The explicit Laplace transform for the Wishart process

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.Comment: Accepted on: Journal of Applied Probability 51(3), 201

The Wishart short rate model

We consider a short rate model, driven by a stochastic process on the cone of positive semidefinite matrices. We derive sufficient conditions ensuring that the model replicates normal, inverse or humped yield curves

Optimal portfolios for financial markets with Wishart volatility

We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain explicitly the optimal portfolio strategy and the value function in some parameter settings. In particular when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is indeed to identify when the solution of the HJB equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the in uence of the investors\u27 risk aversion on the hedging demand

Mathematics of Quantitative Finance

The workshop on Mathematics of Quantitative Finance, organised at the Mathematisches Forschungsinstitut Oberwolfach from 26 February to 4 March 2017, focused on cutting edge areas of mathematical finance, with an emphasis on the applicability of the new techniques and models presented by the participants

The asymptotic behavior of the term structure of interest rates

In this dissertation we investigate long-term interest rates, i.e. interest rates with maturity going to infinity, in the post-crisis interest rate market. Three different concepts of long-term interest rates are considered for this purpose: the long-term yield, the long-term simple rate, and the long-term swap rate. We analyze the properties as well as the interrelations of these long-term interest rates. In particular, we study the asymptotic behavior of the term structure of interest rates in some specific models. First, we compute the three long-term interest rates in the HJM framework with different stochastic drivers, namely Brownian motions, Lévy processes, and affine processes on the state space of positive semidefinite symmetric matrices. The HJM setting presents the advantage that the entire yield curve can be modeled directly. Furthermore, by considering increasingly more general classes of drivers, we were able to take into account the impact of different risk factors and their dependence structure on the long end of the yield curve. Finally, we study the long-term interest rates and especially the long-term swap rate in the Flesaker-Hughston model and the linear-rational methodology

Modeling of volatility-linked financial products

This thesis is the collation of four papers, adapted from their original versions as to form here four distinct chapters. In the first chapter we illustrate and solve the pricing problem of a target volatility option (TVO) using three different methodologies. In the second chapter we study the pricing PDE for a general contingent claim involving an asset and its realized volatility, and then solve it for a variety of actual models and payoffs. The third chapter introduces a class of time-changed stochastic processes based on which a martingale asset price evolution can be devised. Pricing equations for volatility-linked derivatives are also obtained in this framework. In the final chapter we analyze one specific model of this class; we conclude that it does show high flexibility in explaining the forward volatility skew dynamics and that it can capture certain interesting stylized facts