Option pricing with using Levy process and comparison it with Black-Scholes model in Hilbert space

The Black-Scholes model is based on smooth function in continuous time rang, not allowing jumps in stock movements. However in actuality, stock price does jumps, and some risks cannot be handled within continuous-path models. The Exponential Levy model is a choice to include jumps allowing more accurate representation of the market movements. Levy process tenders a more realistic model of price dynamics than Black-Scholes model. It’s obvious that the Levy process model is more difficult to implement and involves more computations compared to the Black-Scholes model. Thus ,the question is whether it is worth to implement a Levy process model.We summarize that the Levy process model does have certain advantages over the Black-Scholes model.