  We introduce, in continuous time, an axiomatic approach to assign to any
financial position a dynamic ask (resp. bid) price process. Taking into account
both transaction costs and liquidity risk this leads to the convexity (resp.
concavity) of the ask (resp. bid) price. Time consistency is a crucial property
for dynamic pricing. Generalizing the result of Jouini and Kallal, we prove
that the No Free Lunch condition for a time consistent dynamic pricing
procedure (TCPP) is equivalent to the existence of an equivalent probability
measure $R$ that transforms a process between the bid process and the ask
process of any financial instrument into a martingale. Furthermore we prove
that the ask price process associated with any financial instrument is then a
$R$-supermartingale process which has a cadlag modification. Finally we show
that time consistent dynamic pricing allows both to extend the dynamics of some
reference assets and to be consistent with any observed bid ask spreads that
one wants to take into account. It then provides new bounds reducing the bid
ask spreads for the other financial instruments.
