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\abstract{\small
In this thesis we present a new model for pricing European call options in presence of proportional transaction costs, 
when the stock price follows a general exponential Lévy process.  
The model is a generalization of the celebrated work of Davis, Panas and Zariphopoulou, where the value of the option is defined as the utility indifference price. 
This approach requires the solution of a stochastic singular control problem in finite time. 
We introduce the general formulation of the problem, and derive the associated Hamilton-Jacobi-Bellman equation (HJB), 
which is a nonlinear partial integro-differential equation, with the form of a variational inequality.  
We prove that the value function of the problem is a solution of the HJB equation in the viscosity sense.
The original problem is then simplified for the specific case of the exponential utility function, under the assumption of absence of default for the investor's portfolio. 
We solve numerically the optimization problems using the Markov chain approximation method. 
We also apply the multinomial method to the Variance Gamma process, which is an alternative and more efficient approach to discretize the continuous time process.  
We provide a numerical scheme and prove that it is monotone, stable and consistent and that the solution converges to the viscosity solution of the original HJB equation. 
Several numerical solutions are presented for both the original problem and the simplified problem. 
Numerical results are obtained for the cases of diffusion, Merton and Variance Gamma processes. 
We provide convergence and time complexity analysis and comparisons with option prices computed using the standard martingale pricing theory.

 \vspace{1em}
 \noindent \textbf{Keywords: }  option pricing, Lévy processes, transaction costs, Markov chain approximation, singular control.
}
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\input{./Cap4_in.tex}
\input{./Cap5_in.tex}
\input{./Cap6_in.tex}	
\input{./Conclusions}




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\bibliographystyle{apalike}
%\bibliography{/home/nicola/Documenti/ISEG/Bibliografia/trans_cost.bib,/home/nicola/Documenti/ISEG/Bibliografia/book.bib,/home/nicola/Documenti/ISEG/Bibliografia/fin_math.bib,/home/nicola/Documenti/ISEG/Bibliografia/viscosity.bib,/home/nicola/Documenti/ISEG/Bibliografia/num_meth.bib,/home/nicola/Documenti/ISEG/Bibliografia/phd_thesis.bib,/home/nicola/Documenti/ISEG/Bibliografia/Levy.bib,/home/nicola/Documenti/ISEG/Bibliografia/control.bib}
\bibliography{Bibliography/trans_cost.bib,Bibliography/book.bib,Bibliography/fin_math.bib,Bibliography/viscosity.bib,Bibliography/num_meth.bib,Bibliography/phd_thesis.bib,Bibliography/Levy.bib,Bibliography/control.bib}
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