Introduction.tex

\chapter{Introduction}\label{Introd}
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The problem of pricing a European call option was first solved mathematically in the paper of \cite{BS73}. 
Even if it is quite evident that this model is too simplistic to represent the real features of the market, it is 
still nowadays one of the most used models to price and hedge options.
The reason for its success is that it gives a closed form solution for the option price, and that the hedging strategy is easily 
implementable.
The Black-Scholes model considers a \emph{complete} market, i.e. a market where it is possible to create a portfolio containing cash 
and shares of the underlying stocks, such that following a particular trading strategy it is always possible to replicate
the payoff of the option. In this framework, this particular portfolio is called \emph{replicating portfolio} and
the trading strategy to hedge the option is called \emph{delta hedge}.
However, this model does not consider many features that characterize the real market. 

In the Black-Scholes model 
the stock price follows a geometric Brownian motion. This is equivalent to assume that the log-returns are 
normally distributed. 
However, a rigorous statistical analysis of financial data
reveals that the normality assumption is not a very good approximation of
reality (see \cite{Cont01}). Indeed, it is easy to see that empirical log-return distributions have
more mass around the origin and along the tails (\emph{heavy tails}).
This means that normal distribution underestimates the probability of large positive and negative
log-returns, and considers them just as rare events. In the real market instead,
log-returns manifest frequently high peaks, that come more and more evident
when looking at short time scales. The log-returns peaks correspond to sudden
large changes in the price, which are called \emph{jumps}. 
There is a huge literature of option pricing models considering an underlying process with discontinuous paths.
Most of these models assume the log-price dynamics to be a \emph{Lévy process}. 
 

A second issue of the Black-Scholes model is that it does not consider the presence of budget constraints and market frictions i.e.
bid/ask spread or transaction fees.
The securities in the market are traded with a bid-ask spread, and this means that there are two prices for the
same security. But the Black-Scholes formula just gives one price.
Moreover, the replicating portfolio cannot be perfectly implemented,
since the delta-hedging strategy involves continuous time trading. 
This is impractical because the presence of transaction costs makes it infinitely costly.
Another feature that needs to be considered is the presence of budget constraints in the real market. 
A limit in the available budget, or (most commonly) a restriction in the possibility of 
selling short, clearly restricts the set of hedging strategies of the investor.

Many authors attempted to include the presence of proportional transaction costs in option pricing models.
In \cite{Le85}, in order to avoid continuous trading, the author specifies 
a finite number of trading dates. He obtains a Black-Scholes-like
nonlinear partial differential equation (PDE) with an adjusted volatility term, that takes into account the transaction costs. 
However, trading at fixed dates is not optimal, and the option price goes to infinity as the number of dates grows.
Further work in this direction has been done by \cite{BoVo92}, which consider a multi-period binomial model (see \cite{CRR79})
with transaction costs. Here again, the cost of the replicating portfolio depends on the number of time periods. 
Recent developments in this direction are for instance \cite{Mocio07}, \cite{FlMaSe14} and \cite{Sengu14} 
who consider different features of the market such as jumps, stochastic volatility and stochastic interest rate respectively.  

A different approach has been introduced by \cite{HoNe89}. The authors use an alternative definition of the option price
called \emph{indifference price}, based on the concepts of \emph{expected utility} and \emph{certainty equivalent}.  
An overview of these concepts applied to several incomplete market models can be found in \cite{Carmona}.
As long as the perfect replicating portfolio is no longer implementable in presence of transaction costs, the 
hedging strategy cannot be anymore riskless. 
The model has to take into account the risk profile of the writer/buyer to describe his trading preferences.
\cite{HoNe89} define the option price as the value that makes an investor indifferent between holding a portfolio with an option
and without, in terms of expected utility of the final wealth.
They show that it is impossible to hedge perfectly the option. The optimal strategy is to keep the portfolio's values within
a band called \emph{no transaction region}. Using numerical experiments, they verify that this strategy outperforms the one 
proposed in \cite{Le85}.
This approach has been further developed in \cite{DaPaZa93}, where the problem is formulated rigorously as a singular 
stochastic optimal control problem. The authors prove that the value function of the optimization problem
can be interpreted as the solution of the associated Hamilton-Jacobi-Bellman (HJB) equation in the viscosity sense. 
They prove also that the solution of the proposed numerical scheme, based on the \emph{Markov chain approximation}, converges to the viscosity solution.
Numerical methods for this model are presented in \cite{DaPa94}, \cite{ClHo97} and \cite{Mon03}, \cite{Mon04}.
In \cite{WhWi97} and \cite{BaSo98} the problem is simplified by using asymptotic analysis methods for small levels of 
transaction costs. The authors, starting from the general HJB variational inequality, derive a simpler non-linear PDE for the option price. 
Further developments are presented in the thesis work of \cite{Damgaard}, where the author 
studies the robustness of the model from a theoretical and numerical point of view. 
He found that under certain conditions the model is quite robust with respect to the choice of the utility function. 

In this thesis the main goal is to develop and analyze a model for pricing options when the market is incomplete due to the presence of jumps in the stock dynamics 
and transaction costs. 
The topics of the thesis are based on the articles 
\cite{Canta2}, \cite{Canta}
and on the contributed chapter \cite{Canta3}, published in the book of \cite{Matthias}.\\
Portfolio selection models with transaction costs and Lévy processes have already been introduced in the financial literature, 
see for instance \cite{OkSu01}, \cite{BKR01} and \cite{Kab16}, 
but these models have never been used to price options.\\  
In this thesis we present a model that is a generalization of \cite{DaPaZa93}. 
We analyze the theoretical properties of the model, and prove the existence of a viscosity solution 
of the nonlinear Partial Integro-Differential Equation (PIDE) associated with the optimization problem. 
We also develop numerical methods to solve the problem under different assumptions, i.e. ignoring the 
default event and not.
We present new numerical results and compare them with numerical values obtained from standard reference models. We also prove that 
the proposed numerical scheme is monotone, stable and consistent, and its solution converges to the viscosity solution of the continuous problem.

In Chapter \ref{Chapter1}, we introduce the general theory of Lévy processes, with a deeper focus on the specific Lévy processes used in this thesis i.e.  
the \emph{Merton} and \emph{Variance Gamma} processes.   
In Chapter \ref{Chapter2} we make a small summary of the main assumptions and theorems of the \emph{No arbitrage theory} for derivative pricing. After that we present 
the most common numerical finite differences methods used to solve the option pricing PIDEs. 
The Chapter \ref{Chapter3} is a digression based on the paper \cite{Canta2} on the applications 
of the multinomial method to solve option pricing problems under the Variance Gamma model. 
In Chapter \ref{Chapter4} we present the optimal control theory for regular and singular controls, and the definition of viscosity solution. 
In Chapter \ref{Chapter5}, we will develop the model for option pricing in presence of proportional transaction costs. This model is a singular stochastic
control problem, which is a generalization of the model proposed in \cite{DaPaZa93}. We derive the HJB equation associated with the optimization problem, 
and prove that the value function can be interpreted as the viscosity solution of this HJB equation.  
In Chapter \ref{Chapter6} we describe the numerical methods used to solve the optimization problems introduced in Chapter \ref{Chapter5}, and present several numerical results. 
In the first part of the chapter, we consider the simplified problem where the number of variables is reduced by one, 
thanks to the assumption of no default and the property of the exponential utility. 
These topics are also analyzed in \cite{Canta} and \cite{Canta3}.\\
In the end of the chapter, we solve the general problem with four variables, considering the possibility of default. 
We also show that the moment matching method developed for 
the Variance Gamma process, can be applied to these kind of problems with good performances. \\
At the end of each chapter we provide a conclusive section containing a summary of the main presented concepts, and their relevance for the thesis.