Financial modeling with jump processes is a fully explored domain. We present in this post some of the theoretical aspects of Lévy processes. We also expose a subset of jump diffusion models available into LexiFi Apropos. Underlying dynamics, European call pricing and calibration procedures are developed for the Merton and Bates models. For the sake of clarity, some theoretical aspects and figures are hidden, don't hesitate to expand them for having more information.
Building the (zero-coupon) yield curve is the first step in most of the financial modeling problems. We will present in this post how the yield curve is computed into LexiFi Apropos, and how we make our algorithm generic so that it can easily be used to compute different kinds of yield curves or solve different problems. Simple formulation of the problem We assume a simplified market where we can observe the Deposit Rates and the Swap Rates.
Classical equity models don't handle very well dividends, in particular constant ones. We present in this post an approach to model equities with (deterministic) dividends. In a first time, we will show the necessary form of an equity in order there is no arbitrage with its forward. Then, we will use this form to extend the classical models to handle constant dividends. To finish, we will see how it work on an important example: the local volatility model.
Poisson distribution is a classical distribution that often appears in mathematical finance, like in jump diffusion. In this article, I will present an efficient simulation method under the constraint to use only one random variate per simulation.
Partial Differential Equation (PDE) solving is an important part of numerical analysis in mathematical finance. The classical approach to solve PDE that appears in finance is to use finite difference method. As most of numerical analysis problems, the goal is to find a solution with an error that converges quickly. The academic way to achieve this goal is to use Crank-Nicolson to solve our PDE, which provides an "order 2"
As my first subject (and the first subject on LexiFi's quantitative blog), I would like to focus on something I particularly appreciate at LexiFi: our formal contract description, that allows one to treat quantitative topics in a surprisingly elegant and easily generalisable way. I will focus on a particular topic: control variates methods. In the first section, I will introduce the underlying theory. Then, I will explain its use in mathematical finance.