This study is originally framed in a static set-up. However with our methods we can define the corresponding concept of no-good-deal bounds for a dynamic set-up (both multiperiod and continuous) and hence study the pricing measures for such frameworks. This extension to dynamic no-good-deal bounds is non-trivial as it requires a new, but equivalent, representation of the Sharpe ratio bounds.
The talk is based on joint work with Jocelyne Bion-Nadal (Ecole Polytechnique).