Asymmetric drift and halo flattening in disk galaxies

The main result in this work is the solution of the Jeans equations for an axisymmetric galaxy model containing a baryonic component (distributed according to a Miyamoto-Nagai profile) and a dark matter halo (described by the Binney logarithmic potential). The velocity dispersion, azimuthal velocity and some other interesting quantities such as the asymmetric drift are studied, along with the influence of the model parameters on these (observable) quantities. We also give an estimate for the velocity of the radial flow, caused by the asymmetric drift. Other than the mathematical beauty that lies in solving a model analytically, the interest of this kind of results can be mainly found in numerical simulations that study the evolution of gas flows. For example, it is important to know how certain parameters such as the shape (oblate, prolate, spherical) of a dark matter halo, or the flattening of the baryonic matter, or the mass ratio between dark and baryonic matter, have an influence on observable quantities such as the velocity dispersion. In the introductory chapter, we discuss the Jeans equations, which provide information about the velocity dispersion of a system. Next we will consider some dynamical quantities that will be useful in the rest of the work, e.g. the asymmetric drift. In Chapter 2 we discuss in some more detail the family of galaxy models we studied. In Chapter 3 we give the solution of the Jeans equations. Chapter 4 describes and illustrates the behaviour of the velocity dispersion, as a function of the several parameters, along with asymptotic expansions. In Chapter 5 we will investigate the behaviour of certain dynamical quantities for this model. We conclude with a discussion in Chapter 6.

Forward implied volatility expansions in LSV models

In this work we address the problem of finding formulas for efficient and reliable analytical approximation for the calculation of forward implied volatility in LSV models, a problem which is reduced to the calculation of option prices as an expansion of the price of the same financial asset in a Black-Scholes dynamic. Our approach involves an expansion of the differential operator, whose solution represents the price in local stochastic volatility dynamics. Further calculations then allow to obtain an expansion of the implied volatility without the aid of any special function or expensive from the computational point of view, in order to obtain explicit formulas fast to calculate but also as accurate as possible.

Orthogonal gamma-based expansions for volatility option prices under jump-diffusion dynamics

In my work I derive closed-form pricing formulas for volatility based options by suitably approximating the volatility process risk-neutral density function. I exploit and adapt the idea, which stands behind popular techniques already employed in the context of equity options such as Edgeworth and Gram-Charlier expansions, of approximating the underlying process as a sum of some particular polynomials weighted by a kernel, which is typically a Gaussian distribution. I propose instead a Gamma kernel to adapt the methodology to the context of volatility options. VIX vanilla options closed-form pricing formulas are derived and their accuracy is tested for the Heston model (1993) as well as for the jump-diffusion SVJJ model proposed by Duffie et al. (2000).