Difficulties in pricing of real options

Transcription

1 Difficulies in pricing of real opions Francis Asu U.U.D.M. Projec Repor 7:5 Examensarbee i maemaik, poäng Handledare och examinaor: Maciej Klimek Januari 7 Deparmen of Mahemaics Uppsala Universiy

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3 Acknowledgemen I would like o express my appreciaion o Professor Maciej Klimek, my supervisor, no only for his excepional help on his projec, bu also for he course (Financial Derivaives) ha he augh which graned me he undersanding of Real Opions and he necessary mahemaical background o come ou wih his piece of wriing. I would also like o hank Professor Johan Tysk, who inroduced me o financial mahemaics a he iniial sage of my sudies and Professor Abrahamsson Leif as a personal course selecion adviser. To he res of he professors in he Financial Mahemaics and Financial Economics programme who provided insrucion, encouragemen and guidance, I would like o say a big hank you o you all. They did no only each me how o learn, hey also augh me how o each, and heir excellence has always inspired me. Finally, I would like o hank my moher, Mary Nambo, for her financial suppor and encouragemen, Zsuzsanna Krisofi of Mahemaics Deparmen, for he help she rendered o me when I firs conaced he Deparmen and he enire members of he Deparmen of Mahemaics. 1

4 Dedicaion I dedicae his piece of wriing o God for His divine suppor, my moher Mary Nambo and sisers Francisca Asufe (win), Nambo Mavis.

5 TABLE OF CONTENTS Absrac 5 Chaper One Background 1.1 Opion Basics 6 1. Types of Opions Exercise Syle European Opion European Call Opion European Pu Opion American Opion Bermudan Opion Asian Opion Barrier Opion Lookback Opion Turbo Warrans Chooser Opion Compound Opion Baske Opion Behaviour of Opions Real Opions Inroducion Definiions Types of Real Opions Abandonmen or Terminaion Opion Swiching Opion Expansion and Conracion Opion Deferral Opion Srucural Differences beween Real and Financial Opions Behaviour of Real Opions 19 Chaper Two Mahemaical Background.1 Noaion 3

6 . Io s Lemma..1 Io s Lemma wih Uncorrelaed Wiener Processes -1.. Io s Lemma wih Correlaed Wiener Processes 1-.3 Black-Scholes Formula dimensional Black-Scholes Formula.3. Assumpions of he Black-Scholes Model -3.4 Definiions 3.5 Girsanov s Theorem Saemen of heorem 4 Chaper Three Insufficiency of No-arbirage Pricing of Real Opions 3.1 Overview 5 3. Saemen of Problem (Hubalek and Schachermayer) Inroducion General Se-up (Hubalek and Schachermayer) Definiions Derivaion of S Theorem (Hubalek and Schachermayer) 8-3 Chaper Four Alernaive Pricing 4.1 Overview Basic Definiions High Correlaion (Non-Perfec Correlaion) Minimizaion of he Variance of he Hedging Error Imiaion or Naïve Sraegy References 4 4

7 Absrac This projec is basically made up of four pars. The firs par reviews various ypes of financial and real opions. The second par presens various mahemaical ools for example Io s lemma, Black-Scholes equaion and Girsanov s heorem which are needed in he res of he paper. The hird par serves as he core of he enire projec. We presen in deail a resul due o Hubalek and Schachermayer showing ha sandard argumens based on he Black-Scholes model and non-arbirage assumpions can be unsuiable for pricing of real opions. More specially, aemps a pricing of an opion wrien on a real non-radable asse, by means of pricing a surrogae financial asse, may fail compleely even if he wo asses are highly correlaed. The las par of his projec presens wo remedies suggesed by Hubalek and Schachermayer: rading sraegy based on minimizaion of he variance of he hedging error and naive rading sraegy in which he surrogae financial asse is considered in place of he real asse 5

8 CHAPTER 1 Background 1.1 Opion Basics Opions are financial insrumens ha gran he owner some righs. Hence an opion is he righ bu no an obligaion o buy or sell some underlying asse under some predefined condiions. The one who issues an opion is called he wrier and he one who acquires he opion is called he holder or he owner. To acquire an opion, he holder pays he wrier a premium. An opion would be exercised only when i is in he ineres of is holder o do so. 1. Types of Opions Theoreically, here is an infinie variey of opions, bu he wo basic and sandard ones are called calls and pus. Complex financial insrumens (see e.g. Lyuu ()) can ofen be made up of calls and pus. A call opion gives is holder he righ o buy an amoun of an underlying asse by paying a specified srike price (or exercise price) K. A pu opion grans is holder he righ o sell a specific amoun of he underlying asse by paying a specific srike price K. The srike price K of a call (pu) opion is he price a which he underlying asse will be bough (sold) a he expiraion daet. Expiraion dae T is he las day or dae on which an opion can be exercised. There are some opions wih no special or unusual feaures, such opions are called Vanillas. Hence he mos common and sandard ype of vanilla opion wih a sandard expiraion dae T and srike price K and no exra feaures is called plain vanilla. There are also some more complex financial opions wih addiional feaures; hey are called exoic opions or pah-dependen opions. 1.3 Exercise Syle Opions normally differ in when hey can be exercised. Some gran he owner he righ o exercise a anyime wihin a specified ime horizon, whereas ohers gran he owner he righ o exercise a only a specific ime poin. Furhermore, some may depend on he hisory of he price process of he underlying asse. There are some opions which may use one or more of he above syles of exercise. 6

9 Examples of he mos common exercise syle of opions are European, American, Bermudan, Asian, Barrier, Lookback, Turbo warrans, Compound, Chooser and Baske European Opion Le he price process of he underlying asse be (). A European opion gives S, [, T] he owner he righ o exercise he opion only on he expiraion daet. Hence he holder receives he amoun ϕ ( ST ( )), where ϕ is a conrac funcion. Moreover, here are wo basic ypes of European opion namely European call Opions and European Pu Opions European Call Opion A European call opion which gives is owner he righ o purchase an underlying asse for a given price (exercise price) on he expiraion daet. A European call will be exercised only if he sock price S is higher han he srike price K. When a call opion is exercised, he holder pays he wrier he srike price in exchange for he sock, and he opion ceases o exis. Hence he value or he payoff of a call a expiraion is ϕ + c( ST ( )) = max( ST ( ) K,) = ( ST ( ) K,) where T is he expiraion dae. ϕc is called he conrac funcion of he call opion. Therefore ϕ ( ST ( )) is an explici formula for he value of a call opion a expiraion T as a funcion of he price process of he underlying asse a ime T as shown below. c ϕ c Figure 1.: The pay-off of call opion K S(T) 7

10 From Figure 1. we could see ha, he opion is valueless if he price of he underlying securiy is less han he srike price ( ST ( ) < K), he opion value increases linearly wih he price when he price of he underlying is greaer han he srike price ST ( ) A any ime a European call is said o be in he money if S ( ) if S ( ) = K, and ou of he money if S ( ) < K. > K. > K, a he money European Pu Opion A European pu opion which gives he owner he righ o sell an underlying asse for a given srike price on he expiraion daet. The underlying asses may be socks, sock indices, fuure conracs, ineres rae, ec. A European pu is exercised only if he sock price S is less han he srike price K. When a pu opion is exercised, he holder receives from he wrier he srike price in exchange for he sock and he opion ceases o exis. Hence, he payoff of a pu a expiraion is ϕ + ( ST ( )) = max( K ST ( ),) = ( K ST ( )). p ϕ p is called he conrac funcion of he pu and i gives an explici formula for he value of he opion a he expiraion dae T, as shown in Figure 1.1. ϕ p K S(T) Figure 1.1: The pay-off of a pu opion 8

11 Figure 1.1 shows ha, he opion is worhless if he price of he underlying asse S is greaer han he srike price K (i.e. ST ( ) > K), bu if he srike price is greaer han he sock price hen he pu opion has value. A any ime a European pu is said o be in he money if S () < K, a he money if S () = K, and ou of he money if S () > K. From Figures 1. and 1.1, we can see ha, he conrac funcion of a call is unbounded whereas he conrac funcion of a pu is bounded. Hence in pracice, he possibiliy of a loss is unbounded when issuing a call opion, bu bounded for a pu (see e.g. Luenberger (1998)) American Opion An American opion gives he owner he righ o exercise he opion on or before he expiraion dae ( T ). So, one has o assume ha, here is a well-defined payoff for before he expiraion dae (also called early exercise). Hence a any ime wihin he specified ime frame, holder of an American opion needs o decide wheher o exercise immediaely or o wai. If he holder decides o exercise a say ϕ ( S ( )) where ϕ is he appropriae conrac funcion. Similarly his opion can also be classified ino wo basic ypes: T, hen he receives American call opion which grans he owner he righ o buy an underlying asse for a given srike price on or before he expiraion dae, and American pu opion which grans he owner he righ o sell an underlying asse for a cerain srike price on or before he expiraion dae. If he underlying sock pays no dividends early exercise of an American call opion is no opimal. On he oher hand early exercise of an American pu opion can be opimal even if he underlying sock pays no dividends. An American opion is worh a leas as much as an idenical European opion because of he early exercise feaure Bermudan Opion This ype of opions lies beween American and European. They can be exercised a cerain discree ime poins 1 < <... < n = T. As a consequence of Bermudan opions being a hybrid of European and American opions, he value of a Bermudan opion is 9

12 greaer han or equal o an idenical European opion bu less han or equal o is American counerpar Asian Opion This ype of opion depends on he average value of he underlying asse over a ime horizon. Therefore an Asian opion is pah dependen. Asian opions are cheaper relaive o heir European and American counerpars because of heir lower volailiy feaure. This ype of opion (see e.g. Global Derivaives) can broadly be classified ino hree basic caegories; Arihmeic average Asians (AV) This is he sum of he sampled asse prices S, S1,..., Sn 1divided by he number of S + S Sn 1 samples: AV = n Geomeric average Asians(GA) The value of his is aken as: GA = n S S1... Sn 1 Average srike Asian (AS) This is he srike average beween arihmeic and geomeric average Asians: AV + GA AS = Asian opions can eiher have European or American exercise syle Barrier Opions Given an opion wih a pay-off funcion φ one can consider barrier versions of his opion. Barrier Opions (see e.g. Björk (1999)) are like he usual opions wih he characerisic ha a predefined acion akes place when he sock assumes a cerain level called he barrier. Typical examples of barrier opions are down-and-ou, down-and -in up-and- ou and up-and-in, conracs. Down and ou conrac is given by: φ( ST ( )), if S ( ) > L,, T Z =, if ST ( ) L,, T 1

13 This means ha he amoun Z is paid o he conrac holder if he sock price lies above he barrier L during he enire life of he conrac or he conrac erminaes and nohing is paid o he holder if a cerain imes before he expiraion T he price of he sock assumes he barrier L. Down and in conrac is defined as: φ( ST ( )), if S ( ) L,, T Z =, if ST ( ) > L,, T The above conrac is defined as, he amoun Z is paid o he conrac holder if a cerain ime before he expiraion T he sock price assumes he barrier L or nohing is paid o he conrac holder if during he enire conrac life of he conrac he sock price lies above barrier L. Up and ou conrac is given by: φ( ST ( )), if S ( ) < L,, T Z =, if ST ( ) L,, T This is defined as follows; he amoun Z is paid o he conrac holder if he during he enire life of he conrac he sock price lies below he barrier L or he conrac erminaes and nohing is paid o he conrac holder if a cerain imes before he expiraion T he sock price assumes values which greaer or equal o he barrier L. Up and in conrac is given by: φ( ST ( )), if S ( ) L,, T Z =, if ST ( ) < L,, T Thus, he amoun Z is paid o he conrac holder if a cerain ime before he expiraion T he sock price assumes he barrier L or he nohing is paid o he conrac holder if during he enire life of he conrac he sock price lies below he barrier L Lookback Opions Lookback Opions are conracs whose pay-offs a expiry T depend on eiher he maximum or minimum price level of he underlying price process achieved during he enire life of he conrac. Examples of pay-off funcions of ypical lookback opions (see. e.g. Björk (1999)) are: 11

14 Lookback Call: ST ( ) min S ( ) T Lookback Pu: max S ( ) ST ( ) T Forward Lookback Call: max max S ( ) K, T Forward Lookback Pu: max K min S( ), T Turbo warrans A warran is a financial insrumen which grans is owner he righ o buy (or sell) a fixed quaniy of socks a a predefined price and condiions. There are wo basic ypes of warrans namely; a call warran and pu warran. A call warran grans is owner he righ o buy a specific number of shares from he wrier a a predefined price on or before a specific dae. A pu warran also grans is owner he righ o sell back a specific amoun of socks (or shares) back o he wrier a a predefined price on or before a predeermine dae. As claimed by Galiz, A few companies acively use warrans as a means of promoing shareholder value, and obaining a seady sream of new invesmen and warrans can also be used in he same way as equiies, o provide a way of invesing in he shares of a specific company, bu wih a relaively low capial oulay a he sar. New shares are normally sold ou when warrans are exercised (see e.g. Musiela/Rukowski (1997)). The name Turbo warrans had been used earlier in Germany for a usual barrier opion of down and ou syle having is srike price as he barrier. The French bank Sociee Generale issued a conrac called a Turbo warran in early 5, and defined i as a barrier opion - down and ou syle having he barrier in he money and ha he owner receives a rebae if he barrier is hi. The rebae can be reconsidered as a new conrac which is a call opion on he realized minimum by he process S ( ). Turbo 1

15 warrans can also be classified ino wo forms namely Turbo warrans calls (urbo-calls) and Turbo warrans pus (Turbo-pus). Turbo-call warran (see e.g. Persson/Eriksson, 5) pays max{ ST ( ) K,} a mauriy T if a predefined barrier L K has no been hi by Sa ( ) any ime < T.This conrac insananeously ceases o exis and a new conac is iniiaed when he sock S ( ) a anyime < T his he barrier. The new conrac having he same srike price is a call opion on he minimum of he process S ( ).This maures a a cerain laer ime. Similarly, a urbo pu pays max{ K S( T),} a mauriy T if a predefined barrier L K has no been hi by Sa ( ) any ime < T. This conrac insananeously ceases o exis and a new conrac is iniiaed when he S ( ) his he barrier. The new conrac having he same srike price is a pu opion on he realized minimum by he process S.This ( ) maures a a cerain laer ime Chooser Opions This is a ype of opion ha grans you (buyer) he righ o deermine he feaures of an opion. Thus he buyer deermines he value of he srike price, wheher he opion is a pu or call and in some special cases even fix he delivery ime T.Therefore, because of hese flexibiliies a he disposal of he buyer, hese ypes of opions are generally quie cosly Compound Opions A compound opion (see e.g. Musiela/Rukowski (1997) is a sandard opion whose underlying is anoher sandard opion. In oher words a compound opion is an opion wrien on exising opion. Typical and basic examples of compound opions are: a call opion on a call opion, a call opion on a pu opion, a pu opion on a call opion and a pu opion on a call opion Baske Opions This is a ype of opion ha grans you (buyer) he righ o creae a self financing porfolio of wo or more asses (a baske of chosen asses). The payoff of hese ypes of 13

16 opions is given by he difference beween a predeermined srike price and he combined weighed level of he baske of asses (a predefined porfolio of asses) chosen a he ouse. Hence he payoff is coningen on he mean of prices of several asses chosen a he ouse. 1.4 Behaviour of Opions A raional invesor would ofen seek o hedge his or her porfolio value agains risk. Ofen, asses conribuing o he value of he porfolio are conracs wrien on various opions. Hence i helps o know he characerisics ha opion values exhibi under cerain condiions. All opion prices reac o he price of he underlying securiy. If he price of an underlying securiy changes (increases or decreases) hen he value of he corresponding opion also changes. Dela ( Δ = ϕ S ) describes he insananeous rae of change of he opion price ϕ as a funcion of he price of he underlying sock S. Furhermore, he value of a European call opion (see e.g. Persson (6)) wih opion price being convex (i.e. S f(, S) is convex for every fixed [, T] ) varies nearly in proporion o he volailiy, which means ha, he value of a call opion increases wih an increase in volailiy and would decrease wih a decrease in volailiy (Volailiy is he sandard deviaion of he sock price). Hence doubling he volailiy of a European call opion ha is near he money would approximaely double he opion s value. Tripling he volailiy would also riple he opion s value and so on. In addiion, erm srucure paern (or prevailing ineres rae) also plays an imporan role because purchasing a call in some sense is a mehod of purchasing he sock a a reduced price (see e.g. Luenberger (1998)). Therefore one saves ineres expenses. In shor, opion prices depend on he prevailing ineres raes. Finally, when here is a posiive ime o expiraion, he value of a call opion decreases as he ime o expiraion decreases. 14

17 1.5 Real Opions Inroducion In recen imes, mos firms are embarking on projecs in environmens of risk and uncerainy, for which he radiional valuaion mehods for insance discouned clash flow (DCF) fail o correcly give he inrinsic value of such projecs. This is because he radiional mehods fail o ake ino accoun risk, uncerainy and managerial flexibiliies. The deficiency of he usual radiional valuaion mehods resuled ino he inroducion of a new mehod called real opion heory which akes ino accoun uncerainies, various risks and managerial flexibiliies. In his mehod, managers and invesors can choose o esimae he value of managerial decisions for insance o abandon, defer, and expand a projec. Moreover, here are hree specific areas in paricular (see e.g. Maubounssin (1999)) where radiional DCF, mos widely expressed as he ne presen value rule (NPV), comes shor versus opion heory (Real Opions) namely flexibiliy, coningency, and volailiy; Firsly, flexibiliy is he abiliy o defer, abandon, expand, or conrac an invesmen. The NPV rule lacks he capabiliy of capuring he correc value of flexibiliy as compared o he real opion heory, because i does no ake ino consideraion he value of uncerainy For example, consider he classical example given by Michael J.Mauboussin a company may choose o defer an invesmen for some period of ime unil i has more informaion on he marke. The NPV rule would value ha invesmen a zero, while he real opion heory would correcly allocae some value o ha invesmen s poenial. Secondly, coningency is a siuaion whereby he success of oday s invesmen is a deermining facor for fuure invesmens. Hence, Mauboussin claims ha managers may make invesmen oday, even hose deemed o be NPV negaive in oher o access fuure invesmen opporuniies. This could simply mean ha, mangers could incur cos oday due o some managerial decisions in oher o have brigher invesmen opporuniies in he fuure where he cos incurred oday could be facored ino he fuure s cos of invesmen. 15

18 Thirdly, volailiy-which is he sandard deviaion of he price process of an underlying asse. In opions pricing heory, business pursuis wih greaer risk have high opion value. Moreover, due o he convexiy naure of mos payoff schemes of opions heir values increase in accordance wih volailiy. Hence he higher (lower) he volailiy he higher (lower) he opion value Hence he radiional mehodologies fail o adequaely value hese opion creaing invesmen opporuniies. In summary, he real opion heory would correcly esimae he inrinsic cos of real invesmen, bu he radiional mehods (for insance NPV rule) would underesimae. The real opion heory employs he regular financial opions heory (for insance Black- Scholes model) o real invesmens such as expansion of producion seings, research and developmen invesmens, esaes developmen ec. The real opion heory (see e.g. Amram - Kulailaka (1999)) can also be seen as a mehodology ha uses he usual opion heory o quanify managerial flexibiliies in a world of uncerainy. The real opions heory performs prey good in business environmens where uncerainy is he order of he day and if here exiss a proacive and smar managerial eam which has he capabiliies of idenifying, creaing and exercising real opion Definiions A real opion can be defined as an opion embedded in he decisions peraining o real asses. Hence a real opion is no financial insrumen. Since his could also be seen as an acion opion in he sense of choice, he heory grans more flexibiliies han he usual financial opions heory. In conras o financial derivaives (opions), a real opion is no radable, for example, a mobile phone manufacuring company (for insance Nokia) may decide o emporally abandon he producion of a paricular model of heir mobile phones if he price of such a model falls below he producion cos. On he oher hand, Nokia has he righ o sar he producion of ha model if i happens ha he price of ha model rises above he producion cos in he near fuure. Nokia can only choose o or no exercise hese righs, 16

19 bu canno sell hem o anoher pary. The erm real, because he opion involves angible or physical asses. 1.6 Types of Real Opions In he business world, here are several managerial flexibiliies (real opions) which are usually embedded in real invesmen opporuniies of mos firms. Hence his would call in for a managerial eam wih he abiliy o idenify and exercise hese opions. Moreover, hese opions (flexibiliies) allow managemen o leverage uncerainy and considerably reduce risk. Therefore, managemen has he opion o abandon, expand, swich use, defer, conrac (jus o menion few) a projec in oher o preserve he value of such firms Abandonmen or Terminaion Opion This is a ype of opion where he managerial eam reserves he righ o abandon a projec whose inrinsic value seems no be promising. This opion is very useful in capial inensive firms (for insance gold mines, airlines, peroleum firms). Consider a mining firm, which has he opion o mine gold over a ime frame, so afer some number of operaions managemen has realized ha, he operaional cos could no even be covered. Therefore he managerial eam would have o exercise he righ of abandoning his projec in order o preserve he firm s value. In his classical example; if he managerial eam is no smar enough o insananeously exercise his opion, hen he firm sands on he verge of collapsing or losing greaer par of is value. Hence he righ iming for exercising his opion make i very useful Swiching Opion This is a ype of opion ha describes he possibiliy o swich beween wo or more opions due o some informaion from he marke which managemen hinks, ha could add value o he firm s wealh. For insance he managerial eam has he opion o swich beween inpus or oupus if he here is a change in demand or prices or beween differen echnologies of producion. 17

20 Inpu mix opions or process flexibiliy -This an opion of using differen inpus o produce he same oupu. These opions are valuable in he uiliy firms or organizaions (see e.g. Harvey (1999)). Consider he hypoheical example; he Sweden Mero Transpor Auhoriy has he opion o swich beween he use of biogas and oher fuel sources for he Mero busses in case of any evenualiy. In paricular, if i happens ha, he Environmenal Hazards Conrol Board discovers ha, coninual release of biproducs of biogas ino he amosphere pose healh reas, hen he Sweden Mero Transpor Auhoriy would have o swich o a differen fuel source which bi-producs are environmenally friendly.. Oupu mix or produc flexibiliy - This as an opion of producing differen oupus from he same producion seing. These opions (see e.g. Harvey (1999)) are paricular used in business environmens where high volailiy in demand is he order of he day. For example, consider a car manufacuring company (for insance VOLVO) has he opion o swich is producion seing in order o manufacure a brand of VOLVO car which is in high demand Expansion and Conracion Opions Expansion opion is he opion where managemen has he righ o expand producion oupu in response o fuure increase in demand. The opion o decrease producion in he fuure in response o drop in demand is referred o as he conracion opion Deferral Opion Deferral opion is an opion where he managerial eam has he righ o defer an acion due o some informaion from he marke (for insance changes in demand and prices). 1.7 Srucural Differences beween Real and Financial Opions In real opion heory, opions reflec he value of he business opporuniies which are a he disposal of firms. For insance, managemen has he opion o defer or abandon a 18

21 projec due o unavailabiliy of enough informaion from he marke. The expiraion dae of real opions are no necessarily explici as financial opions and heir exercise are no necessarily insananeous. Furhermore, real opions have limied liquidiy. The erm liquidiy (see e.g. Wikipedia) refers o he possibiliy o insananeously sell or buy an asse (financial insrumen) wihou causing a significan change in he price. Therefore from our previous definiion of real opions, i is normally no possible o quickly buy or sell hem wihou causing significan movemens in heir price processes. Finally, analogies of ransacional coss for real opions may be higher han hose of financial opions. For example, an opion wrien on a gold mine lease would surely have a higher ransacional cos han hose of regular opions on financial insrumens such as socks, sock indices, ec. 1.8 Behaviour of Real Opions The value of a real opion can be increased (decreased) or driven by he scope of managerial flexibiliies and decisions. Hence, in general, he various behaviour of real opions are coningen o managerial flexibiliies. 19

22 CHAPTER Mahemaical Background.1 Noaion If I is an open inerval and n Ω is an open se where n, hen C 1, ( I Ω) is denoed by he family of all funcions which are coninuously differeniable wih respec o he firs variable and wice coninuously differeniable wih respec o he remaining variables (see e.g. Björk (1998)).. Io s Lemma This lemma was firs developed in he 194 s in he field of sochasic calculus (a branch of applied mahemaics and physics). In recen imes, his has been an indispensable ool in finance (for insance opions pricing). This lemma acs as he basis in he derivaion of Black-Scholes equaion which is one of he mos famous recen ools in he opion heory. There are basically wo forms of Io s lemma; A case where he Wiener processes are independen (Uncorrelaed) A case where he Wiener processes are dependen (Correlaed)..1 Io s Lemma wih Uncorrelaed Wiener Processes Le he random process S = ( S( ), ) (see e.g. Björk (1998)) be defined by sochasic differenial equaion -Io process ds( ) = μ ( Sd, ) + σ ( SdW, ) ( ), 1 i n i i ij j j= 1 m where μ = ( μ μ μ ) is an n-dimensional drif erm and W ( W W W ),,..., n 1 m-dimensional independen Wiener process. =,,..., m is an 1 Le 1, n C ( ). Define F () = f(, S ()), hen f () saisfies he Io equaion f + df = + + c d + dw n n n f f 1 f f μi i, j σi i i 1 Si i, j 1 Si S = = j i= 1 Si

23 Where σ σ... σ... σ = σ σ... σ m n1 n nm and c ij is he i, j-enry of he marix σσ where σ is he ranspose of he marix σ The following formal muliplicaion rules apply ( d ) = ; d dw j = ; ( dw ) = d, j = 1,..., m j dw dw =, i j; i j.. Io s Lemma wih Correlaed Wiener Processes Le us define (see e.g. Björk (1999)) m-dimensional sandard Wiener processes by W1, W,..., W m. We can relae W and W by W = δ W, i = 1,..., n, where δ is a purely deerminisic marix given by i j m i ij j j δ δ... δ... δ = ij δ δ... δ m n1 n nm Define he covariance beween Wi ( ) and W ( ) by m i j δikδ jx k x kx, = 1 cov W( ), W ( ) = cov W ( ), W ( ) j m m δikδ var jk Wk ( ) δikδ jx k= 1 k, x= 1 = = on he assumpion ha var Wk ( ) = 1, define also a correlaion marix ρ = ρ ij = δδ where δ is he ranspose ofδ. If all he above condiions are saisfied, hen we can comforably wrie Io s lemma wih correlaed Wiener processes as 1

24 df = + + d + dw n n n f f 1 f f μi σσ i jρij σi i i 1 Si i, j 1 Si S = = j i= 1 Si Almos all he above convenional muliplicaion rules sill hold excep dw dw = ρ d i j ij.3 Black- Scholes Equaion The Black-Scholes opion pricing model was developed on he basis of he non-arbirage argumen. This implies ha any opion wrien on underlying socks can be replicaed perfecly by an accepable rading sraegy applied o a porfolio of he underlying socks and he risk-free asse. In he classical version of heir model, Black and Scholes assumed ha, he price process of he underlying sock can be described by an appropriae Io s process, namely he Geomeric Brownian Moion dimensional Black-Scholes Equaion Le he price process S of an underlying sock (see e.g. Luenberger (1998)) be modeled by ds = rsd + σ SdW, where W is a Wiener process, r is he ineres rae (he drif erm) 1, and σ he volailiy (he diffusion erm).suppose ha f( S, ) C ( + ) is such ha he price of he opion wrien on he underlying sock is f ( S ( ), ).Then he funcion f has o saisfy he parial differenial equaion f f 1 f + rs + σ S = rf S S where r in his case, is considered as he risk free ineres rae..3. Assumpions of he Black and Scholes Model The following assumpions (see e.g. Rubash-A Sudy of Opion Pricing Models) are normally made in he implemenaion of he basic Black-Scholes pricing model.

25 There are no arbirage opporuniies. Fischer Black and Myron Scholes basically propounded he model on his assumpion. This assumpion is he mos essenial among all oher assumpions, since i is equivalen o he saemen ha any opion wrien on he underlying sock can be perfecly replicaed. The sock pays no dividends and commissions during he opion's life. European exercise syles are used Known and consan ineres raes Reurns are lognormally disribued.4 Definiions 1. Le { X / T} be a sochasic process defined on he measure space ( Ω, F, P) and { / T} F a filraion (an increasing sequence of sigma sub-algebras of F ) where T is linearly ordered subse of wih a minimum adaped o he filraion { } borel se B. Then he process { } F if for each, X is F -measurable: X X is said o be 1 ( B) F for each W W. Le F ( ) be a filraion and f ( ) be an adaped process (i.e. f ( ) F ( ) ), hen y W L [, x y]={ f : E[ f ()] s ds< and f() F ()} x If f L [, x y] (see e.g. Björk (1998)), hen y E f() s dw() s = x E y ( f ( s ) dw y ( s ) = E [ f ( s )] ds x x. This is called Io s isomery. ( ).5 Girsanov s Theorem In is financial applicaions Girsanov s heorem essenially describes how changes of he drif coefficien of a sochasic process (and hence changes of our model of risk) influence he volailiy erm of he process. 3

26 .5.1 Saemen of Theorem: LeT > be fixed and le X be an n-dimensional sochasic process represened by he sochasic differenial equaion dx () = α() d + γ() dw (),, T, driven by an m-dimensional Wiener process W. ( ) Suppose ha here exis processes β () and u () adaped o W () such ha wih probabiliy one, and γ () u () + β() α() =, Define 1 T EP exp u ( ) u( s) ds < ( ) 1 Z = exp u ( s) dw( ) u ( ) u( s) ds,, T W () = usds () + W (),, T, dq = Z dp. Then Z is a maringale wih respec o P, Q is a probabiliy measure equivalen o P, W ( ) is a Wiener process wih respec o Q and he process X ( ) can be represened by he sochasic differenial equaion dx () = β() d + γ() dw (),, T, Noe ha he values of α () and β () are column vecors in γ n and he values of () are n m-marices. The values of W (), u () and W ( ) are column vecors in m. The propery of u described in formula ( ) is known as Novikov s condiion. Girsanov s heorem remains valid wihou i, provided ha Z is assumed o be a P - maringale (see e.g. Oksendal (5)). 4

27 CHAPTER 3 Insufficiency of No-arbirage Pricing of Real Opions 3.1 Overview This chaper is basically an exrac of Hubalek and Schachermayer (1) and Klimek (4), which deals wih he pricing of non-radable asses using he sandard Black- Scholes opion pricing mehod wih some explici condiions for insance preferences and subjecive probabiliies. 3. Saemen of Problem (Hubalek and Schachermayer) We consider an opion which is coningen on an underlying S ha is no a raded asse. We analyze he siuaion when here is a surrogae raded asse S whose price process is highly correlaed wih ha of S wih he assumpion ha, fricionless rading in coninuous ime is possible. We apply he famous Black and Scholes opion pricing model o find he arbirage-free price of he considered opion. 3.3 Inroducion The Black-Scholes opion pricing model is an essenial ool for pricing and hedging derivaive securiies in financial markes, based on he assumpion ha here are no arbirage opporuniies. This implies ha any derivaive securiy can be perfecly replicaed by an appropriae rading sraegy on he underlying asse. We formalize he seing of a real opion (see e.g. Hubalek and Schachermayer (1)) on an underlying S, which is no a raded asse, bu such ha here is a raded asse S whose price process is highly correlaed o ha of S.We do his by modeling S and S as geomeric Brownian moions, correlaed by a correlaion coefficien ρ which is close o one. 5

28 3.4 General se-up (Hubalek-Schachermayer) Consider he following se-up: ( Ω, F, P) - a probabiliy space; WW, -wo independen Brownian moions; T > -ime horizon; (F - he filraion generaed by WW, ; and assume ha W = W = ) T F = { Ω, } and assume ha F = F ; For a fixed ρ ( 1,1) we define ρ ρ W = W + 1 W Then W is also a Brownian moion and corr( W, W ) = ρ T Le μ, μ and r, σ, σ > be fixed. Consider he sandard Black-Scholes model db = rbd ds = μsd + σsdw r This means ha B = Be and S = S exp( X ), where 1 X = ( μ σ ) + σw, and S is he curren price and S is he price for T. 3.5 Definiions. 1. Le M( S) = { Q: Q probabiliy measure on F equivalen o P, S is a Q maringale} B S r where = ( Se ) T is he discouned price process of he raded asse S.The price B process of he bond B acs as he discouning facor.. Le 6

29 ds = μsd + σsdw since W = ρw + 1 ρ W we have dw = ρdw + 1 ρ dw and so ds = μsd + σsρdw + σs 1 ρ dw dw = μsd + S σρ, σ 1 ρ dw 3.6 Derivaion of S : Le F = lns Applying Io s lemma Inegraing from o : S S S 1 = μ σ d + σdw df = + μs σ S d + σs dw 1 F ( ) F() = μ σ d+ σdw S 1 ln = μ σ + σw, W = S 1 S = S exp { μ σ } + σw Consider also he asse S which canno be raded, bu serves as he underlying asse for a + European call opion wih price processc, where C = ( S K) and K > denoes he srike price. We have: 1 S S X X W = exp( ), where = ( μ σ ) + σ Our firs objecive is o find C if only he bond and he surrogae asse can be raded. T T 7

30 3.7 Theorem (Hubalek and Schachermayer) For any numberc (, ), here exiss a probabiliy measure Q M ( S) such ha rt + c = e EQ[( S T K) ]. Hence he exisence of he risk neural probabiliy Q makes he financial marke{( B ),( S ),( C )} T arbirage-free, where C = e E [( S K) F ] rt ( ) + Q T Proof: Choose v. Le s apply Girsanov s heorem o he wo-dimensional process ds μ σ dw d ds = μ + σρ σ 1 ρ dw We wan o change μ o r and μ o v + σ.firs we calculae he funcion u (which in his case reduces o a consan vecor) used o consruc he Radon-Nikodym derivaive in Girsanov s heorem: 1 σ μ r u = σρ σ 1 ρ μ v + σ 1 μ r σ = ρ 1 σ μ v σ 1 ρ σ 1 ρ μ r σ λ = = ρ( r μ) μ v σ λ + σ 1 ρ σ 1 ρ We define he new measure Q M ( S) by dq dp = Z v v T, where λ + ( λ ) Z = exp λw + λ W,, T 8

31 The new wo-dimensional Wiener process (wih independen componens) under W W λ λ Qv is The processes X, X are now given by σ X = r + σ( W λ) X = v + σ ρ( W λ) + 1 ρ ( W λ ) In paricular he random variable U = ρ( W λ) + 1 ρ ( W λ ) is Gaussian wih ( v) T mean and variance T under he new measureq. Hence + X T + Ev () = E ( S T K) = E ( Se K) ( v ) vt + σut + = E ( S e K) vt + σ T z + = ( Se K) ϕ( zdz ) Where ϕ is he PDF for N [,1]. Furhermore where for a fixed v. vt + σ T z E( v) = ( S e K) ϕ( z) dz z v zv K ln vt S = σ T Now for v variable in, i is clear ha lim Ev ( ) = and lim Ev ( ) = v v v Deducions: Since lim z v v ± =, i follows ha and K a lim K ϕ( z) dz v ± = z v a - 9

32 σ Tz a posive consan a lim e ϕ( z) dz v ± = zv a- This, combined wih he fac ha yields he required limis. lim Ev ( ) lim Se e σ T z ϕ( zdz ), v ± vt = v ± zv Now i suffices o observe ha he funcion v E( v) is coninuous and use he inermediae value heorem. To see coninuiy considerv 1 v. Then ( vt 1 + σ Tz vt+ σ Tz ) ( ) Ev ( ) Ev ( ) Se K Se K ϕ( zdz ) ( vt + σ Tz vt 1 + σ Tz ) ( ) + + = Se K Se K ϕ( zdz ) vt Tz vt 1 Tz Se + + σ Se ϕ( zdz ) σ 1 ( ) = e e S e ϕ( z) dz vt vt σ Tz Noe ha we have used he fac ha if f is a real-valued funcion hen f ( v ) f ( v ) f( v ) f( v ) In summary, we can deduce from he heorem ha, when hings are done mechanically, ha is by using only he non-arbirage principles; hen nohing can be said abou he price c of a real opion even if here exis a surrogae which is no perfecly correlaed wih he underlying real asse. Therefore he non-arbirage principle alone is insufficien for pricing of real opions even if he underlying real and surrogae are highly correlaed. 3

33 CHAPTER 4 Alernaive Pricing 4.1 Overview This chaper basically deals wih he various suggesed mehods based upon he following crieria: If here exiss a surrogae whose price process is highly bu no perfecly correlaed wih he price process of he underlying real asse. If here exiss a surrogae whose price process is perfecly correlaed wih he price process of he underlying real asse. 4. Basic Definiions S A porfolio or rading sraegy (see e.g. Björk (1998)) is and F ( ) -adaped n- dimensional process f = ( f1,..., f n ), where fi ( ) denoes he amoun of unis (or shares) of he j h asse conribues o he value of he porfolio a ime T and S ( ) = ( S1( ),..., Sn ( )) is an n-dimensional he price process. If he purchase of a new porfolio, as well as all consumpion, mus be financed solely by selling asses already in he porfolio hen i is called a self financing porfolio. Sharpe index-is an index which is used o measure he performance of a given porfolio over a predefined ime horizon and aking ino accoun he risk of he porfolio. Porfolio local Reurn( μ) Risk free Reurn( r) Sharpe = Volailiy( σ ) which is also called he marke price of risk. A se n Ω is convex if,, xy Ω β x+ (1 β) y Ω β [,1] A real valued funcion f is convex on he convex se Ω if he condiion below is me: f ( β x+ (1 β) y) β f( x) + (1 β) f( x) 31

34 for all, xy Ω and for each [,1] β. A predicable process f () for T is called an admissible rading sraegy for he asse S if he sochasic inegral f () ds () for T is an L - bounded maringale (see e.g. Björk (1998)). T Leφ() = β + f() ds() which is he oucome of a rading sraegy saring wih an iniial capial β a ime = and subsequenly holding f () unis of he asse S a ime (see e.g. Hubalek-Schachermayer (1)). 4.3 High Correlaion (Non -Perfec Correlaion) Under his assumpion of high correlaion beween S and S i is very emping o employ he regular financial opions heory, which could lead o severe mispricing (see e.g. Hubalek and Schachermayer (1)). Hence Hubalek and Schachermayer proposed ha one has o resor o rading sraegies relaed o minimizing he variance of he hedging error Minimizaion of he Variance of he Hedging Error Se-up (Hubalek-Schachermayer) Hubalek-Schachermayer concluded ha, if he real asse S is no perfecly correlaed wih he financial asse S, hen we canno perfecly replicae an opion on S by only rading in S and he risk-free asse (for simplificaion assume ha r = ).Therefore,hey suggesed ha one has o adop a new rading sraegy on he surrogae S so ha he resul is prey close o he payoff funcion max( S K,).The difference beween he new and he old rading sraegies is called he hedging error ε ( T ) for which we seek o minimize i s variance (i.e. var [ ( T )] ε ) We assume ha he srike price, he curren price of he surrogae and he real asse are he same and equal o one (i.e. K = S = S = 1); he risk-neural ineres rae, he expeced 3

35 rae of reurn of he financial asse and he real asse are he same and equal o zero (i.e. r = μ = μ = ); he volailiy of he financial asse and he real asse are he same and equal o one (i.e. σ = σ = 1). The above assumpions were made by Hubalek- Schachermayer purely for he sake of clariy. Hence, from he above assumpions he Io s process ha governs he price process of he financial asse is given by; ds( ) S( ) dw ( ) = ( ) Similarly for he real asse; ( ) ds( ) = S( ) dw ( ) For our new rading sraegy, we look for a predicable process f ( ) for T which is an admissible rading sraegy for he surrogae S. In view of he admissibiliy propery of he predicable process f ( ) for T and he equaion( ); we can now comforably wrie ds( τ ) f( τ) ds( τ) = f( τ) S( τ) S( τ ) = f( τ ) S( τ) dw( τ) On he basis of he above sochasic inegral and he assumpions; Hubalek and Schachermayer formulaed he minimizaion problem below: T minvar[( ST + ( ) K) ( x+ f() ds ())], where he minimum is aken over all x and all admissible rading sraegies f. The soluion o he minimizaion problem is he pair ( xˆ, f ˆ) such ha T xˆ + ˆ f() ds() minimizes he problem. Furhermore, ˆx is obained from he sandard Black-Scholes explici pricing formula of he rading sraegy ( ST ( ) K ) + and is given by; xˆ = c= S N( d ) KN( d ) 1 and f ˆ( ) is he opimal sraegy for new rading skills on S which is given by; ˆ S () f = ρ f( S( ), ) S () 33

36 Hubalek and Schachermayer proved ha; he hedging error for an iniial invesmen of ˆx and rading according o he opimal sraegy is given by; Taking he expecaion of he error: ε. T ( T) = 1 ρ f( ) S( ) dw ( ) ( ) T ε( ) = 1 ρ ( ) ( ) ( ) E T E f S dw T 1 ρ E( f()) E( S()) E( dw ()) = = Since EdW ( ( )) = Similarly, aking he variance of he error: T [ ε T ] = ( ρ f S dw ) ( 1 ) ( ) T () () () 1 T = E ρ f SdW E ρ f () SdW () () T = E ( 1 ρ f ( ) S ( ) dw ( ) ) var ( ) var 1 ( ) ( ) ( ) T (1 ρ ) E f ( ) S ( )( dw ( )) = T ( by Io' s isomery) (1 ρ ) E f ( ) = = ( ST + K ) = ρ (1 ) var ( ( ) ) The explici formula for he random variable var (( ST ( ) K ) + ) Where ( ) ( ) S () d is given by; var ( ST ( ) K) = E ( ST ( ) K) E ( ST ( ) K) d d ( ) T e SN d SKN d1 K N d SN d1 KN d = + ln( S K) + 3T = T 1 = ln( ) + ( μ+ σ ) S K T σ T 34

37 d = d σ T 1 and N denoes he univariae sandard normal cumulaive disribuion funcion We look a he relaionship beween correlaion [ 1,1] ρ and he var [ ( T )] ε graphically using our previous assumpions: σ = σ = 1, K = S = S = 1, r = μ = μ = wih T = 1. Therefore d ln(1) + (3 ) = 1 = 3 d 1 ln(1) + ( + (1 )) = 1 = 1 d = = 1 Subsiuing he values of, 1, d d d ino he explici formula of var (( ST ( ) K ) + ), we obain Hence ( ST + K ) = en N + N[ ] ( N N[ ]) = en [ 3 ] N [ 1 ] + 1 N [ 1 ] ( N [ 1 ] 1) var ( ( ) ) [ 3] [ 1] 4 [ 1] = en 3 N N 1 4N 1 + 4N = en + N N = [ ε T ] ( ρ ) var ( ) = The able and graph below show he relaionship beween he variance of he hedging error var [ ε ( T )] and correlaion ρ [ 1,1] ρ [ T ] var ε ( )

38 1.4 Variance of he Hedging Error agains he Correlaion 1. 1 Variance of he Hedging Error Correlaion The above graph is a parabola and a concave funcion. We can deduce from he above Figure ha;. When ρ =, ha is; here is no correlaion beween he financial asse S and he underlying real asse S.The variance of he hedging error var [ ( T )] ε aains is maximum value a his poin. Hence, using he financial asse S as a surrogae would lead o a severe mispricing. When < ρ < 1, he variance of he hedging error decreases. Hence using S as surrogae for values of ρ which are prey close o 1, he resuls of our new rading sraegy would be prey close o being opimal. Therefore he smaller he variance of he hedging error var [ ε ( T )] opion wrien on he underlying real asse. when 1 ρ = he variance of he hedging error var [ ( T )], he leas he danger of mispricing of he ε is zero, so he financial asse S and he underlying real asse S are perfecly correlaed. In his case one could apply he regular financial opions pricing heory. Therefore any raional invesor would only adop his rading sraegy for values of ρ which are very close o 1 ha is if he surrogae and he underlying real asse are very highly correlaed. 36

39 Noe ha as 1 ρ, var [ ( T )] [ ε T ] = ( ρ ) var ( ) ε ends o zero asympoically in1 ρ : ( ρ )( ρ ) ( ρ ) = Imiaion or Naive Sraegy Hubalek and Schachermayer considered anoher mehod called imiaion (or naive) sraegy. This mehod simply replaces he real asse S wih he raded asse S.The imiaion sraegy replicaes he opion wrien on he raded asse S, which is given by ( ) T max ST ( ) K, = cs (, T) + f( S ( )) ds ( ) ( ) Since he price process of he real asse S is driven by managerial decisions and flexibiliies as posed o he price process of he raded asse S which is purely driven by marke forces, he mehod has several drawbacks which may lead o severe mispricing. Hence when applying his mehod one has o bear in mind ha; because of he sriking differences beween he price processes of he real asse and he raded asse, his mehod would normally fail o be opimal. So more effors have o be made o quanify how far is he oucome from being opimal. Moreover, from ( ), he hedging error ε im ( T ) is; ( ) ( ) U U ( e 1) ( e 1) ε ( T) = max S( T) K, max S( T) K, im = based up he assumpions: μ = μ = r =, σ = σ = 1, S = S = K = T made by Hubalek and Schachermayer which is purely on he grounds of simpliciy. ( UU, ) is a bivariae normal random variable wih mean.5 and uni variance. 37

40 Moreover, Hubalek and Schachermayer came ou wih an explici formula for he variance of he Imiaion Sraegy var [ ( )] as [ ε T ] = im ( f f 1) var ( ) f = en 3 3N 1+ 1 ε using he bivariae Esscher ransformaion im T ρ f1 = e M ρ +, ρ+, ρ M, ρ, ρ + M,, ρ ρ [ 1,1] and M represen he correlaion and he bivariae sandard normal cumulaive disribuion funcion respecively. The symbol N denoes he univariae sandard normal cumulaive disribuion funcion. Using he above formula, Schachermayer proved ha, as ρ 1, he variance of he hedging error of he imiaion sraegy var [ ( )] ε ends o zero asympoically in 1 ρ, im T [ ε T im ] en [ ρ ] 5.734[ 1 ρ] var ( ) 3 1 As ρ 1 he variance of he naïve rading sraegy is approximaely wice he variance of he hedging error of he minimal sraegy. Hence any raional invesor would always prefer he rading sraegy relaing o he minimizaion of he variance of he hedging error o he imiaion sraegy. The graph below shows a comparison beween he asympoic behaviour of var [ ε ( T )] and var [ ( )] ε as ρ 1. The graph is of imporance only for values of ρ prey close im T o one bu makes no meaning for values of ρ prey far from one. 38

41 1 1 Comparison of he Asympoic behaviours of he Variances of he Hedging Error of he above wo rading sraegies as Correlaion approaches 1 imiaion opimal Hence, from he diagram; as ρ 1 he variance of he hedging error of boh rading sraegies urns linearly o zero in( 1 ρ ). The gradien of he imiaion sraegy is approximaely as wice as he gradien of he rading sraegy relaing o he minimizaion of he variance of he hedging. Hence one could deduce ha (see e.g. Hubalek and Schachermayer (1)); he minimum price of being naive is approximaely as wice as being a bi wiser (for insance minimal variance). 39

42 References [ 1 ] Marha Amram and Nalin Kulailaka, Real Opions-managing Sraegic Invesmen in an Uncerain world, Harvard Business School Press, 1999 Tomas Björk. Arbirage Theory in Coninuous Time, Oxford Universiy Press, New York F. Peer Boer, The Real Opions Soluions-Finding Toal Value in a High-Risk World, John Wiley & Sons, Inc., 4 Lawrence Galiz, Financial Engineering-Tools and echniques o Manage Financial Risk, Campbell R. Harvey, Idenifying Real Opions, Lecure Noes 1999, Fuqua School of Business, Duke Universiy 6 F. Hubalek, W. Schachermayer, The Limiaions of No-arbirage Argumens for Real Opions, Inernaional Journal of heoreical and Applied Finance Vol.4, No. (1) Maciej Klimek, Financial Derivaives Lecure Noes, 6 Deparmen of Mahemaics Uppsala Universiy. 8 David G. Luenberger, Invesmen Science, Oxford Universiy Press, 1998 [ 9 ] Yuh-Dauh Lyuu, Financial Engineering-Principles, Mahemaics, Algorihms, Cambridge Universiy Press, 1 Michael J.Mauboussin, Ge Real-Using Real Opions in Securiy Analysis. [ 11 ] Marek Musiela and Marek Rukowski, Maringale Mehods in Financial Modeling, Springer Bern Oksendal, Sochasic Differenial Equaions-An Inroducion Wih Applicaions, Springer 5 13 Jonas Persson, Accurae finie Difference Mehods for Opion Pricing, PhD Thesis Deparmen of Informaion Technology Uppsala Universiy 14 Kevin Rubash-A Sudy of Opion Pricing Models, Lecure Noes, Foser College of Business Adminisraion, Bradley Universiy 15 Global Derivaives, hp:// [ 16 ] Wikipedia, hp://en.wikipedia.org/wiki/liquidiy 4

43 41