Dynamic Asset Allocation with Forwards and Futures

268 Pages·2005·9.33 MB·English

by  Abraham Lioui

DYNAMIC ASSET ALLOCATION WITH
FORWARDS AND FUTURES
DYNAMIC ASSET ALLOCATION
WITH FORWARDS AND FUTURES
By
ABRAHAM LIOUI
Bar Ilan University, Israel
and
PATRICE PONCET
University of Paris I Pantheon-Sorbonne, France
and
ESSEC Business School, France
Springer
Library of Congress Cataloging-in-Publication Data
Lioui, Abraham.
Dynamic asset allocation with forwards and futures / by Abraham Lioui and Patrice Poncet
p. cm.
Includes bibliographical references and index.
1.Capital assets pricing model. 2. Hedging (Finance) 3. Equilibrium (Economics) I.
Poncet, Patrice. II. Title.
HG4515.2.L56 2005
332.64'524—dc22 2004065099
ISBN 0-387-24107-8 e-ISBN 0-387-24106-X Printed on acid-free paper.
© 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without
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To Osnat, Itzhak and Yair
To Marie, Agnes, Caroline and Sophie
TABLE OF CONTENTS
Preface ix
Acknowledgements xiii
Notations xv
Part I: The basics
Chapter 1: Forward and Futures Markets 3
Chapter 2: Standard Pricing Results Under Deterministic
and Stochastic Interest Rates 23
Part II: Investment and Hedging
Chapter 3: Pure Hedging 37
Chapter 4: Optimal Dynamic Portfolio Choice In
Complete Markets 59
Chapter 5: Optimal Dynamic Portfolio Choice In
Incomplete Markets 81
Chapter 6: Optimal Currency Risk Hedging 93
Chapter 7: Optimal Spreading 117
Chapter 8: Pricing and Hedging under Stochastic
Dividend or Convenience Yield 143
Part III: General Equilibrium Pricing
Chapter 9: Equilibrium Asset Pricing In an Endowment
Economy With Non-Redundant Forward or
Futures Contracts 165
Chapter 10: Equilibrium Asset Pricing In a Production
Economy With Non-Redundant Forward or
Futures Contracts 197
Chapter 11: General Equilibrium Pricing of Futures and
Forward Contracts written on the CPI 221
References 251
Subject Index 261
Preface
This book is an advanced text on the theory of forward and futures
markets which aims at providing readers with a comprehensive knowledge
of how prices are established and evolve in time, what optimal strategies one
can expect the participants to follow, whether they pertain to arbitrage,
speculation or hedging, what characterizes such markets and what major
theoretical and practical differences distinguish futures from forward
contracts. It should be of interest to students (MBAs majoring in finance
with quantitative skills and PhDs in finance and financial economics),
academics (both theoreticians and empiricists), practitioners, and regulators.
Standard textbooks dealing with forward and futures markets generally
focus on the description of the contracts, institutional details, and the
effective (as opposed to theoretically optimal) use of these instruments by
practitioners. The theoretical analysis is often reduced to the (undoubtedly
important) cash-and-carry relationship and the computation of the simple,
static, minimum variance hedge ratio. This book proposes an alternative
approach of these markets from the perspective of dynamic asset allocation
and asset pricing theory within an inter-temporal framework that is in line
with what has been done many years ago for options markets.
The main ingredients of this recipe are those of modern finance, namely
the assumed absence of frictions and arbitrage opportunities in financial and
real markets, the uniqueness of the economic general equilibrium (when the
no-arbitrage assumption is not powerful enough and such an equilibrium is
required), and the tools of continuous time finance, namely martingale
theory and stochastic dynamic programming (to keep developments
tractable, we will assume that all stochastic processes are diffusion
processes). Therefore, tribute must be paid to the pioneers of the relevant
fields or techniques: Merton (who introduced continuous time in finance and
whose numerous articles during the seventies dealt with all the major topics
in that field, such as optimal investment and consumption decisions,
contingent claim analysis (an extension of the celebrated Black-Scholes
formula (1973)), and inter-temporal asset pricing), Sharpe (1964), Lintner
(1965) and Breeden (1979) for capital asset pricing models (CAPM),
Harrison and Kreps (1979) and Harrison and Pliska (1981) for the complete
structure of asset pricing theory, Cox, Ingersoll and Ross (1985a) for their
stochastic production economy and their work on the yield curve, Karatzas,
Lehoczky and Shreve (1987) and Cox and Huang (1989, 1991) who showed
under what conditions a dynamic optimisation problem reduces to a simpler,
static, one, and Heath, Jarrow and Morton (1992) for their pioneering model
of stochastic interest rates. Although we will provide a refresher on these
concepts, approaches and models before using them extensively, the reader
should have preferably a basic knowledge of these materials.
The book is neither a streamlined course text nor a research monograph,
but rather stands between the two, as it is the natural extension of one of our
common fields of published research. The reader is referred to Kolb (2002),
Rendleman (2002) or Hull (2003) for very pedagogical textbooks and to
Duffie (1989) for a somewhat more advanced text. The scope of this book is
essentially theoretical. Although technicalities are unavoidable, they are kept
at the lowest possible level (beyond which some substance is lost). Emphasis
is on economic meaning and financial interpretation rather than on
mathematical rigor. No attempts are made at estimating or testing the
empirical validity of the various models that we or others have developed. It
is only by incidence that empirical evidence will be mentioned or discussed.
However, simulations will at times be performed when important insights
can be delivered or when it is important to assess the practical relevance of
some theoretical results. Also, as to the use of forwards and futures for
investment and/or hedging, focus is on optimal strategies rather than on
actual practice. Finally, potentially important aspects of these derivatives
markets are ignored: transaction and information costs, borrowing
constraints, legal and tax considerations, issues (such as liquidity) that are
best analyzed by means of microstructure theory, differences between
forwards and futures other than the marking-to-market mechanism, and so
on. On the other hand, differences due to the nature of the underlying asset
(be it a commodity, a currency, an interest rate, a bond, a stock or stock
index, or an non-tradable asset such as the Consumer Price Index) are
discussed when relevant.
The book is structured as follows. Part I offers a general presentation of
forward and futures markets and should be read first. Chapter 1 presents the
basic economic analysis of forward and futures contracts, some essential
institutional details, such as the marking-to-market mechanism that
characterizes futures, various data as to the size and scope of the relevant
markets, and empirical evidence as to the use and expansion of such
instruments, their price relationships and the usefulness of some institutional
features. Chapter 2 is essential to the understanding of the sequel as it
provides the basic valuation methodology and price formulas we have at our
disposal under both deterministic and stochastic interest rates. Throughout
the remainder of the book, interest rates will obey stochastic processes.
Parts II and III can be read in any order, although it is more logical to
read them in the proposed order. Part II consists of 6 chapters of, very
XI
roughly, increasing generality. In each of them, optimal strategies using
futures are compared to strategies using the forward counterparts. Chapter 3
deals with a pure hedging problem, as this seems to be a main motivation for
market participants. Chapter 4 is more general as it solves the optimal
portfolio problem of an investor endowed with a non-traded cash position.
Chapter 5 is concerned by investment (or speculation) alone, but in an
incomplete (rather than complete as in the previous two chapters) setting.
Chapter 6 is specific to exchange risk. It uses a different methodology and
tackles the problem of a foreign investor who faces a currency risk in
addition to the risks associated with his/her investment abroad and both
domestic and foreign random interest rates. Chapter 7 deals with the
optimality of using a spread (a long position in one contract and a short one
in an other contract of different maturity) and provides the characteristics of
the optimal spread. Chapter 8 finally examines the issue of stochastic
dividend or convenience yield. Although we retain a complete market
setting, this feature alone invalidates most of the results regarding
equilibrium prices and optimal strategies valid when these yields are
deterministic.
Part III is about general equilibrium pricing. When forward or futures
contracts are not redundant instruments, their introduction completes the
financial market. Therefore, the usual no-arbitrage arguments are not
sufficient to price them and a general equilibrium exercise must be
performed. Chapter 9 is set in a pure exchange economy and shows how the
various CAPMs must be amended to take properly into account this
introduction, which modifies all portfolio allocations and all asset prices. In
particular, traditional results regarding the mean-variance efficiency of the
market portfolio become invalid. Chapter 10 extends the analysis to the case
of a production economy a la Cox, Ingersoll and Ross (1985a), which
reshapes the form of the various CAPMs. Also, it is shown that the cash-
and-carry relationship does not hold in general and, when it does, must be
grounded on equilibrium, not absence of arbitrage, considerations. Finally,
Chapter 11 presents the most general framework of all. To the production
economy of the previous chapter, we add a monetary sector in which the
money supply by the Central Bank is an exogenous stochastic process, so
that a genuine monetary economy is obtained. The stochastic process
followed by the Consumer Price Index, CPI, is derived in an endogenous
manner and then the prices of forward and futures contracts written on it.
Since the CPI is not a traded asset, general equilibrium analysis is required.
Acknowledgements
We are grateful to many people, in particular the researchers who
have developed the theories and techniques outlined above, as well as
the editors and anonymous referees whose comments, remarks and
criticisms have often improved substantially the quality of our
published work. We also have a long standing intellectual debt
towards Florin Aftalion, Bernard Dumas, and, especially, Roland
Portait. We have benefited from useful communications and
discussions with Darrell Duffie and Oldrich Vasicek, and a joint work
in a related area with Pascal Nguyen Due Trong. We have also
benefited from stimulating discussions during workshops and
seminars with our respective colleagues at Bar Ilan, the Sorbonne and
the ESSEC Business School, and attendants to various international
conferences or seminars. As is almost always the case, teaching to our
respective students part of the materials that this book is made of was
both a challenge and a reward.
Special thanks are due to David Cella for initiating this project and
Judith Pforr for continuous assistance during the process.
Finally, we cannot be grateful enough to our wives and children
who have had to suffer from often too long intellectual or physical
absences and have nonetheless given us their love and patience
without parsimony.
Naturally, we alone assume full responsibility for any errors that
would have escaped our attention. Readers are welcome to let us
know about any of them as well as to send comments. Our respective
email addresses are [email protected] and [email protected]