Value of Hedging - Final

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THE VALUE OF HEDGING

A DISSERTATIONSUBMITTED TO THE DEPARTMENT OF

MANAGEMENT SCIENCE & ENGINEERINGAND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITYIN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

Thomas C. Seyller March 2008


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I certify that I have read this dissertation and that, in my opinion, it is fully adequate iscope and quality as a dissertation for the degree of Doctor of Philosophy.

_______________________________ (Ronald A. Howard) Principal Adviser


_______________________________ (Ali E. Abbas)


_______________________________ (Samuel S. Chiu)

Approved for the Stanford University Committee on Graduate Studies.


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Abstract

In this dissertation I introduce a definition of hedging which is based on the comparisoof two indifferent buying prices. With that definition, it then becomes possible to ascriba monetary value to the hedging provided by a specific deal with respect to the decisionmakers existing portfolio. The value of hedging concept can also be used to identifwithin a list of several deals the ones that best complement the portfolio.

Next, I present some fundamental properties of the value of hedging, including somwhich only arise if the decision-makers u-curve satisfies the delta property. I also shesome light on the probabilistic phenomena which are at the source of hedging: in that paof the thesis, I show that hedging can be thought of as moment reengineering, in othewords, as an opportunity to favorably reshape the moments of the decision-maker portfolio by adding other deals to it.

The last concept I introduce is that of the value of perfect hedging. While the value ohedging captures how well a specific deal would fit within the existing portfolio, thvalue of perfect hedging captures the decision-makers willingness to pay for the behedges one can construct to complement his portfolio, based on a specific uncertaintySuch an analysis provides three significant benefits to the decision-maker: first, it enablehim to decide how much of his resources he should devote to searching for hedgesecondly, it allows him to identify the uncertainties on which it is most valuable to hedgeand therefore to focus his search on the most promising classes of deals; and finally, thvalue of perfect hedging can help him establish an upper bound on his personaindifferent buying price for any uncertain deal which he might be considering acquiring.

Throughout the dissertation, I illustrate the approach through practical examples anapplications.


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Acknowledgements

I believe that few experiences in someones life can be as instructive and transforming aworking on a doctoral thesis. My deepest gratitude goes to the wonderful people whmade this experience even more enjoyable than I imagined it would be; I especially wisto thank my wife Aditi for her love, patience and constant encouragements, and m parents, my sister and my brother for their support and for all they have taught me.

I am very much indebted to my dissertation advisor, Ronald Howard, for his teachingand his friendship. From him I have learnt the importance of clarity in thought an

communication, and the efficiency and elegance of simple solutions. I would also like tthank Ross Shachter, Ali Abbas, Daphne Koller, Samuel Chiu and Jim Matheson for theguidance and for many thought-provoking classes and discussions. Their love of andedication to teaching has been an inspiring example for me.

I finally wish to thank my friends and colleagues Ibrahim Al Mojel, Somik RahaDebarun Bhattacharjya, Ram Duriseti and Christopher Han for their support and fonumerous suggestions on this research. My years at Stanford University have bee

especially enriched by my experiences as a teaching assistant for decision analysis atheir side. Never let it be said that it is impossible to conciliate an efficient work ethiwith the desire to have fun.


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Table of Contents

Abstract ......................................................................................................... iv

Acknowledgements ........................................................................................ v

Table of Contents ......................................................................................... vi

Index of Tables & Figures ........................................................................... ix

List of otations Used ................................................................................. xi

Chapter 1 Introduction .............................................................................. 1 1. Dissertation Overview ...................................................................................... 1

2. Motivation ......................................................................................................... 2

3. Research Questions ........................................................................................... 5

4. Elements of Decision Analysis ......................................................................... 7

a) The Six Elements of Decision Quality ........................................................... 8

b) The Five Rules of Actional Thought .............................................................. 9

c) U-curves, Indifferent Buying and Selling Prices, and Delta Property ....... 11

d) The Decision Analysis Cycle ....................................................................... 15

5. Hedging in the Financial Literature ................................................................ 32

a) Traditional Definitions of Hedging ............................................................. 33

b) Mean-Variances Approaches ...................................................................... 35

c) Utility-Based Approaches ........................................................................... 39

Chapter 2 Definition & Valuation of Hedging .......................................41 1. Definition of Hedging ..................................................................................... 42

2. Definition of the Value of Hedging ................................................................ 49

a) The Value of Hedging as a Difference of Two PIBPs ................................. 50

b) The Value of Hedging as the PIBP of a Translated Deal ........................... 52

3. Influence Diagram Representation ................................................................. 56

4. Basic Properties of the Value of Hedging ....................................................... 59


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a) Hedging in the Risk-neutral Case ............................................................... 59

b) Existence of egative Hedging ................................................................... 61

c) Sign and Monotonicity of the Value of Hedging in the Risk-Averse Case .. 62

d) Value of Hedging for Two Deals Which Differ by a Constant ................... 65

e) PIBP for an Uncertain Deal and Value of Hedging ................................... 68

5. Extension of the Value of Hedging Concept to the Sale of Deals .................. 69

Chapter 3 Value of Hedging in the Delta Case ......................................72 1. A Simpler Formula to Compute the Value of Hedging .................................. 73

2. Special Properties of the Value of Hedging in the Delta Case ....................... 76

a) Symmetry ..................................................................................................... 76

b) Upper Bound on the Value of Hedging ....................................................... 77

c) Value of Hedging for Multiples of a Deal with Respect to Itself ................ 83 d) Value of Hedging and Irrelevance .............................................................. 91

3. The Chain Rule for the Value of Hedging and its Implications ..................... 95

a) Chain Rule .................................................................................................. 95

b) Toward an Irrelevance-Based Value of Hedging Algebra ....................... 101

Chapter 4 Hedging as Moment Reengineering ....................................106 1. General Principle .......................................................................................... 107

2. Variance Reengineering ................................................................................ 109 3. Reengineering Moments of Higher Order .................................................... 116

Chapter 5 The Value of Perfect Hedging .............................................120 1. Definition ...................................................................................................... 121

2. Basic Properties ............................................................................................ 126

a) Risk Attitude and Sign of the Value of Perfect Hedging ........................... 126

b) PIBP for an Uncertain Deal and Value of Perfect Hedging .................... 127

c)

Joint Value of Perfect Hedging ................................................................. 130

3. Impact of Relevance on the Value of Perfect Hedging ................................. 134

a) Complete Relevance .................................................................................. 135

b) Irrelevance ................................................................................................ 139

c) Incomplete Relevance ............................................................................... 142

d) Relevance-Based Dominance of one Hedge over Another ....................... 146


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e) Summary Impact of Relevance on the Value of Perfect Hedging .......... 150

f) Influence of the U-Curve on the Choice of a Perfect Hedge .................... 151

4. General Upper Bound on the PIBP of an Uncertain Deal ............................ 153

5. Approximation of the Value of Perfect Hedging in the Delta Case ............. 156

6. Similarities between Value of Perfect Hedging and Value of Clairvoyance 166

7. The Value of Perfect Hedging as a New Appraisal Tool .............................. 168

Chapter 6 Conclusions & Future Work ...............................................172 1. Contributions................................................................................................. 172

2. Limitations .................................................................................................... 174

3. Directions for Future Work ........................................................................... 177

List of References .......................................................................................179 Probability, Bayesian Methods and Decision Analysis: ........................................ 179

Finance and Hedging: ............................................................................................ 184

Index of Terms ...........................................................................................188


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Index of Tables & Figures

Figure I.1 Our valuation of Y can be greatly impacted by X ........................................... 2

Figure I.2 The six elements of decision quality ............................................................... 8

Table I.1 Common u-curves and some of their characteristics ..................................... 13

Figure I.3 The decision analysis cycle ........................................................................... 15

Figure I.4 A decision hierarchy ..................................................................................... 16

Table I.2 Influence diagram semantics possible nodes and their meanings ............... 17

Table I.3 Influence diagram semantics possible arrows and their meanings ............. 18

Figure I.5 An influence diagram .................................................................................... 19

Figure I.6 A tornado diagram ........................................................................................ 21

Figure I.7 Cumulative distribution functions for two alternatives ................................ 23

Figure I.8 Sensitivity to the risk-aversion coefficient .................................................... 26

Figure I.9 Open loop and closed loop sensitivity analyses ........................................... 28

Figure I.10 Minimum-variance hedging for foreign currency ...................................... 36

Figure II.1 Hedging as the comparison of two PIBPs ................................................... 43

Figure II.2 Hedging with complete relevance between X and Y .................................... 44

Figure II.3 Hedging with partial relevance between X and Y ....................................... 46

Figure II.4 An equivalent definition of the value of hedging ......................................... 52

Figure II.5 Hedging with partial relevance between X and Y ....................................... 55

Figure II.6 Using influence diagrams to compute the value of hedging ....................... 56

Figure II.7 An example for which VoH(Y | w, X) is not monotonic in ........................ 63

Figure II.8 Sensitivity of VoH(Y | w, X) to .................................................................. 63

Figure II.9 Hedging provided by a deal which differs from Y by a constant ................ 67

Figure III.1 Best hedge for X ......................................................................................... 79 Figure III.2 Computation of an upper bound on the PIBP of Y .................................... 82

Figure III.3 Sensitivity of VoH(X | X) to .................................................................... 90

Figure III.4 Value of hedging for two irrelevant deals .................................................. 94

Figure III.5 Mnemonic for the chain rule for the value of hedging ............................... 98


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List of otations Used

{A | B, &} Probability assigned to A given B and the background state ofinformation & of the individual.

A B | C, & A is irrelevant to (or some might say probabilisticallyindependent of) B given C and the background state of information & of the person. This means that in the opinion of the person making the statement, {A | B = bi, C, &} = {A | B = b j, C,

&} for any possible outcomes bi and b j of B. Relevance statements, just like probability assignments, are matters of information andnot matters of logic two individuals might disagree as to whether two uncertainties are relevant or irrelevant given a third.

X = {(pi, xi)i [1, n]} The uncertain deal X comprises n prospects xi, with their

associated probabilities pi.

Y | & Mean of Y given &.

V Y | & Variance of Y given &.

S~ Y | w, X The decision-makers PISP for deal Y given that he also ownswealth w and a portfolio of deals denoted by X. In the delta case,

we will use the notationS~ X | to refer toS~ X | w , since w does

not have any effect on the value of certain equivalents.

RPS(Y | w, X) The decision-makers selling risk premium for deal Y given that healso owns wealth w and a portfolio of deals denoted by X. It is

defined as Y | & S~ Y | w, X.


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Chapter 1 Introduction

Humanity is constantly struggling with two contradictory processes.

One of these tends to promote unification, while the other aims at

maintaining or re-establishing diversification.

Claude Lvi-Strauss (b. 1908),

Race et Histoire

1. Dissertation Overview

In this dissertation we will study hedging through a decision analytic lens.

I will first introduce a definition of hedging which is entirely grounded on the principleof personal indifferent buying and selling prices. This will also allow me to establish method through which we can assign a monetary value to the hedging that a dea

provides with respect to the decision-makers existing portfolio. Many of the propertieof the so-defined value of hedging will be presented and illustrated through practicaexamples.

In the later parts of the dissertation, I will present a concept which provides furtheinsights to the decision-maker: the value of perfect hedging. Given a specific portfoliothe value of perfect hedging is the amount that the decision-maker should be willing t pay for the most favorable deal that can be constructed by hedging on some particul

uncertainty or set of uncertainties. The value of perfect hedging thus helps us place aupper bound on the resources that should be devoted to hedging the existing portfolio.


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2. Motivation

In decision analysis, we often observe that our valuation of uncertain deals can be greataffected by the existence of other deals we own. It is a common oversight tounderestimate how pervasive the phenomenon actually is; many students in decisioanalysis, including some who have received training in the discipline for several quarteroften believe that it does not hold for decision-makers who follow the delta property. Thstudents intuition is that since the certain equivalents of such decision-makers founcertain deals do not depend on their initial wealth, they should not depend on thunresolved deals which they own either.

A simple example such as the one presented in the next figure is enough for thosstudents to become aware of their mistake. In that example, the decision-maker, whfollows the delta property, owns deal X and has a certain equivalent of $2,000 for it. Thdecision-maker is then offered deal Y for a fee. At first it might be tempting to reasothat the decision-maker should be willing to pay up to $2,000 for Y, because it comprisethe same probabilities of the same prospects as X.

Figure I.1 Our valuation of Y can be greatly impacted by X

However, such logic is flawed; it does not take into account the fact that deals X and Yare probabilistically relevant. In fact, by combining X and Y into the same portfolio, thdecision-maker is assured of a profit of $5,000 once both deals are resolved, irrespectiv

0.5

0.5

s1

s2

$10,000

-$5,000

Deal X

-$5,000

$10,000

Deal Y


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of whether s1 or s2 is realized. The decision-maker should thus be willing to pay as muchas $3,000 for deal Y, since it can help him transition from a situation worth $2,000 to onworth $5,000.

The difference between the two possible answers, the erroneous one ($2,000) and thcorrect one ($3,000), is sizable. It should remind us that, just like information and contrhedging can be regarded as an alternative which might significantly affect the value othe decision-makers portfolio.

Another element which should make that alternative worthy of our consideration is iavailability. Many financial instruments have been used extensively and for severadecades for hedging purposes, from insurance contracts to derivatives such as forwards swaps. Hedging instruments have even been gaining in accessibility in recent years, be for organizations or private investors; for instance, companies such as HedgeStreet havstarted offering financial derivatives which are designed to give individuals morflexibility as they try to manage the risk of their portfolio.

The primary ambition of this dissertation is to raise our sensitivity as decision analysts the value that hedging can bring to decision-makers. I will strive to present enoughevidence for the reader to be able to decide whether he wants to apply those views ohedging in his professional practice; but it is not one of my intentions to dissect thideological differences between the approach to hedging presented here and that whiccan be found in most of the traditional finance literature. It is true that philosophicadifferences between the decision analytic framework and the financial framework abounthe treatment of uncertainty being one of the most notable examples; but I share thopinion of Edwin T. Jaynes [Jaynes, E.T., 1976], who argued that the true test of thmerits of two competing statistical theories should consist in applying them to the samexamples and deciding which one provides the most sensible results:

Let me make what, I fear, will seem to some a radical, shocking suggestion:themerits of any statistical method are not determined by the ideology which led to

it. For, many different, violently opposed ideologies may all lead to the samefinal working equations for dealing with real problems. Apparently, this phenomenon is something new in statistics; but it is so commonplace in physicsthat we have long since learned how to live with it. Today, when a physicist says,


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3. Research Questions

The fundamental research questions which I intend to address in this dissertation ar

congruent with the motivation and objectives I exposed above:[1] What is hedging? What is a clarity-test definition of hedging?

[2] How can I measure the value of hedging?

[2.a] Given the decision-makers current portfolio, and given a specificdeal which he does not own but is considering buying , how can Iascribe a monetary value to the hedging that the deal provides withrespect to the portfolio?

[2.b] How can I extend the value of hedging concept to the case of aspecific dealwhich he owns but is considering selling ?

[2.c] What remarkable properties does the value of hedging have? Doesit have any special properties in the case in which the decision-makers u-curve satisfies the delta property?

[3] Which probabilistic phenomena are at work in hedging? How can the

existence of hedging be explained?[4] Given the decision-makers portfolio, how can I help him identify the

kinds of deals which best complement it?

[4.a] How can I detect the uncertainties on which it is most valuable to perform hedging?

[4.b] Is there a way to quantify the decision-makers willingness to payfor the ability to perform hedging on a given uncertainty or set of uncertainties?

[5] How can analyses related to hedging be incorporated into the existingdecision analysis process? How can decision analysts best apply thosetools and methods to practical situations?


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I will now give an overview of some of the existing research which will help us answethose questions. I will start with a short introduction to decision analysis, the engineerindiscipline whose principles and techniques will serve as a foundation for this dissertatioAfter that, I will provide a succinct account of the works on hedging which can be founin the finance literature.


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4. Elements of Decision Analysis

The term decision analysis was coined by Ronald A. Howard [1966a] and refers tapplied decision theory. The objective of a decision analysis is to help the decisionmaker achieve clarity of action in other words, to help him identify what is the bescourse of action given the information he has, given his preferences, and given thalternatives that are available to him.

Decision analysis is a normative discipline, and not a descriptive discipline; as such, itsole intent is to provide axioms and norms for how people should make decisions, annot to identify the intellectual and psychological processes which account for how peopactually make decisions.

Over the next few pages, I will provide an overview of some of the most fundamentaconcepts of decision analysis, with an emphasis on the concepts which will be moscritical to understanding this dissertation. I recommend to the readers who are leasfamiliar with the field that they consult the works which are cited in the list of referencein particular those of Ronald A. Howard [1964, 1965, 1966a, 1966b, 1968, 1992, 2004Peter C. Fishburn [1964], or Howard Raiffa [1968].


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a) The Six Elements of Decision Quality

It is difficult to articulate what constitutes a good decision in no more than a sentence.

In decision analysis, we consider that there are six elements to decision quality

[Matheson, D., and Matheson, J. E., 1998]. They are enumerated on the figure belowAny decision or decision process can be rated along those six dimensions, from 0% t100%. 100% indicates the point at which trying to do better on that specific dimensiowould yield an improvement which would not be worth the additional expense oresources required to obtain it.

Three of the six elements of decision quality, namely the alternatives, information an preferences of the decision-maker, are also collectively referred to as the decision basi

Figure I.2 The six elements of decision quality

DecisionQuality 0% 100%

1Appropriate

Frame

6Commitment

To Action

5

LogicallyCorrectReasoning

4Clear Values& Trade-offs

3Meaningful, Reliable

Information

2

Creative,DoableAlternatives


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b) The Five Rules of Actional Thought

Like any other normative system, decision analysis is entirely based on a few principlewhich serve as axioms. There are five of them [Howard, R. A., 1992], and they ar

commonly known as the five rules of actional thought: The probability rule requires the decision-maker to be able to characterize every

alternative he faces by a possibility tree that is, a tree showing the differentscenarios which might unfold following the selection of that particular alternativeThe decision-maker should also be able to assign probabilities to all branches othe tree, and he may or may not choose to assign value measures to characterizthe quality of all of the corresponding possible futures. Once the probability rulhas been applied, the decision-maker obtains a list of prospects, in other words list of all of the possible futures he might face as a result of the decision situation.

The order rule requires the decision-maker to order every prospect in the list from best to worst, according to his own preference. Ties are allowed: the decisionmaker might declare that he is indifferent between several prospects.

The equivalence rule states that for any triplet of prospects which are at differen

levels in the ordered list, A, B and C where A> B > C (> denotes the

preference order), the decision-maker should be able to assign a number p,comprised between 0 and 1, such that he would be just indifferent betweenreceiving B for sure and an uncertain deal in which he would receive A with probability p and C with probability 1 p. Such a number p is called a preference probability.

The substitution rule states that for any triplet of prospects A, B and C where A>

B > C, if the decision-maker faces an uncertain deal in which he assigns a

probability p to his receiving A versus 1 p of his receiving C, where p is alsoequal to his preference probability for B in terms of A versus C, then the decisionmaker should be indifferent between keeping the uncertain deal and exchanging ifor B.


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Finally, let us suppose that there are two prospects A and B at different levels in

the ordered list of prospects, and that A> B; let us also suppose that the decision-

maker needs to choose one of two uncertain deals: in the first, he will receiveeither A with probabilityr or B with probability 1 r , while in the other deal hewill receive either A with a different probability s or B with a probability 1 s.Then, the choice rule requires that the decision-maker choose the deal whichoffers the better chance of the prospect he likes better, namely A: he should selecthe first deal if r > s, the second if r < s.

Decision analysis can only be of help to decision-makers who choose to subscribe to thfive rules of actional thought. By successively and carefully applying the rules, one aftethe other, to a particular decision situation, a decision analyst can help a decision-makeinfer what the best alternative available to him is. Conversely, to the decision-maker whdoes not subscribe to at least one of the five rules, counsel from a decision analyst is of little use as are the theorems of Euclidian geometry to someone who rejects one of it postulates.


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c) U-curves, Indifferent Buying and Selling Prices, and Delta Property

It would be a tedious process to repeatedly apply the equivalence rule to elicit from thdecision-maker a preference probability for each prospect of a large-scale decisio

situation: if there are 3 prospects {Pi}i [1, ], P1 being the best and P N the worst, onewould need at least 2 assessments, one for each prospect within the list {Pi}i [2, -1] in

terms of P1 and P N. Fortunately, it is possible to greatly simplify the assessment process by introducing a few more basic concepts.

Let us first observe that in any decision situation, the course of action which irecommended by the five rules of actional thought does not change if all preferenc probabilities are arbitrarily multiplied by the same positive constant, or if any rea

number is added to all of them. We can thus lift the requirement that preference probabilities should be contained within the interval [0, 1]. All we need to do is introduca new term to refer to those unrestricted preference probabilities, as the analogy with probability is lost once we allow those numbers to dwell outside of [0, 1]; we will cathem u-values instead [Howard, R. A., 1998].

We can then simplify the assessment process for preference probabilities by assuminthat the u-values all follow a particular functional form, which we will call u-curve. If thvalue measure which the decision-maker cares about is his wealth, w, we can denote thu-curve by u(w). If in addition the decision-maker prefers more money to less, u(.) shou be an increasing function.

Once it is assessed from the decision-maker, the u-curve can be used to solve decisiosituations without resorting to preference probabilities anymore: it can be shown thasubscribing to the five rules of actional thought is equivalent to making decisions bselecting the alternative which yields the highest expected u-value.

With u-curves, it also becomes possible to compute the decision-makers PersonaIndifferent Buying Price (PIBP) for something he does not own, or his PersonaIndifferent Selling Price (PISP) for something he owns [Howard, R. A., 1998]. The PIBfor an item is defined as the sum of money such that the decision-maker is just indifferen between not acquiring the item at all and acquiring the item at that price; more formally

if we consider a deal X with probabilities pi and prospects xi, i [1, n], which the


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decision-maker is considering acquiring, and if we denote his present wealth by w and hPIBP for X by b, the indifference statement can be translated into the following equation

) b-xu(w pu(w) ii]n,1[i

+=

Similarly, the PISP for an item is defined as the sum of money such that the decisionmaker is just indifferent between retaining the item and selling the item at that price. ThPISP for an uncertain deal is also called certain equivalent. For an uncertain deal wit

probabilities qi and prospects yi, i [1, m] which the decision-maker owns and is

considering selling, if we denote his PISP for Y by s, then:

)yu(ws)u(w i],1[i

+=+ im

q

The decision-makers risk attitude is directly reflected in the shape of his u-curve, anmore specifically in its concavity or convexity. We will call risk-neutral a decision-makewho assigns to an uncertain deal a certain equivalent which is exactly equal to th probability-weighted average of its monetary prospects (an amount usually referred to athe e-value of the monetary prospects of the deal). A risk-averse decision-maker idefined as one whose certain equivalent for an uncertain deal is inferior to the e-value othe monetary prospects of the deal; for example, a decision-maker who has a certai

equivalent of $450 for an uncertain deal in which he assigns a 50% chance to receivin$1,000 and a 50% chance to receiving nothing is risk-averse. Such a decision-maker hasconcave u-curve. In contrast, a risk-seeking decision-maker is one who assigns to auncertain deal a certain equivalent which is greater than the e-value of the monetar prospects of the deal; his u-curve is convex.

It is important to understand that nothing in the five rules of actional thought prohibits ucurves which are concave on some intervals but convex on others. A decision-make

using such a u-curve would exhibit a risk-averse behavior for some deals and a riskseeking behavior for others. Piecewise concave and convex u-curves are seldom used practice, however; it is not surprising if we consider that very few decision-makers, oncthey understand what the term risk-seeking truly means in decision analysis, would wanto adopt that sort of risk attitude in an actual decision situation.


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The next table shows some of the u-curves which can be most commonly found in thliterature [Bickel, E. J., 1999]:

Functional Form ame Decision-Makers Risk Attitude

x bau(x) += ,

where b > 0 Linear Risk-neutral

x-e bau(x) += ,

where b is of the opposite

sign of

Exponential Risk-averse if > 0Risk-seeking if < 0

n(x)lu(x)= Logarithmic Risk-averse2xc-x bau(x) += ,

where b > 0, c > 0,and x < 2b/c

Quadratic Risk-averse

=

x u(x) Power Risk-averse if < 1Risk-neutral if = 1Risk-seeking if > 1

Table I.1 Common u-curves and some of their characteristics

Among the u-curves listed above, two of them, the linear and exponential form u-curve possess a fascinating characteristic called the delta property [Howard, R. A., 1998consider a decision-maker who has a certain equivalent s for an uncertain deal X = {(pi,

xi)i [1, n]}; what will be his certain equivalent s for an uncertain deal X = {(pi, xi + )i [1,n]}, in other words, for a deal which differs from X by a constant ? It is a natura

desideratum for many decision-makers, especially if they regard the magnitude of thmonetary prospects involved as relatively small compared to their total wealth, to declarthat s should be equal to s + . If that is the case, the decision-makers u-curve is said tsatisfy the delta property.


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What makes the delta property particularly remarkable is the fact that it entails manadditional properties for the decision-maker:

The decision-makers valuation of any uncertain deal remains the same whethe

he includes his initial wealth in his characterization of the deals monetary prospects or not. Therefore, it is not necessary to take initial wealth into accounwhen computing his PIBP or PISP for a deal.

For any uncertain deal, the decision-makers PIBP and PISP for it are equal.

The decision-makers u-curve can only be of one of two forms: linear orexponential. Because u-curves are unique up to a positive linear transformationas mentioned earlier, this also implies that the only number which needs to be

assessed to determine the decision-makers u-curve in its entirety is, which isoften called the decision-makers risk-aversion coefficient. A of zero

corresponds to a risk-neutral decision-maker, a positive to a risk-aversedecision-maker, and a negative to a risk-seeking decision-maker.

Therefore, one question is all decision analysts need to ask in order to assess anthereby the whole u-curve at least in theory, since in practice it is still advisable

to proceed to several measurements of in order to assess it more accurately.

The value of any information gathering process can be computed as the differenc between the value of the deal with the help of the additional information and thvalue of the deal without.


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d) The Decision Analysis Cycle

The decision analysis process [Howard, R.A., 1966a] is iterative. It consists of fou phases, which are depicted below:

Figure I.3 The decision analysis cycle

It is another common oversight to believe that the decision analysis process merely boidown to the construction of a mathematical model and its analysis in reality, the entir process can be thought of as a continuous conversation between the decision-maker anthe decision analyst, intended to guide the decision-maker towards clarity of action, andmathematical model is no more than one of the recurrent and most visible constituents othat conversation.

DeterministicAnalysis

ProbabilisticAnalysis AppraisalStructure

Formulation Evaluation Appraisal

Decision


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Formulation Phase

The main objective of the formulation phase is to frame the decision situation: decisionmaker and decision analyst seek to distinguish the issues which should be unde

consideration in the analysis from those which should not.Framing is more of an art than a procedure with strictly established steps and guidelineHowever, an important element of virtually every framing exercise is to determineexactly which decisions need to be made at this epoch in time. In many practicaexamples, especially if many stakeholders are involved in the decision process, it is nsimple question to answer. A decision hierarchy is a tool which helps address the issue; prompts the decision-maker to separate his decisions into three distinct categories: thothat can be taken as givens; those which are important enough to be the object of thcurrent analysis; and, finally, those which are thought to have a small enough impact othe value for them to see their examination safely deferred until later.

Figure I.4 A decision hierarchy

Once all decisions have been sorted into those three separate containers, we can focus ouattention on the middle category. The next important point to ponder is the list of thuncertainties which should be included in the analysis, given the decisions which are t

Decision 1 Decision 2

Decision A Decision B

Decision I Decision II

TAKE AS GIVE

A ALYZE OW

EXAMI E LATER


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be analyzed, and how those decisions and uncertainties all affect one another as well athe decision-makers value. Influence diagrams, which are also referred to as decisiodiagrams in parts of the literature, are powerful tools which assist analyst and decisionmaker as they communicate about such matters [Howard, R. A., and Matheson, J. E1981]. They offer a graphical representation of the structure of the decision situation, anof the relationships which exist between all the issues involved.

The semantics of influence diagrams are outlined in the following tables; a total of foukinds of nodes and four kinds of arrows, each with a specific meaning, can beencountered in influence diagrams:

ode ame Meaning

Uncertainty NodeAn issue which is not under the

decision-makers control

Decision NodeAn issue which is under the decision-

makers control

Deterministic Node

(or Functional Node)

An uncertainty node which is adeterministic function of all of its parents, i.e. which is known withcertainty if all parents are observed

Value Node What the decision-maker cares about

Table I.2 Influence diagram semantics possible nodes and their meanings

F(x,y)

x y


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Arrow ame Meaning

Relevance ArrowA and B are probabilistically

relevant given their parent nodes

InformationalArrow

Decision D1 and the outcome of uncertainty C are both observed

before making decision D2

Functional

Arrow

Deterministic node F is a function

of decision D and the outcome of E

Influence ArrowThe probability distribution over the

degrees of C varies depending onthe decision which is made at D

Table I.3 Influence diagram semantics possible arrows and their meanings

An example of an influence diagram for an oil exploration and drilling decision is showin the next figure; it is based on one of the models which Ross Shachter presents in hiintroduction to the subject [Shachter, R. D., 1997]:

A B

CD2D1

E

D F

CD


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Figure I.5 An influence diagram

It should also be noted that the role of influence diagrams can be extended beyond theuse as a representation of the structure of the decision situation: in fact, it is possible tencode probabilities and value measures into the influence diagram, and to then solve in order to identify the best alternative and its associated certain equivalent [Shachter, RD., 1986 and 1988].

Profit

RevenueTest? Test Results

Amount of OilSeismic Structure

Drill?

Costs of Drilling

Costs of Testing


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Evaluation Phase

During the evaluation phase, a mathematical model of the decision situation, as frameduring the formulation stage, is built and analyzed in order to identify the best alternativ

Deterministic Analysis

Naturally, the first step consists in building such a mathematical model of the decisiosituation. It should include:

all of the alternatives under consideration;

all of the uncertainties listed on the influence diagram; for each of them, the

decision-maker is asked to provide three possible values, either by himself or witthe help of a designated expert: a base value, corresponding to the median of th probability distribution over the uncertainty; a low value, corresponding to th10th percentile of the distribution; and a high value, corresponding to the 90th percentile;

calculations showing the value which any possible scenario, that is to say any possible selection of an alternative followed by any possible realization of th

uncertainties, would yield for the decision-maker. In many cases, it is the net present value of the scenario which is evaluated.

Such models are often built using spreadsheet programs, as they allow for a rapirecalculation of the value for various scenarios a feature which will prove valuable omany occasions throughout the analysis, as we will see later.

The next task for the decision analyst consists in assessing the relative importance of thuncertainties by comparing their individual effects on the value. Tornado diagrams fi

that need [McNamee, P., and Celona, J., 1987]. For a specific alternative, the analyst fircomputes what is called its base value in other words, the value of the alternative wheall uncertainties are set to their base values. Then, the analyst perturbs each uncertaintyone by one and one at a time, swinging it from base to low and from base to high. Thresults are recorded and the uncertainties are sorted, from the one triggering the greate


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swing in value to the one triggering the smallest. A tornado diagram such as the onshown below is built in order to display the results in a way which makes it easier tdiscuss and interpret them. The whole procedure we just described is sometimes alscalled deterministic sensitivity analysis, in recognition of the fact that the probability oeach possible scenario has not yet been taken into account.

Figure I.6 A tornado diagram

The analyst and the decision-maker can then read off the tornado diagram the names othe uncertainties which have the greatest potential impact on value, and decide whic

variables they want to model probabilistically throughout the remainder of the analys[Howard, R. A., 1966; Matheson, J. E., and Howard, R. A., 1968]. Uncertainties whicare not selected will be modeled as deterministic quantities instead. In addition, thanalyst and the decision-maker may choose to exclude an alternative from the analysis, they observe during the deterministic sensitivity phase that it is systematically dominate by at least one of its rivals.

-100.0 -50.0 0.0 50.0 100.0 150.0

Market Size

Revenue per Unit

Efficacy of the Drug

Number of Competitors

Side Effects of the Drug

Phase III Trial Duration

Phase III Trial Cost

Production Cost per Unit

New Plant Construction Cost

Profit ($ million)


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Probabilistic Analysis

A few assessments need to be made before the analyst can proceed with the evaluation othe probabilistic model: the analyst should first elicit probability distributions for th

uncertainties which were selected at the term of the deterministic phase, with the help othe decision-maker himself or of a designated expert.

Probability encoding is a time-consuming and complex procedure [Spetzler, C. S., anStal von Holstein, A. S., 1975]; like anyone else, decision-makers, experts or othestakeholders in the decision-making process are often the victims of biases which cloutheir judgment, and it is crucial that the analyst help them discern their true belief fromthe distortions that those biases impose. I encourage the reader to refer to the works oAmos Tversky and Daniel Kahneman [1974] for an overview of such biases, and to thoof Carl S. Spetzler and Axel S. Stal von Holstein [1975] for an introduction totechniques which decision analysts can use to combat their influence during probabilitencoding.

Armed with those probability distributions, the analyst can then plot for each alternativecumulative distribution function over the value measure. Such a curve shows, for eac possible realization of the value measure, the probability assigned to obtaining a resuwhich would be inferior to that value. An example of two cumulative distributionfunctions corresponding to two different alternatives is shown on the next figure.

Next, the analyst can assess the decision-makers u-curve over the range of prospecinvolved in the decision situation; as mentioned earlier, if the decision-maker icomfortable with following the delta property over that range, the task is considerableasier, since only one parameter needs to be assessed in that case. Once the u-curve ha been elicited, the analyst can compute the certain equivalents of all the alternatives anidentify the best course of action.


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Figure I.7 Cumulative distribution functions for two alternatives

It is at times possible to infer from the cumulative distribution functions of twoalternatives A and B that A will be preferable to B regardless of the degree to which thdecision-maker is risk-averse [Howard, R. A., 1998]; in such cases, it might thus b

unnecessary to assess the decision-makers risk-aversion coefficient with great precision. The situation arises in instances of:

deterministic dominance: when the worst prospect achievable under alternative Ais better than the best prospect achievable under alternative B;

first-order probabilistic dominance: when the difference between the cumulativdistribution functions of A and B keeps a negative sign over the entire range;

or of second-order probabilistic dominance: when the integral of the difference between the cumulative distribution functions of A and B keeps a negative signover the entire range.

In the example shown above, there is no dominance relationship between A and B. It not a surprising conclusion when one considers that B offers a better mean profit than A

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

-$100 -$50 $0 $50 $100 $150 $200

C u m u l a t

i v e

P r o

b a b i l i t y

Mean: $39 M

Mean: $29 M

Alternative AAlternative B

et Present Value ($ Million)


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but at the cost of a substantially larger downside: to a risk-neutral decision-maker, sincthe mean is all that matters, B will thus appear as the better alternative, whereas to sufficiently risk-averse decision-maker, the magnitude of Bs potential downside wiseem so large that A will earn his favors overall.


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Appraisal Phase

The decision analyst identified the best alternative during the evaluation phase ancomputed its certain equivalent. But that does not mark the end of the analysis; in orde

to achieve complete clarity of action, it is helpful to answer two additional questions: Under what conditions should we make a different decision?

Before choosing the alternative which currently appears to be the best availablecan the decision situation be improved by gathering further information, or byexerting some influence over some of the uncertainties involved? How muchshould the decision-maker be willing to pay for each of those activities?

Sensitivity Analysis

Sensitivity analysis [Howard, R. A., 1968] addresses the first of the two issues: by perturbing the variables involved in the model, one by one and over reasonable ranges values, and by tracking the impact that such changes have on the certain equivalent oeach alternative, the analyst can assess the relative importance of different variablessome may be recognized as being incapable of altering the decision-makers choicewhile others may trigger spectacular shifts in the decision. At the term of the analysis, thdecision-maker will thus be able discern the elements which are worthy of his attentiofrom those which are not.

For instance, if there is no dominance between two alternatives up to the second order, might prove helpful to perform a sensitivity analysis on the decision-makers risk

aversion coefficient,. An example follows; a decision-maker needs to choose betweenfour alternatives, A, B, C and D. He follows the delta property over the entire range o prospects involved and during the assessment process for his risk-aversion coefficient, h

answers all implied values of comprised between 0.07 and 0.11. At first, that mightseem to be a wide interval of possible values, and the analyst might think that it will b

indispensable to ask more questions in order to assess more accurately. However,inspection of the sensitivity analysis chart below reveals that alternative B is the besalternative within this entire range by a comfortable margin, and its certain equivalent


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contained in a relatively narrow interval: [$112,000; $121,000]. All of a sudden, wha

seemed to be an uncomfortably wide range of possible values for appears as no morethan an inconsequential annoyance.

Figure I.8 Sensitivity to the risk-aversion coefficient

Sensitivity analysis also helps the decision-maker understand how the best course oaction might change given slightly different circumstances. This is especially importanin business settings, since the results of a market study, an important announcement mad by a competitor or a large-scale economic event often leave little time to react, and

would not be to the decision-makers advantage if he had to commission a freshevaluation of his decision situation every time he receives a new piece of information.

Sensitivity analysis is usually conducted slightly differently in the case of theuncertainties which were modeled probabilistically during the evaluation phase. As reminder, typically, three values are elicited from the decision-maker for each of thos

$0

$20

$40

$60

$80

$100

$120

$140

$160

C e r

t a i n E q u

i v a l e n

t ( $ T h o u s a n

d )

0.000

Alternative C

Alternative D

Alternative B

Alternative A

0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.02


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uncertainties: a low, a base and a high, which respectively correspond to the 10th, 50th and90th percentiles of the distribution. Those three values can then be used to perform whais called open-loop and closed-loop sensitivity analyses [Howard, R. A., 1968]; morspecifically, if we denote by the alternative which was shown to be the best at the termof the evaluation phase, the analyst computes for uncertainty U with fractiles ulow, u base and uhigh:

the certain equivalents of given that U = ulow, given that U = u base, and giventhat U = uhigh. This is called an open-loop sensitivity analysis: the alternative is permanently set at , and the analyst simply computes the fluctuations in itcertain equivalent as U varies;

the certain equivalent of the alternative which is preferred when U = ulow, thecertain equivalent of the alternative which is preferred when U = u base, and thecertain equivalent of the alternative which is preferred when U = uhigh. Thosealternatives need not be the same, nor does any of them need to be . This iscalled a closed-loop sensitivity analysis: this time, the alternative is not rigidlyanchored on ; as the analyst varies U, he also allows himself to change hisdecision in favor of a better alternative, and it is the certain equivalent of thaalternative which is recorded.

The results of the open-loop and closed-loop sensitivity analyses can be plotted aillustrated below. It should be noted that the open-loop curve should not surpass thclosed-loop curve at any point.


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Figure I.9 Open loop and closed loop sensitivity analyses

Open-loop and closed-loop sensitivity analyses are computationally intensive, since therequire three separate and complete reevaluations of the certain equivalent of eachalternative one reevaluation for U = ulow, one for U = u base and one for U = uhigh.However, the results they produce are remarkably compact, since six data points on chart are enough to capture them in their entirety, and admirably powerful: in ou

illustrative example, the analyst can conclude that would remain the best alternativeven if the decision-maker were to obtain additional information which led him to believthat U = ulow with certainty, or that U = u base; indeed, at those two values, the results of the open-loop and closed-loop sensitivity analyses coincide. Conversely, the besalternative would change if the decision-maker became sure that U = uhigh. In that event, is outperformed by at least one alternative by $56 million.

We will soon see that a few more significant insights can be extracted from the open-loo

and closed-loop curves; but before that, we should mention as a conclusion to ouoverview of sensitivity analysis procedures that in many decision situations, it can also billuminating to evaluate the sensitivity of the decision to the relevance of variouuncertainties to one another [Lowell, D. G., 1994]. This helps distinguish the relevancrelationships which need to be elicited and explicitly modeled as conditional probabilitdistributions from those which can be left as marginal distributions without impairing th

$0

$50

$100

$150

$200

$250

$300

u low u base u high

C e r

t a i n E q u

i v a l e n

t ( $

M i l l i o n )

Closed Loop

Open Loop

C low

C base

C high

O low

O base

O high


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quality of the recommendation. Sensitivity analysis to relevance is especially critical idecision situations with large and intricate probabilistic structures, since it is in thosinstances that simplifying the model in order to avoid unessential conditional probabilitassessments will tend to prove most valuable.

Value of Information and Value of Control

As announced earlier, a second objective of the appraisal phase is to determine whethethe decision situation can be improved by gathering further information, or by exertininfluence over some of the uncertainties involved.

The value of information [Howard, R. A., 1966b] addresses the first half of the questio

for any information gathering scheme, be it a clinical test in a medical decision setting omarket studies and R&D experiments in a major corporation, the value of information defined as the decision-makers PIBP for the additional information. In other words, it the price P at which he is just indifferent between facing his present decision situation ais, and facing the same decision situation with the help of the additional information bualso with a bank account which was reduced by P.

For decision-makers who follow the delta property over the entire range of prospect

involved, we observed earlier that the value of information on some uncertainty U can bcomputed as the difference between the value of the deal with free and perfectinformation on U, and the value of the deal without any further information on U. Both othose quantities can easily be calculated based on the results of the open-loop and closeloop sensitivity analyses on U:

The value of the deal with free and perfect information on U corresponds to whawe might call the certain equivalent of the closed-loop curve; it is equal to the

certain equivalent of an uncertain deal in which there are three monetary prospects, Clow, C base and Chigh, as identified on the previous figure, withrespective probabilities {ulow | &}, {u base | &}, and {uhigh | &}.

On the other hand, the value of the deal with no additional information on Ucorresponds to what we might call the certain equivalent of the open-loop curve


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it is equal to the certain equivalent of an uncertain deal in which there are threemonetary prospects, Olow, O base and Ohigh, with respective probabilities {ulow | &},{u base | &}, and {uhigh | &}. It should be noted that the analyst already calculatedthe quantity in question earlier in the analysis it is equal to the certain equivalenof , as computed during the evaluation phase.

Similarly, the value of control over some uncertainty U is defined as the decision-makerPIBP for being able to choose the outcome of U [Matheson, J. E., 1990]. In a businessetting, it is for example possible to try and influence a products market share blaunching a marketing campaign which will increase the products visibility. In the higtechnology and pharmaceutical sectors, firms might also choose to devote more resourcto a particular project or molecular compound in order to increase the likelihood otechnical success.

Just like in the case of the value of information, there exists an easy procedure tocompute the value of control for decision-makers who follow the delta property over thentire range of prospects: then, the value of control is equal to the difference between thvalue of the deal with free and perfect control over the outcome of U, and the value of thdeal without any control over U. Those two numbers can again be calculated with thhelp of the open-loop and closed-loop chart:

The value of the deal with free and perfect control over U corresponds to thehighest point of the closed-loop curve; in other words, it is equal to the maximumof the values Clow, C base and Chigh. That can be explained by the fact that this time,the decision-maker is free to select the outcome of U he prefers, as well as the best alternative under those conditions.

The value of the deal with no control over U corresponds once again to the certaiequivalent of the open-loop curve, or more explicitly to the certain equivalent o as computed during the evaluation phase.


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Conclusion of the Appraisal Phase

By the end of the appraisal phase, the decision-maker has clarity on the following issues

He has identified the course of action which is the best given his current

preferences, the alternatives that are available to him and his current beliefs.

He has identified the variables which may cause him to modify his decision ithey were to change. He also knows for which revised values of those variablehis choice should change, and how large of an impact such an event would haveon his certain equivalent.

He knows how much he should be willing to pay for information on each one othe uncertainties involved in the decision situation, and how much he should bwilling to pay for the ability to exert control over them.

If there exists an information gathering scheme whose cost would be inferior to the valuof information it would provide to the decision-maker, then it should be pursued. Thfirst three steps of the decision analysis process would then be repeated with the help othe additional information, iteratively, until it eventually becomes sensible to stop thanalysis and act. In Ronald A. Howards words [Howard, R. A., 1988]:

At some point, the appraisal step will show that the recommended alternative isso right for the decision-maker that there is no point in continuing the analysisany further.


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5. Hedging in the Financial Literature

In this part of the dissertation, I will provide a succinct account of the views on hedginwhich are most commonly expressed in the financial literature. We will first examinconventional definitions of hedging, before moving on to an exposition of the key poinof the two prevalent normative theories on hedging the family of mean-variancapproaches, and the family of utility-based approaches.

I will not, however, review the descriptive literature on hedging. This is not for lack oresearch on the subject; indeed, a large body of literature studies questions such as thextent to which hedging is used by corporations [Guay, W. R., and Kothari, S. P., 2003or the degree to which corporate hedging is effective [Nance, D. R., 1993, Smith, C. Wand Smithson, C. W., 1993]. But this dissertation exclusively focuses on normativ perspectives, and on issues which are almost entirely disconnected from the descriptivresearch.


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a) Traditional Definitions of Hedging

In much of the financial literature, hedging refers to the use of financial derivativesuch as futures and forwards in order to mitigate the risk that exists in another investmen

The following definition by Kandice H. Kahl [Kahl, K. H., 1983] illustrates that view:The traditional literature on commodity futures markets defined a hedge as afutures market position which is equal but opposite to the individual's cashmarket position.

The fact that trading derivatives and hedging have become synonyms for many in thfinancial community has to do with the long common history that those two ideas sharIt is indeed with the inception of commodity futures that the concept of hedging firsemerged: in late 17th century Japan, a futures market in rice was developed at Dojima,near Osaka, to help suppliers protect themselves against the risks imposed by natur(poor weather) or by man (war and pillage). It was the first recorded instance in whicsuch a market was created. In the U.S., derivatives trading also started with commoditfutures, with the Chicago Board of Trade, which had been created in 1848, allowininvestors to trade wheat, pork belly and copper futures as early as in the 1860s.

That history probably explains, at least in part, why there has been so much interest ihedging in a field such as agricultural economics, and more importantly why in academcircles the term has often been understood to mean the use of forwards or futures. Bumore recently, a more inclusive definition of hedging began to surface: in the financialiterature of the last two decades, more and more frequently, hedging has referred to aactivity which helps reduce or cancel out the risk imposed by another investmentShehzad L. Mian, for example, proposes the following definition [Mian, S. L., 1996]:

Corporations are exposed to uncertainties regarding a variety of prices. Hedgingrefers to activities undertaken by the firm in order to mitigate the impact of these

uncertainties on the value of the firm.In that broader definition, the reference to derivatives was progressively relegated to throle of mere example. For instance, Deana R. Nance and al. observe that there is more thedging than the use of financial derivatives, and they make an explicit distinctio between off-balance-sheet hedging, in other words the use of those derivatives, an


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b) Mean-Variances Approaches

In the 1950s, Harry M. Markowitz developed the first theory on portfolio allocation iwhich there was an explicit treatment of risk [Markowitz, H. M., 1952 and 1959]. Th

theory taught investors how to identify optimal mean-variance portfolios portfolios iwhich no added diversification could lower the portfolios risk for a given expectereturn, and in which no gain in expected return could be achieved without simultaneousconsenting to an increase in the risk of the portfolio.

Markowitz work, which earned him the Nobel Prize in Economics in 1990, has had profound and lasting influence on financial mathematics. Portfolios which offer thhighest expected return for each level of risk are still called Markowitz efficien portfolios, and at a philosophical level, much of financial theory remains grounded on th premise that investors should make trade-offs between a given reward level, as measure by the mean return of the portfolio, and a given risk level, as captured by the variance othe portfolio.

It is thus no surprise that mean-variance thinking gave rise to a normative theory ohedging, and a popular one at that. In what is perhaps its most basic and most radicaform, the method recommends that the investor completely eliminate the variance in hinvestment by taking an equal and opposite position in the futures market [Luenberger, G., 1997]; if it is infeasible in practice, for example because there exists no futurecontract for the asset whose value needs to be hedged, the investor should identify future contract related to a different commodity, but such that the price of the futurecontract is probabilistically relevant to that of the asset that requires hedging. Thinvestor can then compute the number of future contracts he should buy in order tminimize the variance of the resulting portfolio [Brown, S. L., 1985; Luenberger, D. G1997]. For that reason, the approach is often called minimum-variance hedge.

I will now illustrate the minimum-variance approach through a simple example, which loosely based on one discussed by David G. Luenberger [1997]. A U.S. firm will receiv100,000 euros in a month; they decide to hedge the value of that contract, in U.S. dollar by trading future contracts on the Japanese yen: more specifically, the firm will enter a


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agreement now to sellh yens in a month at an exchange rate of 0.88 dollar for 100 yen.The current exchange rate between U.S. dollars and euros is 1.34 dollars for every euro:

Figure I.10 Minimum-variance hedging for foreign currency

The above tree shows the possible evolution of exchange rates over the coming month, awell as the profits the company would be making from the euro contract and the yecontract under different scenarios as a function of h. Minimizing the variance of the

0.3$/ = 1.21

Contract

0.4

0.3

0.7

$/100 = 0.82

0.2

0.1

$/ = 1.34

$/ = 1.42

$/100 = 0.88

$/100 = 0.94

0.3

$/100 = 0.82

0.4

0.3

$/100 = 0.88

$/100 = 0.94

$/100 = 0.82

0.1

0.2$/100 = 0.88

$/100 = 0.94 0.7

$121,000

$121,000

$121,000

$134,000

$134,000

$134,000

$142,000

$142,000

$142,000

h (0.88-0.82)

Contract

h (0.88-0.88)

h (0.88-0.94)

h (0.88-0.82)

h (0.88-0.88)

h (0.88-0.94)

h (0.88-0.82)

h (0.88-0.88)

h (0.88-0.94)


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entire portfolio, we obtainh* = - 8,750,000 as the optimal value of h. But the answer raises some fundamental questions: for example, why would the solution not depend othe decision-makers risk attitude? If we believe that some investors are more risk-aversthan others, why would they all choose the exact same hedging strategy?

In order to remedy to that weakness of the minimum-variance method, some authors fromthe financial literature advocate replacing as the purpose of hedging the minimization ovariance by the optimization of an objective function which encodes the investors meanvariance trade-offs [Heifner, R. G., 1972 and 1973; Kahl, K. H., 1983; Frechette, D. L2000]. For example, Kandice H. Kahl elects to optimize the following function, where designates profit:

&|&|V

In that function, is a positive number which captures the risk-aversion of the investoAs increases, the influence of the variance of the portfolio on the investors decisionwill also increase in other words, larger values of correspond to more risk-aversinvestors.

While the use of such objective functions allows the investor to take into account hi personal risk attitude as he makes hedging decisions, it can be argued that such a

approach is still not entirely satisfactory. In reality, it can sometimes be misleading toevaluate an investment opportunity based solely on the first two moments of it probability distribution over profits.

For instance, let us consider the situation of a risk-averse investor who owns a portfolwith uncertain returns, and who is given the chance to receive for free another financiinstrument which, if added to his present investments, would leave both the mean anvariance unchanged, but would also significantly improve the third central moment of h

profits by shifting the downside and the upside of the distribution to the right, foexample, and the central region of the distribution to the left. Why would a risk-aversinvestor necessarily dismiss such an investment opportunity as uninteresting? Would hnot appreciate this opportunity to reduce the magnitude of the potential downsidewithout altering the mean or variance of his investment?


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We will come back to that criticism of mean-variance methods much later in thedissertation: we will show that moments of order three and above can be of considerablimportance, and we will even be able to quantify their weight in the investors hedgindecisions.


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c) Utility-Based Approaches

In the late 1980s, interest in another family of normative theories for the study of hedginstarted to surface in the financial literature. Those new approaches, which I will ca

utility-based approaches, were originally inspired by a paper written by Stewart DHodges and Anthony Neuberger [1989], in which they adapted the expected utilitframework from decision theory to options pricing: essentially, Stewart D. Hodges anAnthony Neuberger were computing the price at which the investor would be indifferen between having a given option in his portfolio and not having it.

The theory, which was soon extended to a larger class of transactional cost structurethan the one it was originally developed for [Davis, M. H. A., Panas, V. G., andZariphopoulou, T., 1993], became especially popular for the pricing of derivatives iwhat are called incomplete markets in the financial literature. In such markets, the casflows which an option yields cannot always be replicated by a suitable combination oother assets in the market; therefore, the traditional financial paradigms of portfolireplication and risk-neutral valuation are of little help to an investor who is trying to put price tag on an option, and that explains the popularity of utility-based approaches tsolve such problems [Cvitani, J., Schachermayer, W., and Wang, H., 2001; Delbaen, F.Grandits, P., Rheinlnder, T., Samperi, D., Schweizer, M., and Stricker, C., 2002;

Henderson, V., 2002; Musiela, M., and Zariphopoulou, T., 2004]. For the same reasonsutility-based approaches have also been used recently to price highly sophisticated assesuch as volatility derivatives [Friz, P., and Gatheral, J., 2005; Carr, P., Gman, H., MadaD., and Yor, M., 2005; Grasselli, M. R., and Hurd, T. R., 2007].

In most of those papers, the utility function which is used is the exponential form whicwe already discussed in our review of decision analysis concepts; Matheus R. Grasseland Thomas R. Hurd for example suggest [Grasselli, M. R., and Hurd, T. R., 2007]:

x-eU(x) =

Power utility functions have also drawn some interest from the financial communit[Henderson, V., 2002], but their use has been more sporadic:


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0R where,R 1

x U(x)R 1

>

=

Of all types of power utility functions, the quadratic form is one of the most commonlencountered in the literature [Luenberger, D. G., 1997]. However, the use of quadratiutility functions should probably be regarded as a bridge between mean-variance anutility-based approaches rather than as a utility-based approach stricto sensu , sincemaximizing expected utility then becomes equivalent to optimizing a function which onldepends on the mean and variance of the profit distribution. The fact that quadraticutility-function maximizers seldom use the vocabulary of indifference pricing is furthe proof that their thinking is more characteristic of the mean-variance school of thoughrather than its utility-based counterpart.

The normative theory of hedging which I will present and defend in this dissertation much closer in its philosophy to the views developed by the proponents of utility-baseapproaches. This is not surprising if we consider the fact that the financial communityutility-based approaches to hedging are built on a few principles which are similar tthose of decision analysis, such as indifference pricing.


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Chapter 2 Definition & Valuation of Hedging

We think only through the medium of words; languages are true

analytical methods. Algebra, which is adapted to its purpose in every

species of expression, in the most simple, most exact, and best manner

possible, is at the same time a language and an analytical method. The

art of reasoning is nothing more than a language well-arranged.

Antoine-Laurent de Lavoisier (1743-1794),

Trait Elmentaire de Chimie

Antoine-Laurent de Lavoisiers words should remind us that employing a wellconstructed language is indispensable if we want to think clearly about an issue. Beforwe can make any serious attempt at ascribing a monetary value to hedging, it is thuimperative that we define it in the most precise terms there is little point in trying t build a solid edifice on a weak foundation.

Formulating a clear definition of hedging will be our first ambition in this chapter. Wwill then see that with that definition in place, it becomes easy and natural to ascribe monetary value to hedging.


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1. Definition of Hedging

As noted in the previous chapter, hedging is now commonly understood to refer to ainvestment that is taken out specifically to reduce or cancel out the risk in anotheinvestment [Wikipedia]. Such a definition, unfortunately, is not entirely clear, because leaves one important question unanswered what do we mean by risk? Is the worused as shorthand for variance, as suggested by supporters of the mean-variance famiof normative theories on hedging?

As we seek to define hedging, we should avoid relying on terms which would only add our interlocutors confusion, such as risk. In fact, we will abandon the idea of a risreduction characterization of hedging altogether, to focus instead on the effect hedgin produces on the decision-makers preferences with respect to a specific deal: we wispeak of hedging if the decision-maker finds a deal he does not own more attractive whehe values it with his current portfolio in mind than when he values it without considerinhis existing portfolio; more formally:

Definition 2.1 Hedging

Suppose that the decision-maker has wealth w and owns an unresolved portfolio of deaX; suppose also that he is considering purchasing a deal Y.

We will say that Y provides hedging with respect to X given w if B~ Y | w, X is greater

than B~ Y | w + S~ X | w . In other words, if we compare purchasing deal Y in the

following two states:

The decision-maker owns portfolio X and wealth w(State 1) , The decision-maker owns wealth w +S~ X | w , but does not own X(State 2) ,

then purchasing Y is regarded as more valuable by the decision-maker in the first statthan in the second.

Conversely, we will say that Y provides negative hedging with respect to X given w B~ Y | w, X is less thanB~ Y | w +S~ X | w .


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We are comparing the decision-makers PIBP for Y under two different sets ofcircumstances. Interestingly, those two states of the world, (1) and (2), are so defined th

the decision-maker is just indifferent between them; state (2) is nothing else than state (1following the sale of portfolio X for the exact price at which the decision-maker wa

indifferent to parting with it,S~ X | w . And yet, nothing guarantees that the decision-

maker would also be indifferent between adding Y to his portfolio in state 1, and addinY to his portfolio in state 2.

It is the possible difference in the decision-makers valuation of Y, in the presence oabsence of X but all else being equal, which determines whether there is hedging or not.

Figure II.1 Hedging as the comparison of two PIBPs

$

State 1 State 2

X

$

Y

S~ X | w

? ?

- - -

-

-

-

-

-

- - -

-

-

-

-

-

but would he preferto add Y to his portfolioin State 1 or in State 2?

~

The decision-maker is just indifferent between

State 1 and State 2


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Example 2.1:

A decision-maker, who states that he is comfortable following the delta propertywithin the [-$20,000, $20,000] range and who has a risk tolerance of $10,000

already owns deal X and is thinking of acquiring deal Y as shown below:

Figure II.2 Hedging with complete relevance between X and Y

Then:B~ Y | = $138.07

But:B~ Y | X = $147.61

> B~ Y |

In this example, by our definition, Y does provide hedging with respect to Xgiven w.

It is reassuring that such a conclusion is not at odds with our intuition if we hadhad nothing to guide us but the vague notion that hedging qualifies an investmenwhich reduces the risk of another investment, then we would also have declaredthat Y provides hedging with respect to X. Indeed, adding Y to X reduces themagnitude of the possible downside in the decision-makers prospects, from $0 t

0.4

0.6

s1

s2

$1,000

$0

Deal X

-$100

$300

Deal Y

S~ X | = $388.08 B~ Y | = $138.07


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$300, at the cost of a slight reduction in the magnitude of the possible upsidefrom $1,000 to $900.

Furthermore, this example shows that in situations of hedging, not only do we

have the inequalityB~

Y | w, X >B~

Y | w +S~

X | w , but it sometimes eventurns out thatB~ Y | w, X > Y | & : a risk-averse decision-maker can thus be

willing to pay more for a deal Y than a risk-neutral decision-maker would pay foit, even if they both agree on the probabilities and magnitudes of the prospectsand even if they both own the exact same portfolio X and the same wealth w.

Here:

B~ Y | w, X = $147.61

> Y | & (= $140).

Example 2.2:

Hedging does not only arise in situations in which the monetary prospects of thdeals under consideration are determined by the exact same set of uncertainties, ain Example 2.1. In order to demonstrate it, let us consider a decision-maker whohas the exact same u-curve as in our first example, and owns the same unresolveddeal X; this time, he is contemplating acquiring a different deal Y, as described bythe next figure.

We now have:

B~ Y | = $82.02

But the PIBP for Y when we take the existing portfolio X into account is larger:B~ Y | X = $87.71

> B~ Y |


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Figure II.3 Hedging with partial relevance between X and Y

In spite of the fact that the monetary prospects of X and Y are determined bydifferent uncertainties, Y provides hedging with respect to X given w. Also, jus

as in our first example, that conclusion matches our intuitions predictions, sinceY is structured in such a way that it is more likely than not to compensate for

some of the potential losses imposed by X.

0.4

0.6

s1

s2

$1,000

$0

S~ X | = $388.08

Deal X

0.9

0.1

s1

s2

-$100

$300

B~ Y | = $82.02

Deal Y

0.3

0.7

s1

s2

-$100

$300


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To the first time reader of our definition of hedging, it may seem puzzling that we chosto define it by comparing the decision-makers PIBPs for Y in states 1 and 2:

The decision-maker owns portfolio X and wealth w(State 1) ,

The decision-maker owns wealth w +S~

X | w , but does not own X(State 2) ,instead of comparing the decision-makers PIBPs for Y in the following states 1 and 2:

The decision-maker owns portfolio X and wealth w(State 1) , The decision-maker owns wealth w +S~ X | w , but does not own X(State 2) .

In more formal terms, why compareB~ Y | w, X to B~ Y | w + S~ X | w , instead of

comparing it toB~ Y | w? Would the latter not be more convenient, and yet yield the

exact same result?

Sadly, problems with that second possible definition of hedging would arise for u-curve

which do not satisfy the delta property. For such u-curves,B~ Y | w +S~ X | w is not

necessarily equal toB~ Y | w . The discrepancy has to do with a phenomenon which, in

decision analysis, has often been called the wealth effect: for a risk-averse individuaan increase in wealth tends to cause an increase in their PIBP for an uncertain deal.

With that in mind, let us come back to the comparison between states 1 and 2 as a

possible definition of hedging. Consider an existing portfolio X which is deterministand offers a sure profit equal to x: one might conclude from applying that erroneoudefinition of hedging that since the decision-makers PIBPs for Y in states 1 and 2 ardifferent, Y provides hedging or negative hedging with respect to X given w. And yet, seems absurd to think that one can hedge a portfolio which involves no uncertainty.

The next example demonstrates the same point numerically.


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Example 2.3:

A decision-maker owns $1,000 as his initial wealth, as well as an unresolved deaX which has an assured outcome, equal to $1,000. Let us suppose that the

decision-maker can choose to purchase deal Y from Example 2.1. Finally, we wilsuppose that his u-curve is encoded by u(x) = ln(w + x), where w denotes hisinitial wealth so that the decision-maker does not follow the delta property fothe range of prospects under consideration. Then:

B~ Y | w, X = $130.16

B~ Y | w = $119.93

The two PIBPs for Y are different, so if we were to define hedging based on a

comparison of B~ Y | w, X and B~ Y | w we would then infer that there is

hedging in this example. Yet, it seems preposterous to argue that Y compensatesfor any of Xs risk.

We can explain that the difference betweenB~ Y | w, X andB~ Y | w with wealth

effects. In our example, the potential negative consequence of acquiring Y,namely, the possibility of losing $100, is not as much of a concern to a decisionmaker who has an initial wealth of $2,000 (i.e., in our example, owns w = $1,00as well as deal X) as it is for a decision-maker who simply has an initial wealth o$1,000.

It should be noted, however, that applying the correct definition of hedging whicwe proposed earlier in this chapter would lead to the intuitively acceptableconclusion that there is no hedging provided by Y with respect to X:

B~ Y | w, X = $130.16

B~Y | w +

S~X | w = $130.16.


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2. Definition of the Value of Hedging

Now that we have a better understanding of what constitutes hedging, we are ready tmove on to the question of how t

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