Mixtures of compound Poisson processes as models of tick-by-tick financial data

A model for the phenomenological description of tick-by-tick share prices in a stock exchange is introduced. It is based on mixtures of compound Poisson processes. Preliminary results based on Monte Carlo simulation show that this model can reproduce various stylized facts.Comment: 12 pages, 6 figures, to appear in a special issue of Chaos, Solitons and Fractal

Extending Time-Changed Lévy Asset Models Through Multivariate Subordinators

The traditional multivariate Lévy process constructed by subordinating a Brownian motion through a univariate subordinator presents a number of drawbacks, including the lack of independence and a limited range of dependence. In order to face these, we investigate multivariate subordination, with a common and an idiosyncratic component. We introduce generalizations of some well known univariate Lévy processes for financial applications: the multivariate compound Poisson, NIG, Variance Gamma and CGMY. In all these cases the extension is parsimonious, in that one additional parameter only is needed. We characterize first the subordinator, then the time changed processes via their Lévy measure and characteristic exponent. We further study the subordinator association, as well as the subordinated processes linear and non linear dependence. We show that the processes generated with the proposed time change can include independence and that they span the whole range of linear dependence. We provide some examples of simulated trajectories,scatter plots and both linear and non linear dependence measures. The input data for these simulations are calibrated values for major stock indices.Lévy processes, multivariate subordinators, dependence (association, correlation), multivariate asset modelling

The Stochastically Subordinated Log Normal Process Applied To Financial Time Series And Option Pricing

The method of stochastic subordination, or random time indexing, has been recently applied to Wiener process price processes to model financial returns. Previous emphasis in stochastic subordination models has involved explicitly identifying the subordinating process with an observable quantity such as number of trades. In contrast, the approach taken here does not depend on the specific identification of the subordinated time variable, but rather assumes a class of time models and estimates parameters from data. In addition, a simple Markov process is proposed for the characteristic parameter of the subordinating distribution to explain the significant autocorrelation of the squared returns. It is shown in particular, that the proposed model, while containing only a few more parameters than the commonly used Wiener process models, fits selected fmancial time series particularly well, characterising the autocorrelation structure and heavy tails, as well as preserving the desirable self-similarity structure present in popular chaos-theoretic models, and the existence of risk-neutral measures necessary for objective derivative valuation

Subordinated affine structure models for commodity future prices

To date the existence of jumps in different sectors of the financial market is certain and the commodity market is no exception. While there are various models in literature on how to capture these jumps, we restrict ourselves to using subordinated Brownian motion by an α-stable process, α ∈ (0,1), as the source of randomness in the spot price model to determine commodity future prices, a concept which is not new either. However, the key feature in our pricing approach is the new simple technique derived from our novel theory for subordinated affine structure models. Different from existing filtering methods for models with latent variables, we show that the commodity future price under a one factor model with a subordinated random source driver, can be expressed in terms of the subordinator which can then be reduced to the latent regression models commonly used in population dynamics with their parameters easily estimated using the expectation maximisation method. In our case, the underlying joint probability distribution is a combination of the Gaussian and stable densities

Revisiting variance gamma pricing : an application to S&P500 index options

We reformulate the Lévy-Kintchine formula to make it suitable for modelling the stochastic time-changing effects of Lévy processes. Using Variance-Gamma (VG) process as an example, it illustrates the dynamic properties of a Lévy process and revisits the earlier work of Geman (2002). It also shows how the model can be calibrated to price options under a Lévy VG process, and calibrates the model on recent S&P500 index options data. It then compares the pricing performance of Fast Fourier Transform (FFT) and Fractional Fourier Transform (FRFT) approaches to model calibration and investigates the trade-off between calibration performance and required calculation time

A Generalized Normal Mean Variance Mixture for Return Processes in Finance

Time-changed Brownian motions are extensively applied as mathematical models for asset returns in Finance. Time change is interpreted as a switch to trade-related business time, different from calendar time. Time-changed Brownian motions can be generated by infinite divisible normal mixtures. The standard multivariate normal mean variance mixtures assume a common mixing variable. This corresponds to a multidimensional return process with a unique change of time for all assets under exam. The economic counterpart is uniqueness of trade or business time, which is not in line with empirical evidence. In this paper we propose a new multivariate definition of normal mean-variance mixtures with a flexible dependence structure, based on the economic intuition of both a common and an idiosyncratic component of business time. We analyze both the distribution and the related process. We use the above construction to introduce a multivariate generalized hyperbolic process with generalized hyperbolic margins. We conclude with a stock market example to show the ease of calibration of the model.multivariate normal mean variance mixtures, multivariate generalized hyperbolic distributions, Levy processes, multivariate subordinators

A Multivariate Jump-Driven Financial Asset Model

We discuss a Lévy multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility. We use it to describe the behavior of a series of stocks or indexes and to study a multi-firm, value-based default model. Starting from an independent Brownian world, we introduce jumps and other deviations from normality, including non-Gaussian dependence. We use a sto- chastic time-change technique and provide the details for a Gamma change. The main feature of the model is the fact that - opposite to other, non jointly Gaussian settings - its risk neutral dependence can be calibrated from univariate derivative prices, providing a surprisingly good fit.Lévy processes, multivariate asset modelling, copulas, risk neutral dependence.

Recent Developments in Financial and Insurance Mathematics and the Interplay with the Industry

The workshop brought together leading experts from all over the world to exchange and discuss the latest developments in mathematical finance and actuarial mathematics. Researchers from the industry had the opportunity to circulate their problems among mathematicians. The participants gained from a fruitful interaction between mathematical methods and practitioner’s problems as well as from the interaction between finance and actuarial mathematics