Option Pricing with Log-stable Lévy Processes (original) (raw)
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Pricing of options on stocks driven by multi-dimensional operator stable Levy processes
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Time-changed Lévy processes and option pricing
Journal of Financial Economics, 2004
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-changed L! evy processes can simultaneously address these three issues. We show that our framework encompasses almost all of the models proposed in the option pricing literature, and it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
Time-Changed Levy Process and Option Pricing
SSRN Electronic Journal, 2000
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-changed Lévy processes can simultaneously address these three issues. We show that our framework encompasses almost all of the models proposed in the option pricing literature, and it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
Financial market models with Lévy processes and time-varying volatility
Journal of Banking & Finance, 2008
Asset management and pricing models require the proper modeling of the return distribution of financial assets. While the return distribution used in the traditional theories of asset pricing and portfolio selection is the normal distribution, numerous studies that have investigated the empirical behavior of asset returns in financial markets throughout the world reject the hypothesis that asset return distributions are normally distribution. Alternative models for describing return distributions have been proposed since the 1960s, with the strongest empirical and theoretical support being provided for the family of stable distributions (with the normal distribution being a special case of this distribution). Since the turn of the century, specific forms of the stable distribution have been proposed and tested that better fit the observed behavior of historical return distributions. More specifically, subclasses of the tempered stable distribution have been proposed. In this paper, we propose one such subclass of the tempered stable distribution which we refer to as the "KR distribution". We empirically test this distribution as well as two other recently proposed subclasses of the tempered stable distribution: the Carr-Geman-Madan-Yor (CGMY) distribution and the modified tempered stable (MTS) distribution. The advantage of the KR distribution over the other two distributions is that it has more flexible tail parameters. For these three subclasses of the tempered stable distribution, which are infinitely divisible and have exponential moments for some neighborhood of zero, we generate the exponential Lévy market models induced from them. We then construct a new GARCH model with the infinitely divisible distributed innovation and three subclasses of that GARCH model that incorporates three observed properties of asset returns: volatility clustering, fat tails, and skewness. We formulate the algorithm to find the risk-neutral return processes for those GARCH models using the "change of measure" for the tempered stable distributions. To compare the performance of those exponential Lévy models and the GARCH models, we report the results of the parameters estimated for the S&P 500 index and investigate the out-of-sample forecasting performance for those GARCH models for the S&P 500 option prices.
Option pricing with time-changed Lévy processes
Applied Financial Economics, 2013
In this paper, we introduce two new six-parameter processes based on time-changing tempered stable distributions and develop an option pricing model based on these processes. This model provides a good fit to observed option prices. To demonstrate the advantages of the new processes, we conduct two empirical studies to compare their performance to other processes that have been used in the literature.
On the Determination of the Levy Exponent in Asset Pricing Models
International Journal of Theoretical and Applied Finance, 2019
We consider the problem of determining the Lévy exponent in a Lévy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure [Formula: see text], consists of a pricing kernel [Formula: see text] together with one or more non-dividend-paying risky assets driven by the same Lévy process. If [Formula: see text] denotes the price process of such an asset, then [Formula: see text] is a [Formula: see text]-martingale. The Lévy process [Formula: see text] is assumed to have exponential moments, implying the existence of a Lévy exponent [Formula: see text] for [Formula: see text] in an interval [Formula: see text] containing the origin as a proper subset. We show that if the prices of power-payoff derivatives, for which the payoff is [Formula: see text] for some time [Formula: see text], are given at time [Formula: see text] for a range of values of [Formula: see text], where [Formula: see text] is the so-called benchmark portfolio define...
Lévy processes for financial modeling
EIGHTH INTERNATIONAL CONFERENCE NEW TRENDS IN THE APPLICATIONS OF DIFFERENTIAL EQUATIONS IN SCIENCES (NTADES2021), 2022
The article considers different types of Lévy processes, their properties, methodology for their analysis, and their applications for financial modeling. Lévy processes admit jumps. Financial models based on Lévy processes with jumps are mainly two types. In the first type, called jump-diffusion models, the normal evolution of prices is given by a diffusion process, punctuated by jumps at random intervals. Here the jumps represent rare events. The second type consists of models with infinite number of jumps in every interval which we will call infinite activity Lévy processes. In these models, one does not need to introduce a Gaussian (Brownian) component since the dynamics of jumps is rich enough to generate nontrivial small time behavior. So very often the Gaussian part is not considered and the processes are called non-Gaussian Lévy processes. The use of the exponential Lévy processes for option pricing is considered as well as some alternatives based on telegraph processes. Some new proposals, including ST-tempering and ST-subordinating, are presented.
Probability Surveys, 2021
We propose a class of financial models in which the prices of assets are Lévy-Ito processes driven by Brownian motion and a dynamic Poisson random measure. Each such model consists of a pricing kernel, a money market account, and one or more risky assets. The Poisson random measure is associated with an n-dimensional Lévy process. We show that the excess rate of return of a risky asset in a pure-jump model is given by an integral of the product of a term representing the riskiness of the asset and a term representing the level of market risk aversion. The integral is over the state space of the Poisson random measure and is taken with respect to the Lévy measure associated with the n-dimensional Lévy process. The resulting framework is applied to a variety of different asset classes, allowing one to construct new models as well as non-trivial generalizations of familiar models.