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Master’s Thesis

MSc in Finance and Investments

A State-Contingent Model for Applied Capital Budgeting

An extension of the state-contingent capital budgeting model by Banz & Miller

Copenhagen Business School May 14, 2018

Knut Gaute Steen Student Number: 107065

Ilan Isak Hollander Student Number: 53999

Supervisor:

Kristian R. Miltersen

261,027 Characters 115 Pages

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Abstract

This thesis develops a state-contingent capital budgeting model based on the model proposed in Banz and Miller (1978). The thesis aims at developing a theoretically consistent and empiri- cally valid model, with clear relevance for applied capital budgeting. In contrast to the original Banz & Miller model, which uses a state-contingent risk-free rate to model state-contingent preferences, the proposed model uses state-contingent volatility. This is partially motivated by reduced estimation complexity and potential issues stemming from changed dynamics of real government bond yields due to quantitative easing.

The thesis presents step-by-step derivations and estimations of the proposed model, in addi- tion to its theoretical building blocks. The model builds on the Arrow-Debreu framework, the Black-Scholes-Merton framework and some theory of finite Markov chains. Furthermore, the thesis relates countercyclical volatility to newer and older research in the financial literature on topics such as the leverage effect, volatility feedback, liquidity spirals and time-varying risk aversion.

The thesis investigates the proposed model’s applicability in real life valuation scenarios. A case study is conducted with the M&A department of the publicly listed Norwegian company Orkla. The case study reveals that the complexity of the model cannot be justified for all valuation purposes. On the other hand, the use of the model can be motivated by its ease of use in dynamic capital budgeting. The case study further reveals unrealistic assumptions in the proposed model, such as the absence of systematic risk within states of the market and non- increasing systematic risk with time. To counter these issues, the thesis proposes two possible solutions, namely the certainty equivalent approach to valuation and a method of valuation using a non-recombining trinomial tree.

The thesis concludes that the proposed model can be adapted to applied capital budgeting and that this can be done with relatively little compromise of theoretical consistency and empirical validity. However, a key criterion for the applicability of the model is that it is used for the right purposes. Furthermore, the presented extensions of the model may also be necessary.

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Acknowledgements

We would like to express our very great appreciation to Kristian Miltersen for his valuable and constructive counseling during the planning and writing of this thesis. We would also like to thank him for introducing us to the Banz & Miller model and state-contingent capital budgeting.

His willingness to give his time so generously has been very much appreciated. We would also like to offer our special thanks to Orkla and its M&A department for letting us conduct a case study with them. We are particularly grateful for the feedback given by Magnus Lerkerød, which has given us valuable insights into how practitioners work with capital budgeting. The cooperation with Orkla has provided us with a unique opportunity to relate theory to practice, and for that we are thankful.

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Contents

1 Introduction 4

1.1 Research Question . . . 6

1.2 Delimitation . . . 6

1.3 Thesis Structure . . . 7

2 Theoretical Framework 8 2.1 Arrow-Debreu Securities . . . 8

2.1.1 The Arrow-Debreu Economy . . . 9

2.1.2 General Arrow-Debreu Securities . . . 10

2.1.3 Utility and State Prices I . . . 11

2.1.4 Relating Utility to the CAPM and CCAPM . . . 13

2.1.5 Utility and State Prices II . . . 15

2.1.6 Motivation for Modeling State-Contingent Preferences . . . 17

2.2 Finite Markov Chains . . . 17

2.2.1 Building Blocks for Finite Markov Chains . . . 17

2.2.2 The Invariant Probability Vector . . . 19

2.3 Brownian Motion . . . 20

2.3.1 Brownian Motion . . . 20

2.3.2 Brownian Motion with Drift . . . 21

2.3.3 Geometric Brownian Motion . . . 22

2.3.4 Stock Prices as Geometric Brownian Motion . . . 22

2.4 Itô’s Lemma . . . 23

2.5 The Black-Scholes-Merton Differential Equation . . . 24

2.6 The Black-Scholes-Merton Formula (Binary Options) . . . 25

2.7 The Banz & Miller Model . . . 25

2.7.1 Pricing of State-Contingent Claims . . . 26

2.7.2 The Banz & Miller Single-Period Model . . . 27

2.7.3 The Banz & Miller Multi-Period Model . . . 28

3 Choice of Underlying and Risk-Free Rate 30 3.1 Proxy for Aggregate Consumption . . . 30

3.2 The Market Portfolio . . . 31

3.2.1 Choosing the Proxy for the Market Portfolio . . . 32

3.3 Choosing a Proxy for the Risk-Free Rate . . . 34

3.3.1 Separation of the Risk-Free Rate . . . 36

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CONTENTS

3.4 Concluding Remarks . . . 38

4 State-Contingent Parameters 39 4.1 State-Contingent Risk-Free Rate . . . 39

4.2 State-Contingent Volatility . . . 41

4.2.1 Volatility and the Market Price of Risk . . . 43

4.2.2 Countercyclical Risk Premiums and Volatility . . . 44

4.2.3 European Stock Market Volatility and the Real Economy . . . 48

4.2.4 Premise for Modeling State-Contingent Volatility . . . 49

4.3 Concluding Remarks . . . 50

5 The Proposed Model 51 5.1 Estimation of the Model . . . 51

5.1.1 Data and Preprocessing . . . 51

5.1.2 Estimation of Unconditional Parameters . . . 52

5.1.3 Definition of States . . . 54

5.1.4 Realized Yearly Returns and Data Subsetting . . . 57

5.1.5 Estimation of State-Contingent Volatility . . . 58

5.1.6 The Risk-Free Rate . . . 59

5.1.7 The State Price Matrix . . . 59

5.2 Stylized Valuation Example . . . 64

5.3 Concluding Remarks . . . 65

6 Exploratory Analysis of the Model 66 6.1 Comparison of State Prices . . . 66

6.1.1 Comparing State Prices for the European and US market . . . 67

6.1.2 Comparison with Banz & Miller . . . 69

6.2 State Price Sensitivity . . . 73

6.2.1 The Low State . . . 74

6.2.2 The Mid State . . . 75

6.2.3 The High State . . . 76

6.3 Analysis of Fat Tail Properties . . . 77

6.3.1 Sampling From a Mixture Distribution . . . 77

6.3.2 Estimating and Deriving the Invariant Probability Vector . . . 79

6.3.3 Comparison of Distributions . . . 82

6.4 Concluding Remarks . . . 83

7 Case Study with Orkla 84 7.1 Orkla in Brief . . . 84

7.2 An Introduction to the Case Study . . . 85

7.3 A Summary of the Case Study . . . 86

7.4 Key Findings . . . 87

7.4.1 Sequential Thinking . . . 88

7.4.2 Ease of Communication . . . 88

7.4.3 Forecasting State-Contingent Cash Flows . . . 89

7.4.4 Forecasting in Real Terms and Nominal Terms . . . 90

7.4.5 Combining the Model with the CAPM . . . 91

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CONTENTS

7.4.6 Using the Model for More Specific Purposes . . . 92

7.5 Concluding Remarks . . . 92

8 Model Extensions 94 8.1 Implementing Real Options . . . 94

8.1.1 A State-Contingent Option to Expand . . . 95

8.2 Extended Modeling of Systematic Risk . . . 98

8.2.1 A Comparative Analysis of Systematic Uncertainty . . . 99

8.2.2 The Certainty Equivalent Approach . . . 101

8.2.3 Non-Recombining Trinomial Tree . . . 103

8.3 Concluding Remarks . . . 109

9 Conclusion 110 Appendices 117 .1 Simulation Algorithm 1 . . . 118

.2 Simulation Algorithm 2 . . . 119

.3 State Price Matrices (STOXX) . . . 120

.4 Banz & Miller’s State Price Matrices (S&P) . . . 121

.5 Our State Price Matrices (S&P) . . . 122

.6 Sensitivity Analysis State Prices . . . 123

.7 1-Year State Price Matrix for Perpetual Matrix . . . 124

.8 State Price Matrices 1% Risk-Free rate (STOXX) . . . 125

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1 | Introduction

In a world where capital is limited, and investors require a return on their invested capital, there is a need for tools that can quantify the allocation of capital and guide investment decisions. It is in this context that capital budgeting has gained its relevance and popularity. As part of the framework that dictates investment decisions, capital budgeting models guide the allocation of capital, and therefore have a fundamental influence on the evolution of the aggregate economy and society as a whole.

Furthermore, a key area in economics and finance is utility theory, which ambitiously aims to quantify the preferences of individuals. One of the central concepts in utility theory is the law of diminishing marginal utility. Simply put, this ’law’ means that individuals have a decreasing appetite for consumption with the level of their consumption. For example, the first ice cream tastes better than the second one. In similarity, an umbrella has a higher value when it is rain- ing, as opposed to when it is sunny and umbrellas are superfluous. Analogous to this, capital has a higher value when the economy is in a recession and capital is scarce, which exemplifies the underlying link between utility theory and capital budgeting.

One of the most embraced asset pricing models used for capital budgeting, is the capital asset pricing model (CAPM). In the journal article Prices for State-Contingent Claims: Some Es- timates and Applications from 1978, Rolf W. Banz and Merton H. Miller proposed a capital budgeting model aimed at remedying some of the shortcomings of the CAPM. For instance, their model allowed for the valuation of projects with highly asymmetric cash flow distributions, and incorporated the notion of changing preferences across time and states of the economy, which stands in contrast to the CAPM. As such, their proposed model was a state-contingent capital budgeting model. Banz & Miller’s aim with the article was to present a capital budgeting ap- proach that could be used by both business managers and economist. Hence, a focus point in their article was the use of their proposed model in applied capital budgeting, which will also be the case for this thesis. Despite this, their model has received little attention, especially amongst practitioners.

Practitioners sometimes use the CAPM and other asset pricing models without offering too much consideration to the underlying assumptions of the models. For example, the CAPM was originally developed as a one-period model, but is often used in a multi-period setting in practice. Therefore, practitioners probably use models for purposes that they are less suitable for. Different tasks require different tools, and it is in this context that a state-contingent capital budgeting model finds its relevance. Banz & Miller intended to construct "a basis set by which all ordinary capital assets can be priced unambiguously once their payoffs relative to the market portfolio have been specified". This may have contributed to limiting the use of their

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CHAPTER 1. INTRODUCTION

proposed model. In our opinion, a state-contingent capital budgeting model does not need to be a universal pricing model, but merely be a suitable alternative that can remedy some shortcomings of other models.

Another factor that may have contributed to the limited use of Banz & Miller’s model is the fact that their model builds on certain concepts and methods that are unfamiliar and complex for most individuals with a business education. This compromises the model’s applicability.

Obviously, a key property of any applied capital budgeting model is that it is applicable.

Accordingly, a theoretically consistent and empirically valid model is of lower relevance if prac- titioners cannot utilize it. On the other hand, a model cannot price assets correctly if it is not theoretically consistent and empirically valid. Hence, for any applied capital budgeting model, theoretical consistency, empirical validity, and applicability must be in balance.

One of the complex aspects in Banz & Miller’s model is the use of the predictive distributions of Ibbotson and Sinquefield (1976, 1977), to model the level of the real risk-free rate in different states of the market. Or in other words, a state-contingent risk-free rate. Unfortunately, the complexity of these predictive distributions makes the model less applicable. We note that Banz

& Miller probably not intended for business managers to re-estimate their model, but merely to use their estimates. However, by making one of their key parameters reliant on complicated empirical research, they effectively rendered their model to become out-dated at some point.

Nonetheless, the state-contingent risk-free rate was introduced to reflect that investor prefer- ences vary across states, much like in the umbrella example. Due to the complex estimation method, we believe that it is a need for other ways to capture state-contingent preferences.

It is well documented that volatility is countercyclical. Furthermore, there are several possible explanations of why. For example, the countercyclicality of volatility could be related to the real business cycle, liquidity spirals, and volatility feedback (see Fornari and Mele (2013), Brun- nermeier and Pedersen (2007) and Bekaert and Wu (2000)). This gives reason to suspect that volatility might be used to model state-contingent preferences. In fact, Banz & Miller motivate the inclusion of a state-contingent volatility parameter in their model. In addition, the fact that the environment of real government bond yields may have changed during the recent years due to quantitative easing, gives reason to investigate whether or not modeling of a state-contingent risk-free rate is still feasible. If state-contingent volatility is less complex to estimate than a state-contingent risk-free rate, it could make a state-contingent capital budgeting model à la Banz & Miller more relevant and more applicable.

Based on the considerations outlined in the previous paragraphs, we find it interesting and relevant to investigate the following research question.

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CHAPTER 1. INTRODUCTION

1.1 Research Question

How can a state-contingent capital budgeting model, as proposed in Banz and Miller (1978), be adapted to applied capital budgeting, while preserving empirical validity and theoretical consistency?

In our investigation of this research question, we will develop and propose a state-contingent capital budgeting model, which models state-contingent preferences using state-contingent volatility. In relation to our research question and the model we will propose, we also pose the following sub-questions:

• Can state-contingent volatility be an efficient alternative to a state-contingent real risk- free rate?

• Can the proposed model be an alternative to conventional models in applied capital budgeting?

1.2 Delimitation

Although the discounting of cash flows is done in a non-conventional manner in the model we will propose, the model is a discounted cash flow (DCF) model. Hence, we will not discuss other valuation methods than DCF analysis, e.g. relative valuation or asset based valuation.

Furthermore, we will leave out discussions on capital structure. As opposed to other DCF methods, capital structure is not a direct input in the model we will propose. Instead, capital structure is implicitly taken into account in the cash flow estimates that go into the model.

Thus, even though capital structure has important implications for the value of a company, it is of less relevance for the investigation of our research question. Moreover, since the Black- Scholes-Merton model will play a significant part in the framework of the model we will propose, it could have been relevant to discuss other option pricing models that deal with some of its shortcomings, for example the variance gamma option pricing model. However, such models are an entire topic of their own, so we will not discuss them. Lastly, we mention that the data used is limited to the European and US economies. We therefore do not discuss our findings in relation to other geographical regions.

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CHAPTER 1. INTRODUCTION

1.3 Thesis Structure

Chapter 2: Theoretical framework

The theoretical building blocks for the proposed model are presented. This includes the most important fundamental theory on state-contingent capital budgeting and some mathematical concepts that are necessary for the derivations and estimations in our model.

Chapter 3: Choice of underlying and risk-free rate

We discuss how to choose the underlying asset and risk-free rate in our proposed model. The main discussion is related to choosing an underlying asset that serves as a proxy for aggregate consumption, and how to choose the real risk-free rate.

Chapter 4: State-contingent parameters

A discussion on state-contingent parameters is conducted. Empirical analyses and economic and financial theories are used to justify the modelling of state-contingent volatility instead of a state-contingent risk-free rate.

Chapter 5: The proposed model

The proposed model is derived and estimated based on the theoretical framework and the choices made in the two previous chapters. This chapter provides a ’recipe’ on how to imple- ment our proposed model, and includes our estimation method for state-contingent volatility.

Chapter 6: Exploratory analysis of the model

Further analyses are conducted based on our proposed model. We compare the findings/results of implementing the model in the American market. The proposed model is also compared to the original model by Banz & Miller. Furthermore, a sensitivity analysis is conducted. Lastly, the distributional implications of time-varying volatility are investigated through a sampling experiment from a mixture distribution. The aim of the chapter is to investigate the reliability and validity of our model, as well as provide the reader with a deeper understanding of the proposed model.

Chapter 7: Case study with Orkla

A case study where our proposed model is applied in a realistic capital budgeting scenario is presented. Both the findings and the procedure of the case study are described. The case study is conducted with the M&A department of the publicly listed Norwegian company Orkla. The aim of the chapter is to investigate the applicability of our model in the real world.

Chapter 8: Model extensions

Some extensions of the model are presented. This includes implementation of real options and two extensions that investigate different ways to account for systematic risk in the proposed model. The aim of the chapter is to refine our model based on the findings from the case study.

Data processing

The main sources for the empirical data that will be used throughout the thesis are Thomson Reuters Datastream, the OECD and the Center for Research in Security Prices. All the different

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2 | Theoretical Framework

As mentioned in the introduction, we will propose a model for applied capital budgeting, which is based upon the model in Banz and Miller (1978). This chapter introduces the theoretical framework that both the Banz & Miller model and our proposed model builds on. We note that this chapter is quite extensive at times, as it is aimed at giving readers that are unfamiliar with the model from Banz and Miller (1978) the necessary prerequisites for understanding it.

Hence, certain sections might be skipped by the reader which is familiar with the subjects or model. The chapter can also be used as a reference throughout the thesis when necessary. Fur- thermore, we emphasize that a comprehensive understanding of the subjects we will present, have been important for the choices we have made when writing our thesis. We begin by exploring the topic of Arrow-Debreu securities, which aims at explaining the theoretical moti- vation for modeling state-contingent preferences. We then continue with the topic of stochastic processes, where we describe finite Markov chains, Brownian motion and geometric Brownian motion. Furthermore, we relate our discussion of geometric Brownian motion to the pricing of state-contingent claims in the Black-Scholes-Merton framework. Lastly, we bring the preceding theoretical topics together when we present the Banz and Miller (1978) model, and show how it can be used to price an asset, both in a single-period and multi-period setting.

2.1 Arrow-Debreu Securities

The time-state preference approach to general equilibrium in an economy as developed by Arrow (1964) and Debreu (1959) is one of the most general frameworks available for the theory of finance under uncertainty. Given the prices of primitive securities (a security that pays $1.00 contingent upon a given state of the world at a given date, and zero otherwise, is a primitive security), the value of any uncertain stream of cash flows is easily calculated.

—Breeden and Litzenberger (1978)

So-called Arrow-Debreu securities constitute a foundational part of our theoretical framework.

We will in this section present what we believe are the most important takeaways from the Arrow-Debreu framework. We warn readers that we will not present all the necessary theoretical conditions to give a holistic description of the framework. In similarity to Banz and Miller (1978), we do not aim at building a complete state world, à la Arrow-Debreu. Nevertheless, we do find it informative to conduct a discussion of the properties of the Arrow-Debreu framework, to give readers an intuition for the most important parts of Arrow and Debreu (1954) and Debreu (1959). This will establish some principles that will be useful for understanding the different discussions throughout the thesis. We start by presenting an outline of the Arrow- Debreu economy, and then show a one-period model for Arrow-Debreu securities, where we

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CHAPTER 2. THEORETICAL FRAMEWORK

explain their relation to risk-neutral pricing. Furthermore, we will provide some explanation to the relation between utility and state prices, which will give an economic intuition to why so-called state prices are suitable for describing investor preferences, in addition to establishing a relation between a state price model and other asset pricing models, such as the CAPM and the CCAPM.

2.1.1 The Arrow-Debreu Economy

We start by defining a basic economy. Let I be the number of individuals in the economy and S be the number of states the economy can take on. The future state of the economy is uncertain, but each possible state s has some physical probability P{s}, which follows from the probability distribution fS. Furthermore, assume that in this economy there is a set of C commodities that individuals would like to consume different amounts of in different states of the economy. We let the amount produced of commodity c in state s be denoted by xsc, and the amount claimed of commodity c in state s by individual i be denoted by xisc. Since the aggregate claim of a commodity cannot exceed the available resources, and individuals are assumed to always desire more of a given commodity, we have

I

X

i=1

xisc=xsc

We let ui(xi11, ..., xi1C, xi21, ..., xiSC) denote the utility of individual i, if he/she is assigned a claim of xisc for commodity c if state s occurs (c = 1, ..., C;s = 1, ..., S). For any given in- dividual, the set of claims for commodities across all states will correspond to some amount of risk-bearing. Thus, there is a problem of optimal allocation of risk-bearing (with utility maximizing individuals), meaning that there is a problem of choosing the magnitudes xisc in such a way that no other choice will make every individual better off. Or in other words, there is a problem of choosing the magnitudes xiscsuch that no other choice can make one individual better of without making another individual worse off, i.e pareto optimality.

Assuming that there exists a system of perfectly competitive markets for the claims on com- modities (individuals can trade the claims without frictions and are price takers), and that the utility functions of all individuals are strictly quasi-concave1 (For example in the case of constant relative risk aversion (CRRA) utility functions for risk-averse individuals), then the chosen values of xisc (chosen by individuals themselves) will be given the optimal allocation xisc. Note that the choice of each individual i is restricted by their income yi. Thus, if we let vsc be the price of a unit claim on commoditycin statesof the economy, we have the following restriction on the optimal allocation

S

X

s=1 C

X

c=1

vscxisc =yi (2.1)

The prices of the unit claims on commodities are state prices, and the claims that are being traded are Arrow-Debreu securities. Since the commodities in the economy is what individuals consume, we can also view Arrow-Debreu securities as claims on consumption. The important

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CHAPTER 2. THEORETICAL FRAMEWORK

takeaway from this example is the principle of a set of claims on consumption, which will be central in some of the sections to come. Moving away from our outline of a basic Arrow-Debreu economy, we now look at an informal one-period model for Arrow-Debreu securities.

2.1.2 General Arrow-Debreu Securities

In similarity to the example in the previous section, assume that there are S possible states of the economy. However, assume that we are now standing at time t = 0 and that we are interested in what will happen in one period, i.e. at time t = 1. Furthermore, assume that we have a set of V = S Arrow-Debreu securities, where the price of these securities is the price at time t = 0 of receiving a unit payoff in state s at time t = 1 and zero otherwise. Let the vector vdenote the set of prices for these securities, such that the state price for state s,vs, is an element of v:vT = [v1, ..., vS]. For the sake of simplicity we assume that what state we are in at t = 0 is irrelevant. Compared to the previous example we also relax the assumption of what type of goods are being consumed in the economy, we will simply refer to them as assets.

The important point is that if we can use consumption as the numéraire, i.e. quote all assets in terms of consumption, we can price all assets through claims on consumption. The reason is that our claims on consumption are the most primitive form of securities possible, which allows us to replicate all other payoffs with them. Hence, assuming absence of arbitrage, we have that att = 0 the price of a security which pays offd :dT = [d1, ..., dS] at timet= 1 is

P0 =dTv=d1v1+· · ·+dSvS (2.2) Moreover, note that since the elements of v correspond to the elements of our state space exactly, and each security gives a certain unit payoff, owning all of the securities ensures a unit payoff in all states at time t = 1. In other words, owning all of the state-contingent claims corresponds to owning a risk-free asset. Hence, the price of the risk-free asset is B0 =PS

i=1vi, and the continuously compounded risk-free rate is

r = ln 1

B0

which rearranged yields,

B0 = exp(−r)

The Arrow-Debreu securities can also be used to define so-called risk-neutral probabilities, which are frequently used in pricing of options and derivatives. To obtain the risk-neutral probabilities we divide each state price by the price of the risk-free asset, which following the definition of the risk-free asset above corresponds to scaling state prices by their sum. This ensures

S

X

i=1

vi B0 = 1

B0

S

X

i=1

vi = 1

B0B0 = 1

such that the scaled state price space actually suffices as a probability measure (each scaled state price is bounded in [0,1] and a value of 1 is assigned to the entire probability space2).

2This also implies an assumption of non-negative state prices to begin with.

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CHAPTER 2. THEORETICAL FRAMEWORK

We also have that

S

X

i=1

vi <1

which must be true if the individuals in the market should be compensated for postponing consumption. Given that B0 = exp(−r)we can define the risk-neutral probability of state ias

qi =vi×exp(r) (2.3)

Since (2.3) impliesvi =qi×exp(−r), we can reexpress (2.2) as

P0 =d1×q1 ×exp(−r) +· · ·+dS×qS ×exp(−r) (2.4) which can be rewritten as

P0 = (d1q1+· · ·+dSqS)×exp(−r) = EQ[d]×exp(−r)

Here EQ[d] denotes the expected value of future payoffs (i.e. at t = 1) under the risk-neutral probability measure Q. In addition, since the pricing equation is just a rewritten version of (2.2), we know that the pricing under the risk-neutral measure must be equal to pricing under the physical probability measure P. This leads to

EQ[d]×exp(−r) = EP[d]×exp(−k)

where k is the required return for the investment under the physical probability measure P.

2.1.3 Utility and State Prices I

In the previous section, we used a no-arbitrage argument to justify why we can use Arrow- Debreu securities to price other assets. However, there is actually an economic justification as well. To show this, we will revisit the example from section 2.1.1, albeit a more generalized version. As shown in section 2.1.1, the optimal allocationxisc will be a solution to the problem of optimal allocation of risk bearing. Moreover, it was also stated that the economy consisted of utility maximizing individuals, such that the optimal allocation of risk bearing was chosen in a way that no other choice would make every individual better off. Hence, we can say that even though individuals are price takers in such an economy, there is a clear relation between the price of a state-contingent claim and the utility of individuals. To give an explanation of this relation, we show some derivations and statements inspired by Breeden and Litzenberger (1978), Arrow (1964) and Ireland (2018).

In every states investori would want to choose the bundle of commodities that maximizes his or hers utility, subject to the income constraint

C

X

c=1

vscxisc=yis (2.5)

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CHAPTER 2. THEORETICAL FRAMEWORK

Moreover, we assume that individuals’ preferences for state-contingent commodity allocations are representable by the expected value a von Neumann-Morgenstern utility function

Ui(xi11, ..., xiSC) =

S

X

s=1

ˆ

pisui(xis1, ..., xisC)

Here pˆis represents individual i’s subjective probability of state s. Furthermore, we re-express the utility function by considering consumption as the only ’real’ good, such that a bundle of different commodity allocations corresponds to some amount of the good consumption, denoted by g. As in the one-period general model, we now have consumption as the numéraire. This gives

Ui(gi1, ...giS) =

S

X

s=1

ˆ

pisui(gis) (2.6)

We do the same for (2.5), but generalized over all states such that we get

S

X

s=1

vsgi =Gi (2.7)

Where vs is now the price of a unit claim on consumption in state s and Gi is individual i’s total income quoted in consumption. Individual i’s problem is thus to maximize Ui(gi1, ...giS) with respect to (2.7). Reexpressing (2.7) as

Gi

S

X

s=1

vsgi = 0 we get the following Lagrangian function

L(gi1, ..., giS, λi) =

S

X

s=1

ˆ

pisui(gis) +λi

Gi

S

X

s=1

vsgi

This yields the following first order conditions

∂L

∂gis = ˆpisu0i(gis)−λivs, ∀s

Setting the first order conditions equal to zero and solving for vs gives vs= pˆisu0i(gis)

λi , ∀s (2.8)

From (2.8) we can see that a state price tends to be higher when marginal utility of consumption is higher, and that a state price tends to be higher when the probabilities of a state is higher.

Note that we do not claim that higher subjective probabilities of a state or higher individual marginal utility in a state leads to a higher state price, which is an important distinction as individuals are assumed to be price takers in our economy. Specifically, this means that indi- viduals cannot move prices, but that prices are set by the market as a whole. Nonetheless, the economic implication is that assets that give a relatively higher payoff in states where marginal utility is relatively greater tend to be relatively more worth. Since individuals are assumed to

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CHAPTER 2. THEORETICAL FRAMEWORK

have strictly quasi-concave utility functions, and thereby are risk averse, the marginal utility of consumption will always be highest in states where aggregate consumption is lowest, as the consumption of individuals is limited by the available resources. Hence, assets that give rela- tively higher payoffs in states with relatively lower aggregate consumption are relatively more worth.

2.1.4 Relating Utility to the CAPM and CCAPM

The concept of a price premium for assets that give high payoffs when resources are scarce can be related to the Capital Asset Pricing Model (CAPM) and the concept of non-diversifiable risk. In the CAPM, a risk premium, i.e. price discount, is given to assets with high amounts of non-diversifiable risk. The non-diversifiable risk is so-called market risk, which is determined by an asset’s co-movement in returns with the market portfolio, also known as the beta. This leads to the well-known equation

E[rj] =rfj(E[rm]−rf) (2.9) where rj is the rate of return on asset j, rm is the rate of return on the market portfolio, βj is the measure of co-movement between the returns of assetj and returns of the market portfolio, andrf is the risk-free rate of return. Since the returns of the market portfolio should be related to the economy, we can see why there should be a relationship between Arrow-Debreu securities and the CAPM. However, note that in the case of the CAPM, the link between the asset prices and the economy is less explicit than in the case of Arrow-Debreu securities. On the other hand, the link is made clearer if we look at a modification of the CAPM, the Consumption- based Capital Asset Pricing Model (CCAPM), derived by Rubinstein (1976), Lucas (1978) and Breeden (1979).

In similarity to the Arrow-Debreu framework, the CCAPM explicitly assumes that individuals are concerned about consumption, and thus the assets they value highest are assets that give relatively higher payoffs in relatively worse economic states (in terms of consumption). The relation between the CCAPM and Arrow-Debreu securities can be seen through the problem any individual faces according to the CCAPM, in a general one-period model. The problem can be summarized as follows

x1,...,xmaxN

ui(gi0) + exp(−δi)E[ui(gi1)]

s.t. g0 =wi0

N

X

j=1

xjPj0

g1 =wi1

N

X

j=1

xj(Dj1+Pj1)

Herexj is some asset in individual i’s portfolio ofN assets,git is individuali’s consumption at time t, wit is individual i’s wealth at time t, exp(−δi) is the impatience factor of individual i and ui is the utility function of individual i. Moreover, Pjt is the price of security j at time t

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CHAPTER 2. THEORETICAL FRAMEWORK

this problem is characterized by the following:

If(gi0, gi1)is the consumption plan corresponding to the optimal portfolio decision for individual i, then for any asset j

Pj0 =E

"

exp(−δi)u0i(gi1)

u0i(gi0) (Dj1+Pj1)

#

(2.10) and

E[rj] =rf −Cov[rj, u0i(gi1)]

E[u0i(gi1)] =rf − Cov

"

rj,u0i(gi1) u0i(gi0)

#

E

"

rj,u0i(gi1) u0i(gi0)

# (2.11)

Where rj = Pj1P+Dj1

j0 −1is the rate of return on asset j. In relation to Arrow-Debreu securities, the pricing equation (2.10) is perhaps of most interest. In the next section, we will in fact derive a similar expression for the price of an Arrow-Debreu security. Nevertheless, from (2.10) we can see that the price of an asset consist of two main parts. To make it a little bit more clear, we will assume that consumption at timet= 0 is known, such that we can rewrite (2.10) as

Pj0 = exp(−δi)E[u0i(gi1)]

u0i(gi0) ×E[Dj1+Pj1] (2.12) We can now see that the price of an asset j at time t= 0 is related to:

1. The expected payoff from the asset at timet= 1

2. The relationship between the marginal utility of consumption at time t = 0 and the expected marginal utility of consumption at time t= 1.

3. The willingness to postpone consumption

The economic implication is as follows. Firstly, prices tend to be higher when the expected payoff at time t= 1 is higher. Moreover, prices tend to be higher when the expected marginal utility of consumption at time t = 1 is higher. Conversely, prices tend to be lower the higher marginal utility of consumption is at time t = 0. Relating this to the concluding remarks about aggregate consumption from the previous section (and assuming the same type of utility functions as we have assumed so far, i.e. strictly quasi-concave utility functions), we can say that the market price of an asset depends on the current scarcity of resources, the expectations of the future scarcity of resources, and the expected payoff of the asset. In other words, the market price of an asset depends on the current possibility of consumption, the expectations of future possibilities of consumption and the consumption possibilities an asset provides. Lastly, individuals also tend to prefer consumption at their current point in time, which means that assets giving opportunities of future consumption are relatively less worth than assets giving opportunities of current consumption, everything else held equal.

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CHAPTER 2. THEORETICAL FRAMEWORK

2.1.5 Utility and State Prices II

In the last section we showed how in the CCAPM, the price of an asset is related to impatience, current marginal utility and expectations of future marginal utility. We will now show that this is also the case for state prices. To do so, we revisit the example from section 2.1.3, but modify it slightly. Firstly, we make it less general by assuming a one-period economy, and introduce a notation for time. More specifically, we have two points in time in our economy: Time t = 0 and time t = 1. Secondly, we now assume that an individual i in our economy receives a state dependent income at both timet = 0and timet = 1, which has the general notationyist, which is Gist if quoted in terms of consumption. Thirdly, we introduce the concept of impatience explicitly, such thatexp(−δi)is the impatience factor of individual i. The problem individual i faces is now to choose the optimal bundle of goods in terms of consumption at both timet= 0 and time t= 1. Therefore, it is natural to introduce a time notation in (2.6), which gives

Ui(gi1t, ...giSt) =

S

X

s=1

ˆ

pistui(gist)

Heregist corresponds to an amount of consumption for individualiin statesat pointtin time.

Moreover, since an individual’s choice of consumption is restricted by his/hers income, and the state of the economy at time t = 0 is assumed to be known, we have the following restriction on consumption

gi0+

S

X

s=1

vsgis1 =Gi0+

S

X

s=1

vsGis1

Note that the state-subscript is dropped in all parts of the expression corresponding to time t = 0, since there at time t = 0 can be only one state, and the subscript is therefore redun- dant. Moreover, the time-subscript is dropped on the state price itself, since in our one-period economy, the state price will always be the time t = 0 price of a unit claim on consumption at time t = 1. Further, the implication of using an equality for the restriction (instead of an inequality) maintains our previous assumption of unsaturated desires (individuals will use their complete income on consumption during the existence of the economy). Since all time t = 0 realizations are known, the utility at time t = 0 is also known. Hence, we can summarize the problem individual i faces at time t= 0 as the following maximization problem

gi0,gmaxi1t,...gi1S

ui(gi0) + exp(−δi)

S

X

s=1

ˆ

pis1ui(gis1)

(2.13)

s.t. Gi0+

S

X

s=1

vsGis1−gi0

S

X

s=1

vsgis1 = 0 (2.14)

Note that even thoughgi0is known, it is still a variable in the maximization problem, as different amounts of gi0 will determine future consumption opportunities and total utility during the existence of the economy. If we write the problem as a Lagrangian function we get

L=ui(gi0) + exp(−δi)

S

Xpˆis1ui(gis1) +λi

Gi0+

S

XvsGis1−gi0

S

Xvsgis1

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CHAPTER 2. THEORETICAL FRAMEWORK

which yields the following first order conditions

∂L

∂gi0 =u0i(gi0)−λi (2.15)

∂L

∂gis1 = exp(−δi)ˆpis1u0i(gis1)−λivs, ∀s (2.16) We start by setting (2.16) equal to zero and solve for vs, as done in section 2.1.3. This gives

vs= exp(−δi)pˆis1u0i(gis1)

λi (2.17)

We can proceed equivalently with (2.15), but solve forλi

λi =u0i(gi0) (2.18)

Using (2.18) in (2.17) we get

vs= exp(−δi)pˆis1u0i(gis1)

u0i(gi0) (2.19)

Hence, we can see that the general economic implications for the price of an asset, which were discussed in the previous section, must also be true for the price of an Arrow-Debreu security. In addition, the expression above gives us the theoretical motivation for modeling state-contingent preferences, which will be established in the next section. We can use (2.19) to see that the Arrow-Debreu framework is consistent with the CCAPM in the case of a risk-free asset. Utilizing the definition of a risk-free asset from section 2.1.2, we write

B0 =

S

X

s=1

vs =

S

X

s=1

exp(−δi)pˆis1u0i(gis1) u0i(gi0) The expression can be further rewritten as

B0 = exp(−δi) u0i(gi0)

S

X

s=1

ˆ

pis1u0i(gis1) = exp(−δi)

u0i(gi0) E[u0i(gi1)] = exp(−δi)E[u0i(gi1)]

u0i(gi0)

Notice that this is the same expression which is obtained when substituting the expected payoff term in (2.12) with a certain unit payoff. In addition, if we utilize the assumption of investors as price takers, but the market as price setter, we can write the general expression

B0 = exp(−δ)E[u0(g1)]

u0(g0) = exp(−r)

where everything on the right-hand side symbolizes the aggregate preferences of individuals (i.e.

the market), and thus determines the price of the risk-free asset (the left-hand side). Lastly, we can use this to obtain a utility-based expression for the risk-free rate

r= ln 1 exp(−δ)

u0(g0) E[u0(g1)]

!

(2.20) We will return to this equation in section 3.3, where we discuss the choice of risk-free rate in

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CHAPTER 2. THEORETICAL FRAMEWORK

our proposed model.

2.1.6 Motivation for Modeling State-Contingent Preferences

Based on our previous discussions, we will now establish the theoretical motivation for state- contingent preferences. To do so, we consider equation (2.19), but with the right-hand side as the aggregate preferences of individuals. More specifically, we consider

vs= exp(−δ)pˆs1u0(gs1)

u0(g0) (2.21)

From (2.21), we can see that the state price is determined by impatience, current marginal utility and the expected future marginal utility of the state that the state-contingent claim corresponds to. The important thing to note is that since current marginal utility matters for the state price, what state we are in now matters for the state price. Furthermore, in a more dynamic world than the one we have outlined in our examples, we could have that both expectations of future marginal utility and degree of impatience could be contingent on the current state. In other words, if we have a set of state prices that covers all future states, this set will differ depending on what state we are standing in today. This is the theoretical motivation for modeling state-contingent preferences.

2.2 Finite Markov Chains

Section 2.1 introduced the concept of state prices in a one-period setting. However, we will in this thesis use the concept of state prices in a multi-period setting. Moreover, we will model state-contingent preferences, as in the framework of Banz and Miller (1978). Since this framework builds on the theory of finite Markov chains (FMCs), we find it relevant to introduce some fundamental properties of FMCs before introducing the framework from Banz and Miller (1978). Furthermore, we will also utilize some theory of finite Markov chains in our exploratory analysis in section 6.3.3, which further calls for some theoretical clarifications.

2.2.1 Building Blocks for Finite Markov Chains

Let us begin with an explanation of some of the building blocks for finite Markov Chains, which follows the contents of chapter 1 in Lawler (2006).

State Space and Transition Probabilities

ForXt, which is a stochastic process in discrete time, wheret ={0,1,2, ...}, andXttakes values in the finite set S ={1, ..., N} or S ={0, ..., N −1}, we say that the possible realizations for Xt is the states of the system. We say that S is the state space. Given the initial probability distribution (which can be written as a vector with non-negative entries summing to unity, i.e.

a probability vector)

φ(i) = P{X0 =i}, i= 1, ...N (2.22)

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CHAPTER 2. THEORETICAL FRAMEWORK

and the ’transition probabilities’

qt(it|i0, ..., it−1) = P{Xt=it|X0 =i0, ..., Xt−1 =it−1} we have the simultaneous distribution

P{X0 =i0, ..., Xt=it}=φ(i0)q1(i1|i0)q2(i2|i0, i1)· · ·qt(it|i0, ..., it−1) The Markov Property

If a stochastic process such as the one just described exhibits the Markov property, then we have that for predictions of the future state of the chain, we need only to consider the current state.

All previous realizations of the chain (i.e. excluding the current one) are therefore irrelevant.

In other words, it does not matter how the chain got to the state it is currently in, onlywhat state the chain is currently in. Formally, we write

P{Xt=it|X0 =i0, ..., Xt−1 =it−1}=P{Xt=it|Xt−1 =it−1}

Furthermore, we assume that transition probabilities are constant over time, so-called time homogeneity. Which means that

P{Xt =it|X0 =i0, ..., Xt−1 =it−1}=p(it−1, it)

for some function p:S×S →[0,1], and that P{Xt=it|Xt−1 =it−1}are transition probabili- ties.

The Transition Matrix and the Chapman-Kolmogorov Equation

We say that the transition matrix P for the FMC is the N ×N matrix whose (i, j) entry Pij is p(i, j). The matrix P is a stochastic matrix, which means that it satisfies

0≤Pij ≤1, 1≤i, j ≤N and

N

X

j=1

Pij = 1, 1≤i≤N

For example, the following matrix satisfies the criteria above (and has the 3×3 shape, which will be the case for the model presented in this thesis)

P=

0 1 2

0 1−p p 0

1 q(1−p) 1−q(1−p)−p(1−q) p(1−q)

2 0 q 1−q

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CHAPTER 2. THEORETICAL FRAMEWORK

The matrix contains the transition probabilities over a single time interval. The rows of the matrix denote the ’departure’ states, and the columns of the matrix denote the ’arrival’ states.

We can think about multiple ’trips’ through the matrix as the chain’s development over time.

Note that even though there are certain states of the chain that have a transition probability of zero between them (0 and 2), all states communicate, meaning that there is a positive probability of reaching every other state from any state, over some number of time periods.

The notion of moving to other states through other states can be seen from the Chapman- Kolmogorov equation

P{Xt+1 =j|X0 =i}=X

k∈S

P{Xt =k|X0 =i}P{Xt+1 =j|Xt =k}=X

k∈S

pt(i, k)p(k, j) Which means that pt+1(i, j)is an element in

PtP=Pt+1 which can be generalized to

Pt+k=PtPk

The Chapman-Kolmogorov equation also lets us say something about the marginal distribution at any time t in the future. Specifically, if φ0 is an initial probability vector, with elements as defined in (2.22), we have

φt0Pt

2.2.2 The Invariant Probability Vector

We proceed by using our building blocks to describe the large-time behavior of a FMC. The invariant probability vector is a vector of transition probabilities describing the large-time behavior of a FMC. To characterize the invariant probability vector we say that there exists a matrix Π = limt→∞Pt, where we for every probability vector ρ have limt→∞ρPt = π. The implication is that the invariant probability vector is

π = lim

t→∞ρPt+1 = ( lim

t→∞ρPt)P=πP which further implies

π =πP

such that π is a left eigenvector to P (with eigenvalue 1). However, we cannot always be sure that the invariant probability vector exists. Nor can we always be sure that it is unique. From Lawler (2006) we have the following fact.

*If P is a stochastic matrix such that for some t, Pt has all entries strictly positive, then P satisfies the following:

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CHAPTER 2. THEORETICAL FRAMEWORK

2. The eigenvalue 1 is simple and all other eigenvalues have absolute value less than 1 Furthermore, we know that a stochastic matrixPsatisfies * if it isirreducibleandaperiodic.

We will now give an explanation of these terms.

Reducibility

We say that a matrix is irreducible if all states communicate, and that it is reducible if all states do not communicate. Formally we define communication as follows: We say thati, j ∈S communicate (denoted by i↔j), if there exists k, t≤0 such thatpk(i, j)>0 and pt(j, i)>0.

Periodicity

The periodd(i)of a state iis defined as the largest common divisor ofJi ={t≥0|pt(i, i)>0}.

If i ↔ j then d(i) = d(j). Hence, if a matrix is irreducible, we have the same period for all states. An irreducible matrixP is called aperiodic if d(i) = 1,∀i

2.3 Brownian Motion

In addition to the FMC, the model that will be presented in this thesis also builds extensively on another stochastic process, Brownian motion. Brownian motion is a stochastic process that models random continuous motion. It is widely used in finance when modelling how asset returns (especially stock returns) evolve, for example in the Black-Scholes-Merton (BSM) model, which is the model used for calculating state prices in Banz and Miller (1978). The popularity of Brownian motion in finance can be largely attributed to the Efficient Market Hypothesis (EMH) and the random walk theory, which in their essence state that stock prices incorporate all available information. This follows from the fact that Brownian motion is a Markov process. We will in this section describe geometric Brownian motion (based on parts of chapter 8 in Lawler (2006)), which will be the stochastic process used for modelling the underlying asset in our state price model. Furthermore, we will also briefly discuss some real world inconsistencies when using geometric Brownian motion in our modelling of the underlying asset.

2.3.1 Brownian Motion

We start by defining a simple process without drift. We say that a Brownian motion with variance parameter σ2 is a stochastic process Xt taking values in the real numbers satisfying:

1. X0 = 0

2. For anys1 ≤t1 ≤s2 ≤t2 ≤ · · · ≤sn≤tn, the random variables Xt1−Xs1, ..., Xtn−Xsn are independent

3. For any s < t, the random variable Xt−Xs has a normal distribution with mean 0 and variance (t−s)σ2

4. The paths are continuous, i.e., the functiont7−→Xt is a continuous function of t

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CHAPTER 2. THEORETICAL FRAMEWORK

We will now show that Brownian motion exhibits the Markov property. We start by considering a standard Brownian motion, i.e. a Brownian motion with σ2 = 1, denoted by Xt. Firstly, we letFt represent the information contained inXs, s≤t, such thatFt is all the information that can be obtained from watching the Brownian motion up through timet. We then assume s < t and consider the conditional expectation E[Xt|Fs]. We can write E[Xt|Fs]as

E[Xt|Fs] =E[Xs|Fs] +E[Xt−Xs|Fs]

Note that if all information inFs is already contained inXs. This means thatXscan be written as a function of Fs and we say thatXs is Fs-measurable. Since Xs is already a function ofFs, we have that E[Xs|Fs] = Xs. Furthermore, by the definition of Brownian motion and since Xt−Xs is independent of Fs, we have E[Xt−Xs|Fs] = E[Xt−Xs] = 0. Therefore, we can write

E[Xt|Fs] =Xs=E[Xt|Xs]

The left-hand side and right-hand side of the equation shows the Markov property of Brownian motion, i.e. that to predict the future we need only to consider the current value of the process.

More generally, the Markov property implies that for functions f, E[f(Xt)|Fs] =E[f(Xt)|Xs]

This follows from an even stronger property of Brownian motion, which is that ifYt=Xs+t−Xs, then Yt is a Brownian motion independent of Fs, which means that Zt = Xs+t is a Brownian motion starting at the (random) starting point Xs (which we do not show).

2.3.2 Brownian Motion with Drift

As mentioned, a Brownian motion has a mean of zero. If returns on risky assets, such as stocks, were in fact distributed with a mean of zero, this would be an appropriate model for returns on risky assets. However, we know that since investors are risk averse, they require a risk premium for holding risky assets, which then again implies that the mean return for risky assets is positive, i.e. expected return is positive. Hence, it is natural to include a drift parameter when modeling returns. We can extend the previous definition of a Brownian motion, to obtain a Brownian motion with drift. Consider a Brownian motion Xt with variance parameter σ2, starting at x. Let µ denote the drift parameter. We say that Yt is a Brownian motion with drift if

Yt=µt+Xt where Yt satisfies:

1. Y0 =x

2. ifs1 ≤t1 ≤s2 ≤t2 ≤ · · · ≤sn ≤tn, thenYt1 −Ys1, ..., Ytn−Ysn are independent 3. Yt−Ys has a normal distribution with mean µ(t−s) and varianceσ2(t−s) 4. Y is a continuous function of t.

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CHAPTER 2. THEORETICAL FRAMEWORK

2.3.3 Geometric Brownian Motion

In the Black-Scholes-Merton model (see section 2.5), the price of the underlying follows a lognormal distribution. This is convenient, as it ensures that the value of an asset is never negative. Given that taking the natural logarithm of a lognormally distributed variable gives a normally distributed variable, we also have that the continuously compounded returns of the underlying asset are normally distributed. If we consider a Brownian motion with drift

Xt=µt+σWt

whereWt is a standard Brownian motion. Then we say thatYtis a geometric Brownian motion if

Yt = exp(Xt) = exp(µt+σWt)

Note that here Y0 = 1. However, if we used such a process to model the price of our underlying asset, the expectation of our asset value would be

E[Yt] = exp

µ+1 2σ2

t

which follows from the expectation of a lognormal variable. If µ is the expected return of the asset and σ is the return volatility of the asset, and all compensation for risk bearing should be accounted for in expected returns, this does not make sense economically. In fact, from Jensen’s inequality we can see that

E[exp(Xt)]≥exp(E[Xt]) = exp(µt)

where the right-hand side of the inequality gives the economically reasonable unconditional expectation for our underlying asset. We can see that the term on the right-hand side of the inequality will be our expectation of the asset value if

Yt = exp

µ− 1 2σ2

t+σWt

(2.23) since the expectation is then

E[Yt] = exp

µ− 1

2+ 1 2σ2

t

= exp(µt)

Hence, we will use (2.23) to model the price of our underlying asset.

2.3.4 Stock Prices as Geometric Brownian Motion

When using geometric Brownian motion to model the market price of our underlying asset, we make some assumptions about the real world that are inconsistent with reality. We will briefly address what we believe are the most important inconsistencies. As we will explain further in section 3.2.1, we use a European stock market index as our underlying asset. Therefore, it makes sense to limit the inconsistencies to stock prices.

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