Indifference Pricing: Theory and Applications - Rilegato

Rilegato

ISBN 10: 0691138834 ISBN 13: 9780691138831

Casa editrice: Princeton Univ Pr, 2009

Riassunto

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 Pricing offers 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 pricing


• Explains the fundamentals of indifference pricing, from simple models to the most technical ones


• Goes beyond utility functions to analyze optimal risk transfer and the theory of dynamic risk measures


• Covers non-Markovian and partially observed models and applications to portfolio optimization, defaultable securities, static and quadratic hedging, weather derivatives, and commodities


• Includes extensive bibliography and indexes


• Provides essential reading for PhD students, researchers, and professionals

Su questo libro

Informazioni sull'autore

René Carmona is the Paul M. Wythes '55 Professor of Engineering and Finance in the Department of Operations Research and Financial Engineering at Princeton University. His books include Interest Rate Models and Statistical Analysis of Financial Data in S-Plus.

Dalla quarta di copertina

"This book sets out to elucidate various conceptual and methodological aspects of indifference pricing, and it succeeds with flying colors. Indifference Pricing gives an interesting overview of this new field and is written in a careful, professional, and clear manner. It will be of interest to graduate student's in mathematics, finance, and economics, as well as mathematicians working in mathematical finance and quantitatively minded economists."--Gordan Zitkovic, University of Texas, Austin

Dal risvolto di copertina interno

"This book sets out to elucidate various conceptual and methodological aspects of indifference pricing, and it succeeds with flying colors. Indifference Pricing gives an interesting overview of this new field and is written in a careful, professional, and clear manner. It will be of interest to graduate student's in mathematics, finance, and economics, as well as mathematicians working in mathematical finance and quantitatively minded economists."--Gordan Zitkovic, University of Texas, Austin

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Indifference Pricing

Theory and Applications

PRINCETON UNIVERSITY PRESS

Copyright © 2009 Princeton University Press
All right reserved.
ISBN: 978-0-691-13883-1

Contents

Preface.............................................................................................................................ixPART 1. FOUNDATIONS.................................................................................................................1Chapter 1. The Single Period Binomial Model Marek Musiela and Thaleia Zariphopoulou................................................31.1 Introduction...................................................................................................................31.2 The Incomplete Model...........................................................................................................5Chapter 2. Utility Indifference Pricing: An Overview Vicky Henderson and David Hobson..............................................442.1 Introduction...................................................................................................................442.2 Utility Functions..............................................................................................................452.3 Utility Indifference Prices: Definitions.......................................................................................482.4 Discrete Time Approach to Utility Indifference Pricing.........................................................................512.5 Utility Indifference Pricing in Continuous Time................................................................................522.6 Applications, Extensions, and a Literature Review..............................................................................652.7 Related Approaches.............................................................................................................682.8 Conclusion.....................................................................................................................72PART 2. DIFFUSION MODELS............................................................................................................75Chapter 3. Pricing, Hedging, and Designing Derivatives with Risk Measures Pauline Barrieu and Nicole El Karoui.....................773.1 Indifference Pricing, Capital Requirement, and Convex Risk Measures............................................................783.2 Dilatation of Convex Risk Measures, Subdifferential and Conservative Price.....................................................933.3 Inf-Convolution................................................................................................................983.4 Optimal Derivative Design......................................................................................................1053.5 Recalls on Backward Stochastic Differential Equations..........................................................................1183.6 Axiomatic Approach and g-Conditional Risk Measures.............................................................................1203.7 Dual Representation of g-Conditional Risk Measures.............................................................................1283.8 Inf-Convolution of g-Conditional Risk Measures.................................................................................1363.9 Appendix: Some Results in Convex Analysis......................................................................................141Chapter 4. From Markovian to Partially Observable Models Ren Carmona..............................................................1474.1 A First Diffusion Model........................................................................................................1474.2 Static Hedging with Liquid Options.............................................................................................1544.3 Non-Markovian Models with Full Observation.....................................................................................1594.4 Optimal Hedging in Partially Observed Markets..................................................................................1694.5 The Conditionally Gaussian Case................................................................................................174PART 3. APPLICATIONS................................................................................................................181Chapter 5. Portfolio Optimization Aytac Ilhan, Mattias Jonsson, and Ronnie Sircar..................................................1835.1 Introduction...................................................................................................................1835.2 Indifference Pricing and the Dual Formulation..................................................................................1865.3 Utility Indifference Pricing...................................................................................................1905.4 Stochastic Volatility Models...................................................................................................197Chapter 6. Indifference Pricing of Defaultable Claims Tomasz R. Bielecki and Monique Jeanblanc.....................................2116.1 Preliminaries..................................................................................................................2116.2 Indifference Prices Relative to the Reference Filtration.......................................................................2166.3 Optimization Problems and BSDEs................................................................................................2226.4 Quadratic Hedging..............................................................................................................230Chapter 7. Applications to Weather Derivatives and Energy Contracts Ren Carmona...................................................2417.1 Application I: Temperature Options.............................................................................................2417.2 Application II: Rainfall Options...............................................................................................2497.3 Application III: Commodity Derivatives.........................................................................................256PART 4. COMPLEMENTS.................................................................................................................265Chapter 8. BSDEs and Applications Nicole El Karoui, Said Hamadne, and Anis Matoussi...............................................2678.1 General Results on Backward Stochastic Differential Equations..................................................................2698.2 Applications to Optimization Problems..........................................................................................2798.3 Markovian BSDEs................................................................................................................2858.4 BSDEs with Quadratic Growth with Respect to Z..................................................................................2968.5 Reflected Backward Stochastic Differential Equations...........................................................................303Chapter 9. Duality Methods Robert J. Elliott and John van der Hoek.................................................................3219.1 Introduction...................................................................................................................3219.2 Model..........................................................................................................................3229.3 Utility Functions..............................................................................................................3259.4 Pricing Claims.................................................................................................................3269.5 The Dual Cost Function.........................................................................................................3339.6 The Minimum of [[??].sub.G](y) and [[??].sub.0](y).............................................................................3419.7 The Calculation of [V.sub.0](x)................................................................................................3469.8 The Indifference Asking Price for Claims.......................................................................................3489.9 The Indifference Bid Price.....................................................................................................3559.10 Examples.......................................................................................................................3569.11 Properties of v................................................................................................................3619.12 Numerical Methods..............................................................................................................3649.13 Approximate Formulas...........................................................................................................3749.14 An Alternative Representation for [V.sub.G](x).................................................................................381Bibliography........................................................................................................................387List of Contributors................................................................................................................405Notation Index......................................................................................................................409Author Index........................................................................................................................410Subject Index.......................................................................................................................413

Chapter One

The Single Period Binomial Model Marek Musiela Thaleia Zariphopoulou

1.1 INTRODUCTION

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-called sell 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-called buy side of 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 such models complete. The option price is calculated as the cost of this replication. Adjustments to the price are later made to cover for risks due to the unrealistic representation of reality.

More accurate description of the market is given by the so-called incomplete models in which not all risk in a derivative product can be eliminated by dynamic hedging. However, this potential model advantage is hampered by another difficulty. Namely, the concept of price for a derivative contract is not uniquely defined. Many approaches have been proposed and extensively studied; however, until now no clear consensus has emerged.

On the other side of the spectrum of financial markets there are wealth managers. They have developed their own methodology for implementation of their investment decisions. They may use derivative products to improve their performance; however, their focus is on investment strategy with a view to optimize returns rather than on risk replication. Therefore, it should not come as a surprise that the models they use are very different from the models used in derivatives pricing.

The main aim of this chapter is to work toward convergence of the methodologies used in these apparently quite distant areas. The idea is to associate the concept of price for a derivative contract with a rather natural, to a wealth manager, constraint, that is, maximization of expected utility of wealth. We choose to work with exponential utility and a very simple model structure, namely, the classical single period binomial model. We do so in order to eliminate all technical difficulties, explain the fundamental ideas and compare them with the classical arbitrage free theory, and concentrate exclusively on the most important links between the two areas.

The chapter is organized as follows. In the next section we introduce and analyze in detail the single period binomial model. In particular, we derive an intuitively appealing formula for the indifference price of a general claim. Then, we study the various properties of the indifference price and exhibit the connection with convex risk measures. The link with the classical methodology of pricing by replication is analyzed next. It turns out that the analogue of the so-called delta retains its natural interpretation as the sensitivity of the price with respect to the movement of the instrument used for hedging. Moreover, it also appears that the components of risk that are left unhedged, in our incomplete model setup, have zero value from the perspective of valuation by indifference.

Another important observation about the nature of pricing by the indifference is exposed in the subsection dedicated to relative pricing. Namely, this type of pricing scheme is relative to the agent's portfolio in contrast to the arbitrage-free pricing scheme which is relative to the market portfolio. When interpreted this way, the indifference valuation can be viewed as linear, while, of course, when seen as a functional over a set of random variables it is not.

Going deeper into the comparisons with the pricing by arbitrage, we then investigate the issue of unit choice and the necessary consistency with the static no-arbitrage constraint. We show, in particular, that in order to eliminate static arbitrage one needs to relate the risk aversion parameter with the unit of wealth. To our knowledge, this is the first time that modeling issues pertinent to consistency across units have been identified and addressed.

To accommodate more general situations, we allow for the risk aversion to be modified according to our local in time views about anticipated performance of the traded securities. Specifically, we study the case when the risk aversion parameter depends on the future value of the traded stock. It turns out that the pricing rule retains its intuitive form. Moreover, the associated value function exhibits an interesting relationship between the risk tolerance (the reciprocal of risk aversion) at the end and at the beginning of a time period. Namely, the end of a period of risk tolerance can be viewed as an option payoff, and the consistent risk tolerance for the beginning of the period is its arbitrage-free price.

Motivated by the general need for absence of static arbitrage, and by the above observation in particular, we conclude introducing the notions of the utility normalization and the concepts of the backward and forward utilities.

1.2 THE INCOMPLETE MODEL

We introduce a simple one-period binomial model with one riskless and two risky assets, of which only one is traded. By construction, the model is incomplete and our aim is to develop a coherent approach for investment management and derive from it a pricing methodology for derivative contracts. Optimal investment management is based on maximization of expected utility of wealth. There are a number of constraints we want to impose on our investment decision process and on the derivatives valuation method. To mention just two, we want our investment decisions not to depend on units in which the wealth is expressed. This is mainly because we also need to ensure that our pricing method is consistent with the absence of arbitrage and that it is also numeraire independent. We also want our pricing concept to have a clear intuitive meaning, so an effort is made to interpret the results and, whenever possible, to draw analogies with the classical arbitrage-free theory of complete markets.

1.2.1 Indifference Price Representation

Consider a single period model in a market environment with one riskless and two risky assets. The riskless asset is assumed to offer zero interest rate. Only one of the traded assets can be traded, taken to be a stock. The current values of the traded and nontraded risky assets are denoted, respectively, by [S.sub.0] and [Y.sub.0]. At the end of the period T , the value of the traded asset is [S.sub.T] with [S.sub.T] = [S.sub.0.[xi]], where the random variable [xi] = [[xi].sup.d], [[xi].sup.u] and 0 < [[xi].sup.d] < 1 < [[xi].sup.u]. Similarly, the value of the nontraded asset Y.sub.T satisfies [Y.sub.T] = [Y.sub.0.[eta]], with [eta] = [[eta].sup.d], [[eta].sup.u], with [[eta].sup.d] < [[eta].sup.u], ([Y.sub.0], [Y.sub.T] [not equal to] 0).

We introduce randomness into our single-period model by means of the probability space ([OMEGA], [F.sub.T], P), where [OMEGA] = {[[omega].sub.1], [[omega].sub.2], [[omega].sub.3], [[omega].sub.4]} and P is a probability measure on the [sigma] - algebra [F.sub.T] = [2.sup.[OMEGA]] of all subsets of [OMEGA]. For each i = 1, ..., 4, we assume that [p.sub.i] = P {[[omega].sub.i]} > 0 and we model the upwards and the downwards movement of the two risky assets ST and YT by setting their values as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The measure P represents the so-called historical measure.

Observe that the [sigma]-algebra [F.sub.T] coincides with the [sigma]-algebra [F.sup.(S,Y).sub.T] generated by the random variables [S.sub.T] and [Y.sub.T]. In what follows we will also need the [sigma]-algebra [F.sup.S.sub.T] generated exclusively by the random variable [S.sub.T].

Consider a portfolio consisting of [alpha] shares of the traded asset and the amount invested in the riskless one. Its current value [X.sub.0] = x is equal to + [alpha][S.sub.0] = x, while its wealth [X.sub.T], at the end of the period [0, T], is given by

[X.sub.T] = + [alpha][S.sub.T] = x + [alpha]([S.sub.T] - [S.sub.0]). (1.1)

Now introduce a claim, settling at time T and yielding payoff [C.sub.T]. In pricing of [C.sub.T], we need to specify our risk preferences. We choose to work with the exponential utility

U(x) = [-e.sup.-[gamma]x], x [member of] R and [gamma] > 0. (1.2)

Optimality of investments, which will ultimately yield the indifference price of the claim, is examined via the value function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

Below, we recall the definition of indifference prices.

Definition 1.1 The indifference price of the claim [C.sub.T] = c([S.sub.T], [Y.sub.T]) is defined as the amount ([C.sub.T]) for which the two value functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [V.sub.0], defined in (1.3) and corresponding, respectively, to the claims [C.sub.T] and 0, coincide. Namely, v([C.sub.T]) is the amount which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

for all initial wealth levels x [member of] R.

Looking at the classical arbitrage-free pricing theory, we recall that derivative valuation has two fundamental components which do not depend on specific model assumptions. Namely, the price is obtained as a linear functional of the (discounted) payoff representable via the (unique) risk neutral equivalent martingale measure.

Our goal is to understand how these two components, namely, the linear valuation operator and the risk neutral pricing measure, change when markets become incomplete. In the context of pricing by indifference, we will look for a valuation functional and a naturally related pricing measure under which the price is given as

v([C.sub.T]) = [epsilon]Q([C.sub.T]). (1.5)

Before we determine the fundamental features that E and Q should have, let us look at some representative cases.

Examples:

i) First, we consider a claim of the form [C.sub.T] = c([S.sub.T]). Intuitively, the indifference price should coincide with the arbitrage-free price, for there is no risk that cannot be hedged. Indeed, one can construct a nested complete one-period binomial model and show that

v(c([S.sub.T])) = [E.sub.Q*] (c([S.sub.T])), (1.6)

with [Q.sup.*] being the relevant risk neutral measure. The indifference price mechanism reduces to the arbitrage-free one and any effect on preferences dissipates.

ii) Next, we look at a claim of the form [C.sub.T] = c([Y.sub.T]) and assume for simplicity that the random variables [S.sub.T] and [Y.sub.T] are independent under the measure P. In this case, intuitively, the presence of the traded asset should not affect the price. Indeed, working directly with the value function (1.3) and Definition 1.1, it is straightforward to deduce that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.7)

The indifference price coincides with the classical actuarial valuation principle, the so-called certainty equivalent value, which is nonlinear in the payoff and uses as pricing measure the historical one.

iii) Finally, we examine a claim of the form [C.sub.T] = [c.sub.1]([S.sub.T]) + [c.sub.2]([Y.sub.T]). One could be, wrongly, tempted to price [C.sub.T] by first pricing [c.sub.1]([S.sub.T]) by arbitrage, next pricing [c.sub.2]([Y.sub.T]) by certainty equivalent, and adding the results. Intuitively, this should work when [S.sub.T] and [Y.sub.T] are independent. However, this cannot possibly work under strong dependence between the two variables, for example, when [Y.sub.T] is a function of [S.sub.T]. In general,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The above illustrative examples indicate certain fundamental characteristics E and Q should have. First of all, we observe that a nonlinear valuation functional must be sought. Clearly, any effort to represent indifference prices as expected payoffs under an appropriately chosen universal measure should be abandoned. Indeed, no linear pricing mechanism can be compatible with the concept of indifference based valuation as defined in (1.4). Note that this fundamental observation comes in contrast to the central direction of existing approaches in incomplete models that yield prices as expected payoffs under an optimally chosen measure.

We also see that risk preferences may affect the valuation device given their inherent role in price specification. However, intuitively speaking, we would prefer to specify the pricing measure independently on the risk preferences. Finally, the pricing measure and the valuation device should ideally be the same for all claims to be priced.

The next proposition yields the indifference price in the desired form (1.5).

Proposition 1.1 Let Q be a measure under which the traded asset is a martingale and, at the same time, the conditional distribution of the nontraded asset, given the traded one, is preserved with respect to the historical measure P, i.e.,

Q([Y.sub.T]|[S.sub.T]) = P([Y.sub.T]|[S.sub.T]). (1.8)

Let [C.sub.T] = c([S.sub.T], [Y.sub.T]) be the claim to be priced under exponential preferences with risk aversion coefficient [gamma]. Then, the indifference price of [C.sub.T] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.9)

Proof. We prove the above result by constructing the indifference price via its definition (1.4). We start with the specification of the value functions [V.sub.0] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We represent the payoff [C.sub.T] as a random variable defined on [OMEGA] with values [C.sub.T]([[omega].sub.i]) = [c.sub.i] [member of] R, for i = 1, ..., 4. Elementary arguments lead to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Maximizing over [alpha] leads to the optimal number of shares [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Further straightforward, albeit tedious, calculations yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.11)

where

q = [1 - [[xi].sup.d]]/[[[xi].sup.u] - [[xi].sup.d]]. (1.12)

For [C.sub.T] = 0, the value function takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.13)

From the definition of the indifference price (1.4) and the representations (1.11), (1.13) of the relevant value functions, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.14)

Next, we show that the above price admits the probabilistic representation (1.9). We first consider the terms involving the historical probabilities in (1.14) and we note that they can be actually written in terms of the conditional historical expectations, namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where A = {[[omega].sub.1], [[omega].sub.2]} = {[omega] : [S.sub.T]([omega]) = [S.sub.0] [[xi] .sup.u]}. It is important to observe that conditioning is taken with respect to the terminal values of the traded asset.

We continue with the specification of the pricing measure defined in (1.12). For this, we denote (with a slight abuse of notation) by [q.sub.1], [q.sub.2], [q.sub.3], [q.sub.4] the elementary probabilities of the sought measure Q. Straightforward calculations yield that

[q.sub.1] + [q.sub.2] = q, (1.15)

(Continues...)

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Dettagli dell'edizione

• Editore: Princeton Univ Pr
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 0691138834
• ISBN 13: 9780691138831
• Rilegatura: Copertina rigida
• Numero di pagine: 414
• Redattore: Carmona Rene
• Contatto del produttore: Princeton University Press; https://press.princeton.edu/books; ; 7th Floor, New South Building; PRINCETON; NJ; 8544; U.S.A.
• Persona responsabile: gpsr@libri.de; gpsr@libri.de; ; Europaallee 1; Bad Hersfeld; 36244; Germania