Tesi di LAUREA SPECIALISTICA Titolo Calcolo di Malliavin e Copertura di Opzioni Asiatiche Data 2011-03-31 Autore/i Papagno, Marco Relatore Sgarra, C. Full text non disponibile Abstract The aim of this paper is to propose a method to cover and pricing Asian options, using the Malliavin calculus. Before doing so, we will show various methods developed over the years to cover Asian Options. In this overview, chapter one, we would prefer to focus on its strengths and weaknesses, limitations and possible developments of the methods listed, rather than a rigorous formalism to demonstrate results. The first class of methods that we analyze will be Monte-Carlo methods for derivation of which see differential methods, methods based on the path, and using maximum likelihood estimators. Another class of methods that we shall see, the analytic, which divide into comonotonic approximation, integral representation and approximation of Taylor. The last class of methods that we find in this chapter are those based on finite differences. More precisely, we will discuss semi-analytical method of Zhang. In chapter two we will enter instead into a detailed discussion of the Malliavin calculus, in particular, show the concept of Malliavin derivative, and demonstrate some of its properties that will serve us later in our work. The most important result, that we will meet in the chapter, will be the Clark-Ocone formula. We will play as an example of application, the calculation of the Greeks of an Asian options with arithmetic average. In the third chapter we will enter into the heart of the work. Apply the results previously obtained to be able to reach a quasi-explicit formulation of price and hedging strategy in arithmetic average Asian options. We will find two possible ways: a formula solved by using a triple integral, and another by solving a partial differential equation. In the fourth chapter we will implement some programs with which we will discuss which method seems best. To solve the EDP we will use a finite difference method: the techniques we will apply to our equation for calculating are UPWIND and THETA-METHOD. In the fifth chapter retrace the same steps carried out in chapter three, this time trying to arrive at a method for pricing and hedging Asian option with geometric average. In this case, however, we will encounter more obstacles that force us to abandon the way of analytical methods and exploit the Monte-Carlo. This is because we are not able to express a probability explicitly or quasi-explicitly. We will use the antithetic variable method to reduce the variance of the problem. In the end we will be able to assess and compare between them, dynamic hedging strategies for Asian options with average rate.