Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

The aim of this paper is to compute the quadratic error of a discrete time-hedging strategy in a complete multidimensional model. This result extends that of Gobet and Temam (2001) and Zhang (1999). More precisely, our basic assumption is that the asset prices satisfy the d-dimensional stochastic differential equation dXit=Xit(bi(Xt)dt+σi,j(Xt)dWjt). We precisely describe the risk of this strategy with respect to n, the number of rebalancing times. The rates of convergence obtained are for any options with Lipschitz payoff and 1/n1/4 for options with irregular payoff.

Analysis Of Error With Malliavin Calculus: Application To Hedging

Wir ubertragen in diesem Kapitel die Satze von Mittag-Leffler und Weierstras auf beliebige offene Mengen in der komplexen Ebene. Die Hauptschwierigkeit besteht dabei in der Konstruktion konvergenzerzeugender Summanden — im Fall der komplexen Ebene konnten wir solche Summanden sofort durch Potenzreihenentwicklung finden. Diese Schwierigkeit wird durch die Approximationssatze von § 1 behoben.

Funktionentheorie auf beliebigen Bereichen

In this note we consider approximate pricing of volatility options on an underlying which follows an exponential Levy process. More specifically, we study call options on the realized variance. The key idea of our approach is to interpret the compensated quadratic variation of the Levy process as a perturbed Brownian motion. The approximation involves even cumulants of the Levy process and option price sensitivities (greeks) in the limiting Bachelier model. We illustrate numerically that our formulas work well if the cumulants of the Levy process are not too large.

Approximate Pricing of Call Options on the Quadratic Variation in Lévy Models

This thesis deals with two basic problems of Mathematical Finance, namely the pricing and hedging of European-style derivatives. We analyze these questions in models that describe the asset underlying the derivative by a process with jumps and stochastic volatility. However, we are not interested in exact solutions to the mentioned problems but in reasonable approximations that allow for a better insight into the structure of the respective question.

More precisely, we consider hedging problems in geometric Levy models, i.e., in models where the logarithmic price process of the underlying follows a process with independent and stationary increments. In this kind of models, there typically exist no perfect hedging strategies. We quantify the remaining risk of a self-financing trading strategy by its mean squared hedging error, i.e., the second moment of the difference between the payoff of the derivative and the terminal wealth of the hedging portfolio.

We study the question of derivative pricing in a comprehensive model class that encompasses geometric Levy models and several stochastic volatility models from the literature. In doing so, we consider prices that are compatible with the absence of arbitrage, i.e., prices that do not allow for riskless gains.

For several hedging strategies, for their hedging errors, as well as for derivative prices in the described framework, the literature provides semi-explicit representations that can be efficiently evaluated numerically for many parametric models. However, these representations admit little understanding, e.g., of the determining factors of the respective quantity. We develop approximate solutions that provide more insight in this respect. To this end, we interpret the complex model with jumps and stochastic volatility as a perturbed Black-Scholes model, and we compute correction terms of second order. Our approach differs from traditional perturbation techniques in the sense that in our case, there is no univariate problem-inherent parameter that quantifies the amount of perturbation. Therefore, we develop a general framework for perturbation approaches in this situation, and we apply this approach in the models under consideration.

The approximate solutions obtained in this way consist of few moments of components of the asset price process as well as of sensitivities (greeks) of the Black-Scholes derivative price. In particular, the formulas do not depend on the fine structure of the considered model and are robust in this sense. We show in detailed numerical experiments that our approximations yield satisfactory results in several parametric models from the literature.

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Stephan Denkl

Papers

Second-order approximations to pricing and hedging in presence of jumps and stochastic volatility