This paper gives a theorem for the continuous time super-replication cost of European options where the stock price follows an exponential Levy process. Under a mild assumption on the legend transform of the trading cost function, the limit of the sequence of the discrete super-replication cost is proved to be greater than or equal to an optimal control problem. The main tool is an approximation multinomial scheme based on a discrete grid on a finite time interval [0,1] for a pure jump Levy model. This multinomial model is constructed similar to that proposed by (Szimayer & Maller, Stoch. Proce. & Their Appl., 117, 1422-1447, 2007). Furthermore, it is proved that the existence of a liquidity premium for the continuous-time model under a Levy process. This paper concentrates on the Levy processes with infinitely many jumps in any finite time interval. The approach overcomes some difficulties that can be encountered when the Levy process has infinite activity.
Xing, Mei, "Liquidity Premiums in a Levy Market" (2015). CUNY Academic Works.