Lévy processes in Asset Pricing
نویسنده
چکیده
The main empirical motivation of using Lévy processes in finance comes from fitting asset return distributions. Consider the daily (either continuous or simple) returns of S&P 500 index (SPX) from Jan 2, 1980 to Dec 31, 2005. We plot the histogram of normalized (mean zero and variance one) daily simple returns in Figure 1, along with the standard normal density function. The max and min (which all occurred in 1987) of the normalized daily returns are about 7.9967 and -21.1550. Note that for a standard normal random variable Z, P (Z < −21.1550) ≈ 1.4∗10−107; as a comparison the whole universe is believed to have existed for 15 billion years or 5 ∗ 1017 seconds.
منابع مشابه
Option pricing using multivariate Lévy processes
For d-dimensional Lévy models we provide a method for Finite Element-based asset pricing. We derive the partial integrodifferential pricing equation and prove that the corresponding variational problem is well-posed. Hereto, an explicit characterization of the domain of the bilinear form is given. For the numerical implementation the problem is discretized by sparse tensor product Finite Elemen...
متن کاملPure Jump Lévy Processes for Asset Price Modelling
The goal of the paper is to show that some types of Lévy processes such as the hyperbolic motion and the CGMY are particularly suitable for asset price modelling and option pricing. We wish to review some fundamental mathematic properties of Lévy distributions, such as the one of infinite divisibility, and how they translate observed features of asset price returns. We explain how these process...
متن کاملTime-Changed Lévy Processes and Option Pricing
The classic Black-Scholes option pricing model assumes that returns follow Brownian motion, but return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. Time-change...
متن کاملAn equilibrium asset pricing model based on Lévy processes: relations to stochastic volatility, and the survival hypothesis
This paper presents some security market pricing results in the setting of a security market equilibrium in continuous time. The model consists in relaxing the distributional assumptions of asset returns to a situation where the underlying random processes modeling the spot prices of assets are exponentials of Lévy processes, the latter having normal inverse Gaussian marginals, and where the ag...
متن کاملModel-based Reinforcement Learning in Modified Lévy Jump-Diffusion Markov Decision Model and Its Financial Applications
This thesis intends to address an important cause of the 2007-2008 financial crisis by incorporating prediction on asset pricing jumps in asset pricing models, the non-normality of asset returns. Several different machine learning techniques, including the Unscented Kalman Filter and Approximate Planning are used, and an improvement in Approximate Planning is developed to improve algorithm time...
متن کامل