Abstract

This paper gives a theorem for the continuous time super-replication cost of European options where the stock price follows an exponential L\'{e}vy 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 L\'{e}vy 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 L\'{e}vy process.
This paper concentrates on the L\'{e}vy processes with infinitely many jumps in any finite time interval.
The approach overcomes some difficulties that can be encountered when the L\'{e}vy process has infinite activity.

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