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Indifference Pricing

Indifference Pricing: Theory and Applications

Edited by René Carmona
Copyright Date: 2009
Pages: 440
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  • Book Info
    Indifference Pricing
    Book Description:

    This is the first book about the emerging field of utility indifference pricing for valuing derivatives in incomplete markets. René Carmona brings together a who's who of leading experts in the field to provide the definitive introduction for students, scholars, and researchers. Until recently, financial mathematicians and engineers developed pricing and hedging procedures that assumed complete markets. But markets are generally incomplete, and it may be impossible to hedge against all sources of randomness.Indifference Pricingoffers cutting-edge procedures developed under more realistic market assumptions.

    The book begins by introducing the concept of indifference pricing in the simplest possible models of discrete time and finite state spaces where duality theory can be exploited readily. It moves into a more technical discussion of utility indifference pricing for diffusion models, and then addresses problems of optimal design of derivatives by extending the indifference pricing paradigm beyond the realm of utility functions into the realm of dynamic risk measures. Focus then turns to the applications, including portfolio optimization, the pricing of defaultable securities, and weather and commodity derivatives. The book features original mathematical results and an extensive bibliography and indexes.

    In addition to the editor, the contributors are Pauline Barrieu, Tomasz R. Bielecki, Nicole El Karoui, Robert J. Elliott, Said Hamadène, Vicky Henderson, David Hobson, Aytac Ilhan, Monique Jeanblanc, Mattias Jonsson, Anis Matoussi, Marek Musiela, Ronnie Sircar, John van der Hoek, and Thaleia Zariphopoulou.

    The first book on utility indifference pricingExplains the fundamentals of indifference pricing, from simple models to the most technical onesGoes beyond utility functions to analyze optimal risk transfer and the theory of dynamic risk measuresCovers non-Markovian and partially observed models and applications to portfolio optimization, defaultable securities, static and quadratic hedging, weather derivatives, and commoditiesIncludes extensive bibliography and indexesProvides essential reading for PhD students, researchers, and professionals

    eISBN: 978-1-4008-3311-5
    Subjects: Finance, Economics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Preface
    (pp. ix-xii)
    René Carmona

    • Chapter One The Single Period Binomial Model
      (pp. 3-43)
      Marek Musiela and Thaleia Zariphopoulou

      Derivatives pricing and investment management seem to have little in common. Even at the organizational level, they belong to two quite separate parts of financial markets. The so-calledsell side,represented mainly by the investment banks, among other things offers derivatives products to their customers. Some of them are wealth managers, belonging to the so-calledbuy sideof financial markets.

      So far, the only universally accepted method of derivative pricing is based upon the idea of risk replication. Models have been developed which allow for perfect replication of option payoffs via implementation of a replicating and self-financing strategy. We call...

    • Chapter Two Utility Indifference Pricing: An Overview
      (pp. 44-74)
      Vicky Henderson and David Hobson

      The idea of gamblers ranking risky lotteries by their expected utilities dates back to Bernoulli [22]. An individual’s certainty equivalent amount is the certain amount of money that makes them indifferent between the return from the gamble and this amount, as described in Chapter 6 of Mas-Colell et al. [184]. Certainty equivalent amounts and the principle of equi-marginal utility (see Jevons [141]) have been used by economists for many years.

      More recently, these concepts have been adapted for derivative security pricing. Consider an investor receiving a particular derivative or contingent claim offering payoff CTat future time T>0. When there...


    • Chapter Three Pricing, Hedging, and Designing Derivatives with Risk Measures
      (pp. 77-146)
      Pauline Barrieu and Nicole El Karoui

      The question of pricing and hedging a given contingent claim has a unique solution in a complete market framework. When some incompleteness is introduced, the problem becomes, however, more difficult. Several approaches have been adopted in the literature to provide a satisfactory answer to this problem for a particular choice criterion. Among them, Hodges and Neuberger [131] proposed in 1989 a method based on utility maximization. The price of the contingent claim is then obtained as the smallest (resp. largest) amount leading the agent indifferent between selling (resp. buying) the claim and doing nothing. The price obtained is the indifference...

    • Chapter Four From Markovian to Partially Observable Models
      (pp. 147-180)
      René Carmona

      In this chapter, we compute the value functions needed for indifference pricing in models of continuous time finance of increasing generality. We first implement the indifference pricing paradigm for the exponential utility function in a Markovian setting motivated by an application to weather derivatives, which will be discussed in Chapter 7. As an added bonus to our results we consider static hedging with liquid options in the spirit of the analysis of Chapter 5, and we connect our results to the “maximum entropy philosophy” often used as a basic principle when nothing else can help with the calibration of a...


    • Chapter Five Portfolio Optimization
      (pp. 183-210)
      Aytac Ilhan, Mattias Jonsson and Ronnie Sircar

      We study the problem of portfolio optimization in an incomplete market using derivatives as well as basic assets such as stocks. In such markets, an investor may want to use derivatives, as a proxy for trading volatility, for instance, but they should be traded statically, or relatively infrequently, compared with assumed continuous trading of stocks, because of the much larger transaction costs. We discuss the computational tractability obtained by assuming exponential utility, and connection to the method of utility-indifference pricing. In particular, we show that the optimal number of derivatives to invest in is given by the optimizer in the...

    • Chapter Six Indifference Pricing of Defaultable Claims
      (pp. 211-240)
      Tomasz R. Bielecki and Monique Jeanblanc

      Our goal in this chapter is to give an application of the theory of indifference prices in the context of defaultable claims within thereduced-form approach. In this approach the defaultable market is incomplete and there does not exist a (perfect) hedging strategy for claims that depend on the occurrence of the default. An important issue is the choice of relevant information.

      The chapter is organized as follows. Section 6.1 contains a brief description of the basic concepts of default risk that are used in the chapter. The second section is devoted to indifference pricing in the filtration of default-free...

    • Chapter Seven Applications to Weather Derivatives and Energy Contracts
      (pp. 241-264)
      René Carmona

      In this chapter, we consider three applications of the theory developed in this book. We bring to bear some of the theoretical concepts studied earlier in rather concrete settings. Indifference pricing has recently been applied to many incomplete market models. We discuss only a small sample, and we refer the reader interested in insurance problems to [189] and [140] and in pricing catastrophic bonds to [269]. The numerical results presented in this chapter are borrowed from [43] and [45].

      The crux of option pricing theory in complete markets is that every contingent claim can be perfectly replicated, and in the...


    • Chapter Eight BSDEs and Applications
      (pp. 267-320)
      Nicole El Karoui, Said Hamadène and Anis Matoussi

      Backward stochastic differential equations (BSDEs) were first introduced by J. M. Bismut in 1973 [28] as equations for the adjoint processes in the stochastic version of the Pontryagin maximum principle. Pardoux and Peng [216] generalized the notion in 1990 and were the first to consider general BSDEs and to solve the questions of existence and uniqueness.

      A solution for a BSDE associated with a coefficientg(t, ω, y, z)and a terminal valueξTis a pair of square integrable, adapted (w.r.t. the Brownian filtration) processes${\left( {{Y_t},{Z_t}} \right)_{t \leqslant T}}$such that

      ${Y_t}\; = \;{\xi _T} + \;\int\limits_t^T {\;g\left( {s,\omega ,{Y_s},{Z_s}} \right)} ds - \;\int\limits_t^T {\;{Z_s}d{B_s}} ,\quad t \leqslant T$. (8.1)

      When the functiongis Lipschitz continuous with respect...

    • Chapter Nine Duality Methods
      (pp. 321-386)
      Robert J. Elliott and John van der Hoek

      We shall consider a simple one-period model that includes both tradeable and nontradable assets. Our results generalize some of the results presented by Musiela and Zariphopoulou in chapter 1. When there are no nontradable assets and sufficiently many tradable assets to span the market one can invest to replicate future risk. The initial value of the portfolio must then equal the arbitrage-free price of the future position.

      This method does not work when certain assets cannot be traded, giving rise to positions that cannot be hedged.

      To price such assets and related claims, we extend the development of Chapter 1...

  8. Bibliography
    (pp. 387-404)
  9. List of Contributors
    (pp. 405-408)
  10. Notation Index
    (pp. 409-409)
  11. Author Index
    (pp. 410-412)
  12. Subject Index
    (pp. 413-414)