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We introduce a new class of dynamic utilities that are generated forward in time. We discuss the associated value functions, optimal investments, and indifference prices and we compare them with their traditional counterparts, implied by backward dynamic utilities. [ABSTRACT FROM AUTHOR]

We study the effect of risk aversion on the valuation of credit derivatives. Using the technology of utility-indiffierence valuation in intensity-based models of default risk, we analyze resulting yield spreads for single-name defaultable bonds and a simple representative two-name credit derivative. The impact of risk averse valuation on prices and yield spreads is expressed in terms of "effective correlation." [ABSTRACT FROM AUTHOR]

The one-factor Gaussian model is well known not to fit the prices of the different tranches of a collateralized debt obligation (CDO) simultaneously, leading to the implied correlation smile. Recently, other one-factor models based on different distributions have been proposed. Moosbrucker [12] used a one-factor Variance-Gamma (VG) model, Kalemanova et al. [7] and Guégan and Houdain [6] worked with a normal inverse Gaussian (NIG) factor model, and Baxter [3] introduced the Brownian Variance-Gamma (BVG) model. These models bring more flexibility into the dependence structure and allow tail dependence. We unify these approaches, describe a generic one-factor Lévy model, and work out the large homogeneous portfolio (LHP) approximation. Then we discuss several examples and calibrate a battery of models to market data. [ABSTRACT FROM AUTHOR]

A new approach to credit risk modelling is introduced that avoids the use of inaccessible stopping times. Default events are associated directly with the failure of obligors to make contractually agreed payments. Noisy information about impending cash flows is available to market participants. In this framework, the market filtration is modelled explicitly, and is assumed to be generated by one or more independent market information processes. Each such information process carries partial information about the values of the market factors that determine future cash flows. For each market factor, the rate at which true information is provided to market participants concerning the eventual value of the factor is a parameter of the model. Analytical expressions that can be readily used for simulation are presented for the price processes of defaultable bonds with stochastic recovery. Similar expressions can be formulated for other debt instruments, including multi-name products. An explicit formula is derived for the value of an option on a defaultable discount bond. It is shown that the value of such an option is an increasing function of the rate at which true information is provided about the terminal payoff of the bond. One notable feature of the framework is that it satisfies an overall dynamic consistency condition that makes it suitable as a basis for practical modelling situations where frequent recalibration may be necessary. [ABSTRACT FROM AUTHOR]

The goals of the paper are as follows: (i) review some qualitative properties of oil and gas prices in the last 15 years; (ii) propose some mathematical elements towards a definition of mean reversion that would not be reduced to the form of the drift in a stochastic differential equation; (iii) conduct econometric tests in order to conclude whether mean reversion still exists in the energy commodity price behavior. Regarding the third point, a clear "break" in the properties of oil and natural gas prices and volatility can be exhibited in the period 2000-2001. [ABSTRACT FROM AUTHOR]

We derive forward partial integrodifferential equations (PIDEs) for pricing up-and-out and down-and-out call options when the underlying is a jump diffusion. We assume that the jump part of the returns process is an additive process. This framework includes the Variance-Gamma, finite moment logstable, Merton jump diffusion, Kou jump diffusion, Dupire, CEV, arcsinh normal, displaced diffusion, and Black-Scholes models as special cases. [ABSTRACT FROM AUTHOR]

In an affine term structure framework with stochastic volatility, we derive the characteristic function of the log swap rate. Having the characteristic function, we employ the fast Fourier transform (FFT) to price swaptions. Using ten years of swap rates and swaption premiums, model parameters are estimated using a square-root unscented Kalman filter. We investigate the relationship between model premiums and interest rate factors, as well as between market premiums and interest factors, to conclude that long-dated swaptions are highly correlated to the shape of the curve. [ABSTRACT FROM AUTHOR]

We review the Lévy-driven interest rate theory that has been developed in recent years. The intimate relations between the various approaches, as well as the differences, are outlined. The main purpose of this paper is to elaborate on calibration in the real world as well as in the risk-neutral setting. [ABSTRACT FROM AUTHOR]

Most financial asset-pricing models assume frictionless competitive markets that imply the absence of arbitrage opportunities. Given the absence of ar-bitrage opportunities and complete asset markets, there exists a unique martingale measure that implies martingale pricing formulae and replicating asset portfolios. In incomplete markets, or markets with transaction costs, these results must be modified to admit nonunique measures and the possibility of imperfectly replicating portfolios. Similar di3culties arise in markets with taxation. Some theoretical research has argued that some taxation functions will imply arbitrage opportunities and the nonexistence of a competitive asset economy. In this paper we construct a multiperiod, discrete time/state general equilibrium model of asset markets with transaction costs and taxes. The transaction cost technology and the tax system are quite general, so that we can include most discrete time/state models with transaction costs and taxation. We show that a competitive equilibrium exists. Our results require careful modeling of the government budget constraints to rule out tax arbi-trage possibilities. [ABSTRACT FROM AUTHOR]

This paper reviews and extends the mathematical finance literature on bubbles in complete markets. We provide a new characterization theorem for bubbles under the standard no-arbitrage framework, showing that bubbles can be of three types. Type 1 bubbles are uniformly integrable martingales, and these can exist with an infinite lifetime. Type 2 bubbles are nonuniformly integrable martingales, and these can exist for a finite, but unbounded, lifetime. Last, Type 3 bubbles are strict local martingales, and these can exist for a finite lifetime only. When one adds a no-dominance assumption (from Merton [24]), only Type 1 bubbles remain. In addition, under Merton's no-dominance hypothesis, put-call parity holds and there are no bubbles in standard call and put options. Our analysis implies that if one believes asset price bubbles exist and are an important economic phenomena, then asset markets must be incomplete. [ABSTRACT FROM AUTHOR]

This paper provides a brief tutorial on the notions of a zero volatility (ZV) spread and an option adjusted spread (OAS), as applied to fixed income securities. Using the standard definitions, it is shown that the zero volatility spread measures the percentage of a security's spread due to any embedded options and any mispricings. The mispricings could be due to either market or model error. In contrast, the OAS only measures the percentage of the security's spread due to mis-pricings. Refinements and alternative measures of a bond's embedded optionality and mispricings are also provided. [ABSTRACT FROM AUTHOR]

This article reviews the theory of fractional Brownian motion (fBm) in the white noise framework, and we present a new approach to the proof of Itô-type formulas for the stochastic calculus of fractional Brownian motion. [ABSTRACT FROM AUTHOR]

To prove their formulae for the moments of the characteristic polynomial of the generic matrix of U(N), Keating and Snaith [8] (see also Keating [7]) use Selberg's integrals as a ‘black box.' In this note, we point out some identities in law which are equivalent to the expressions of Selberg's integrals and which involve beta, gamma, and normal variables. However, this is a mere probabilistic translation of Selberg's results, and does not provide an independent proof of them. An outcome of some of these translations is that certain logarithms of (Vandermonde) random discriminants are self-decomposable, which hinges on the self-decomposability of the logarithms of the beta (a, b) (2a +b = ≥ 1) and gamma (a > 0) variables. Such self-decomposability properties have been of interest in some joint papers with D. Madan. [ABSTRACT FROM AUTHOR]

A number of remarkable properties of gamma processes are gathered in this paper, including realisation of their bridges, absolute continuity relationships, realisation of a gamma process as an inverse local time, and the effect of a gamma process as a time change. Some of them are put in perspective with their Brownian counterparts. [ABSTRACT FROM AUTHOR]

The Variance-Gamma (VG) process was introduced by Dilip B. Madan and Eugene Seneta as a model for asset returns in a paper that appeared in 1990, and subsequently used for option pricing in a 1991 paper by Dilip and Frank Milne. This paper serves as a tutorial overview of VG and Monte Carlo, including three methods for sequential simulation of the process, two bridge sampling methods, variance reduction via importance sampling, and estimation of the Greeks. [ABSTRACT FROM AUTHOR]

Dilip Madan and I worked on stochastic process models with stationary independent increments for the movement of log-prices at the University of Sydney in the period 1980-1990, and completed the 1990 paper [21] while respectively at the University of Maryland and the University of Virginia. The (symmetric) Variance- Gamma (VG) distribution for log-price increments and the VG stochastic process first appear in an Econometrics Discussion Paper in 1985 and two journal papers of 1987. The theme of the pre-1990 papers is estimation of parameters of log-price increment distributions that have real simple closed-form characteristic function, using this characteristic function directly on simulated data and Sydney Stock Exchange data. The present paper reviews the evolution of this theme, leading to the definitive theoretical study of the symmetric VG process in the 1990 paper. [ABSTRACT FROM AUTHOR]

The Balian-Low Theorem is one of many manifestations of the uncertainty principle in harmonic analysis. Originally stated as a result on the poor time-frequency localization of generating functions of Gabor orthonormal bases, it has become a synonym for many general and abstract problems in time-frequency analysis. In this chapter we present some of the directions in which the Balian-Low Theorem has been extended in recent years. [ABSTRACT FROM AUTHOR]

For a separated sequence Λ={λk}k∈z of real numbers there is a close link between the lower and upper densities D−(Λ), D+(Λ) and the frame properties of the exponentials $$ \{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}:} $$ in fact, $$ \{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}} $$ is a frame for its closed linear span in L2(−ν, ν) for any ν ∈ (0, πD-(Λ)) ∪ (πD+(Λ),∞) . We consider a classical example presented already by Levinson [11] with D-(Λ) = D+(Λ) = 1; in this case, the frame property is guaranteed for all ν ∈ (0; ∞) ∖ {π}. We prove that the frame property actually breaks down for ν = π. Motivated by this example, it is natural to ask whether the frame property can break down on an interval if D−(Λ) ≠ D+(Λ). The answer is yes: We present an example of a family Λ with D−(Λ) ≠ D+(Λ) for which $$ \{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}} $$ has no frame property in L2(−ν, ν) for any ν ∈ (π D−(Λ), π D+(Λ)). [ABSTRACT FROM AUTHOR]

A description of the fine structure of a refinable, shift-invariant sub-space of L2(ℝ) is presented. This fine structure is exhibited through the existence of a canonical frame of functions in such a space, and a related notion of frequency content in these frame elements uniquely determines a multiplicity function that quantifies a redundancy of the frequencies. The refinability of the subspace can then be described by a pair of matrices of periodic functions that satisfy a set of equations, related to the multiplicity function, which play the role of high-dimensional filter equations. [ABSTRACT FROM AUTHOR]

In this chapter we discuss the problem of finding the shift-invariant space model that best fits a given class of observed data F. If the data is known to belong to a fixed—but unknown—shift-invariant space V(Φ) generated by a vector function Φ, then we can probe the data F to find out whether the data is sufficiently rich for determining the shift-invariant space. If it is determined that the data is not sufficient to find the underlying shift-invariant space V, then we need to acquire more data. If we cannot acquire more data, then instead we can determine a shift-invariant subspace S ⊂ V whose elements are generated by the data. For the case where the observed data is corrupted by noise, or the data does not belong to a shift-invariant space V(Φ), then we can determine a space V(Φ) that fits the data in some optimal way. This latter case is more realistic and can be useful in applications, e.g., finding a shift-invariant space with a small number of generators that describes the class of chest X-rays. [ABSTRACT FROM AUTHOR]

We give a complete solution to the problem of sampling and interpolation of functions in PWσ(R) on finite unions of shifted lattices in R of the form $$ \Lambda j = \frac{1} {{2\sigma _j }}Z + \alpha _j ,j = 1, \ldots ,m $$ where σj > 0, αj ∈ R, and ∑j σj =σ. At points where more than one lattice intersect, we sample the function and its derivatives. None of the results or techniques employed is new, but a systematic and elementary treatment of this situation does not seem to exist in the literature. Sampling on unions of shifted lattices includes classical sampling, bunched or periodic sampling, and sampling with derivatives. Such sampling sets arise in deconvolution, tomography, and in the theory of functions bandlimited to convex regions in the plane. [ABSTRACT FROM AUTHOR]

This chapter reviews several ideas that grew out of observations of Djokovic and Vaidyanathan to the effect that a generalized sampling method for bandlimited functions, due to Papoulis, could be carried over in many cases to the spline spaces and other shift-invariant spaces. Papoulis' method is based on the sampling output of linear, time-invariant systems. Unser and Zerubia formalized Papoulis' approach in the context of shift-invariant spaces. However, it is not easy to provide useful conditions under which the Unser-Zerubia criterion provides convergent and stable sampling expansions. Here we review several methods for validating the Unser-Zerubia approach for periodic nonuniform sampling, which is a very special case of generalized sampling. The Zak transform plays an important role. [ABSTRACT FROM AUTHOR]

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L2(ℝn) under the action of lattice translations and dilations by products of elements drawn from non-commuting sets of matrices A and B. Typically, the members of B are matrices whose eigenvalues have magnitude one, while the members of A are matrices expanding on a proper subspace of ℝn. The theory of these systems generalizes the classical theory of wavelets and provides a simple and flexible framework for the construction of orthonormal bases and related systems that exhibit a number of geometric features of great potential in applications. For example, composite wavelets have the ability to produce "long and narrow" window functions, with various orientations, well-suited to applications in image processing. [ABSTRACT FROM AUTHOR]

We consider two classes of actions on ℝn—one continuous and one discrete. For matrices of the form A = eB with B ∈ Mn(ℝ), we consider the action given by γ → γAt. We characterize the matrices A for which there is a cross-section for this action. The discrete action we consider is given by γ → γAk, where A ∈ GLn(ℝ). We characterize the matrices A for which there exists a cross-section for this action as well. We also characterize those A for which there exist special types of cross-sections; namely, bounded cross-sections and finite-measure cross-sections. Explicit examples of cross-sections are provided for each of the cases in which cross-sections exist. Finally, these explicit cross-sections are used to characterize those matrices for which there exist minimally supported frequency (MSF) wavelets with infinitely many wavelet functions. Along the way, we generalize a well-known aspect of the theory of shift-invariant spaces to shift-invariant spaces with infinitely many generators. [ABSTRACT FROM AUTHOR]

This chapter is an introduction to an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L2 function. Background and motivation for the conjecture are provided in the form of a survey of related ideas, results, and open problems in frames, Gabor systems, and other aspects of time-frequency analysis, especially those related to independence. The partial results that are known to hold for the conjecture are also presented and discussed. [ABSTRACT FROM AUTHOR]

Pseudodifferential operators are an indispensable tool for the study of partial differential equations and are therefore a branch of classical analysis. In this chapter we offer an approach using time-frequency methods. In this approach time-frequency representations that are standard in signal analysis are used to set up the formalism of pseudodifferential operators, and certain classes of function spaces and symbols, the modulation spaces, arise naturally in the investigation. Although the approach is "pedestrian" and based more on engineering intuition than on "hard" analysis, strong results on boundedness and Schatten class properties are within its scope. [ABSTRACT FROM AUTHOR]

Gabor duality studies have resulted in a number of characterizations of dual Gabor frames, among which the Wexler-Raz identity and the operator approach reformulation by Janssen and by Daubechies, Landau, and Landau are well known. A concise overview of existing Gabor duality characterizations is presented. In particular, we demonstrate that the Gabor duality conditions by Wexler and Raz [23] and by Daubechies, Landau, and Landau [6], and the parametric dual Gabor formula of [15] are equivalent. [ABSTRACT FROM AUTHOR]

We define the distributional Zak transform and study some of its properties. We show how the distributional Zak transform can be used as an effective tool in the theory of Gabor systems where the window function belongs to the Schwartz class S(ℝ) and where the product of the parameters defining the Gabor system is rational. In particular, we obtain a necessary and sufficient condition for the linear span of such a Gabor system to be dense in S(ℝ) in the topology of S(ℝ) and, if this is the case, we show that a dual window in the Schwartz class can be constructed. We also characterize when such a Gabor system satisfies the Riesz property. [ABSTRACT FROM AUTHOR]

We characterize the existence of finite tight frames whose frame elements are of predetermined length. In particular, we derive a "fundamental inequality" which completely characterizes those sequences which arise as the lengths of a tight frame's elements. Furthermore, using concepts from classical physics, we show that this characterization has an intuitive physical interpretation. [ABSTRACT FROM AUTHOR]

A semidiscrete multiplier is an operator between a space of functions or distributions on a locally compact Abelian group G on the one hand, and a space of sequences on a discrete subgroup H of G on the other hand, with the property that it commutes with shifts by H. We describe the basic form of such operators and show a number of representation theorems for classical spaces like Lp, C0, etc. We also point out parallels to representation theorems for multipliers. [ABSTRACT FROM AUTHOR]

We derive from Fourier inequalities between weighted Lebesgue spaces, weighted Sobolev gradient inequalities for a wide range of indices. The weight functions for which these inequalities hold are easily computable, but the norm constants are not optimal. [ABSTRACT FROM AUTHOR]

The concept of star discontinuity is defined for functions of several variables. A star discontinuity in dimension one is simply a jump discontinuity. It is then shown that in arbitrary dimensions the Gibbs phenomenon for square convergence occurs for periodic functions satisfying appropriate hypotheses at star discontinuities. [ABSTRACT FROM AUTHOR]