# Which infinite activity Levy process is the most popular for option pricing

Hey I heard about different Levy processes with infinite activity like VG, NIG, Meixner or CGMY process, but which proccesses are the most popular? And which processes can be simulated (as simple as possible) so that it is possible to price options using monte carlo simulations?

• If you use the representation via a gamma process, the VG process is the easiest one to simulate. It's a special case of the CGMY process. John Hull describes how you can simulate paths in Excel, so it can't be that difficult :) – Kevin Aug 31 '20 at 12:36
• @Kevin I see that you are good at Levy processes in finance. I have problem with defining stochastic integral with respect to stock price process $S_t=e^{X_t}$ where $X_t$ is a Levy process (I need it to self-financing strategy definition in my thesis). Protter's book has veeery strange definition of semimartingale and I have problem with understanding his idea :( So far I only know stochastic integral with respect to brownian motion and martingale, but semimartingale is a sum of local martingale and FV process. I have to define integral w.r.t local martingale or is there easiest approach? – Mr.Price Feb 21 at 18:59
• In general, every Lévy process $X_t$ is a semimartingale. Thus, so is $S_t=e^{X_t}$. To understand the meaning of $\text{d}S_t$ (for a general Lévy process), we need to know how to generalise the Itô integral to semimartingales. This is not a trivial task, but the aim of any stochastic calculus course. I don't think there's an easier way around it and you need to look at the sum of a local martingale and FV process if you want to dig into the theory :/ Is this still for your thesis? How far are you? :) – Kevin Feb 21 at 19:07
• Yes I need this for my thesis, could you recommend me some books in which stochastic integration w.r.t local martingales are well described? My thesis is done in 80% (I have done numeric part which was the most time demanding part), now I have to define this stochastic integral. – Mr.Price Feb 21 at 20:31
• Have a look at Cont and Tankov’s (2004?) book on financial modelling with Lévy professes. They describe it in detail and the book is well-written. I hope I get to read your MSc thesis after all the comments, questions and answers we exchanged! – Kevin Feb 21 at 23:00