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Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an L^p-L^q bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

After a short excursion from the discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. Further development on this idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As a particular case we obtain the fractional Fokker–Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally, we discuss the results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained by different approaches.

The movement of organisms and cells can be governed by occasional long distance runs, according to an approximate Lévy walk. For T cells migrating through chronically-infected brain tissue, runs are further interrupted by long pauses and the aim here is to clarify the form of continuous model equations that describe such movements. Starting from a microscopic velocity-jump model based on experimental observations, we include power-law distributions of run and waiting times and investigate the relevant parabolic limit from a kinetic equation for resting and moving individuals. In biologically relevant regimes we derive nonlocal diffusion equations, including fractional Laplacians in space and fractional time derivatives. Its analysis and numerical experiments shed light on how the searching strategy, and the impact from chemokinesis responses to chemokines, shorten the average time taken to find rare targets in the absence of direct guidance information such as chemotaxis.

The origin of fractal patterns is a fundamental problem in many areas of science. In ecological systems, fractal patterns show up in many subtle ways and have been interpreted as emergent phenomena related to some universal principles of complex systems. Recently, Lévy-type processes have been pointed out as relevant in large-scale animal movements. The existence of Lévy probability distributions in the behavior of relevant variables of movement, introduces new potential diffusive properties and optimization mechanisms in animal foraging processes. In particular, it has been shown that Lévy processes can optimize the success of random encounters in a wide range of search scenarios, representing robust solutions to the general search problem. These results set the scene for an evolutionary explanation for the widespread observed scale-invariant properties of animal movements. Here, it is suggested that scale-free reorientations of the movement could be the basis for a stochastic organization of the search whenever strongly reduced perceptual capacities come into play. Such a proposal represents two new evolutionary insights. First, adaptive mechanisms are explicitly proposed to work on the basis of stochastic laws. And second, though acting at the individual-level, these adaptive mechanisms could have straightforward effects at higher levels of ecosystem organization and dynamics (e.g. macroscopic diffusive properties of motion, population-level encounter rates). Thus, I suggest that for the case of animal movement, fractality may not be representing an emergent property but instead adaptive random search strategies. So far, in the context of animal movement, scale-invariance, intermittence, and chance have been studied in isolation but not synthesized into a coherent ecological and evolutionary framework. Further research is needed to track the possible evolutionary footprint of Lévy processes in animal movement.

Motivated by the interplay between structural and reduced form credit models, we propose to model the firm value process as a time-changed Brownian motion that may include jumps and stochastic volatility effects, and to study the first passage problem for such processes. We are lead to consider modifying the standard first passage problem for stochastic processes to capitalize on this time change structure and find that the distribution functions of such "first passage times of the second kind" are efficiently computable in a wide range of useful examples. Thus this new notion of first passage can be used to define the time of default in generalized structural credit models. Formulas for defaultable bonds and credit default swaps are given that are both efficiently computable and lead to realistic spread curves. Finally, we show that by treating joint firm value processes as dependent time changes of independent Brownian motions, one can obtain multifirm credit models with rich and plausible dynamics and enjoying the possibility of efficient valuation of portfolio credit derivatives.

We examine how to approximate a Lévy process by a hyperexponential jump-diffusion (HEJD) process, composed of Brownian motion and of an arbitrary number of sums of compound Poisson processes with double exponentially distributed jumps. This approximation will facilitate the pricing of exotic options since HEJD processes have a degree of tractability that other Lévy processes do not have. The idea behind this approximation has been applied to option pricing by Asmussen et al. (2007) and Jeannin and Pistorius (2008). In this paper we introduce a more systematic methodology for constructing this approximation which allow us to compute the intensity rates, the mean jump sizes and the volatility of the approximating HEJD process (almost) analytically. Our methodology is very easy to implement. We compute vanilla option prices and barrier option prices using the approximating HEJD process and we compare our results to those obtained from other methodologies in the literature. We demonstrate that our methodology gives very accurate option prices and that these prices are more accurate than those obtained from existing methodologies for approximating Lévy processes by HEJD processes.

We present a very fast and accurate algorithm for calculating prices of finite lived double barrier options with arbitrary terminal payoff functions under regime-switching hyper-exponential jump-diffusion (HEJD) models, which generalize the double-exponential jump-diffusion model pioneered by Kou and Lipton. Numerical tests demonstrate an excellent agreement of our results with those obtained using other methods, as well as a significant increase in computation speed (sometimes by a factor of 5). The first step of our approach is Carr's randomization, whose convergence we prove for barrier and double barrier options under strong Markov processes of a wide class. The resulting sequence of perpetual option pricing problems is solved using an efficient iteration algorithm and the Wiener-Hopf factorization.

Dollar cost averaging (DCA) is a widely employed investment strategy in financial markets. At the same time it is also well documented that such gradual policy is sub-optimal from the point of view of risk averse decision makers with a fixed investment horizon T > 0. However, an explicit strategy that would be preferred by all risk averse decision makers did not yet appear in the literature. In this paper, we give a novel proof for the suboptimality of DCA when (log) returns are governed by Lévy processes and we construct a dominating strategy explicitly. The optimal strategy we propose is static and consists in purchasing a suitable portfolio of path-independent options. Next, we discuss a market governed by a Brownian motion in more detail. We show that the dominating strategy amounts to setting up a portfolio of power options. We provide evidence that the relative performance of DCA becomes worse in volatile markets, but also give some motivation to support its use. We also analyse DCA in presence of a minimal guarantee, explore the continuous setting and discuss the (non) uniqueness of the dominating strategy.

While the proportional hazard model is recognized to be statistically meaningful for analyzing and estimating financial event risks, the existing literature that analytically deals with the valuation problems is very limited. In this paper, adopting the proportional hazard model in continuous time setting, we provide an analytical treatment for the valuation problems. The derived formulas, which are based on the generalized Edgeworth expansion and give approximate solutions to the valuation problems, are widely useful for evaluating a variety of financial products such as corporate bonds, credit derivatives, mortgage-backed securities, saving accounts and time deposits. Furthermore, the formulas are applicable to the proportional hazard model having not only continuous processes (e.g., Gaussian, affine, and quadratic Gaussian processes) but also discontinuous processes (e.g., Lévy and time-changed Lévy processes) as stochastic covariates. Through numerical examples, it is demonstrated that very accurate values can be quickly obtained by the formulas such as a closed-form formula.

We present new approximation formulas for local stochastic volatility models, possibly including Lévy jumps. Our main result is an expansion of the characteristic function, which is worked out in the Fourier space. Combined with standard Fourier methods, our result provides efficient and accurate formulas for the prices and the Greeks of plain vanilla options. We finally provide numerical results to illustrate the accuracy with real market data.

We study the behavior of the long-term yield in a HJM setting for forward rates driven by Lévy processes. The long-term rates are investigated by examining continuously compounded spot rate yields with maturity going to infinity. In this paper, we generalize the model of Karoui et al. (1997) by using Lévy processes instead of Brownian motions as driving processes of the forward rate dynamics, and analyze the behavior of the long-term yield under certain conditions which encompass the asymptotic behavior of the interest rate model's volatility function as well as the variation of the paths of the Lévy process. One of the main results is that the long-term volatility has to vanish except in the case of a Lévy process with only negative jumps and paths of finite variation serving as random driver. Furthermore, we study the required asymptotic behavior of the volatility function so that the long-term drift exists.

In this paper, the option hedging problem for a Markov-modulated exponential Lévy model is examined. We use the local risk-minimization approach to study optimal hedging strategies for Europeans derivatives when the price of the underlying is given by a regime-switching Lévy model. We use a martingale representation theorem result to construct an explicit local risk minimizing strategy.

This paper proposes a generalization of the Barndorff-Nielsen and Shephard model, in which the log return on an asset is governed by a Lévy process with stochastic volatility modeled by a non-Gaussian Ornstein–Uhlenbeck process. Under the generalized model, we derive a closed-form expression of the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

In this paper, a compact scheme with three time levels is proposed to solve the partial integro-differential equation that governs the option prices in jump-diffusion models. In the proposed compact scheme, the second derivative approximation of the unknowns is approximated using the value of these unknowns and their first derivative approximations, thereby allowing us to obtain a tridiagonal system of linear equations for a fully discrete problem. Moreover, the consistency and stability of the proposed compact scheme are proved. Owing to the low regularity of typical initial conditions, a smoothing operator is employed to ensure the fourth-order convergence rate. Numerical illustrations concerning the pricing of European options under the Merton’s and Kou’s jump-diffusion models are presented to validate the theoretical results.

The aim of this paper is to study the small time to maturity of the behavior of the geometric Asian option price and implied volatility under a general stochastic volatility model with Lévy process. The volatility process does not need to be a diffusion or a Markov process, but the future average volatility in the model is a nonadapted process. An anticipating Itô formula for Lévy process and the decomposition of the price (Hull–White formula) are obtained using the Malliavin calculus techniques. The decomposition formula is applied to find the small-time limit of the geometric Asian option price and the implied volatility for the model in at-the-money and out-of-the-money cases.

We establish several closed pricing formulas for various path-independent payoffs, under an exponential Lévy model driven by the Variance Gamma process. These formulas take the form of quickly convergent series and are obtained via tools from Mellin transform theory as well as from multidimensional complex analysis. Particular focus is made on the symmetric process, but extension to the asymmetric process is also provided. Speed of convergence and comparison with numerical methods (Fourier transform, quadrature approximations, Monte Carlo simulations) are also discussed; notable feature is the accelerated convergence of the series for short-term options, which constitutes an interesting improvement of numerical Fourier inversion techniques.

In this paper, we propose an innovative VIX model which takes future market information available to the traders into account. The future information is modeled by an initially enlarged filtration in our setup. We derive an explicit representation for the anticipative VIX process and obtain the associated time dynamics. We also investigate the pricing of variance swaps under both backward- and forward-looking information. We finally deduce the optimal mean variance hedging portfolio in a financial market consisting of a bank account and a VIX futures. In order to have some benchmark model available, we introduce a non-anticipative stochastic volatility stock price model right at the beginning and infer representations for the related VIX index, the VIX futures and a VIX call option.

In this paper, we extend the popular Barndorff–Nielsen–Shephard stochastic volatility model to the case of a pure-jump Ornstein–Uhlenbeck equation with non-vanishing stochastic mean-reversion level. Based on this setup, we derive representations for the squared VIX process and related VIX futures prices. Having these results at hand, we introduce an initially enlarged filtration which models the view of a VIX market insider who has knowledge about the future behavior of the stochastic mean-reversion level of the squared volatility process available. In this enlarged filtration framework, we infer an explicit representation for the anticipative VIX process and obtain the associated time dynamics. We finally investigate the pricing of variance swaps under both backward- and forward-looking information.

With view on global warming and the ongoing climate change, weather derivatives play an increasingly important role for many companies and financial investors, as they constitute useful hedging instruments against disadvantageous weather conditions. In this paper, we present a new temperature model based on generalized Langevin equations driven by Lévy processes. The proposed arithmetic approach captures numerous stylized facts of empirical temperature behavior like seasonal variations, time-dependent volatilities, memory effects, heavy tails and skewness. We further derive a representation for the related meteorological temperature forecast curve and infer the risk-neutral price dynamics of temperature derivatives like CAT, CDD and HDD futures. We finally deduce the minimal variance hedging portfolio in a specific temperature futures market by an application of a stochastic maximum principle and present several practical examples.

In the paper, we develop a very fast and accurate method for pricing double barrier options with continuous monitoring in wide classes of Lévy models; the calculations are in the dual space, and the Wiener–Hopf factorization is used. For wide regions in the parameter space, the precision of the order of $10^{-15}$ is achievable in seconds, and of the order of $10^{-9}$–$10^{-8}$ — in fractions of a second. The Wiener–Hopf factors and repeated integrals in the pricing formulas are calculated using sinh-deformations of the lines of integration, the corresponding changes of variables and the simplified trapezoid rule. If the Bromwich integral is calculated using the Gaver–Wynn Rho acceleration instead of the sinh-acceleration, the CPU time is typically smaller but the precision is of the order of $10^{-9}$–$10^{-6}$, at best. Explicit pricing algorithms and numerical examples are for no-touch options, digitals (equivalently, for the joint distribution function of a Lévy process and its supremum and infimum processes), and call options. Several graphs are produced to explain fundamental difficulties for accurate pricing of barrier options using time discretization and interpolation-based calculations in the state space.