Difficulties in pricing of real options

Difficulties in pricing of real options

Francis Atsu

U.U.D.M. Project Report 2007:5

Examensarbete i matematik, 20 poäng Handledare och examinator: Maciej Klimek

Januari 2007

Department of Mathematics

Uppsala University

Acknowledgement

I would like to express my appreciation to Professor Maciej Klimek, my supervisor, not only for his exceptional help on this project, but also for the course (Financial Derivatives) that he taught which granted me the understanding of Real Options and the necessary mathematical background to come out with this piece of writing.

I would also like to thank Professor Johan Tysk, who introduced me to financial mathematics at the initial stage of my studies and Professor Abrahamsson Leif as a personal course selection adviser.

To the rest of the professors in the Financial Mathematics and Financial Economics programme who provided instruction, encouragement and guidance, I would like to say a big thank you to you all. They did not only teach me how to learn, they also taught me how to teach, and their excellence has always inspired me.

Finally, I would like to thank my mother, Mary Nambo, for her financial support and encouragement, Zsuzsanna Kristofi of Mathematics Department, for the help she rendered to me when I first contacted the Department and the entire members of the Department of Mathematics.

Dedication

I dedicate this piece of writing to God for His divine support, my mother Mary Nambo and sisters Francisca Atsufe (twin), Nambo Mavis.

TABLE OF CONTENTS

Abstract

5

Chapter One

Background

1.1 Option Basics 6

1.2 Types of Options 6

1.3 Exercise Style 6-7

1.3.1 European Option 7

1.3.1.1 European Call Option 7-8

1.3.1.2 European Put Option 8-9

1.3.2 American Option 9 1.3.3 Bermudan Option 9-10 1.3.4 Asian Option 10 1.3.5 Barrier Option 10-11 1.3.6 Lookback Option 11-12 1.3.7 Turbo Warrants 12-13 1.3.8 Chooser Option 13 1.3.9 Compound Option 13 1.3.10 Basket Option 13-14 1.4 Behaviour of Options 14 1.5 Real Options 15 1.5.1 Introduction 15-16 1.5.2 Definitions 16-17

1.6 Types of Real Options 17

1.6.1 Abandonment or Termination Option 17

1.6.2 Switching Option 17-18

1.6.3 Expansion and Contraction Option 18

1.6.4 Deferral Option 18

1.7 Structural Differences between Real and Financial Options 18-19

1.8 Behaviour of Real Options 19

Chapter Two

Mathematical Background

2.2 Ito’s Lemma 20 2.2.1 Ito’s Lemma with Uncorrelated Wiener Processes 20-21 2.2.2 Ito’s Lemma with Correlated Wiener Processes 21-22

2.3 Black-Scholes Formula 22

2.3.1 1-dimensional Black-Scholes Formula 22 2.3.2 Assumptions of the Black-Scholes Model 22-23

2.4 Definitions 23

2.5 Girsanov’s Theorem 23

2.5.1 Statement of theorem 24

Chapter Three

Insufficiency of No-arbitrage Pricing of Real Options

3.1 Overview 25

3.2 Statement of Problem (Hubalek and Schachermayer) 25

3.3 Introduction 25

3.4 General Set-up (Hubalek and Schachermayer) 26

3.5 Definitions 26-27

3.6 Derivation of S 27

3.7 Theorem (Hubalek and Schachermayer) 28-30

Chapter

Four

Alternative

Pricing

4.1 Overview 31

4.2 Basic Definitions 31-32

4.3 High Correlation (Non-Perfect Correlation) 32 4.3.1 Minimization of the Variance of the Hedging Error 32-37 4.4 Imitation or Naïve Strategy 37-39

Abstract

This project is basically made up of four parts.

The first part reviews various types of financial and real options. The second part presents various mathematical tools for example Ito’s lemma, Black-Scholes equation and Girsanov’s theorem which are needed in the rest of the paper. The third part serves as the core of the entire project. We present in detail a result due to Hubalek and

Schachermayer showing that standard arguments based on the Black-Scholes model and non-arbitrage assumptions can be unsuitable for pricing of real options. More specially, attempts at pricing of an option written on a real non-tradable asset, by means of pricing a surrogate financial asset, may fail completely even if the two assets are highly correlated. The last part of this project presents two remedies suggested by Hubalek and

Schachermayer: trading strategy based on minimization of the variance of the hedging error and naive trading strategy in which the surrogate financial asset is considered in place of the real asset

CHAPTER 1

Background

1.1 Option Basics

Options are financial instruments that grant the owner some rights. Hence an option is the right but not an obligation to buy or sell some underlying asset under some predefined conditions. The one who issues an option is called the writer and the one who acquires the option is called the holder or the owner. To acquire an option, the holder pays the writer a premium.

An option would be exercised only when it is in the interest of its holder to do so.

1.2 Types of Options

Theoretically, there is an infinite variety of options, but the two basic and standard ones are called calls and puts. Complex financial instruments (see e.g. Lyuu (2002)) can often be made up of calls and puts.

A call option gives its holder the right to buy an amount of an underlying asset by paying a specified strike price (or exercise price)K . A put option grants its holder the right to sell a specific amount of the underlying asset by paying a specific strike price K. The strike price K of a call (put) option is the price at which the underlying asset will be bought (sold) at the expiration dateT . Expiration date T is the last day or date on which an option can be exercised. There are some options ‘with no special or unusual features’, such options are called Vanillas. Hence the most common and standard type of vanilla option with a standard expiration date T and strike price K and no extra features is called plain vanilla. There are also some more complex financial options with additional features; they are called exotic options or path-dependent options.

1.3 Exercise Style

Options normally differ in when they can be exercised. Some grant the owner the right to exercise at anytime within a specified time horizon, whereas others grant the owner the right to exercise at only a specific time point.

Furthermore, some may depend on the history of the price process of the underlying asset. There are some options which may use one or more of the above styles of exercise.

Examples of the most common exercise style of options are European, American, Bermudan, Asian, Barrier, Lookback, Turbo warrants, Compound, Chooser and Basket.

1.3.1 European Option

Let the price process of the underlying asset be ( )S t ,t∈

[ ]

0,T . AEuropean option gives the owner the right to exercise the option only on the expiration dateT . Hence the holder receives the amountϕ( ( ))S T , where ϕ is a contract function.

Moreover, there are two basic types ofEuropean option namely European call Options and European Put Options.

1.3.1.1 European Call Option

A European call option which gives its owner the right to purchase an underlying asset for a given price (exercise price) on the expiration dateT . A European call will be exercised only if the stock price S is higher than the strike price K. When a call option is exercised, the holder pays the writer the strike price in exchange for the stock, and the option ceases to exist. Hence the value or the payoff of a call at expiration is

( ( )) max( ( ) , 0) ( ( ) , 0)

c S T S T K S T K

ϕ _{= − = −} +

where T is the expiration date. ϕ_cis called the contract function of the call option. Therefore ϕ_c( ( ))S T is an explicit formula for the value of a call option at expiration T as a function of the price process of the underlying asset at time T as shown below.

K S(T)

Figure 1.0: The pay-off of call option c

From Figure 1.0 we could see that, the option is valueless if the price of the underlying security is less than the strike price ( ( )S T <K ), the option value increases linearly with the price when the price of the underlying is greater than the strike price ( )S T >K . At any time t a European call is said to be in the money if ( )S t > , at the money K if ( )S t = , and out of the money if ( )K S t < . K

1.3.1.2 European Put Option

A European put option which gives the owner the right to sell an underlying asset for a given strike price on the expiration dateT . The underlying assets may be stocks, stock indices, future contracts, interest rate, etc. A European put is exercised only if the stock price S is less than the strike price K . When a put option is exercised, the holder receives from the writer the strike price in exchange for the stock and the option ceases to exist. Hence, the payoff of a put at expiration isϕ_p( ( ))S T =max(K−S T( ), 0)=(K−S T( ))+.

p

ϕ is called the contract function of the put and it gives an explicit formula for the value of the option at the expiration date T , as shown in Figure 1.1.

K S(T)

Figure 1.1: The pay-off of a put option

p

Figure 1.1 shows that, the option is worthless if the price of the underlying asset S is greater than the strike price K (i.e. ( )S T >K), but if the strike price is greater than the stock price then the put option has value. At any time t a European put is said to be in

the money ifS t( )< , at the money if ( )K S t = , and out of the money if ( )K S t > . K From Figures 1.0 and 1.1, we can see that, the contract function of a call is unbounded whereas the contract function of a put is bounded. Hence in practice, the possibility of a loss is unbounded when issuing a call option, but bounded for a put (see e.g. Luenberger (1998)).

1.3.2 American Option

AnAmerican option gives the owner the right to exercise the option on or before the expiration date (t ≤T ). So, one has to assume that, there is a well-defined payoff for before the expiration date (also called early exercise). Hence at any time within the specified time frame, holder of an American option needs to decide whether to exercise immediately or to wait. If the holder decides to exercise at sayt≤T , then he receives

( ( ))S t

ϕ where ϕ is the appropriate contract function.

Similarly this option can also be classified into two basic types:

American call option which grants the owner the right to buy an underlying asset for a given strike price on or before the expiration date, and American put option which grants the owner the right to sell an underlying asset for a certain strike price on or before the expiration date. If the underlying stock pays no dividends early exercise of an American call option is not optimal. On the other hand early exercise of an American put option can be optimal even if the underlying stock pays not dividends.

An American option is worth at least as much as an identical European option because of the early exercise feature.

1.3.3 Bermudan Option

This type of options lies between American and European. They can be exercised at certain discrete time pointst₁< < < = . As a consequence of Bermudan options t₂ ... t_n T being a hybrid of European and American options, the value of a Bermudan option is

greater than or equal to an identical European option but less than or equal to its American counterpart.

1.3.4 Asian Option

This type of option depends on the average value of the underlying asset over a time horizon. Therefore an Asian option is path dependent. Asian options are cheaper relative to their European and American counterparts because of their lower volatility feature. This type of option (see e.g. Global Derivatives) can broadly be classified into three basic categories;

• Arithmetic average Asians (AV)

This is the sum of the sampled asset prices S S₀, ₁,...,S_n−1divided by the number of samples: _AV S0 S1 ... Sn 1

n

−

+ + + =

• Geometric average Asians(GA)

The value of this is taken as: n _{0 1} ... ₁ n GA= S ⋅ ⋅ ⋅S S ₋ • Average strike Asian (AS)

This is the strike average between arithmetic and geometric average Asians:

2 AV GA AS = +

Asian options can either have European or American exercise style.

1.3.5 Barrier Options

Given an option with a pay-off function φ one can consider barrier versions of this option. Barrier Options (see e.g. Björk (1999)) are like the usual options with the characteristic that a predefined action takes place when the stock assumes a certain level called the barrier. Typical examples of barrier options are down-and-out, down-and -in up-and- out and up-and-in, contracts.

Down and− −out contract is given by:

[ ]

[ ]

( ( )), if ( ) , 0, 0, if ( ) , 0, S T S t L t T Z S T L t T φ ⎧ > ∈ ⎪ = ⎨ _{≤ ∈} ⎪⎩

This means that the amount Z is paid to the contract holder if the stock price lies above the barrier L during the entire life of the contract or the contract terminates and nothing is paid to the holder if at certain times before the expiration T the price of the stock assumes the barrier L .

Down and− −in contract is defined as:

[ ]

[ ]

( ( )), if ( ) , 0, 0, if ( ) , 0, S T S t L t T Z S T L t T φ ⎧ ≤ ∈ ⎪ = ⎨ _{> ∈} ⎪⎩

The above contract is defined as, the amount Z is paid to the contract holder if at certain time before the expiration T the stock price assumes the barrier L or nothing is paid to the contract holder if during the entire contract life of the contract the stock price lies above barrier L .

Up and− −out contract is given by:

[ ]

[ ]

( ( )), if ( ) , 0, 0, if ( ) , 0, S T S t L t T Z S T L t T φ ⎧ < ∈ ⎪ = ⎨ _{≥ ∈} ⎪⎩

This is defined as follows; the amount Z is paid to the contract holder if the during the entire life of the contract the stock price lies below the barrier L or the contract terminates and nothing is paid to the contract holder if at certain times before the expiration T the stock price assumes values which greater or equal to the barrier L . Up−and− contract is given by: in

[ ]

[ ]

( ( )), if ( ) , 0, 0, if ( ) , 0, S T S t L t T Z S T L t T φ ⎧ ≥ ∈ ⎪ = ⎨ _{< ∈} ⎪⎩

Thus, the amount Z is paid to the contract holder if at certain time before the expiration T the stock price assumes the barrier L or the nothing is paid to the contract holder if during the entire life of the contract the stock price lies below the barrierL

1.3.6 Lookback Options

Lookback Options are contracts whose pay-offs at expiry T depend on either the maximum or minimum price level of the underlying price process achieved during the entire life of the contract.

• Lookback Call: ( ) min ( ) t T S T S t ≤ − • Lookback Put: max ( ) ( ) t T S t S T ≤ −

• Forward Lookback Call:

max max ( ) , 0 t T S t K ≤ ⎡ ₋ ⎤ ⎢ ⎥ ⎣ ⎦

• Forward Lookback Put:

max min ( ), 0 t T K S t ≤ ⎡ ₋ ⎤ ⎢ ⎥ ⎣ ⎦

1.3.7 Turbo warrants

A warrant is a financial instrument which grants its owner the right to buy (or sell) a fixed quantity of stocks at a predefined price and conditions. There are two basic types of warrants namely; a call warrant and put warrant.

A call warrant grants its owner the right to buy a specific number of shares from the writer at a predefined price on or before a specific date. A put warrant also grants its owner the right to sell back a specific amount of stocks (or shares) back to the writer at a predefined price on or before a predetermine date. As claimed by Galitz, ‘A few companies actively use warrants as a means of promoting shareholder value, and obtaining a steady stream of new investment and warrants can also be used in the same way as equities, to provide a way of investing in the shares of a specific company, but with a relatively low capital outlay at the start’. New shares are normally sold out when warrants are exercised (see e.g. Musiela/Rutkowski (1997)).

The name Turbo warrants had been used earlier in Germany for a usual barrier option of down and− −out style having its strike price as the barrier. The French bank Societe Generale′ issued a contract called a Turbo warrant in early 2005, and defined it as a barrier option - down and− −out style having the barrier in the money− − and that the owner receives a rebate if the barrier is hit. The rebate can be reconsidered as a new

warrants can also be classified into two forms namely Turbo warrants calls (turbo-calls) and Turbo warrants puts (Turbo-puts).

Turbo-call warrant (see e.g. Persson/Eriksson, 2005) pays max{ ( )S T −K, 0} at maturity T if a predefined barrier L≥K has not been hit by ( )S t at any time 0≤ <t T.This contract instantaneously ceases to exist and a new contact is initiated when the stock ( )S t at anytime 0≤ <t T hits the barrier. The new contract having the same strike price is a call option on the minimum of the process ( )S t .This matures at a certain later time. Similarly, a turbo put pays max{K−S T( ), 0} at maturity T if a predefined barrier L≤K has not been hit by ( )S t at any timet<T. This contract instantaneously ceases to exist and a new contract is initiated when the ( )S t hits the barrier. The new contract having the same strike price is a put option on the realized minimum by the process ( )S t .This matures at a certain later time.

1.3.8

Chooser Options

This is a type of option that grants you (buyer) the right to determine the features of an option. Thus the buyer determines the value of the strike price, whether the option is a put or call and in some special cases even fix the delivery time T .Therefore, because of these flexibilities at the disposal of the buyer, these types of options are generally quite costly.

1.3.9

Compound Options

A compound option (see e.g. Musiela/Rutkowski (1997) is a standard option whose underlying is another standard option. In other words a compound option is an option written on existing option. Typical and basic examples of compound options are: a call option on a call option, a call option on a put option, a put option on a call option and a put option on a call option.

1.3.10

Basket Options

This is a type of option that grants you (buyer) the right to create a self financing portfolio of two or more assets (a basket of chosen assets). The payoff of these types of

options is given by ‘the difference between a predetermined strike price and the combined weighted level of the basket of assets (a predefined portfolio of assets) chosen at the outset. Hence the payoff is contingent on the mean of prices of several assets chosen at the outset.

1.4

Behaviour of Options

A rational investor would often seek to hedge his or her portfolio value against risk. Often, assets contributing to the value of the portfolio are contracts written on various options. Hence it helps to know the characteristics that option values exhibit under certain conditions. All option prices react to the price of the underlying security. If the price of an underlying security changes (increases or decreases) then the value of the corresponding option also changes. Delta (Δ = ∂ ∂ϕ S) describes the instantaneous rate of change of the option price ϕ as a function of the price of the underlying stockS.

Furthermore, the value of a European call option (see e.g. Persson (2006)) with option price being convex (i.e. S f t S is convex for every fixed( , ) t∈[0, ]T ) varies nearly in proportion to the volatility, which means that, the value of a call option increases with an increase in volatility and would decrease with a decrease in volatility (Volatility is the standard deviation of the stock price). Hence doubling the volatility of a European call option that is near the money would approximately double the option’s value. Tripling the volatility would also triple the option’s value and so on.

In addition, term structure pattern (or prevailing interest rate) also plays an important role because purchasing a call in some sense is a method of purchasing the stock at a reduced price (see e.g. Luenberger (1998)). Therefore one saves interest expenses. In short, option prices depend on the prevailing interest rates.

Finally, when there is a positive time to expiration, the value of a call option decreases as the time to expiration decreases.

1.5

Real

Options

1.5.1 Introduction

In recent times, most firms are embarking on projects in environments of risk and uncertainty, for which the traditional valuation methods for instance discounted clash flow (DCF) fail to correctly give the intrinsic value of such projects. This is because the traditional methods fail to take into account risk, uncertainty and managerial flexibilities. The deficiency of the usual traditional valuation methods resulted into the introduction of a new method called real option theory which takes into account uncertainties, various risks and managerial flexibilities. In this method, managers and investors can choose to estimate the value of managerial decisions for instance to abandon, defer, and expand a project.

Moreover, there are three specific areas in particular (see e.g. Maubounssin (1999)) where traditional DCF, most widely expressed as the net present value rule (NPV), comes short versus option theory (Real Options) namely flexibility, contingency, and volatility; Firstly, flexibility is the ability to defer, abandon, expand, or contract an investment. The NPV rule lacks the capability of capturing the correct value of flexibility as compared to the real option theory, because it does not take into consideration the value of uncertainty For example, consider the classical example given by Michael J.Mauboussin ‘a company may choose to defer an investment for some period of time until it has more information on the market. The NPV rule would value that investment at zero, while the real option theory would correctly allocate some value to that investment’s potential’.

Secondly, contingency is a situation whereby the success of today’s investment is a determining factor for future investments. Hence, Mauboussin claims that ‘managers may make investment today, even those deemed to be NPV negative in other to access future investment opportunities’. This could simply mean that, mangers could incur cost today due to some managerial decisions in other to have brighter investment opportunities in the future where the cost incurred today could be factored into the future’s cost of investment.

Thirdly, volatility-which is the standard deviation of the price process of an underlying asset. In options pricing theory, business pursuits with greater risk have high option value. Moreover, due to the convexity nature of most payoff schemes of options their values increase in accordance with volatility. Hence the higher (lower) the volatility the higher (lower) the option value

Hence the traditional methodologies fail to adequately value these option creating investment opportunities. In summary, the real option theory would correctly estimate the intrinsic cost of real investment, but the traditional methods (for instance NPV rule) would underestimate.

The real option theory employs the regular financial options theory (for instance Black-Scholes model) to real investments such as expansion of production settings, research and development investments, estates development etc. The real option theory (see e.g. Amram - Kulatilaka (1999)) can also be seen as a methodology that uses the usual option theory to quantify managerial flexibilities in a world of uncertainty.

The real options theory performs pretty good in business environments where uncertainty is the order of the day and if there exists a proactive and smart managerial team which has the capabilities of identifying, creating and exercising real option.

1.5.2 Definitions

A real option can be defined as an option embedded in the decisions pertaining to real assets. Hence a real option is not financial instrument. Since this could also be seen as an action option in the sense of choice, the theory grants more flexibilities than the usual financial options theory.

In contrast to financial derivatives (options), a real option is not tradable, for example, a mobile phone manufacturing company (for instance Nokia) may decide to temporally abandon the production of a particular model of their mobile phones if the price of such a model falls below the production cost. On the other hand, Nokia has the right to start the production of that model if it happens that the price of that model rises above the production cost in the near future. Nokia can only choose to or not exercise these rights,

but cannot sell them to another party. The term ‘real’, because the option involves tangible or physical assets.

1.6 Types of Real Options

In the business world, there are several managerial flexibilities (real options) which are usually embedded in real investment opportunities of most firms. Hence this would call in for a managerial team with the ability to identify and exercise these options. Moreover, these options (flexibilities) allow management to ‘leverage uncertainty’ and considerably reduce risk. Therefore, management has the option to abandon, expand, switch use, defer, contract (just to mention few) a project in other to preserve the value of such firms.

1.6.1 Abandonment or Termination Option

This is a type of option where the managerial team reserves the right to abandon a project whose intrinsic value seems not be promising. This option is very useful in capital intensive firms (for instance gold mines, airlines, petroleum firms). Consider a mining firm, which has the option to mine gold over a time frame, so after some number of operations management has realized that, the operational cost could not even be covered. Therefore the managerial team would have to exercise the right of abandoning this project in order to preserve the firm’s value. In this classical example; if the managerial team is not smart enough to instantaneously exercise this option, then the firm stands on the verge of collapsing or losing greater part of its value. Hence the right timing for exercising this option make it very useful.

1.6.2 Switching Option

This is a type of option that describes the possibility to switch between two or more options due to some information from the market which management thinks, that could add value to the firm’s wealth. For instance the managerial team has the option to switch between inputs or outputs if the there is a change in demand or prices or between different technologies of production.

Input mix options or process flexibility -This an option of using different inputs to produce the same output. These options are valuable in the utility firms or organizations (see e.g. Harvey (1999)). Consider the hypothetical example; the Sweden Metro Transport Authority has the option to switch between the use of biogas and other fuel sources for the Metro busses in case of any eventuality. In particular, if it happens that, the Environmental Hazards Control Board discovers that, continual release of bi-products of biogas into the atmosphere pose health treats, then the Sweden Metro Transport Authority would have to switch to a different fuel source which bi-products are environmentally friendly..

Output mix or product flexibility - This as an option of producing different outputs from the same production setting. These options (see e.g. Harvey (1999)) are particular used in business environments where high volatility in demand is the order of the day. For example, consider a car manufacturing company (for instance VOLVO) has the option to switch its production setting in order to manufacture a brand of VOLVO car which is in high demand.

1.6.3 Expansion and Contraction Options

Expansion option is the option where management has the right to expand production output in response to future increase in demand. The option to decrease production in the future in response to drop in demand is referred to as the contraction option.

1.6.4 Deferral Option

Deferral option is an option where the managerial team has the right to defer an action due to some information from the market (for instance changes in demand and prices).

1.7 Structural Differences between Real and Financial Options

In real option theory, options reflect the value of the business opportunities which are at the disposal of firms. For instance, management has the option to defer or abandon a

project due to unavailability of enough information from the market. The expiration date of real options are not necessarily explicit as financial options and their exercise are not ‘necessarily instantaneous’.

Furthermore, real options have limited liquidity. The term liquidity (see e.g. Wikipedia) refers to the possibility to instantaneously sell or buy an asset (financial instrument) without causing a significant change in the price. Therefore from our previous definition of real options, it is normally not possible to quickly buy or sell them without causing significant movements in their price processes.

Finally, analogies of transactional costs for real options may be higher than those of financial options. For example, an option written on a gold mine lease would surely have a higher transactional cost than those of regular options on financial instruments such as stocks, stock indices, etc.

1.8 Behaviour of Real Options

The value of a real option can be increased (decreased) or driven by the scope of managerial flexibilities and decisions. Hence, in general, the various behaviour of real options are contingent to managerial flexibilities.

CHAPTER 2

Mathematical Background

2.1 Notation

If I ⊂ is an open interval and Ω ⊂ n is an open set wheren∈ , then C1,2(I× Ω) is denoted by the family of all functions which are continuously differentiable with respect to the first variable and twice continuously differentiable with respect to the remaining variables (see e.g. Björk (1998)).

2.2 Ito’s Lemma

This lemma was first developed in the 1940’s in the field of stochastic calculus (a branch of applied mathematics and physics). In recent times, this has been an indispensable tool in finance (for instance options pricing). This lemma acts as the basis in the derivation of Black-Scholes equation which is one of the most famous recent tools in the option theory. There are basically two forms of Ito’s lemma;

• A case where the Wiener processes are independent (Uncorrelated) • A case where the Wiener processes are dependent (Correlated)

2.2.1 Ito’s Lemma with Uncorrelated Wiener Processes

Let the random process S=(S(t),t≥0) (see e.g. Björk (1998)) be defined by stochastic differential equation -Ito process

1 ( ) ( , ) ( , ) ( ) , 1 m i i ij j j dS t μ S t dt σ S t dW t i n = = +

∑

≤ ≤

where μ=

(

μ μ1, 2,...,μn

)

is an n-dimensional drift term and W =

(

W W1, 2,...,Wm

)

is an m-dimensional independent Wiener process.

Let f ∈C1,2( ₊× n). DefineF t( )= f t S t( , ( )), then f(t) satisfies the Ito equation 2 , 1 , 1 1 1 2 n n n i i j i i i i i j i j i i f f f f dF c dt dW t = μ S = S S = Sσ ⎛_{∂ ∂ ∂} ⎞ _∂ =⎜_⎜ + + ⎟_⎟ + ∂ ∂ ∂ ∂ ∂ ⎝

∑

∑

⎠

∑

Where 11 12 1 1 2 . . . . . . = . . . . . . . . . m n n nm σ σ σ σ σ σ σ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

and c is the ,_ij i j -entry of the matrix σσ∗

where σ∗ is the transpose of the matrix σ The following formal multiplication rules apply

• 2 ( )dt = 0; • dt dW_j = 0; • 2 (dW_j) =dt j, 1,..., = m • dW dW_{i j} =0, ;i≠ j

2.2.2 Ito’s Lemma with Correlated Wiener Processes

Let us define (see e.g. Björk (1999)) m-dimensional standard Wiener processes by

1 , 2 ,..., m

W⊥ W⊥ W⊥. We can relate W_i and W_j⊥by , 1,..., m

i ij j

j

W =

∑

δ W⊥ i= n, where δ is a purely deterministic matrix given by

11 12 1 1 2 . . . . . . = . . . . . . . . . m ij n n nm δ δ δ δ δ δ δ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

Define the covariance between W t and i( ) W t by j( ) , 1 cov ( ), ( ) cov ( ), ( ) m i j ik jx k x k x W t W t δ δ W⊥ t W⊥ t = ⎡ ⎤ ⎡ ⎤ = ⎣ ⎦

∑

⎣ ⎦ 1 , 1 var ( ) m m ik jk k ik jx k k x W t δ δ ⊥ δ δ = = ⎡ ⎤ =

∑

_{⎣ ⎦}=

∑

on the assumption that var_⎣⎡W_k⊥( )t _⎦⎤ =1, define also a correlation matrix ρ=⎡ ⎤_{⎣ ⎦}ρ_ij =δδ′ where δ′ is the transpose ofδ . If all the above conditions are satisfied, then we can comfortably write Ito’s lemma with correlated Wiener processes as

2 1 , 1 1 1 2 n n n i i j ij i i i i i j i j i i f f f f dF dt dW t = μ S =σ σ ρ S S = Sσ ⎛_{∂ ∂ ∂} ⎞ _∂ =⎜_⎜ + + ⎟_⎟ + ∂ ∂ ∂ ∂ ∂ ⎝

∑

∑

⎠

∑

Almost all the above conventional multiplication rules still hold except dW dW_i⋅ _j =ρ_ijdt

2.3 Black- Scholes Equation

The Black-Scholes option pricing model was developed on the basis of the non-arbitrage argument. This implies that any option written on underlying stocks can be replicated perfectly by an acceptable trading strategy applied to a portfolio of the underlying stocks and the risk-free asset. In the classical version of their model, Black and Scholes assumed that, the price process of the underlying stock can be described by an appropriate Ito’s process, namely the Geometric Brownian Motion.

2.3.1 1-dimensional Black-Scholes Equation

Let the price process S of an underlying stock (see e.g. Luenberger (1998)) be modeled bydS=rSdt+σSdW, where W is a Wiener process, r is the interest rate (the drift term) and σ the volatility (the diffusion term).Suppose that f S t( , )∈C1,2( ₊× ) is such that the price of the option written on the underlying stock is f S t t .Then the function ( ( ), )

f has to satisfy the partial differential equation 2 2 2 2 1 2 f f f rS S rf t S σ S ∂ ₊ ∂ ₊ ∂ ₌ ∂ ∂ ∂

where r in this case, is considered as the risk free interest rate.

2.3.2 Assumptions of the Black and Scholes Model

The following assumptions (see e.g. Rubash-A Study of Option Pricing Models) are normally made in the implementation of the basic Black-Scholes pricing model.

• There are no arbitrage opportunities.

Fischer Black and Myron Scholes basically propounded the model on this assumption. This assumption is the most essential among all other assumptions, since it is equivalent to the statement that any option written on the underlying stock can be perfectly replicated.

• The stock pays no dividends and commissions during the option's life. • European exercise styles are used

• Known and constant interest rates • Returns are lognormally distributed

2.4 Definitions

1. Let

{

X_t/t∈T

}

be a stochastic process defined on the measure space ( , , )Ω F P and

{

F_t/t∈T

}

a filtration (an increasing sequence of sigma sub-algebras ofF ) where T is linearly ordered subset of with a minimumt . Then the process ₀

{ }

X_t is said to be adapted to the filtration

{ }

F_t if for eacht≥ , t₀ X is _t F -measurable: _t 1

( ) t

X− B ∈F for each _t borel set B⊆

2. Let F_tW( ) t be a filtration and ( )f t be an adapted process (i.e. ( )f t ∈F_tW( )t ), then 2[ , ]={ : y [ 2( )] and ( ) _tW( )}

x

L x y f

∫

E f s ds< ∞ f t ∈F t If f ∈L x y2[ , ] (see e.g. Björk (1998)), then

• y ( ) ( ) 0 x E⎡_⎢ f s dW s ⎤ =_⎥ ⎣

∫

⎦

•

(

)

2 2 ( ( ) ( ) [ ( )] y y x x E⎡_⎢ f s dW s ⎤_⎥= E f s ds

⎣

∫

⎦

∫

. This is called Ito’s isometry.

2.5 Girsanov’s Theorem

In its financial applications Girsanov’s theorem essentially describes how changes of the drift coefficient of a stochastic process (and hence changes of our model of risk) influence the volatility term of the process.

2.5.1 Statement of Theorem:

LetT >0 be fixed and let X be an n-dimensional stochastic process represented by the stochastic differential equation

dX t( )=α( )t dt+γ( )t dW t( ), t∈

[ ]

0,T ,

driven by an m-dimensional Wiener process W t . Suppose that there exist processes ( ) ( )t

β and u t adapted to ( ) W t such that ( ) γ( ) ( )t u t +β( )t −α( )t = , 0 with probability one, and

0 1 exp ( ) ( ) ( ) 2 T P E ⎡_⎢ ⎛_⎜ u t u s ds∗ ⎞ < ∞_⎟⎤_⎥ ◊ ⎝ ⎠ ⎣

∫

⎦ Define

[ ]

0 0 1 exp ( ) ( ) ( ) ( ) , 0, 2 t t t Z = ⎛_⎜− u s dW t∗ − u t u s ds∗ ⎞_⎟ t∈ T ⎝

∫

∫

⎠

[ ]

0 ( ) t ( ) ( ), 0, , W t =

∫

u s ds W t+ t∈ T dQ=Z dP_t .

Then Z is a martingale with respect to_t P , Q is a probability measure equivalent to P ,W t is a Wiener process with respect to Q and the process ( ) X t can be represented ( ) by the stochastic differential equation

dX t( )=β( )t dt+γ( )t dW t( ), t∈

[ ]

0,T ,

Note that the values of α( )t and β( )t are column vectors in n and the values of γ( )t are n m× -matrices. The values of W t u t and ( ), ( ) W t are column vectors in( ) m.

The property of u described in formula

( )

◊ is known as Novikov’s condition. Girsanov’s theorem remains valid without it, provided that Z is assumed to be a P -_t martingale (see e.g. Oksendal (2005)).

CHAPTER 3

Insufficiency of No-arbitrage Pricing of Real Options

3.1 Overview

This chapter is basically an extract of Hubalek and Schachermayer (2001) and Klimek (2004), which deals with the pricing of non-tradable assets using the standard Black-Scholes option pricing method with some explicit conditions for instance preferences and subjective probabilities.

3.2 Statement of Problem (Hubalek and Schachermayer)

We consider an option which is contingent on an underlying S that is not a traded asset. We analyze the situation when there is a surrogate traded asset S whose price process is highly correlated with that of S with the assumption that, frictionless trading in continuous time is possible.

We apply the famous Black and Scholes option pricing model to find the arbitrage-free price of the considered option.

3.3 Introduction

The Black-Scholes option pricing model is an essential tool for pricing and hedging derivative securities in financial markets, based on the assumption that there are no arbitrage opportunities. This implies that any derivative security can be perfectly replicated by an appropriate trading strategy on the underlying asset.

We formalize the setting of a real option (see e.g. Hubalek and Schachermayer (2001)) on an underlying S, which is not a traded asset, but such that there is a traded asset S whose price process is highly correlated to that of S.We do this by modeling S and S as geometric Brownian motions, correlated by a correlation coefficient ρ which is close to one.

3.4 General set-up (Hubalek-Schachermayer)

Consider the following set-up:

• ( , , )Ω F P - a probability space;

• W W, ⊥-two independent Brownian motions; • T >0-time horizon;

• (F_t)_{0≤ ≤t T}- the filtration generated by W W, ⊥; and assume that W0 W0 0 ⊥

= =

• F₀ = Ω ∅{ , } and assume that F F ; _T = • For a fixed ρ∈ −

(

1,1

)

we define W =ρW+ 1−ρ2W⊥

Then W is also a Brownian motion and corr W W( _t, _t)= ρ

Let ,μ μ∈ and , ,r σ σ > be fixed. Consider the standard Black-Scholes model 0 dB rBdt

dS μSdt σSdW =

= +

This means that 0

rt t B =B e and S_t =S₀exp(X_t), where ( 1 2) 2 t t X = μ− σ t+σW ,

and S is the current price and ₀ S is the price for _t t≤T .

3.5 Definitions.

1. Let ( )S { :Q Q probability measure on equivalent to , P S is a Q martingale} B = − M F where S (S e₀ rt)_{0 t T} B − ≤ ≤

= is the discounted price process of the traded asset S.The price process of the bond B acts as the discounting factor.

dS =μSdt+σSdW since 2 1 W =ρW + −ρ W⊥ we have 2 1 dW =ρdW + −ρ dW⊥ and so 2 2 1 , 1 dS Sdt S dW S dW dW Sdt S dW μ σ ρ σ ρ μ σρ σ ρ ⊥ ⊥ = + + − ⎡ ⎤ ⎡ ⎤ = + _⎣ − _{⎢ ⎥} ⎦ ⎣ ⎦

3.6 Derivation of

S

:

Let F =lnS

Applying Ito’s lemma

2 2 2 2 1 1 1 1 0 2 1 2 t t dF S S dt S dW S S S dt dW μ σ σ μ σ σ ⎛ ⎞ =_⎜ + − _⎟ + ⎝ ⎠ ⎛ ⎞ =_⎜ − _⎟ + ⎝ ⎠ Integrating from 0 to t : 2 0 0 2 0 0 2 0 1 ( ) (0) 2 1 , 0 2 1 exp { } 2 t t t t t t t F t F dt dW S ln t W W S S S t W μ σ σ μ σ σ μ σ σ ⎛ ⎞ − = _⎜ − _⎟ + ⎝ ⎠ ⎛ _{⎞ ⎛ ⎞} = − + = ⎜ ⎟ ⎜_⎝ ⎟_⎠ ⎝ ⎠ ⎛ ⎞ = _⎜ − + _⎟ ⎝ ⎠

∫

∫

Consider also the asset S which cannot be traded, but serves as the underlying asset for a European call option with price processC , where _t C_T =(S_T −K) and + K >0 denotes the strike price. We have: 0 2 1 exp( ), where ( ) 2 t t t t S =S X X = μ− σ t+σW

3.7 Theorem (Hubalek and Schachermayer)

For any numberc₀∈(0, )∞ , there exists a probability measure Q∈ M( )S such that c₀ =e−rTE_Q[(S_T −K) ]+ .

Hence the existence of the risk neutral probability Q makes the financial market{(B_t), ( ), (S_t C_t)}_{0≤ ≤t T} arbitrage-free, where

C_t =e−r T t( −)E_Q[(S_T −K) | ]+ F _t

Proof

:

Choosev∈ . Let’s apply Girsanov’s theorem to the two-dimensional process 2 0 1 dS dW dt dS dW σ μ μ σρ σ ρ ⊥ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − ⎣ ⎦ ⎣ ⎦ ⎢_⎣ ⎥_⎦⎣ ⎦

We want to changeμ to r and μ to v+σ2 2.First we calculate the function u (which in this case reduces to a constant vector) used to construct the Radon-Nikodym derivative in Girsanov’s theorem: 1 2 2 2 2 2 2 2 2 0 2 1 1 0 1 2 1 1 ( ) 2 1 1 r u v r v r r v σ _μ μ σ σρ σ ρ μ σ σ ρ _μ σ ρ σ ρ μ σ λ ρ μ μ σ _λ σ ρ σ ρ − ⊥ ⎡ ⎤ ⎡⎛ ⎞ ⎛ ⎞⎤ =⎢ ⎥ ⎢⎜ ⎟ ⎜− ₊ ⎟⎥ − ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = _{− ⎢ ⎥} ⎢ ⎥_⎢ − − _⎥ ⎣ ⎦ ⎢ _{− −} ⎥ ⎣ ⎦ − ⎡ ⎤ ⎢ ⎥ _{⎡ ⎤} ⎢ ⎥ = _{− − − = − ⎢ ⎥} ⎢ ₊ ⎥ _{⎣ ⎦} ⎢ _{− −} ⎥ ⎣ ⎦

We define the new measureQv∈M( ) by S dQv dP=ZT , where

[ ]

2 2 ( ) exp , 0, 2 t t t Z λW λ W λ λ t t T ⊥ ⊥ ⊥ ⎡ + ⎤ = _⎢ + − _⎥ ∈ ⎣ ⎦

The new two-dimensional Wiener process (with independent components) under Q is v t t W t W t λ λ ⊥ ⊥ − ⎡ ⎤ ⎢ ₋ ⎥ ⎣ ⎦

The processes X_t, X are now given by _t

2 2 ( ) 2 ( ) 1 ( ) t t t t t X r t W t X vt W t W t σ _{σ λ} σ ρ λ ρ ⊥ λ⊥ ⎛ ⎞ =_⎜ − _⎟ + − ⎝ ⎠ ⎡ ⎤ = + _⎣ − + − − _⎦

In particular the random variable UT( )v ρ(Wt λt) 1 ρ2(Wt λ t)

⊥ ⊥

= − + − − is Gaussian with mean 0 and variance T under the new measureQ . Hence _v

( ) 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) T v T X T vT U vT T z E v E S K E S e K E S e K S e K z dz σ σ _ϕ + + + + ∞ _{+ +} −∞ ⎡ ⎤ ⎡ ⎤ = _⎣ − _⎦= _⎣ − _⎦ ⎡ ⎤ = _⎣ − _⎦ =

_∫

−

Where ϕ is the PDF forN[0,1]. Furthermore ( ) ( ₀ ) ( ) v vT T z z E v =

∫

∞ S e +σ −K ϕ z dz where 0 v K ln vT S z T σ − = for a fixedv.

Now for v variable in , it is clear that lim ( ) 0 and lim ( )

v→−∞E v = v→∞E v = ∞

Deductions:

Since lim _v v→±∞z = ∞∓ , it follows that lim ( ) at 0 at -v z v K K ∞ϕ z dz →±∞ ∞ ⎧ = ⎨ _∞ ⎩

∫

and

lim ( ) a postive constant at 0 at- v T z z v e z dz σ _ϕ ∞ →±∞ ∞ ⎧ = ⎨ _∞ ⎩

∫

This, combined with the fact that

0 lim ( ) lim ( ) , v vT T z z v E v v S e e z dz σ _ϕ ∞ →±∞ = →±∞

∫

yields the required limits.

Now it suffices to observe that the function v E v is continuous and use the ( ) intermediate value theorem. To see continuity considerv₁≤ . Then v₂

(

) (

)

(

) (

)

1 2 2 1 1 2 0 0 0 0 ( ) ( ) ( ) ( ) v T T z v T T z v T T z v T T z E v E v S e K S e K z dz S e K S e K z dz σ σ σ σ ϕ ϕ + + ∞ _{+ +} −∞ + + ∞ _{+ +} −∞ − ≤ − − − ⎡ ⎤ = _⎢ − − − _⎥ ⎣ ⎦

∫

∫

(

)

2 1 2 1 0 0 0 ( ) ( ) v T T z v T T z v T v T T z S e S e z dz e e S e z dz σ σ σ ϕ ϕ ∞ _{+ +} −∞ ∞ −∞ ⎡ ⎤ ≤ _⎣ − _⎦ = −

∫

∫

Note that we have used the fact that if f is a real-valued function then

2 1 2 1

( ) ( ) ( ) ( ) f+ v − f+ v ≤ f v − f v

In summary, we can deduce from the theorem that, when things are done mechanically, that is by using only the non-arbitrage principles; then nothing can be said about the price

0

c of a real option even if there exits a surrogate which is not perfectly correlated with the underlying real asset. Therefore the non-arbitrage principle alone is insufficient for pricing of real options even if the underlying real and surrogate are highly correlated.

CHAPTER 4

Alternative Pricing

4.1 Overview

This chapter basically deals with the various suggested methods based upon the following criteria:

• If there exists a surrogate whose price process is highly but not perfectly correlated with the price process of the underlying real asset.

• If there exists a surrogate whose price process is perfectly correlated with the price process of the underlying real asset.

4.2 Basic

Definitions

•

A portfolio or trading strategy (see e.g. Björk (1998)) is and F_tS( )t -adapted n-dimensional process f =( ,..., ) f₁ f_n , where f t denotes the amount of units (or _i( ) shares) of the j asset contributes to the value of the portfolio at timeth 0≤ ≤t T andS t( )=( ( ),...,S t₁ S t_n( )) is an n-dimensional the price process. If ‘the purchase of a new portfolio, as well as all consumption, must be financed solely by selling assets already in the portfolio’ then it is called a self financing portfolio.

•

Sharpe index-is an index which is used to measure the performance of a given portfolio over a predefined time horizon and taking into account the risk of the portfolio.

( ) ( )

( )

Portfolio local Return Risk free Return r Sharpe Volatility μ σ − =

which is also called the market price of risk. • A set Ω ⊂ n

is convex if, ∀x y, ∈Ω

βx+ −(1 β)y∈Ω ∀ ∈β [0,1]

• A real valued function f is convex on the convex set Ω if the condition below is met:

for all ,x y∈Ω and for each β∈

[ ]

0,1 .

• A predictable process ( )f t for t≤Tis called an admissible trading strategy for the asset S if the stochastic integral

0 ( ) ( )

t

f t dS t

∫

for t≤T is an 2

L -bounded martingale (see e.g. Björk (1998)).

• Let

0

( )t T f t dS t( ) ( )

φ = +β

∫

which is the outcome of a trading strategy starting with an initial capitalβ at time t=0and subsequently holding f t units of ( ) the asset Sat time t (see e.g. Hubalek-Schachermayer (2001)).

4.3

High Correlation (Non -Perfect Correlation)

Under this assumption of high correlation between S and S it is very tempting to employ the regular financial options theory, which could lead to severe mispricing (see e.g. Hubalek and Schachermayer (2001)). Hence Hubalek and Schachermayer proposed that one has to resort to ‘trading strategies related to minimizing the variance of the hedging error’.

4.3.1 Minimization of the Variance of the Hedging Error

Set-up (Hubalek-Schachermayer)

Hubalek-Schachermayer concluded that, if the real asset S is not perfectly correlated with the financial asset S, then we cannot perfectly replicate an option on Sby only trading in S and the risk-free asset (for simplification assume that r=0).Therefore ,they suggested that one has to adopt a new trading strategy on the surrogate Sso that the result is pretty close to the payoff function max(S−K, 0).The difference between the new and the old trading strategies is called the hedging error ( )ε T for which we seek to minimize it’s variance (i.e. var

[

ε( )T

]

)

We assume that the strike price, the current price of the surrogate and the real asset are the same and equal to one (i.e.K =S₀ =S₀ =1); the risk-neutral interest rate, the expected

rate of return of the financial asset and the real asset are the same and equal to zero (i.e.r= = = ); the volatility of the financial asset and the real asset are the same and μ μ 0 equal to one (i.e.σ σ= =1). The above assumptions were made by Hubalek-Schachermayer purely for the sake of clarity. Hence, from the above assumptions the Ito’s process that governs the price process of the financial asset is given by;

dS t( )=S t dW t( ) ( )

( )

† Similarly for the real asset;

dS t( )=S t dW t( ) ( )

( )

†† For our new trading strategy, we look for a predictable process ( )f t for t≤T which is an admissible trading strategy for the surrogateS.

In view of the admissibility property of the predictable process ( )f t for t≤T and the equation

( )

† ; we can now comfortably write

0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t dS f dS f S S f S dW τ τ τ τ τ τ τ τ τ = =

∫

∫

∫

On the basis of the above stochastic integral and the assumptions; Hubalek and Schachermayer formulated the minimization problem below:

0

min var[( ( )S T −K)+− +(x

∫

T f t dS t( ) ( ))],

where the minimum is taken over all x∈ and all admissible trading strategies f . The solution to the minimization problem is the pair ( , )x f such that ˆ ˆ

0

ˆ

ˆ T ( ) ( ) x+

∫

f t dS t minimizes the problem.

Furthermore, ˆx is obtained from the standard Black-Scholes explicit pricing formula of the trading strategy ( ( )S T −K)+ and is given by;

xˆ= =c S N d₀ ( )₁ −KN d( ₂)

and ˆ ( )f t is the optimal strategy for new trading skills on S which is given by; ˆ ( ( ), ) ( ) ( ) S t f f S t t S t ρ =

Hubalek and Schachermayer proved that; the hedging error for an initial investment of ˆx and trading according to the optimal strategy is given by;

2

0

( )T 1 T f t S t dW( ) ( ) ( )t

ε _{= −}ρ ⊥

∫

.

Taking the expectation of the error:

[

]

(

2

)

0 2 0 ( ) 1 ( ) ( ) ( ) 1 ( ( )) ( ( )) ( ( )) 0 T T E T E f t S t dW t E f t E S t E dW t ε ρ ρ ⊥ ⊥ = − = − =

∫

∫

Since (E dW⊥( ))t = 0 Similarly, taking the variance of the error:

[

]

(

)

(

)

(

)

(

)

2 2 0 2 2 2 2 0 0 2 2 0 2 2 2 0 2 2 var ( ) var 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) (1 ) ( ) ( )( ( )) ( ' ) (1 ) ( ) T T T T T T f t S t dW t E f t S t dW t E f t S t dW t E f t S t dW t E f t S t dW t by Ito s isometry E f t ε ρ ρ ρ ρ ρ ρ ⊥ ⊥ ⊥ ⊥ ⊥ = − ⎡ _{⎤ ⎡ ⎤} = _⎢ − _{⎥ ⎢}− − _⎥ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ = _⎢ − _⎥ ⎣ ⎦ ⎡ ⎤ = − _{⎢ ⎥} ⎣ ⎦ = = −

∫

∫

∫

∫

∫

(

)

2 0 2 ( ) (1 ) var ( ( ) ) T S t dt S T K ρ + ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ = − −

∫

The explicit formula for the random variable var ( ( )

(

S T −K)+

)

is given by;

(

)

(

)

[ ]

[ ]

[ ]

(

[ ]

[ ]

)

2 2 2 2 2 0 0 0 1 2 0 1 2 var ( ( ) ) ( ( ) ) ( ( ) ) 2 T S T K E S T K E S T K e S N d S KN d K N d S N d KN d + ⎡ + ⎤ ⎡ _⎡ +_⎤⎤ − = _⎢ − _⎥−_{⎣ ⎣} − _⎦⎦ ⎣ ⎦ = − + − − Where 0 0 ln(S K) 3T 2 d T + = 2 0 1 ln(S K) ( 2)T d T μ σ σ + + =

d₂ =d₁−σ T

and N denotes the univariate standard normal cumulative distribution function

We look at the relationship between correlation ρ∈ −[ 1,1] and the var

[

ε( )T

]

graphically using our previous assumptions: σ σ= =1,K = = =S S 1,r= = = with μ μ 0 T = . 1 Therefore 0 (1) (3 2) 1 3 2 ln d = + = 1 ln(1) (0 (1 2)) 1 1 2 d = + + = 2 1 2 1 1 1 2 d = − = −

Substituting the values of d d d into the explicit formula of₀, ₁, ₂ var ( ( )

(

S T −K)+

)

, we obtain

(

)

[ ]

[ ]

[

]

(

[ ]

[

]

)

[ ]

[ ]

[ ]

(

[ ]

)

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

2 2 2 2 var ( ( ) ) 3 2 2 1 2 1 2 1 2 1 2 3 2 2 1 2 1 1 2 2 1 2 1 3 2 2 1 2 1 1 2 4 1 2 4 1 2 1 3 2 1 2 4 1 2 1.3157 S T K eN N N N N eN N N N eN N N N N eN N N + − = − + − − − − = − + − − − = − + − − + + = + − = Hence var

[

ε( )T

]

=1.3157 1

(

−ρ2

)

.

The table and graph below show the relationship between the variance of the hedging error var

[

ε( )T

]

and correlationρ∈ −

[

1,1

]

ρ _{-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0}

[

]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Correlation

Variance of the Hedging Error against the Correlation

V a ri an c e of t he H e dg in g E rr o r

The above graph is a parabola and a concave function. We can deduce from the above Figure that;

• Whenρ = , that is; there is no correlation between the financial asset 0 S and the underlying real asset S.The variance of the hedging errorvar

[

ε( )T

]

attains its maximum value at this point. Hence, using the financial asset S as a surrogate would lead to a severe mispricing.

• When 0< < , the variance of the hedging error decreases. Hence using ρ 1 S as surrogate for values of ρ which are pretty close to 1, the results of our new trading strategy would be pretty close to being optimal. Therefore the smaller the variance of the hedging errorvar

[

ε( )T

]

, the least the danger of mispricing of the option written on the underlying real asset.

• when ρ = the variance of the hedging error 1 var

[

ε( )T

]

is zero, so the financial asset S and the underlying real asset S are perfectly correlated. In this case one could apply the regular financial options pricing theory.

.

Therefore any rational investor would only adopt this trading strategy for values of ρ which are very close to 1 that is if the surrogate and the underlying real asset are very

Note that asρ→ ,1 var

[

ε( )T

]

tends to zero asymptotically in1− : ρ

[

]

(

)

(

)(

)

(

)

2 var ( ) 1.3157 1 1.3157 1 1 2.6313 1 T ε ρ ρ ρ ρ = − = − + ≈ −

4.4 Imitation or Naive Strategy

Hubalek and Schachermayer considered another method called imitation (or naive) strategy. This method simply replaces the real asset Swith the traded asset S.The imitation strategy replicates the option written on the traded assetS, which is given by

(

)

₀

0

max S T( )−K, 0 =c S T( , )+

∫

T f S t dS t( ( )) ( ) ( † ) Since the price process of the real asset S is driven by managerial decisions and flexibilities as posed to the price process of the traded asset Swhich is purely driven by market forces, the method has several drawbacks which may lead to severe mispricing. Hence when applying this method one has to bear in mind that; because of the striking differences between the price processes of the real asset and the traded asset, this method would normally fail to be optimal. So more efforts have to be made to quantify how far is the outcome from being optimal.

Moreover, from († ), the hedging error ε_im( )T is;

(

)

(

)

(

)

(

)

( ) max ( ) , 0 max ( ) , 0 1 1 im U U T S T K S T K e e ε = − − − = − − −

based up the assumptions: μ μ= = =r 0,σ σ= =1,S₀ =S₀ =K=T made by Hubalek and Schachermayer which is purely on the grounds of simplicity.

(

U U is a bivariate ,

)

normal random variable with mean 0.5 and unit variance.

Moreover, Hubalek and Schachermayer came out with an explicit formula for the variance of the Imitation Strategy var

[

ε_im( )T

]

using the bivariate Esscher transformation as var

[

εim( )T

]

=2

(

f2− f1

)

[ ]

[ ]

2 3 2 3 1 2 1 f =eN − N + 1 1 1 1 1 1 1 , , 2 , , , , 2 2 2 2 2 2 f =e Mρ ⎛_⎜ρ+ ρ+ ρ⎞_⎟− M⎛_⎜ ρ− ρ⎞_⎟+M_⎜⎛− − ρ⎞_⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

[

1,1

]

ρ∈ − and M represent the correlation and the bivariate standard normal cumulative distribution function respectively. The symbol N denotes the univariate standard normal cumulative distribution function.

Using the above formula, Schachermayer proved that, asρ→ , the variance of the 1 hedging error of the imitation strategy var

[

ε_im( )T

]

tends to zero asymptotically in 1− , ρ

[

]

[ ][

]

[

]

var ( ) 2 3 2 1 5.0734 1 im T eN ε ρ ρ − ≈ − ∼

As 1ρ→ the variance of the naïve trading strategy is approximately twice the variance of the hedging error of the minimal strategy. Hence any rational investor would always prefer the trading strategy relating to the minimization of the variance of the hedging error to the imitation strategy.

The graph below shows a comparison between the asymptotic behaviour of var

[

ε( )T

]

and var

[

ε_im( )T

]

asρ→ . The graph is of importance only for values of 1 ρ pretty close to one but makes no meaning for values of ρ pretty far from one.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12

Comparison of the Asymptotic behaviours of the Variances of the Hedging Error of the above two trading strategies as Correlation approaches 1 imitation optimal

Hence, from the diagram; as ρ → the variance of the hedging error of both trading 1 strategies turns linearly to zero in

(

1−ρ

)

. The gradient of the imitation strategy is approximately as twice as the gradient of the trading strategy relating to the minimization of the variance of the hedging. Hence one could deduce that (see e.g. Hubalek and Schachermayer (2001)); the minimum price of being naive is approximately as twice as being a bit wiser (for instance minimal variance).