  We prove limit theorems for the super-replication cost of European options in
a Binomial model with friction. The examples covered are markets with
proportional transaction costs and the illiquid markets. The dual
representation for the super-replication cost in these models are obtained and
used to prove the limit theorems. In particular, the existence of the liquidity
premium for the continuous time limit of the model proposed in [6] is proved.
Hence, this paper extends the previous convergence result of [13] to the
general non-Markovian case. Moreover, the special case of small transaction
costs yields, in the continuous limit, the $G$-expectation of Peng as earlier
proved by Kusuoka in [14].
