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The backbone of financial risk modeling in finance over a long time period of more than a century, the random walk hypothesis has shown substantial variations in its structure throughout its history. In this article, I revisit the history of the random walk model in finance by introducing a new way of describing what a random walk is, based on the Levy measure in the Fourier space, a tool that has not yet been used in the history of financial thought. With this lens, we are able to understand the overview of the life of this model in finance over the entire 20th century, including the precursors of the 19th century.

The paper suggest a new approach to pricing barrier options under pure non-Gaussian Levy processes with jumps of finite variation. The key idea behind the method to represent the process under consideration as a difference between subordinators (increasing Levy processes). Such splitting rule applied to the process at exponentially distributed randomized time points gives us the possibility to find the option price by analytically solving a sequence of simple Wiener-Hopf equations.

We show that the quotient of Levy processes of jump-diffusion type has a fat-tailed distribution. An application is to price theory in economics. We show that fat tails arise endogenously from modeling of price change based on an excess demand analysis resulting in a quotient of arbitrarily correlated demand and supply whether or not jump discontinuities are present. The assumption is that supply and demand are described by drift terms, Brownian (i.e., Gaussian) and compound Poisson jump processes. If $P^{-1}dP/dt$ (the relative price change in an interval $dt$) is given by a suitable function of relative excess demand, $\left( \mathcal{D}% -\mathcal{S}\right) /\mathcal{S}$ (where $\mathcal{D}$ and $\mathcal{S}$ are demand and supply), then the distribution has tail behavior $F\left( x\right) \sim x^{-\zeta}$ for a power $\zeta$ that depends on the function $G$ in $P^{-1}dP/dt=G\left( \mathcal{D}/\mathcal{S}\right) $. For $G\left( x\right) \sim\left\vert x\right\vert ^{1/q}$ one has $\zeta=q.$ The empirical data for assets typically yields a value, $\zeta\tilde{=}3,$ or $\ \zeta \in\left[ 3,5\right] $ for some markets. The discrepancy between the empirical result and theory never arises if one models price dynamics using basic economics methodology, i.e., generalized Walrasian adjustment, rather than the usual starting point for classical finance which assumes a normal distribution of price changes. The function $G$ is deterministic, and can be calibrated with a smaller data set. The results establish a simple link between the decay exponent of the density function and the price adjustment function, a feature that can improve methodology for risk assessment. The mathematical results can be applied to other problems involving the relative difference or quotient of Levy processes of jump-diffusion type.

In this paper, enlightened by the asymptotic expansion methodology developed by Li [(2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380] and Li and Chen [(2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 51–70], we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Lévy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein–Uhlenbeck model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method.

Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime switching Levy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines Continuous-Time Markov Chain (CTMC) approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi, alpha-Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a unified approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.

In this paper, we present several optimal control approaches which are designed to control and regulate the evolution of infection numbers in the COVID-19 disease. In our setting, the number of infected people at a certain location is modeled by a continuous-time stochastic process which can be affected by a related stochastic control process. We use mathematical tools from stochastic analysis and optimal control theory. In particular, we prove an innovative stochastic maximum principle for continuous-state branching processes with immigration (so-called CBI processes) and apply the result to a stochastic control problem stemming from epidemiology.

I use coarse Bayesian updating to explain three stylized facts: trading anomalies around major stock market milestones, excess price volatility and trade volume, and heavy-tailed prices. When expectations are coarse and traders are heterogeneous, traders make heterogeneous expectation errors. This disagreement generates substantial trade volume and I show that the ensuing price discovery process converges in distribution to an empirically relevant class of Levy processes. Notably, I obtain this result without any exogenous shocks or processes; the only exogenous random variable is a heterogeneity parameter drawn at time zero. I then establish a perfectly non-revealing equilibrium: because traders are different, their aggregated trades generate volatile prices, yet this volatility impedes traders from learning their differences.

I present a complete numerical study of the implicit-explicit finite difference method used to solve partial integro-differential equations (PIDEs) arising in mathematical finance. The study considers the specific case of the Merton jump-diffusion model, which is used to compute prices of European call and put options. These PIDE prices are then compared with prices obtained by closed formula, Fourier inversion, and Monte Carlo methods.

The present article deals with intra-horizon risk in models with jumps. Our general understanding of intra-horizon risk is along the lines of the approach taken in [BRSW04], [Ro08], [BMK09], [BP10], and [LV20]. In particular, we believe that quantifying market risk by strictly relying on point-in-time measures cannot be deemed a satisfactory approach in general. Instead, we argue that complementing this approach by studying measures of risk that capture the magnitude of losses potentially incurred at any time of a trading horizon is necessary when dealing with (m)any financial position(s). To address this issue, we propose an intra-horizon analogue of the expected shortfall for general profit and loss processes and discuss its key properties. Our intra-horizon expected shortfall is well-defined for (m)any popular class(es) of Levy processes encountered when modeling market dynamics and constitutes a coherent measure of risk, as introduced in [CDK04]. On the computational side, we provide a simple method to derive the intra-horizon risk inherent to popular Levy dynamics. Our general technique relies on results for maturity-randomized first-passage probabilities and allows for a derivation of diffusion and single jump risk contributions. These theoretical results are complemented with an empirical analysis, where popular Levy dynamics are calibrated to the S&P 500 index and the Brent crude oil data, and an analysis of the resulting intra-horizon risk is presented.

This paper proposes a market consistent valuation framework for variable annuities with guaranteed minimum accumulation benefit, death benefit and surrender benefit features. The setup is based on a hybrid model for the financial market and uses time-in homogeneous Levy processes as risk drivers. Further, we allow for dependence between financial and surrender risks. Our model leads to explicit analytical formulas for the quantities of interest, and practical and efficient numerical procedures for the evaluation of these formulas. We illustrate the tractability of this approach by means of a detailed sensitivity analysis of the fair value of the variable annuity and its components with respect to the model parameters. The results highlight the role played by the surrender behaviour and the importance of its appropriate modelling.

The Paris agreement in 2016 marks a global effort to limit the increase in temperature. In that spirit, the Federal Government of Canada introduced a carbon tax to reduce greenhouse gas emissions. The main goal of this paper is to define the correct approach to carbon pricing. Following the method, introduce by Goutte and Chevalier (2015), we define the carbon price as the necessary tax to incite electricity producers to switch from coal to natural gas. The novelty of this paper is that we use this method for Alberta and North America. In addition, we consider the case of switching from natural gas to wind as a potential new approach to carbon pricing. After reviewing the two methods, we model prices under three stochastic procedures: Levy Normal Inverse Gaussian (NIG), Levy Normal and Heston model. Finally, we generalize our empirical technique to oil, natural gas and coal individually. The main finding of this article is that the Levy NIG outperforms the Levy Normal and Heston as it is able to take into account the jumpy and volatile nature of energy prices.

In this paper, we investigate a two-factor VIX model with infinite-activity jumps, which is a more realistic way to reduce errors in pricing VIX derivatives, compared with Mencia and Sentana (2013). Our two-factor model features central tendency, stochastic volatility and infinite-activity pure jump Levy processes which include the variance gamma (VG) and the normal inverse Gaussian (NIG) processes as special cases. We find empirical evidence that the model with infinite-activity jumps is superior to the models with finite-activity jumps, particularly in pricing VIX options. As a result, infinite-activity jumps should not be ignored in pricing VIX derivatives.

The aim of this study is to present algorithms for the backward simulation of standard processes that are commonly used in financial applications. We extend the works of Ribeiro and Webber and Avramidis and L’Ecuyer on gamma bridge and obtain the backward construction of a Gamma process. Moreover, we are able to write a novel acceptance-rejection algorithm to simulate Inverse Gaussian (IG) processes backward in time. Therefore, using the time-change approach, we can easily get the backward generation of the Compound Poisson with infinitely divisible jumps, the Variance–Gamma the Normal–Inverse–Gaussian processes and then the time-changed version of the OU process (SubOU) introduced by Li and Linetsky. We then compare the computational costs of the sequential and backward path generation of such processes and show the advantages of adopting the latter one, in particular in the context of pricing American options or energy facilities like gas storages.

We consider the problem of quantifying credit and funding risks in the presence of initial margin calculated by dynamically updated risk measures, such as Value-at-Risk and Expected Shortfall. The analytic scaling approach proposed in Andersen et al. [2] is generalized from a system driven by Brownian motion to an arbitrary radially symmetric (or ‘isotropic’) Levy process, permitting application to models possessing fat-tailed market movements during the margin period of risk (MPoR). Our mathematical results are applied to derive a closed-form representation for the credit valuation adjustment (CVA) and margin valuation adjustment (MVA) for centrally cleared portfolios in an arbitrage-free, continuous-time model driven by an isotropic Levy process. Our results cover both the exposures arising from client clearing and from participation in the loss mutualization of clearing member defaults. The latter is a particularly vexing modeling problem due to strong limitations on observable CCP data; our model gives rise to a compact valuation expression depending only on a clearing member’s own portfolio, and certain intuitive macroscopic measures capturing the gross risk and its concentration within the CCP.

Energy sources and commodities exhibit high price risk. This risk is thus an important feature of operational models of the value chains for these goods. These models typically employ Gaussian-based representations of the evolution of this uncertainty. This approach facilitates the optimization of operational policies but is at odds with empirical facts about energy and commodity prices, which are better captured by non-Gaussian processes. We discuss this alternative modeling strategy, focusing on Levy processes. As an illustration, we show that it substantially increases the optimal policy value in a simplified merchant natural gas storage setting. Further, we highlight potential implications of using this approach to formulate realistic energy and commodity operations models. Our work has broader relevance for modeling the dynamics of both other market variables and operational quantities, such as exchange rates and demand forecasts. The study of how the adoption of non-Gaussian processes may impact energy and commodity operations is an appealing area for future research.

We introduce a three-factor model of electricity spot prices, consisting of a deterministic seasonality and trend function as well as short- and long-term stochastic components, and derive a formula for futures prices. The long-term component is modelled as a Levy process with increments belonging to the class of generalised hyperbolic distributions. We describe the short-term factor by Levy semistationary processes: we start from a CARMA(2,1), i.e. a continous-time ARMA model, and generalise it by adding a short-memory stochastic volatility. We further modify the model by including the information about the wind energy production as an exogenous variable. We fit our models to German and Austrian data including spot and futures prices as well as the wind energy production and total load data. Empirical studies reveal that taking into account the impact of the wind energy generation on the prices improves the goodness of fit.

We propose a PIDE and closed-form Fourier Pricing formula for floating type Look-back option when stock price follows exponential Levy Process. We first developed a PIDE based on Martingale method and derived a closed-form Fourier formula for pricing contracts. The formula is simple, easy to compute and can work for any class of Levy Process.

Spread options are multi-asset options whose payoffs depend 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 non-trivial. We consider several such non-trivial cases and explore the performance of the highly efficient numerical technique of Hurd and Zhou (2010), comparing this with Monte Carlo simulation and the lower bound approximation formula of Caldana and Fusai (2013). We show that the former is in essence an application of the two–dimensional Parseval 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 threefactor stochastic volatility model, as well as in examples of models driven by other popular multivariate Levy 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. In terms of FFT computation, we have used the FFTW library (see Frigo and Johnson (2010)) and we document appropriate usage of this library to reconcile it with the MATLAB ifft2 counterpart.

In this paper we propose a derivative valuation framework based on Levy processes which takes into account the possibility that the underlying asset is subject to information-related trading halts/suspensions. Since such assets are not traded at all times, we argue that the natural underlying for derivative risk-neutral valuation is not the asset itself, but rather a contract that, when the asset is in trade suspensions upon maturity, cash-settles the last quoted price plus the interests accrued since the last quote update. Combining some elements from semimartingale time changes and potential theory, we devise martingale dynamics and no-arbitrage relations for such a price process, provide Fourier transform-based pricing formulae for derivatives, and study the asymptotic behavior of the obtained formulae as a function of the halt parameters. The volatility surface analysis reveals that the short term skew of our model is typically steeper than that of the underlying Levy models, indicating that the presence of a trade suspension risk is consistent with the well-documented stylized fact of volatility skew persistence/explosion.

We derive stochastic Leibniz formulas for Volterra integrals under enlarged filtrations. We investigate both pure-jump and Brownian Volterra processes under diverse initially enlarged filtration approaches. In these settings, we compare the ordinary with the stochastic (Doleans-Dade) exponential of a Volterra process and provide the corresponding martingale conditions. We also consider backward stochastic Volterra integral equations (BSVIEs) under enlarged filtrations and obtain the related solution formulas. We finally propose an anticipative Heath Jarrow Morton (HJM) forward rate model of Volterra-type. In an introductory section, we compile various facts on deterministic and stochastic Leibniz formulas for parameter integrals.

In a general jump-diffusion Radon–Nikodym setup with stochastic Girsanov processes, we derive optimal equivalent probability measures. Optimality is measured in terms of minimum relative entropy and also by more general divergence concepts. We further provide an anticipative sufficient stochastic minimum principle and derive optimal equivalent probability measures under various enlarged filtration approaches.

This paper investigates the performance of different parametric models, stable and tempered stable distributions, for capturing the tail behaviour of the log-returns. We first define and discuss the properties of the stable and tempered stable random variables. We then show how to estimate their parameters and simulate them from their characteristic functions. Finally, as an illustration, we perform an empirical experiment to explore the performance of different models representing the distribution of log-returns for the S&P500 and DAX indexes

We provide solution formulas for (linear and non-linear) jump-diffusion backward stochastic differential equations (BSDEs) under diverse enlarged filtration approaches. We also derive a comparison theorem for BSDEs in an enlarged filtration framework and present several applications of our theoretical results. For example, we study temperature derivatives under weather forecasts modeled by an enlarged filtration BSDE, investigate portfolio selection and hedging for an insider with additional information on future price behavior and solve a particular predictive mean-field BSDE in an enlarged filtration setup.

Carr and Wu (2004), henceforth CW, developed a framework that encompasses almost all of the continuous-time models proposed in the option pricing literature. Their framework hinges on the stopping time property of the time changes. By ana- lyzing the measurability of the time changes with respect to the underlying filtration, we show that many asset price models using correlated time changes fail to satisfy this assumption.

In this paper, we develop a framework for discretely compounding interest rates that is based on the forward price process approach. This approach has a number of advantages, in particular in the current market environment. Compared to the classical as well as the Levy Libor market model, it allows in a natural way for negative interest rates and has superb calibration properties even in the presence of extremely low rates. Moreover, the measure changes along the tenor structure are significantly simplified. These properties make it an excellent base for a postcrisis multiple curve setup. Two variants for multiple curve constructions based on the multiplicative spreads are discussed. Time‐inhomogeneous Levy processes are used as driving processes. An explicit formula for the valuation of caps is derived using Fourier transform techniques. Relying on the valuation formula, we calibrate the two model variants to market data.

The objective of this paper is to approximate the probability density functions for Ornstein-Uhlenbeck processes that are driven by Levy processes. This is achieved by considering a truncated set of cumulants relating to the Levy measure and expressing these as moments of Gaussian random variables. This can be thought of as a weaker version of Central Limit Theorem. We illustrate the use of these densities via Brownian motion (a benchmark) and Meixner process.

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend 2014a), to pricing/hedging European-type, early-exercise and discrete- monitored barrier options under a Levy process. The paper employs Chebfun (cf. Trefethen et al. 2014) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing and hedging when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price/hedge options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.

We investigate the pricing of cliquet options in a jump-diffusion model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a drifted Levy process entailing a Brownian diffusion component as well as compound Poisson jumps. We also derive representations for the density and distribution function of the emerging Levy process. In this setting, we infer semi-analytic expressions for the cliquet option price by two different approaches. The first one involves the probability distribution function of the driving Levy process whereas the second draws upon Fourier transform techniques. With view on sensitivity analysis and hedging purposes, we eventually deduce representations for several greeks while putting emphasis on the vega.

We formulate a flexible generalised hyperbolic (GH) option pricing model, which unlike the version proposed by Eberlein and Prause (2002), has all four of its parameters free to be estimated. We also present six three-parameter special cases: a variance gamma (VG), t, hyperbolic, normal inverse Gaussian, reciprocal hyperbolic and normal reciprocal inverse Gaussian option pricing model. Using S&P 500 Index options, we compare the flexible GH, VG, t and Black-Scholes models. The flexible GH model offers the best out-of-sample pricing overall, while the t special case outperforms the VG for both in-sample and out-of-sample pricing. All three models also improve the orthogonality of implied volatility compared to the Black-Scholes model.

We give an explicit formula for the probability distribution based on a relativistic extension of Brownian motion. The distribution 1) is properly normalized and 2) obeys the tower law (semigroup property), so we can construct martingales and self-financing hedging strategies and price claims (options). This model is a 1-constant-parameter extension of the Black-Scholes-Merton model. The new parameter is the analog of the speed of light in Special Relativity. However, in the financial context there is no "speed limit" and the new parameter has the meaning of a characteristic diffusion speed at which relativistic effects become important and lead to a much softer asymptotic behavior, i.e., fat tails, giving rise to volatility smiles. We argue that a nonlocal stochastic description of such (Levy) processes is inadequate and discuss a local description from physics. The presentation is intended to be pedagogical., Comment: 32 pages; a trivial typo corrected in footnote 1 on the title page, no other changes; to appear in Physica A

We advocate the use of an Indirect Inference method to estimate the parameter of a COGARCH(1,1) process for equally spaced observations. This requires that the true model can be simulated and a reasonable estimation method for an approximate auxiliary model. We follow previous approaches and use linear projections leading to an auxiliary autoregressive model for the squared COGARCH returns. The asymptotic theory of the Indirect Inference estimator relies on a uniform SLLN and asymptotic normality of the parameter estimates of the auxiliary model, which require continuity and differentiability of the COGARCH process with respect to its parameter and which we prove via Kolmogorov’s continuity criterion. This leads to consistent and asymptotically normal Indirect Inference estimates under moment conditions on the driving Levy process. A simulation study shows that the method yields a substantial finite sample bias reduction compared to previous estimators.

Tick-by-tick asset price data exhibit a number of empirical regularities, including discreteness, long periods where prices are flat, periods of price moves of alternating plus and minus one tick, periods of rapid successive price moves of the same sign, and others. This paper proposes a framework to examine whether and how these microscopic features of the tick data are compatible with the typical macroscopic continuous-time models, based on Ito semimartingales, that are employed to represent asset prices. We construct in particular tick-by-tick models that deliver by scaling macroscopic semimartingale models with stochastic volatility and jumps.

We prove an anticipative sufficient stochastic minimum principle in a jump process setup with initially enlarged filtrations. We apply the result to several portfolio selection problems like mean and minimal variance hedging under enlarged filtrations. We also investigate utility maximizing portfolio selection under future information. Contrarily to classical optimization methods like dynamic programming, our stochastic minimum principle likewise applies to non-Markovian setups. On the mathematical side, we are concerned with jump processes, forward and backward stochastic differential equations and forward integrals.

Force of mortality is defined using an exponential function of Legendre polynomials, as in Renshaw et al. (1996), plus an extra term which captures mortality shocks. For the extra term Ballotta Haberman (2006) and Ahmadi et al. (2015) consider an Ornstein-Uhlenbeck while we suggest using Levy Continuous Autoregressive Moving Average (CARMA) models. The proposed model encompasses as special cases the existing approaches. We present some extensions in the theory of Levy CARMA models that enter in the construction of the model estimation procedure. Males life tables for USA, Taiwan and Japan for the time period 1998-2013 are used for the presentation of fitting and projection results.

An exact closed-form pricing formula was derived for American options when stock returns follow a normal distribution or Levy processes. It uses a new non-homogeneous partial differential equation for American options, the condition from optimal early exercise general solution, and hence a Feynman Kac martingale expectation valuation with a known exercise boundary value at expiry. It is Kim's (1990) solution, under instantaneous information. By modifying the pricing formula, it solves closed-form pricing problems for American (put or call) options with discrete dividend payouts and Bermudan options without the curse of dimensionality.

We develop a multivariate Levy model for the pricing of quanto options that captures three characteristics observed in real-world market for stock prices and currencies: jumps, heavy tails and skewness. The model is developed by using a bottom-up approach from a subordinator. We do so by replacing the time of a Brownian motion with a non-decreasing Levy process, rapidly decreasing subordinator. We refer to this model as a multivariate rapidly decreasing Levy process. We consider two benchmarks: Black-Scholes and normal tempered stable process, the later constructed using a classical tempered stable subordinator. We then compare using a time series of daily log-returns and market prices of European-style quanto options the relative performance of the rapidly decreasing Levy process to that of Black-Scholes and the normal tempered stable process. We find that the proposed modeling process is superior to the other two processes for pricing quanto options.

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.

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.

This paper proposes a dynamic equilibrium model that can provide a unified explanation for the stylized facts observed in stock index markets such as the fat tails of risk-neutral return distribution relative to physical distribution, negative expected returns on deep OTM call options, and negative realized variance risk premiums. In particular, we focus on the U-shaped pricing kernel against the stock index return, which is closely related to the negative call returns. We assume that the stock index return follows the time-changed Levy process and that a representative investor has power utility over the aggregate consumption that forms a linear regression of the stock index return and its stochastic activity rate. This model offers a macroeconomic interpretation of the stylized facts from the perspective of the sensitivity of the activity rate and stock index return on the aggregate consumption as well as the investor's risk aversion.

We propose a new overarching interpretation of multidimensional information flows and their relation to market movements. The new conceptualization hinges on results of two distinct mathematical theories, Levy processes and marked Poisson processes, bridged in Jevtic et al. (2016) and applied here in the context of finance, specifically in multivariate modelling of asset returns. Specifically, in this paper we construct a class of multivariate Gaussian marked Poisson processes to model asset returns. The model proposed accommodates the cross section properties of trades, allows for returns to be correlated conditional on trading activity, and preserves normality of returns conditional on trading activity. We specify a process of normal inverse Gaussian type and show that, under suitable conditions, we find as subcases some of the well known multivariate processes recently introduced in the financial literature. As a first application example we estimate a two dimensional price process on daily log-returns and perform a sensitivity analysis of the model parameters to show the model dependence structure flexibility.

We study the consumption-portfolio allocation problem in continuous time when asset prices follow Levy processes and the investor is concerned about potential model misspecification. We derive optimal consumption and portfolio policies that are robust to uncertainty about the hard-to-estimate drift rate, jump intensity and jump size parameters. We also provide a semi-closed form formula for the detection-error probability and compare various portfolio holding strategies, including robust and non-robust policies. Our quantitative analysis shows that ignoring uncertainty leads to significant wealth loss for the investor.

In this paper we present new pricing formulas for some Power style contracts of European type when the underlying process is driven by an important class of L´evy processes, which includes CGMY model, generalized hyperbolic Model and Meixner Model, when no symmetry properties are assumed, extending and complementing in this way previous findings in the literature. Also, we show how to implement our new formulas.

This document contains complementary material of section 6 of the article "Marshall-Olkin Distributions, Subordinators, Efficient Simulation, and Applications to Credit Risk". In particular, we provide the time-inhomogeneous extensions of Theorems 4.1 and 4.2 of the main document. These results are given in Theorems A.1 and A.2 where Levy subordinators are replaced with additive subordinators. We also present additional extensions to the MO distribution which incorporate stochastic dynamics and extend the traditional default intensity factor model. The paper, "Marshall-Olkin Distributions, Subordinators, Efficient Simulation, and Applications to Credit Risk" can be found at: http://ssrn.com/abstract=1702087.

This paper constructs a theoretical framework for multi-population mortality modelling by introducing generalized linear models and L evy stochastic perturbations driven by a common and an idiosyncratic factor. To accommodate the evolution of factor loadings across age and calendar time, we specify the loading coefficients on common factors to be deterministic non-linear functions of age and calendar time. Different model formulations are explored by using L evy processes such as Gamma process and Variance Gamma process, and their in-sample fitting and out-of-sample forecasting performances are evaluated. In addition, we systematically investigate the dependence structure of both the inter-population and cross-population kind. In our empirical investigations, the mortality experiences of male and female lives in UK over the period 1964-2013 are used.

This paper considers the valuation of equity-linked life insurance contracts that offer an annually guaranteed minimum return. The policy premiums are invested in a reference portfolio that is modeled by means of a regime switching Levy process where the model parameters depend on a continuous, finite state Markov chain. Thereby, we can take into account persistent changes in the underlying (macro)economic conditions of financial markets and depart from the unsatisfactory assumption of stationary and independent increments in Levy models. While Levy models turn out to be satisfactory for the valuation of short-term financial contracts, the inclusion of macroeconomic changes is a relevant risk driver for (long-term) insurance contracts. This paper demonstrates that a second advantage of regime switching Levy models is their mathematical tractability: In contrast to a large part of the related literature, the results in this paper do not rely on simulation but are easy-to-implement and closed-form expressions based on (fast) Fourier techniques. A numerical example demonstrates the impact of regime switching on the fair value of cliquet-style return guarantees.

Simple analytical solutions for the prices of discretely monitored barrier options do not yet exist in the literature. This paper presents a semi-analytical and fully explicit solution for pricing discretely monitored barrier options when the underlying asset is driven by a general Levy process. The explicit formula only involves elementary functions, and the Greeks are also explicitly available with little additional computation. By performing a Z-transform, we reduce the valuation problem to an integral equation. This equation is solved analytically with the solution expressed in terms of a Fourier cosine series. We then manage to analytically carry out the Z-transform inversion, and obtain a semi-analytical formula for pricing discrete barrier options. We establish the theoretical error bound and analyze the convergence order of our method. Numerical implementation demonstrates that our numerical results are accurate and efficient, and match up with the results from the benchmark methods in the literature.

In this paper we present two (semi)-analytic synthetic CDO tranche pricing formulas using a subordinator Levy Marshall-Olkin credit correlation model. These formulas can be easily evaluated in terms of machine computational time, therefore they are particularly suitable for the correlation model calibration. To compute the first pricing formula, we assume that the recovery rate and the survival probability is the same for each name of the CDO pool (homogeneous pool approximation). We derive a second pricing formula, under the additional assumption that the number of names in the pool is infinite (large homogeneous pool approximation). Both the two formulas are (semi)-analytic and hold for every class of subordinator Levy process. Finally, the computational cost does not increase with the number of subordinators.

We establish an explicit pricing formula for the class of L\'evy-stable models with maximal negative asymmetry (Log-L\'evy model with finite moments and stability parameter $1, Comment: v2-2 (nov 17): theoretical details and other numerical tests added

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Levy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Levy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points.