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Journal articles on the topic 'Multivariate Lévy models'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Ballotta, Laura, and Efrem Bonfiglioli. "Multivariate asset models using Lévy processes and applications." European Journal of Finance 22, no. 13 (April 10, 2014): 1320–50. http://dx.doi.org/10.1080/1351847x.2013.870917.

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2

Panov, Vladimir. "Series Representations for Multivariate Time-Changed Lévy Models." Methodology and Computing in Applied Probability 19, no. 1 (August 29, 2015): 97–119. http://dx.doi.org/10.1007/s11009-015-9461-8.

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3

Jacod, Jean, and Mark Podolskij. "On the minimal number of driving Lévy motions in a multivariate price model." Journal of Applied Probability 55, no. 3 (September 2018): 823–33. http://dx.doi.org/10.1017/jpr.2018.52.

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Abstract In this paper we consider the factor analysis for Lévy-driven multivariate price models with stochastic volatility. Our main aim is to provide conditions on the volatility process under which we can possibly reduce the dimension of the driving Lévy motion. We find that these conditions depend on a particular form of the multivariate Lévy process. In some settings we concentrate on nondegenerate symmetric α-stable Lévy motions.

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4

Avanzi, Benjamin, Jamie Tao, Bernard Wong, and Xinda Yang. "Capturing non-exchangeable dependence in multivariate loss processes with nested Archimedean Lévy copulas." Annals of Actuarial Science 10, no. 1 (December 11, 2015): 87–117. http://dx.doi.org/10.1017/s1748499515000135.

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AbstractThe class of spectrally positive Lévy processes is a frequent choice for modelling loss processes in areas such as insurance or operational risk. Dependence between such processes (e.g. between different lines of business) can be modelled with Lévy copulas. This approach is a parsimonious, efficient and flexible method which provides many of the advantages akin to distributional copulas for random variables. Literature on Lévy copulas seems to have primarily focussed on bivariate processes. When multivariate settings are considered, these usually exhibit an exchangeable dependence structure (whereby all subset of the processes have an identical marginal Lévy copula). In reality, losses are not always associated in an identical way, and models allowing for non-exchangeable dependence patterns are needed. In this paper, we present an approach which enables the development of such models. Inspired by ideas and techniques from the distributional copula literature we investigate the procedure of nesting Archimedean Lévy copulas. We provide a detailed analysis of this construction, and derive conditions under which valid multivariate (nested) Lévy copulas are obtained. Our results are discussed and illustrated, notably with an example of model fitting to data.

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5

Fasen, Vicky. "Limit Theory for High Frequency Sampled MCARMA Models." Advances in Applied Probability 46, no. 3 (September 2014): 846–77. http://dx.doi.org/10.1239/aap/1409319563.

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We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {hn, 2hn,…, nhn}, where hn ↓ 0 and nhn → ∞ as n → ∞, or at a constant time grid where hn = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

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6

Moser, Martin, and Robert Stelzer. "Tail behavior of multivariate lévy-driven mixed moving average processes and supOU Stochastic Volatility Models." Advances in Applied Probability 43, no. 4 (December 2011): 1109–35. http://dx.doi.org/10.1239/aap/1324045701.

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Multivariate Lévy-driven mixed moving average (MMA) processes of the type Xt = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

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7

Moser, Martin, and Robert Stelzer. "Tail behavior of multivariate lévy-driven mixed moving average processes and supOU Stochastic Volatility Models." Advances in Applied Probability 43, no. 04 (December 2011): 1109–35. http://dx.doi.org/10.1017/s0001867800005322.

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Multivariate Lévy-driven mixed moving average (MMA) processes of the type X t = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

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8

Fasen, Vicky. "Limit Theory for High Frequency Sampled MCARMA Models." Advances in Applied Probability 46, no. 03 (September 2014): 846–77. http://dx.doi.org/10.1017/s0001867800007400.

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We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {h n , 2h n ,…, nh n }, where h n ↓ 0 and nh n → ∞ as n → ∞, or at a constant time grid where h n = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

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9

Ballotta, Laura, Gianluca Fusai, Angela Loregian, and M. Fabricio Perez. "Estimation of Multivariate Asset Models with Jumps." Journal of Financial and Quantitative Analysis 54, no. 5 (September 28, 2018): 2053–83. http://dx.doi.org/10.1017/s0022109018001321.

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We propose a consistent and computationally efficient 2-step methodology for the estimation of multidimensional non-Gaussian asset models built using Lévy processes. The proposed framework allows for dependence between assets and different tail behaviors and jump structures for each asset. Our procedure can be applied to portfolios with a large number of assets because it is immune to estimation dimensionality problems. Simulations show good finite sample properties and significant efficiency gains. This method is especially relevant for risk management purposes such as, for example, the computation of portfolio Value at Risk and intra-horizon Value at Risk, as we show in detail in an empirical illustration.

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10

JEVTIĆ, PETAR, MARINA MARENA, and PATRIZIA SEMERARO. "MULTIVARIATE MARKED POISSON PROCESSES AND MARKET RELATED MULTIDIMENSIONAL INFORMATION FLOWS." International Journal of Theoretical and Applied Finance 22, no. 02 (March 2019): 1850058. http://dx.doi.org/10.1142/s0219024918500589.

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The class of marked Poisson processes and its connection with subordinated Lévy processes allow us to propose a new interpretation of multidimensional information flows and their relation to market movements. The new approach provides a unified framework for multivariate asset return models in a Lévy economy. In fact, we are able to recover several processes commonly used to model asset returns as subcases. We consider a first application example using the normal inverse Gaussian specification.

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11

Luciano, Elisa, and Patrizia Semeraro. "Multivariate time changes for Lévy asset models: Characterization and calibration." Journal of Computational and Applied Mathematics 233, no. 8 (February 2010): 1937–53. http://dx.doi.org/10.1016/j.cam.2009.08.119.

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12

Fink, Holger. "Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk." Journal of Applied Probability 50, no. 4 (December 2013): 983–1005. http://dx.doi.org/10.1239/jap/1389370095.

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Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

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13

Fink, Holger. "Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk." Journal of Applied Probability 50, no. 04 (December 2013): 983–1005. http://dx.doi.org/10.1017/s0021900200013759.

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Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on ann-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

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14

MICHAELSEN, MARKUS. "INFORMATION FLOW DEPENDENCE IN FINANCIAL MARKETS." International Journal of Theoretical and Applied Finance 23, no. 05 (July 25, 2020): 2050029. http://dx.doi.org/10.1142/s0219024920500296.

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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 nondecreasing pure-jump Lévy processes. A Lévy 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.

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15

MARENA, MARINA, ANDREA ROMEO, and PATRIZIA SEMERARO. "MULTIVARIATE FACTOR-BASED PROCESSES WITH SATO MARGINS." International Journal of Theoretical and Applied Finance 21, no. 01 (February 2018): 1850005. http://dx.doi.org/10.1142/s021902491850005x.

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We introduce a class of multivariate factor-based processes with the dependence structure of Lévy [Formula: see text]-models and Sato marginal distributions. We focus on variance gamma and normal inverse Gaussian marginal specifications for their analytical tractability and fit properties. We explore if Sato models, whose margins incorporate more realistic moments term structures, preserve the correlation flexibility in fitting option data. Since [Formula: see text]-models incorporate nonlinear dependence, we also investigate the impact of Sato margins on nonlinear dependence and its evolution over time. Further, the relevance of nonlinear dependence in multivariate derivative pricing is examined.

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16

GUILLAUME, FLORENCE. "MULTIVARIATE OPTION PRICING MODELS WITH LÉVY AND SATO VG MARGINAL PROCESSES." International Journal of Theoretical and Applied Finance 21, no. 02 (March 2018): 1850007. http://dx.doi.org/10.1142/s0219024918500073.

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Pricing and hedging of financial instruments whose payoff depends on the joint realization of several underlyings (basket options, spread options, etc.) require multivariate models that are, at the same time, computationally tractable and flexible enough to accommodate the stylized facts of asset returns and of their dependence structure. Among the most popular models one finds models with VG marginals. The aim of this paper is to compare four multivariate models that are characterized by VG laws at unit time and to assess their performance by considering the flexibility they offer to calibrate the dependence structure for fixed marginals.

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17

MARFÈ, ROBERTO. "A MULTIVARIATE PURE-JUMP MODEL WITH MULTI-FACTORIAL DEPENDENCE STRUCTURE." International Journal of Theoretical and Applied Finance 15, no. 04 (June 2012): 1250028. http://dx.doi.org/10.1142/s0219024912500288.

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In this work we propose a new approach to build multivariate pure jump processes. We introduce linear and nonlinear dependence, without restrictions on marginal properties, by imposing a multi-factorial structure separately on both positive and negative jumps. Such a new approach provides higher flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. Using the notion of multivariate subordinator, this modeling approach can be applied to the class of univariate Lévy processes which can be written as the difference of two subordinators. A common example in the financial literature is the variance gamma process, which we extend to the multivariate (multi-factorial) case. The model is tractable and a straightforward multivariate simulation procedure is available. An empirical analysis documents an accurate multivariate fit of stock index returns in terms of both linear and nonlinear dependence. An example of multi-asset option pricing emphasizes the importance of the proposed multivariate approach.

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18

Ivanov, Roman V., and Katsunori Ano. "On exact pricing of FX options in multivariate time-changed Lévy models." Review of Derivatives Research 19, no. 3 (February 11, 2016): 201–16. http://dx.doi.org/10.1007/s11147-016-9120-4.

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19

Gardini, Matteo, Piergiacomo Sabino, and Emanuela Sasso. "Correlating Lévy processes with self-decomposability: applications to energy markets." Decisions in Economics and Finance 44, no. 2 (October 8, 2021): 1253–80. http://dx.doi.org/10.1007/s10203-021-00352-9.

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AbstractBased on the concept of self-decomposability, we extend some recent multidimensional Lévy models built using multivariate subordination. Our aim is to construct multivariate Lévy processes that can model the propagation of the systematic risk in dependent markets with some stochastic delay instead of affecting all the markets at the same time. To this end, we extend some known approaches keeping their mathematical tractability, study the properties of the new processes, derive closed-form expressions for their characteristic functions and detail how Monte Carlo schemes can be implemented. We illustrate the applicability of our approach in the context of gas, power and emission markets focusing on the calibration and on the pricing of spread options written on different underlying commodities.

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20

MARAZZINA, DANIELE, OLEG REICHMANN, and CHRISTOPH SCHWAB. "hp-DGFEM FOR KOLMOGOROV–FOKKER–PLANCK EQUATIONS OF MULTIVARIATE LÉVY PROCESSES." Mathematical Models and Methods in Applied Sciences 22, no. 01 (January 2012): 1150005. http://dx.doi.org/10.1142/s0218202512005897.

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We analyze the discretization of nonlocal degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes. Such equations arise in option pricing problems when the stochastic dynamics of the markets is modeled by Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes, in particular the discontinuous Galerkin Finite Element Methods (DG-FEM). In the DG-FEM, a new regularization of hypersingular integrals in the Dirichlet form of the pure jump part of infinite variation processes is proposed, allowing in particular a stable DG discretization of hypersingular integral operators. Robustness of the stabilized discretization with respect to various degeneracies in the characteristic triple of the stochastic process is proved. We provide in particular an hp-error analysis of the DG-FEM. Numerical experiments for model equations confirm the theoretical results.

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21

Lee, Mei-Ling Ting, and George A. Whitmore. "Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property." Stats 5, no. 1 (February 11, 2022): 172–89. http://dx.doi.org/10.3390/stats5010012.

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The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.

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22

Schlemm, Eckhard, and Robert Stelzer. "Quasi maximum likelihood estimation for strongly mixing state space models and multivariate Lévy-driven CARMA processes." Electronic Journal of Statistics 6 (2012): 2185–234. http://dx.doi.org/10.1214/12-ejs743.

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23

Mai, Jan-Frederik. "The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay." Dependence Modeling 8, no. 1 (October 1, 2020): 210–20. http://dx.doi.org/10.1515/demo-2020-0012.

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AbstractWe derive a sufficient condition on the symmetric norm ||·|| such that the probability distribution associated with the density function f (x) ∝exp(−λ ||x||) is conditionally independent and identically distributed in the sense of de Finetti’s seminal theorem. The criterion is mild enough to comprise the ℓp-norms as special cases, in which f is shown to correspond to a polynomially tilted stable mixture of products of transformed Gamma densities. In another special case of interest f equals the density of a time-homogeneous load sharing model, popular in reliability theory, whose motivation is a priori unrelated to the concept of conditional independence. The de Finetti structure reveals a surprising link between time-homogeneous load sharing models and the concept of Lévy subordinators.

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24

ALFEUS, MESIAS, and ERIK SCHLÖGL. "ON SPREAD OPTION PRICING USING TWO-DIMENSIONAL FOURIER TRANSFORM." International Journal of Theoretical and Applied Finance 22, no. 05 (August 2019): 1950023. http://dx.doi.org/10.1142/s0219024919500237.

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Spread options are multi-asset options with payoffs dependent on the difference of two underlying financial variables. In most cases, analytically closed form solutions for pricing such payoffs are not available, and the application of numerical pricing methods turns out to be nontrivial. We consider several such nontrivial cases and explore the performance of the highly efficient numerical technique of Hurd & Zhou[(2010) A Fourier transform method for spread option pricing, SIAM J. Financial Math. 1(1), 142–157], comparing this with Monte Carlo simulation and the lower bound approximation formula of Caldana & Fusai[(2013) A general closed-form spread option pricing formula, Journal of Banking & Finance 37, 4893–4906]. We show that the former is in essence an application of the two-dimensional Parseval’s Identity. As application examples, we price spread options in a model where asset prices are driven by a multivariate normal inverse Gaussian (NIG) process, in a three-factor stochastic volatility model, as well as in examples of models driven by other popular multivariate Lévy processes such as the variance Gamma process, and discuss the price sensitivity with respect to volatility. We also consider examples in the fixed-income market, specifically, on cross-currency interest rate spreads and on LIBOR/OIS spreads.

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25

Yoshioka, Hidekazu. "Fitting a superposition of Ornstein–Uhlenbeck process to time series of discharge in a perennial river environment." ANZIAM Journal 63 (June 28, 2022): C84—C96. http://dx.doi.org/10.21914/anziamj.v63.16985.

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Classical Ornstein–Uhlenbeck (ou) processes are Lévy-driven linear stochastic models with exponentially decaying autocorrelation functions which do not always fit more slowly decaying real time series data. A superposition of ou processes (known as a supou process) is proposed to overcome this issue for application to river discharge time series data. The discharge data has a sub-exponential autocorrelation function and this is captured by the supou process based on the mean reversion speed generated by a Gamma distribution. All the parameters of the supou process are identified by matching the autocorrelation and the first to fourth statistical moments of the discharge data. The empirical and modelled histograms of the discharge data are comparable with each other. References O. E. Barndorff-Nielsen. Superposition of Ornstein–Uhlenbeck type processes. Theory Prob. Appl. 45.2 (2001), pp. 175–194. doi: 10.1137/S0040585X97978166 O. E. Barndorff-Nielsen, F. E. Benth, and A. E. D. Veraart. Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes. Bernoulli 19.3 (2013), pp. 803–845. doi: 10.3150/12-BEJ476 O. E. Barndorff-Nielsen and N. N. Leonenko. Burgers’ turbulence problem with linear or quadratic external potential. J. Appl. Prob. 42.2 (2001), pp. 550–565. url: http://www.jstor.org/stable/30040809 J. Beran, Y. Feng, S. Ghosh, and R. Kulik. Long-Memory Processes. Springer-Verlag, Berlin, Heidelberg, 2016. doi: 10.1007/978-3-642-35512-7 F. Fuchs and R. Stelzer. Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model. ESAIM: Prob. Stat. 17 (2013), pp. 455–471. doi: 10.1051/ps/2011158 Y. Kabanov and S. Pergamenshchikov. Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process. Fin. Stoch. 24.1 (2020), pp. 39–69. doi: 10.1007/s00780-019-00413-3 R. Kawai and H. Masuda. On simulation of tempered stable random variates. J. Comput. Appl. Math. 235.8 (2011), pp. 2873–2887. doi: 10.1016/j.cam.2010.12.014 S. Pelacani and F. G. Schmitt. Scaling properties of the turbidity and streamflow time series at two different locations of an intra-Apennine stream: Case study. J. Hydro. 603.B (2021), p. 126943. doi: 10.1016/j.jhydrol.2021.126943 R. Stelzer, T. Tosstorff, and M. Wittlinger. Moment based estimation of supOU processes and a related stochastic volatility model. Stat. Risk Model. 32.1 (2015), pp. 1–24. doi: 10.1515/strm-2012-1152 S. Suweis, E. Bertuzzo, G. Botter, A. Porporato, I. Rodriguez-Iturbe, and A. Rinaldo. Impact of stochastic fluctuations in storage-discharge relations on streamflow distributions. Water Resource. Res. 46.3 (2010), W03517. doi: 10.1029/2009WR008038 M. Tamborrino and P. Lansky. Shot noise, weak convergence and diffusion approximations. Physica D: Nonlinear Phenomena 418 (2021), p. 132845. doi: 10.1016/j.physd.2021.132845 E. Taufer and N. Leonenko. Simulation of Lévy-driven Ornstein–Uhlenbeck processes with given marginal distribution. In: Comput. Stat. Data Anal. 53.6 (2009), pp. 2427–2437. doi: 10.1016/j.csda.2008.02.026 C. Van Den Broeck. On the relation between white shot noise, Gaussian white noise, and the dichotomic Markov process. J. Stat. Phys. 31 (1983), pp. 467–483. doi: 10.1007/BF01019494 H. Yoshioka and Y. Yoshioka. Designing cost-efficient inspection schemes for stochastic streamflow environment using an effective Hamiltonian approach. Opt. Eng. (2021), pp. 1–33. doi: 10.1007/s11081-021-09655-7

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26

MAI, JAN-FREDERIK, and MATTHIAS SCHERER. "A TRACTABLE MULTIVARIATE DEFAULT MODEL BASED ON A STOCHASTIC TIME-CHANGE." International Journal of Theoretical and Applied Finance 12, no. 02 (March 2009): 227–49. http://dx.doi.org/10.1142/s0219024909005208.

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A stochastic time-change is applied to introduce dependence to a portfolio of credit-risky assets whose default times are modeled as random variables with arbitrary distribution. The dependence structure of the vector of default times is completely separated from its marginal default probabilities, making the model analytically tractable. This separation is achieved by restricting the time-change to suitable Lévy subordinators which preserve the marginal distributions. Jump times of the Lévy subordinator are interpreted as times of excess default clustering. Relevant for practical implementations is that the parameters of the time-change allow for an intuitive economical explanation and can be calibrated independently of the marginal default probabilities. On a theoretical level, a so-called time normalization allows to compute the resulting copula of the default times. Moreover, the exact portfolio-loss distribution and an approximation for large portfolios under a homogeneous portfolio assumption are derived. Given these results, the pricing of complex portfolio derivatives is possible in closed-form. Three different implementations of the model are proposed, including a compound Poisson subordinator, a Gamma subordinator, and an Inverse Gaussian subordinator. Using two parameters to adjust the dependence structure in each case, the model is capable of capturing the full range of dependence patterns from independence to complete comonotonicity. A simultaneous calibration to portfolio-CDS spreads and CDO tranche spreads is carried out to demonstrate the model's applicability.

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27

Rüschendorf, Ludger, and Viktor Wolf. "Cost-efficiency in multivariate Lévy models." Dependence Modeling 3, no. 1 (April 16, 2015). http://dx.doi.org/10.1515/demo-2015-0001.

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AbstractIn this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence and uniqueness results for the Esscher parameter and determine cost efficient strategies in particular in the case of price processes driven by multivariate NIG- and VG-processes. From a monotonicity characterization of efficient payoffs we obtain that basket options are generally inefficient in Lévy markets when pricing is based on the Esscher measure.We determine efficient versions of the basket options in real market data and show that the proposed cost efficient strategies are also feasible from a numerical viewpoint. As a result we find that a considerable efficiency loss may arise when using the inefficient payoffs.

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28

Semeraro, Patrizia. "Multivariate tempered stable additive subordination for financial models." Mathematics and Financial Economics, July 13, 2022. http://dx.doi.org/10.1007/s11579-022-00321-9.

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AbstractWe study a class of multivariate tempered stable distributions and introduce the associated class of tempered stable Sato subordinators. These Sato subordinators are used to build additive inhomogeneous processes by subordination of a multiparameter Brownian motion. The resulting process is additive and time inhomogeneous and it is a generalization of multivariate Lévy processes with good fit properties on financial data. We specify the model to have unit time normal inverse Gaussian distribution and we discuss the ability of the model to fit time inhomogeneous correlations on real data.

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29

Gonon, Lukas, and Christoph Schwab. "Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models." Finance and Stochastics, August 31, 2021. http://dx.doi.org/10.1007/s00780-021-00462-7.

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AbstractWe study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of $d$ d risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process $X$ X that ensure $\varepsilon $ ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to ${\mathcal{O}}(\varepsilon ^{-1})$ O ( ε − 1 ) , and with constants implied in ${\mathcal{O}}(\, \cdot \, )$ O ( ⋅ ) which grow polynomially in $d$ d , thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps.In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide $\varepsilon $ ε expression error in DNN size ${\mathcal{O}}(|\log (\varepsilon )|^{a})$ O ( | log ( ε ) | a ) with exponent $a$ a proportional to $d$ d , but with constants implied in ${\mathcal{O}}(\, \cdot \, )$ O ( ⋅ ) growing exponentially with respect to $d$ d . Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the presented results.

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Bianchi, Michele Leonardo, Asmerilda Hitaj, and Gian Luca Tassinari. "A welcome to the jungle of continuous-time multivariate non-Gaussian models based on Lévy processes applied to finance." Annals of Operations Research, September 20, 2022. http://dx.doi.org/10.1007/s10479-022-04970-3.

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Brück, Florian, Jan-Frederik Mai, and Matthias Scherer. "Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes." Extremes, December 17, 2022. http://dx.doi.org/10.1007/s10687-022-00450-w.

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AbstractWe establish a one-to-one correspondence between (i) exchangeable sequences of random variables whose finite-dimensional distributions are minimum (or maximum) infinitely divisible and (ii) non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of an exchangeable minimum infinitely divisible sequence is shown to be the sum of a very simple “drift measure” and a mixture of product probability measures, which uniquely corresponds to the Lévy measure of a non-negative and non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. In probabilistic terms, the aforementioned infinitely divisible process is equal to the conditional cumulative hazard process associated with the exchangeable sequence of random variables with minimum (or maximum) infinitely divisible marginals. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many interesting classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.

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