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Lévy Process (lévy + process)

Distribution by Scientific Domains

Business, Economics, Finance and Accounting	73%

Selected Abstracts

Time Changes for Lévy Processes

MATHEMATICAL FINANCE, Issue 1 2001
Hélyette Geman
The goal of this paper is to consider pure jump Lévy processes of finite variation with an infinite arrival rate of jumps as models for the logarithm of asset prices. These processes may be written as time-changed Brownian motion. We exhibit the explicit time change for each of a wide class of Lévy processes and show that the time change is a weighted price move measure of time. Additionally, we present a number of Lévy processes that are analytically tractable, in their characteristic functions and Lévy densities, and hence are relevant for option pricing. [source]

Specification Analysis of Option Pricing Models Based on Time-Changed Lévy Processes

THE JOURNAL OF FINANCE, Issue 3 2004
Jing-zhi Huang
We analyze the specifications of option pricing models based on time-changed Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. Our estimation of a variety of model specifications indicates that to better capture the behavior of the S&P 500 index options, we need to incorporate a high frequency jump component in the return process and generate stochastic volatilities from two different sources, the jump component and the diffusion component. [source]

LOCAL WELL-POSEDNESS OF MUSIELA'S SPDE WITH LÉVY NOISE

MATHEMATICAL FINANCE, Issue 3 2010
Carlo Marinelli
We determine sufficient conditions on the volatility coefficient of Musiela's stochastic partial differential equation driven by an infinite dimensional Lévy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight. [source]

OPTIMAL CONTINUOUS-TIME HEDGING WITH LEPTOKURTIC RETURNS

MATHEMATICAL FINANCE, Issue 2 2007

We examine the behavior of optimal mean,variance hedging strategies at high rebalancing frequencies in a model where stock prices follow a discretely sampled exponential Lévy process and one hedges a European call option to maturity. Using elementary methods we show that all the attributes of a discretely rebalanced optimal hedge, i.e., the mean value, the hedge ratio, and the expected squared hedging error, converge pointwise in the state space as the rebalancing interval goes to zero. The limiting formulae represent 1-D and 2-D generalized Fourier transforms, which can be evaluated much faster than backward recursion schemes, with the same degree of accuracy. In the special case of a compound Poisson process we demonstrate that the convergence results hold true if instead of using an infinitely divisible distribution from the outset one models log returns by multinomial approximations thereof. This result represents an important extension of Cox, Ross, and Rubinstein to markets with leptokurtic returns. [source]

A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELS

MATHEMATICAL FINANCE, Issue 4 2006
Ross A. Maller
This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity. [source]

Optimal auditing in the banking industry

OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 2 2008
T. Bosch
Abstract As a result of the new regulatory prescripts for banks, known as the Basel II Capital Accord, there has been a heightened interest in the auditing process. Our paper considers this issue with a particular emphasis on the auditing of reserves, assets and capital in both a random and non-random framework. The analysis relies on the stochastic dynamic modeling of banking items such as loans, reserves, Treasuries, outstanding debts, bank capital and government subsidies. In this regard, one of the main novelties of our contribution is the establishment of optimal bank reserves and a rate of depository consumption that is of importance during an (random) audit of the reserve requirements. Here the specific choice of a power utility function is made in order to obtain an analytic solution in a Lévy process setting. Furthermore, we provide explicit formulas for the shareholder default and regulator closure rules, for the case of a Poisson-distributed random audit. A property of these rules is that they define the standard for minimum capital adequacy in an implicit way. In addition, we solve an optimal auditing time problem for the Basel II capital adequacy requirement by making use of Lévy process-based models. This result provides information about the optimal timing of an internal audit when the ambient value of the capital adequacy ratio is taken into account and the bank is able to choose the time at which the audit takes place. Finally, we discuss some of the economic issues arising from the analysis of the stochastic dynamic models of banking items and the optimization procedure related to the auditing process. Copyright © 2007 John Wiley & Sons, Ltd. [source]

A risk model driven by Lévy processes

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 2 2003
Manuel Morales
Abstract We present a general risk model where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a Lévy process into the model. This seeks to account for the discrete nature of claims and asset prices. We give several explicit examples of Lévy processes that can be used to drive a risk model. This allows us to incorporate aggregate claims and premium fluctuations in the same process. We discuss important features of such processes and their relevance to risk modeling. We also extend classical results on ruin probabilities to this model. Copyright © 2003 John Wiley & Sons, Ltd. [source]

THE VALUE OF INFORMATION IN STOCHASTIC CONTROL AND FINANCE

AUSTRALIAN ECONOMIC PAPERS, Issue 4 2005
BERNT ØKSENDAL
We present an optimal portfolio problem with logarithmic utility in the following three cases: (i),The classical case, with complete information from the market available to the agent at all times. Mathematically this means that the portfolio process is adapted to the filtration of the underlying Brownian motion (or, more generally, the underlying Lévy process). (ii),The partial observation case, in which the trader has to base her portfolio choices on less information than . Mathematically this means that the portfolio process must be adapted to a filtration for all t. For example, this is the case if the trader can only observe the asset prices and not the underlying Lévy process. (iii),The insider case, in which the trader has some inside information about the future of the market. This information could for example be the price of one of the assets at some future time. Mathematically this means that the portfolio process is allowed to be adapted to a filtration for all t. In this case the associated stochastic integrals become anticipating, and it is necessary to explain what mathematical model it is appropriate to use and to clarify the corresponding anticipating stochastic calculus. We solve the problem in all these three cases and we compute the corresponding maximal expected logarithmic utility of the terminal wealth. Let us call these quantities , respectively. Then represents the loss of value due the loss of information in (ii), and is the value gained due to the inside information in (iii). [source]

MSM Estimators of European Options on Assets with Jumps

MATHEMATICAL FINANCE, Issue 2 2001
João Amaro de Matos
This paper shows that, under some regularity conditions, the method of simulated moments estimator of European option pricing models developed by Bossaerts and Hillion (1993) can be extended to the case where the prices of the underlying asset follow Lévy processes, which allow for jumps, with no losses on their asymptotic properties, still allowing for the joint test of the model. [source]