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

1 Introduction 2 The Binomial Model 3 A More General One Period Model 4 Stochastic Integrals 5 Differential Equations 6 Portfolio Dynamics 7 Arbitrage Pricing 8 Completeness and Hedging 9 Parity Relations and Delta Hedging 10 The Martingale Approach to Arbitrage Theory (For advanced readers) 11 The Mathematics of the Martingale Approach (For advanced readers) 12 Black-Scholes from a Martingale Point of View (For advanced readers) 13 Multidimensional Models: Classical Approach 14 Multidimensional Approach: Martingale Approach (For advanced readers) 15 Incomplete Markets 16 Dividends 17 Currency Derivatives 18 Barrier Options 19 Stochastic Optimal Control 20 Bonds and Interest Rates 21 Short Rate Models 22 Martingale Models for the Short Rate 23 Forward Rate Models 24 Change of Numeraire (For advanced readers) 25 LIBOR and Swap Market Models 26 Forwards and Futures Appendix A Measure and Integration (For advanced readers) Appendix B Probability Theory (For advanced readers) Appendix C Martingales and Stopping Times (For advanced readers) References Index

http://www.gbv.de/dms/hebis-mainz/toc/071897542.pdf

Arbitrage Theory in Continuous Time

Brownian motion and normal distribution have been widely used in the Black--Scholes option-pricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called "volatility smile" in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jump-diffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of option-pricing problems, including call and put options, interest rate derivatives, and path-dependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.

/pdf/a-jump-diffusion-model-for-option-pricing-596r0ztdlt.pdf

A Jump-Diffusion Model for Option Pricing

We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and Ornstein-Uhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.

/pdf/affine-processes-and-applications-in-finance-24n8fyyqfj.pdf

Affine Processes and Applications in Finance

The paper studies the problem of maximizing the expected utility of terminal wealth in the framework of a general incomplete semimartingale model of a financial market. We show that the necessary and sufficient condition on a utility function for the validity of several key assertions of the theory to hold true is the requirement that the asymptotic elasticity of the utility function is strictly less then one. (author's abstract)

/pdf/the-asymptotic-elasticity-of-utility-functions-and-optimal-5arhjhe8g4.pdf

The asymptotic elasticity of utility functions and optimal investment in incomplete markets

We investigate the term structure of zero coupon bonds when interest rates are driven by a general marked point process as well as by a Wiener process. Developing a theory that allows for measure–valued trading portfolios, we study existence and uniqueness of a martingale measure. We also study completeness and its relation to the uniqueness of a martingale measure. For the case of a finite jump spectrum we give a fairly general completeness result and for a Wiener–Poisson model we prove the existence of a time–independent set of basic bonds. We also give sufficient conditions for the existence of an affine term structure.

/pdf/bond-market-structure-in-the-presence-of-marked-point-1i7c4tnw5o.pdf

Bond Market Structure in the Presence of Marked Point Processes

We consider a general semimartingale model of a currency market with transaction costs and give a description of the initial endowments which allow to hedge a contingent claim in various currencies by a self-financing portfolio. As an application we obtain a result on the structure of optimal strategies for the problem of maximizing expected utility from terminal wealth.

Hedging and liquidation under transaction costs in currency markets

To the memory of our friend and colleague Oliviero Lessi. Abstract. The main purpose of the paper is to provide a mathematical back- ground for the theory of bond markets similar to that available for stock markets. We suggest two constructions of stochastic integrals with respect to processes taking values in a space of continuous functions. Such integrals are used to define the evolution of the value of a portfolio of bonds corresponding to a trad- ing strategy which is a measure-valued predictable process. The existence of an equivalent martingale measure is discussed and HJM-type conditions are derived for a jump-diffusion model. The question of market completeness is considered as a problem of the range of a certain integral operator. We introduce a concept of approximate market completeness and show that a market is approximately complete iff an equivalent martingale measure is unique.

/pdf/towards-a-general-theory-of-bond-markets-1gkvdrfzir.pdf

Towards a general theory of bond markets

Markets with transaction costs

Let \({\cal Q}\) be the set of equivalent martingale measures for a given process \(S\), and let \(X\) be a process which is a local supermartingale with respect to any measure in \({\cal Q}\). The optional decomposition theorem for \(X\) states that there exists a predictable integrand \(\varphi\) such that the difference \(X-\varphi\cdot S\) is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.

/pdf/optional-decomposition-and-lagrange-multipliers-2hfh6w3y8j.pdf

Yuri Kabanov

Papers

Bond Market Structure in the Presence of Marked Point Processes

Hedging and liquidation under transaction costs in currency markets

Towards a general theory of bond markets

Markets with transaction costs

Optional decomposition and Lagrange multipliers