Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Subordinating a multivariate Levy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new Levy process, provided the subordinator and the subordinate are independent processes. The variance-gamma model in finance was generated accordingly from a Brownian motion and a gamma process. Alternatively, multivariate subordination can be used to create Levy processes, but this requires the subordinate to have independent components. In this paper, we show that there exists another operation acting on pairs $(T,X)$ of Levy processes which creates a Levy process $X\odot T$. Here, $T$ is a subordinator, but $X$ is an arbitrary Levy process with possibly dependent components. We show that this method is an extension of both univariate and multivariate subordination and provide two applications. We illustrate our methods giving a weak formulation of the variance-$\boldsymbol{\alpha}$-gamma process that exhibits a wider range of dependence than using traditional subordination. Also, the variance generalised gamma convolution class of Levy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.

/pdf/weak-subordination-of-multivariate-levy-processes-and-93su87rzy2.pdf

Weak subordination of multivariate Lévy processes and variance generalised gamma convolutions

Based on the concept of self-decomposability, we extend some recent multivariate Levy models built using multivariate subordination with the aim of capturing situations in which a sudden event in one market is propagated onto related markets after a certain stochastic time delay. Consequently, we study the properties of such processes, derive closed form expressions for the characteristic function and detail how a Monte Carlo scheme can be easily implemented. We illustrate the applicability of our approach in the context of gas and power Energy markets focusing on the calibration and on the pricing of spread options written on different underlying assets using simulations techniques.

Correlating L\'evy processes with Self-Decomposability: Applications to Energy Markets

The weak variance-alpha-gamma process is a multivariate Levy process constructed by weakly subordinating Brownian motion, possibly with correlated components with an alpha-gamma subordinator. It generalises the variance-alpha-gamma process of Semeraro constructed by traditional subordination. We compare three calibration methods for the weak variance-alpha-gamma process, method of moments, maximum likelihood estimation (MLE) and digital moment estimation (DME). We derive a condition for Fourier invertibility needed to apply MLE and show in our simulations that MLE produces a better fit when this condition holds, while DME produces a better fit when it is violated. We also find that the weak variance-alpha-gamma process exhibits a wider range of dependence and produces a significantly better fit than the variance-alpha-gamma process on a S&P500-FTSE100 data set, and that DME produces the best fit in this situation.

/pdf/calibration-for-weak-variance-alpha-gamma-processes-4ul1uf6srf.pdf

Calibration for Weak Variance-Alpha-Gamma Processes

Subordinating a multivariate L\\'evy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new L\\'evy process provided the subordinator and the subordinate are independent processes. The Variance Gamma model in finance was generated accordingly from a Brownian motion and a Gamma process. Alternatively, multivariate subordination can be used to create L\\'evy processes, but this requires the subordinate to have independent or indistinguishable components. In this paper, we show that there exists another operation acting on pairs (T,X) of L\\'evy processes which creates a L\\'evy process X \\.o T. Here, T is a subordinator, but X is an arbitrary L\\'evy process with possibly dependent coordinates. We show that this method is an extension of both univariate and multivariate subordination and provide two applications. Firstly, we give a weak formulation of the Variance-alpha-Gamma-process that exhibits a wider range of dependence than using traditional subordination, and fit this model to a S&P500-FTSE100 data set. Secondly, the Variance Generalised Gamma Convolution class of L\\'evy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.

/pdf/weak-subordination-of-multivariate-levy-processes-3nminlacpm.pdf

Weak Subordination of Multivariate Lévy Processes

We present an approach for modelling dependencies in exponential Levy market models with arbitrary margins originated from time changed Brownian motions. Using weak subordination of Buchmann et al. [Bernoulli, 2017], we face a new layer of dependencies, superior to traditional approaches based on pathwise subordination, since weakly subordinated processes are not required to have independent components considering multivariate stochastic time changes. We apply a subordinator being able to incorporate any joint or idiosyncratic information arrivals. We emphasize multivariate variance gamma and normal inverse Gaussian processes and state explicit formulae for the Levy characteristics. Using maximum likelihood, we estimate multivariate variance gamma models on various market data and show that these models are highly preferable to traditional approaches. Consistent values of basket-options under given marginal pricing models are achieved using the Esscher transform, generating a non-flat implied correlation ...

Marginal consistent dependence modelling using weak subordination for Brownian motions

We present an approach for modeling dependencies in exponential Levy market models with arbitrary margins originated from time changed Brownian motions. Using weak subordination of Buchmann et al. (2016), we face a new layer of dependencies, superior to traditional approaches based on pathwise subordination, since weakly subordinated processes are not required to have independent components considering multivariate stochastic time changes. We apply a subordinator being able to incorporate any joint or idiosyncratic information arrivals. We emphasize multivariate variance gamma and normal inverse Gaussian processes and state explicit formulae for the Levy characteristics. Using maximum likelihood, we estimate a multivariate variance gamma model on various market data and show that the model is highly preferable to traditional approaches. Consistent values of basket-options under given marginal pricing models are achieved using Esscher transform, generating a non-flat implied correlation surface.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID3079478_code2755371.pdf?abstractid=3079478&mirid=1

Marginal Consistent Dependence Modeling Using Weak Subordination for Brownian Motions

In response to empirical evidence, we propose a continuous-time model for multivariate asset returns with a two-layered dependence structure. The price process is subject to multivariate informatio...

Information flow dependence in financial markets

In response to empirical evidence, we propose a continuous-time model for multivariate asset returns with a two-layered dependence structure. The price process is subject to multivariate information arrivals driving the market activity modeled by non-decreasing pure-jump Levy processes. A Levy copula determines the jump dependence and allows for a generic multivariate information flow with a flexible structure. Conditional on the information flow, asset returns are jointly normal. Within this setup, we provide an estimation framework based on maximum simulated likelihood. We apply novel multivariate models to equity data and obtain estimates which meet an economic intuition with respect to the two-layered dependence structure.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID3433032_code2755371.pdf?abstractid=3051180&mirid=1

Information Flow Dependence in Financial Markets

We develop a multivariate return and trading volume model, where each stock’s system is driven by latent information arrivals in continuous time. The arrivals contain idiosyncratic and cross-relevant information, which provides both return and trading volume dependence. Conditional on the accumulated information, returns are jointly normal and correlated, which implies a second layer of dependence in the return dimension. Using a sample of nine common stocks, we show that trading volume significantly adds to the operationalization of the latent information flow process driving the contemporaneous return distribution. The dependence parameter estimates provide significant and interpretable degrees of information flow dependence across all results. Portfolio risk measurement applications are extended by conditioning on the level of trading volume, e.g. reflecting stress, leading to an accurate risk quantification.

Markus Michaelsen

Papers

Marginal consistent dependence modelling using weak subordination for Brownian motions

Marginal Consistent Dependence Modeling Using Weak Subordination for Brownian Motions

Information flow dependence in financial markets

Information Flow Dependence in Financial Markets

Information Flow Dependence in Return and Trading Volume Across Different Stocks