Kernel-Correlated Lévy Field Driven Forward Rate and Application to Derivative Pricing

We propose a term structure of forward rates driven by a kernel-correlated Lévy random field under the HJM framework. The kernel-correlated Lévy random field is composed of a kernel-correlated Gaussian random field and a centered Poisson random measure. We shall give a criterion to preclude arbitrage under the risk-neutral pricing measure. As applications, an interest rate derivative with general payoff functional is priced under this pricing measure.     

Lévy White Noise Calculus Based on Interaction Exponents

Consider the Lévy white noise space where is the space of Schwartz tempered distributions over and μ is a Lévy white noise measure lifted from a one-dimensional infinitely divisible distribution with finite moments. The classical polynomials of Meixner's type are distinguished through a special form of their generating functions. By lifting the generating function of Meixner orthogonal polynomials, we construct the renormalization kernels explicitly in a unified way. Moreover, we define inner products in n-particle spaces in terms of traces on the ‘diagonals’ and obtain a unified explicit chaotic representation of Lévy–Meixner white noise functionals in terms of interacting Fock spaces. The interacting feature is completely determined by a function g which is referred to as ‘interaction exponent’. This method enables us to easily recapture the general form of Lévy–Meixner field operators. ★Project 10171035 and 10401011 Supported by NSFC.     

PIDE and Solution Related to Pricing of Lévy Driven Arithmetic Type Floating Asian Options

We propose a stochastic model to develop a pricing partial integro-differential equation (PIDE) and its Fourier transform expression for floating Asian options based on the Itô-Lévy calculus. The stock price is driven by a class of infinite activity Lévy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for floating Asian options, and apply the Fourier transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Lévy processes. Finally, the model is calibrated with the market data and its accuracy is presented.     

Pricing currency derivatives with Markov-modulated Lévy dynamics

Using a Lévy process we generalize formulas in Bo et al. (2010) for the Esscher transform parameters for the log-normal distribution which ensure that the martingale condition holds for the discounted foreign exchange rate. Using these values of the parameters we find a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson process intensity with respect to this measure. The formulas for a European call foreign exchange option are also derived. We apply these formulas to the case of the log-double exponential distribution of jumps. We provide numerical simulations for the European call foreign exchange option prices with different parameters.     

Reflected BSDE Driven by a Lévy Process

In this paper, we study the reflected solution of one-dimensional backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. We prove the existence and uniqueness of the solution using a penalization method combined with Snell envelope theory.      

Solutions to Integro-differential Problems Arising on Pricing Options in a Lévy Market

We study an integro-differential parabolic problem arising in Financial Mathematics. Under suitable conditions, we prove the existence of solutions for a multi-asset case in a general domain using the method of upper and lower solutions and a diagonal argument. We also model the jump in the related integro differential equation and give a solution procedure for that model assuming that the brownian motions are not correlated. For a bounded domain, this model for the jump gives an elegant expression of the solution in terms of hyper-spherical harmonics.     

The Jacobi Field of a Lévy Process

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ( -valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy–Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only Lévy process whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.     

Pricing American currency options in an exponential Lévy model

In this article the problem of the American option valuation in a Lévy process setting is analysed. The perpetual case is first considered. Without possible discontinuities (i.e. with negative jumps in the call case), known results concerning the currency option value as well as the exercise boundary are obtained with a martingale approach. With possible discontinuities of the underlying process at the exercise boundary (i.e. with positive jumps in the call case), original results are derived by relying on first passage time and overshoot associated with a Lévy process. For finite life American currency calls, the formula derived by Bates or Zhang, in the context of a negative jump size, is tested. It is basically an extension of the one developed by Mac Millan and extended by Barone‐Adesi and Whaley. It is shown that Bates' model generates pretty good results only when the process is continuous at the exercise boundary.     

Option Pricing for Time-Change Exponential Lévy Model Under Memm

The purpose of this article is to study the rational evaluation of European options price whenthe underlying price process is described by a time-change Lévy process.European option pricing formula isobtained under the minimal entropy martingale measure(MEMM)and applied to several examples of particulartime-change Lévy processes.It can be seen that the framework in this paper encompasses the Black-Scholesmodel and almost all of the models proposed in the subordinated market.     

Pricing industry loss warranties in a Lévy–Frailty framework

We propose a novel risk-neutral pricing approach for industry loss warranties. In doing so, we explicitly take into account the statistical dependence of the losses on individual policies in the underlying insurance portfolio, caused by the occurrence of a natural catastrophe. Inspired by recent advances in the structured credit literature, we model joint claim events in a Lévy–Frailty framework with a stochastic time change. Event time is driven by rare and large jumps of a compound Poisson subordinator and thus elapses more quickly when a natural catastrophe has struck, leading to a clustering of losses. We estimate the model on historical ILW quotes and obtain encouraging fit statistics.     

Quantum Lévy processes on dual groups

We present a theory of quantum (non-commutative) Lévy processes on dual groups which generalizes the theory of Lévy processes on bialgebras. It follows from a result of N. Muraki that there exist exactly 5 notions of non-commutative ‘positive’ stochastic independence. We show that one can associate a commutative bialgebra with each pair consisting of a dual group and one of the 5 notions of independence. This construction is related to a construction of U. Franz. Our construction has the advantage that the important case of free independence is included. We show that Lévy processes are given by their generators which are precisely the conditonally positive linear functionals on the dual group.Supported by the European Research Training Network “Quantum Probability with Applications to Physics, Information Theory and Biology”     

A note on branching Lévy processes

We show for the branching Lévy process that it is possible to construct two classes of multiplicative martingales using stopping lines and solutions to one of two source equations. The first class, similar to those martingales of Chauvin (1991, Ann. Probab. 30, 1195–1205) and Neveu (1988, Seminar on Stochastic Processes 1987, Progress in Probability and Statistics, vol. 15, Birkhaüser, Boston, pp. 223–241) have a source equation which provides travelling wave solutions to a generalized version of the K-P-P equation. For the second class of martingales, similar to those of Biggins and Kyprianou (1997, Ann. Probab. 25, 337–360), the source equation is a functional equation. We show further that under reasonably broad circumstances, these equations share the same solutions and hence the two types of martingales are one and the same. This conclusion also tells us something more about the nature of the solutions to the first of our two equations.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

Reflected Generalized Backward Doubly SDEs Driven by Lévy Processes and Applications

In this paper, we study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with Lévy process (RGBDSDELs in short) with one continuous barrier. Under uniformly Lipschitz coefficients, we prove an existence and uniqueness result by means of the penalization method and the fixed-point theorem. As an application, this study allows us to give a probabilistic representation for the solutions to a class of reflected stochastic partial differential integral equations (SPDIEs in short) with a nonlinear Neumann boundary condition.     

Derivative Formula and Harnack Inequality for SDEs Driven by Lévy Processes

By using lower bound conditions of the Lévy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Lévy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Lévy processes is derived.     

A Class of Lévy Driven SDEs and their Explicit Invariant Measures

We prove non-explosion results for Schrödinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.     

两指标Lévy过程的Lévy Markov性

关于两指标过程的Lévy Markov性,[2]证明了:对于广义Brownian Sheet和广义OUP_2,对适当的DR_+,有: 那里充分利用了过程的轨道连续性及正态系的一个性质:独立性等价于不相关性,[2]的这个结果使[1]中结果 (对一般的两指标Markov过程成立)对此特殊过程得到改进,本文的结果是:对于随机连续的独立增量过程(即两指标Lévy过程),对具有分段光滑边界的D∈B_+,有:由于两指标Lévy过程以广义Brownian Sheet,广义OUP_2及Poisson单为特例,故此结果推广了[2]的结果,而方法不同于[2]     

Hyperfinite Lévy Processes

A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula.     

Variational Solutions of the Pricing PIDEs for European Options in Lévy Models

Abstract

One of the fundamental problems in financial mathematics is to develop efficient algorithms for pricing options in advanced models such as those driven by Lévy processes. Essentially there are three approaches in use. These are Monte Carlo, Fourier transform and partial integro-differential equation (PIDE)-based methods. We focus our attention here on the latter. There is a large arsenal of numerical methods for efficiently solving parabolic equations that arise in this context. Especially Galerkin and Galerkin-inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

The contribution of this paper is therefore to analyse weak solutions of the Kolmogorov backward equations which are related to prices of European options in (time-inhomogeneous) Lévy models and to establish a precise link between the prices and the weak solutions of these equations. The resulting relation is a Feynman–Kac representation of the solution as a conditional expectation. Our special concern is to provide a framework that is able to cover both, the common types of European options and a wide range of advanced models in which these derivatives are priced.

An application to financial models requires in particular to admit pure jump processes such as generalized hyperbolic processes as well as unbounded domains of the equation. In order to deal at the same time with the typical pay-offs that can arise, the weak formulation of the equation is based on exponentially weighted Sobolev–Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.     

A Radial Basis Function Scheme for Option Pricing in Exponential Lévy Models

Abstract

We use Radial Basis Function (RBF) interpolation to price options in exponential Lévy models by numerically solving the fundamental pricing PIDE (Partial integro-differential equations). Our RBF scheme can handle arbitrary singularities of the Lévy measure in 0 without introducing further approximations, making it simpler to implement than competing methods. In numerical experiments using processes from the CGMY-KoBoL class, the scheme is found to be second order convergent in the number of interpolation points, including for processes of unbounded variation.