This paper gives a theorem for the continuous time super-replication cost of European options in an unbounded multinomial market. An approximation multinomial scheme is put forward on a finite time interval [0,1] corresponding to a pure jump Levy model with unbounded jumps. Under the assumption that the expected underlying stock price at time 1 is bounded, the limit of the sequence of the super-replication cost in a multinomial model is proved to be greater than or equal to an optimal control problem. Furthermore, it is discussed that the existence conditions of a super-replication cost and a liquidity premium for the multinomial model. This paper concentrates on a multinomial tree with unbounded jumps, which can be seen as an extension of the work of(Xing, 2015). The super-replication cost and the liquidity premium under the variance gamma model and the normal inverse Gaussian model are calculated and illustrated.
Xing, Mei, "Existence Conditions of Super-Replication Cost in a Multinomial Model" (2017). CUNY Academic Works.