Risk-Neural Probabiliies Explained Nicolas Gisiger MAS Finance UZH ETHZ, CEMS MIM, M.A. HSG E-Mail: nicolas.s.gisiger @ alumni.ehz.ch Absrac All oo ofen, he concep of risk-neural probabiliies in mahemaical finance is poorly explained, and misleading saemens are made. The aim of his paper is o provide an inuiive undersanding of risk-neural probabiliies, and o explain in an easily accessible manner how hey can be used for arbirage-free asse pricing. The paper is mean as a sepping-sone o furher reading for he beginning graduae suden in finance. Version: 1 h of Ocober 21; Firs working paper version: 22 nd of April 29 Acknowledgmen: I am graeful o Jakob Gisiger and Sephen McKeon for valuable feedback on earlier versions of his paper. 1 Elecronic copy available a: hp://ssrn.com/absrac=139539
Genlemen, you are now abou o embark on a course of sudies which will occupy you for wo years. Togeher, hey form a noble advenure. Bu I would like o remind you of an imporan poin. Nohing ha you will learn in he course of your sudies will be of he slighes possible use o you in afer life, save only his, ha if you work hard and inelligenly you should be able o deec when a man is alking ro, and ha, in my view, is he main, if no he sole, purpose of educaion. - John Alexander Smih, Professor of Moral Philosophy, Oxford Universiy, 1914 1. Inroducion Mahemaical finance is concerned wih he pricing of redundan securiies. A securiy is called redundan if is payoff can be mached by holding oher securiies ha already exis in he marke, so-called primary securiies. The process of creaing a porfolio in order o mach he payoff of a redundan securiy is called replicaion. The redundan securiy and he replicaion porfolio will have exacly he same payoff. Two posiions wih he same payoff mus also have an idenical marke value. If he value was differen, hen his difference could be locked-in as a risk-free profi by engaging in arbirage. Arbirage means o buy he cheaper posiion and o sell he more expensive posiion. A profi is made immediaely and no risky exposure is lef for he fuure since he payoffs of he wo posiions will precisely cancel each oher. Mahemaical finance ries o esablish precise relaionships beween differen securiies by assuming ha arbirage aciviies do no exis. I is abou relaive pricing, i.e. he price of a securiy is always expressed in erms of oher securiies. The prices of he primary securiies hemselves are assumed o be given by he markes and are no explained by mahemaical finance. Financial economics, on he oher hand is a wider field, i ries o explain he pricing of he primary securiies as well, via conceps such as endowmens, preference funcions, ec. Mahemaical finance is herefore a subfield of financial economics. Mahemaical finance makes in is effors exensive use of he risk-neural probabiliy concep. This concep is so widely used, ha an inuiive undersanding of i should no be avoided. In order o creae his inuiion and allow for a deeper undersanding, we have o sar exploring he concep from a financial economics perspecive. Once we undersand he economic inerpreaion of risk-neural 2 Elecronic copy available a: hp://ssrn.com/absrac=139539
probabiliies, however, we will accep ha he prices of he primary securiies are deermined by he marke, and we will proceed wih relaive pricing. Secions 2.1 2.6 inroduce all he basic noions in a one-period model. The relaionship beween arbirage-freeness and he uniqueness of sae prices is explained. [Noe: all noions will be inroduced sep-by-sep hroughou he paper.] Risk-neural probabiliies are defined in erms of sae prices, and ineres raes are inroduced. Secion 2.7 moves o a muli-period model, which allows secion 2.8 o show ha every normalised expeced asse price pah is a maringale, and he implicaions hereof. Secions 2.9 2.1 inroduce geomeric Brownian moion as a coninuous-ime price process example, and ouline how an arbirage-free pricing formula can be obained by moving from a variance esimae o he risk-neural probabiliy disribuion, and from here o a sae price disribuion. The main resul is ha he drif componen of he original geomeric Brownian moion is no par of he final pricing equaion, bu subsiued wih he risk free rae; his is of significan help when rying o calculae he arbirage-free price of a replicable asse. 2. Risk-neural probabiliies explained 2.1 Basic framework A very simple framework is sufficien o undersand he concep of risk-neural probabiliies. Imagine an economy which is in a known sae a ime, and which can move o a number of possible, muually exclusive saes a ime 1. There is only ime and ime 1. For example, imagine an economy which is now, a ime, in sae #3 (where he number 3 indicaes he level of economic aciviy) and which can move o sae #1, 2, 3, 4 or 5 a ime 1. Sae #5 represens he highes amoun of economic aciviy, and #1 he lowes. The economy can only be in one of hese five saes a ime 1, bu i is unknown a ime, which sae i will urn ou o be. Imagine ha here exiss a homogenous view on he probabiliy for hese saes o happen. We herefore have five probabiliies, one for each sae. 1 The probabiliies sum up o 1, because we know ha a ime 1 he economy mus be in one of hese five saes. This is our iniial framework. I is easy o generalise he framework o [n] possible saes, bu his would no add any clariy o our explanaion. Five saes are easier o visualise. We firs assume ha ineres raes are zero, allowing ineremporal ransfer of wealh a no cos or benefi in dollar erms. This assumpion will laer be relaxed. Figure 1 illusraes our basic framework. 1 These are no risk-neural probabiliies, bu real probabiliies. 3
p(5) p(4) Sae #5 Sae #4 Sae #3 p(3) p(2) p(1) Sae #3 Sae #2 Sae #1 Time p( i) = 1 Time 1 Figure 1 Sae space 5 i= 1 2.2 Arrow securiies Le us define five ypes of securiies in his framework. For each sae, we have one securiy wih a coningen payoff of $1 if ha paricular sae is reached, and $ payoff oherwise. This is a so-called Arrow securiy. Figure 2 illusraes, as an example, he payoff of he Arrow securiy coningen on reaching sae #4. Sae #5 Sae #4 $1 payoff Sae #3 Sae #3 Sae #2 Sae #1. Time Time 1 Payoff a ime 1 Figure 2 An Arrow securiy 4
Therefore, if we hold all five Arrow securiies a ime 1, we are sure o receive a payoff of $1, since exacly one of he securiies will have a payoff coningen on he reached sae, and all he ohers will expire worhless. The value of a porfolio of all five securiies is $1 a ime 1. We presume he exisence of a bank accoun wih overdraf possibiliy. Since we assume ineres rae for ha bank accoun o be % in equilibrium, he arbirage-free price of he whole Arrow porfolio mus also be $1 a ime. If he price of he porfolio was more han $1, say $(1+d), hen we could sell such a porfolio in he marke, hereby receiving $(1+d), and keep a sure profi of $d, since we will have o pay ou exacly $1 a ime 1. If he price was lower han $1, say $(1-d), we could buy such a porfolio a ime, 2 hereby again locking in a sure profi of $d a ime 1. By assuming ha he equilibrium ineres rae is %, hese arbirage aciviies will always drive he price of he complee se of Arrow securiies owards $1. Wha can we say abou he price of each individual Arrow securiy? Each price will be deermined by he supply and demand in he marke. Relevan deerminans of he supply and demand are: he preferences of he marke paricipans wih respec o holding money in one sae versus anoher a ime 1, he preferences wih respec o holding money a ime versus ime 1, and he esimaed probabiliy of a sae acually occurring a ime 1. If he paricipans deem $1 o be more valuable in sae #1 han in sae #5, 3 and boh saes are equally likely o occur, hen he price of securiy #1 will be higher han he price of securiy #5. If he wo prices were he same, hen he paricipans would ry o sell some amoun of securiy #5 and buy securiy #1, because gaining some possible amoun in sae #1 would be worh more han giving up he same possible amoun in sae #5. This will push he marke o price hese wo securiies differenly unil he marke is in equilibrium, i.e. quaniy demanded of each securiy is equal o is quaniy offered. If here was no preponderance of he value of dollars in one sae over he oher, bu he probabiliies of he wo saes were differen, he same process would occur. If we do no care wheher we have a possible dollar more in sae #5 by giving up one possible dollar in sae #4, and vice versa, bu he probabiliy of sae #4 occurring was wice as high as sae #5, hen we would be willing o sell some amoun of securiy #5 o ge some more of securiy #4. I is he same idea as doubling he chances of winning in he loery wihou incurring any addiional cos. The perceived probabiliy of each sae will herefore posiively impac he price of is linked Arrow securiy a ime. Finally, by assuming ha ineres raes are zero, we are effecively implying ha marke paricipans have no preference of having $1 a ime over ime 1, i.e. hey are assumed o be indifferen wih respec o ime. 2 Wih funding from he bank accoun 3 I would no be surprising o value $1 more in imes of a recession han in imes of rapid economic expansion. 5
All we wan o keep in mind a his poin is ha (1) he arbirage-free price of he complee collecion of Arrow securiies a ime is $1, ha (2) an Arrow securiy can command a premium over anoher securiy in line wih he preferences of he marke paricipans, and ha (3) he more likely an Arrow securiy is o have a posiive payoff, he higher is price a ime. However, we do no have o calculae he individual prices bu can simply observe hem in he marke, and specify a pricing vecor a wih he observed marke prices. The price of securiy #1 a ime is a(1), he price of securiy #2 is a(2), ec. We now move on o see if we can say more abou a marke where redundan securiies are raded. A redundan securiy is a claim ha has a payoff a ime 1 which can be replicaed by holding a linear combinaion of differen Arrow securiies. We need o disinguish complee markes from incomplee markes, since we have sronger resuls for complee markes. 2.3 Complee marke A complee marke is a marke where all Arrow securiies can be raded. I does no maer wheher here are any real Arrow securiies or wheher hey can be consruced (i.e. replicaed via a linear combinaion of oher raded securiies) 4. As soon as we can consruc every possible Arrow securiy, he marke is complee. Remember ha we know he price of each Arrow securiy, because we have observed all of hem in he marke. We can now define a paricular redundan securiy. A securiy is always defined by is, possibly sochasic, payoff. By sochasic, we mean ha he payoff migh depend on he sae ha he economy reaches a ime 1. A risk-free zero coupon bond, for example, has a sure payoff of is noional [$n] a ime 1, regardless of he economy s sae. In order o replicae he bond, we need o buy [n]-imes he whole se of Arrow securiies, as illusraed in figure 3. Since we already know ha one whole se of Arrow securiies coss $1 a ime, he replicaion mus cos $n. This is he only arbirage free price of he bond. If he bond coss less han $n, say $(nd), we could lock in a profi of $d by buying he bond and shoring [n]-imes he whole se of Arrow securiies. If he bond cos more han $n, say $(n+d), we could lock in a profi of $d by selling he bond and buying [n]-imes he whole se of Arrow securiies. 5 4 Such replicaion migh involve long, as well as shor posiions, i.e. besides buying, we migh have o sell some borrowed securiies. 5 In order o sell somehing we do no already own, we have o borrow i firs. We ener ino a repo ransacion (repurchase agreemen) wih anoher pary, i.e. we borrow he bond a ime and hand i back a ime 1, including is payoff. This allows us o sell he borrowed bond a ime, bu we mus buy i back in he marke a ime 1, including is payoff. 6
Sae #5 Sae #4 Sae #3 Sae #3 $1 payoff Sae #2 Sae #1. Time Time 1 Payoff a ime 1 Figure 3 A risk-free zero coupon bond As a second example, a risky zero coupon bond has a sochasic payoff. Le us say, for example, ha he risky zero coupon bond pays $1 in each sae, excep in sae #1 where i pays nohing due o defaul. This is illusraed in figure 4. Sae #5 Sae #3 Sae #4 Sae #3 $1 payoff Sae #2 Sae #1. Time Time 1 Payoff a ime 1 Figure 4 A risky zero coupon bond 7
The only arbirage free price of his bond is he sum of he prices of securiies #2, 3, 4 and 5. If he bond was cheaper, say by $d, hen we could buy i, sell Arrow securiies #2, 3, 4 and 5, hereby locking in a sure profi of $d, because he long and shor posiions exacly cover each oher a ime 1, no maer in which sae he economy will urn ou o be. Similarly, if he bond was more expensive, hen we would do he reverse operaion; locking in he profi already a ime by maching he payoff of he long and shor posiions a ime 1. Such arbirage opporuniies offer free profi; our assumpion here is ha such opporuniies vanish because heir pursui pushes he marke price owards he arbirage-free level. The idea seems very sraighforward. Any possible payoff can be replicaed via a linear combinaion of Arrow securiies; and his paricular replicaion porfolio imposes a unique arbirage-free price. The arbirage-free price is equal o he sum of all payoff-weighed Arrow securiy prices. If θ is he arbiragefree price of a securiy a ime, and X (i) is he payoff of he securiy when sae i is reached, hen we have: 5 θ = a( i) X ( i) (1) i= 1 Le us assume he following observed Arrow securiy prices: a(1) = 3 cens, a(2) = 25 cens, a(3) = 2 cens, a(4) = 15 cens, and a(5) = 1 cens, i.e. a securiy which pays $1 only if sae #5 is reached a ime 1, for example, coss 1 cens a ime. Any payoff srucure can be weighed wih hese prices in order o ge he arbiragefree price of he securiy a ime. For example, he risky zero coupon bond ha we presened before, has accordingly a unique arbirage-free price of 7 cens, since i is composed of Arrow securiies #2, 3, 4 and 5. We can replace he cens in he Arrow securiy prices wih [%] if he payoff srucure X(i) is already expressed in dollar erms. These Arrow securiy prices are so-called risk-neural probabiliies; hey are exacly he same hing. I is surprising ha his is no always poined ou very clearly. Noe ha we are currenly sill assuming ineres raes o be zero. Why should hese prices be called probabiliies? For a mahemaician, a probabiliy measure fulfils he crieria ha he sum of all probabiliies (of disjoin evens) mus equal 1, and ha he probabiliy of a paricular even canno be negaive. Since he sysem of Arrow securiy prices oulined so far fulfils hese condiions, i can echnically be called a probabiliy measure. 6 6 A probabiliy measure is simply a mapping of oucomes o cerain probabiliies. In our example, each sae of he economy has an assigned probabiliy. 8
We have: 5 i= 1 a ( i) = 1 (2) and a( i), i 7 (3) There are frequen saemens abou risk neural probabiliies which we can now easily idenify as misconcepions. A risk-neural probabiliy is NOT he real probabiliy of an even happening, bu should raher be inerpreed as a price. I is NOT INDEPENDENT of he real even probabiliy, since he probabiliy posiively impacs he price of a sae-coningen payoff. I does NOT assume ha we are in a risk-free world, quie he opposie, he world is assumed o have an unpredicable fuure. I does NOT assume ha marke paricipans are risk-neural, hey can price differen sae-coningen payoffs in line wih heir risk preferences, be hey risk averse, risk neural or risk seeking. I is simply he only arbirage free price in a complee marke. Risk-neural probabiliies (i.e. Arrow securiy prices or sae prices) 8 simply enforce linear pricing consisency beween all raded securiies wih regards o heir payoff componens. The price for receiving $1 if sae #3 occurs, for example, canno be incorporaed in a securiy s price a 25 cens in one case, and a 2 cens in anoher case. If wo such securiies would coexis, hen his price difference could be singled-ou by sripping boh securiies off all oher sochasic payoffs; 9 and hen rade hese wo sochasic payoffs (coningen on reaching sae #3) agains each oher. Arbirage aciviies are assumed o eliminae such pricing discrepancies. 2.4 Equivalen probabiliy measures The risk-neural probabiliy measure is equivalen o he real probabiliy measure. A mahemaician alks of he equivalence of wo probabiliy measures if boh of hem agree on he possible and he impossible oucomes. If one probabiliy measure allocaes a posiive probabiliy o a cerain oucome (meaning ha i is a possible oucome), hen he oher probabiliy measure also allocaes a posiive probabiliy o he same oucome. However, if he firs measure allocaes zero probabiliy o an oucome (meaning ha i is impossible), hen he second measure also allocaes zero probabiliy. If hese wo condiions are saisfied, he wo probabiliy measures are called equivalen. This propery of probabiliy measures will be imporan when we ry o derive a pricing formula laer in his paper. We discuss i here already, 7 i means for all i. 8 We will use Arrow securiy price and sae price inerchangeably. 9 Wih he help of Arrow securiies. 9
since, if we hink in erms of sae prices, he equivalence of he wo probabiliy measures is easy o see. If an oucome is impossible (according o he real probabiliy measure) hen he corresponding Arrow securiy canno cos anyhing (risk-neural probabiliy). Similarly, if an oucome is possible (according o he real probabiliy measure), hen he price of he corresponding Arrow securiy mus be more han zero (risk-neural probabiliy); oherwise we could ge a probabilisic payoff for free. Arbirage aciviies are assumed o push such a free securiy ino a posiive price range. 2.5 Incomplee marke In an incomplee marke, no all Arrow securiies can be consruced. A securiy which would be redundan in a complee marke migh no be so in an incomplee marke. For example, we migh no have enough raded securiies in our marke in order o rade or consruc Arrow securiies #1 and 2. All we can say is ha a porfolio which conains securiies #1 and 2 mus have an arbirage free price of [$1 - a(3) - a(4) - a(5)], because we know ha he arbirage-free price of all Arrow securiies ogeher sums up o $1. Now le us assume ha a new securiy is inroduced ino he marke which pays $1 if sae #2, 3 or 4 is reached, bu nohing in sae #1 or 5. The payoff of his securiy is illusraed in figure 5. Sae #5 Sae #4 Sae #3 Sae #3 $1 payoff Sae #2 Sae #1. Time Time 1 Payoff a ime 1 Figure 5 A securiy wih sochasic payoff 1
This securiy is no redundan, because we canno srip-off all payoffs wih a hedging porfolio. 1 The price of his securiy mus be higher han [a(3) + a(4)], and we know ha i mus be less han [1 - a(5)]. Any price ha is assumed wihin his range is arbirage-free. This is he same as saying ha he arbirage-free price of Arrow securiy #2 is no unique; which in urn is he same as saying ha he riskneural probabiliy of sae #2 is no uniquely deermined. Several differen riskneural probabiliy measures can be accommodaed by he marke prices wihou leading o any arbirage possibiliies. Once his paricular securiy is rading in he marke and a price is esablished, he marke will be complee in our example. Any oher new securiy would now be redundan and is price sricly deermined by he oher rading securiies. Once we have a price for he securiy inroduced before, we can deermine a(2), 11 and herefore also a(1). 12 We noe ha an incomplee marke seing allows a range of arbirage-free prices for cerain securiies. This is he same as saying ha a range of risk-neural probabiliies exiss wih respec o he saes for which no Arrow securiies exis ye. Arbirage-free pricing offers only limied guidance in such a seing. We will no look ino incomplee markes any furher. 2.6 Relaxing he ineres rae assumpion So far we have assumed ineres raes o be zero. We will now relax his assumpion and observe wha happens o sae prices and risk-neural probabiliies, respecively. Assuming an ineres rae differen from zero implies ha he marke allocaes differen uiliy o possessing $1 now versus laer. $1 a ime can be compounded o [$1 * (1 + r)] a ime 1. Similarly, $1 a ime 1 can be discouned o [$1 / (1 + r)] a ime. This ineres rae is applied o our risk-free bank accoun. We saw ha $1 a ime 1 is he same as he payoff of he complee se of Arrow securiies. The price of he complee se of Arrow securiies a ime mus herefore be equal o [$1 / (1 + r)]; oherwise, one can make a risk-free profi by rading he complee se via funding hrough he bank accoun. We have: 5 ( i) = 1 r a (4) i= 1 1+ 1 The reverse posiion of he replicaion porfolio is called hedging porfolio. By reverse, we mean a long posiion is flipped o a shor posiion, and vice versa. 11 a(2) = [a(2) + a(3) + a(4)] a(3) a(4) 12 a(1) = $1 a(2) a(3) a(4) a(5) 11
This discouning can be exended o all individual Arrow securiies, i.e. he former price of each securiy is divided by [1 + r] o ge he new price when inroducing an ineres rae differen from zero. We assume ha he risk preferences, sae preferences and probabiliy esimaions remain he same; all ha has changed in he marke is ime preference. Of course i could be envisaged ha ime preference is inerrelaed wih oher preferences, and ha herefore sae prices do no only change by he discoun facor, bu go hrough an addiional disorion. All we can really be sure abou is ha he price of he whole se of Arrow securiies is [$1 / (1 + r)] a ime. In any case, we do no need o calculae he individual Arrow securiy prices; hey are deermined for us by he supply and demand in he marke. Wha happened o risk-neural probabiliies along he way? We know ha he sum of he probabiliies of all oucomes mus be equal o 1, if we sill wan o mainain a mahemaical probabiliy concep. This is no he same as [1 / (1 + r)], if r is differen from zero. We mus herefore muliply each sae price wih [1 + r] in order o ge he risk-neural probabiliy q i for each sae. As a sum, hey again add up o 1. We have: () q( i) = a( i) (1 + r), i (5) and 5 i= 1 q ( i) = 1 (6) One migh argue ha his sars o become an arificial consruc; which is indeed rue. A risk-neural probabiliy is simply somehing we define. However, as long as we remember ha risk-neural probabiliies are nohing else han compounded sae prices, we are fine. The reason for doing his consruc is ha here is value in keeping he condiion alive ha he sum of all individual probabiliies adds up o 1. This means we can apply all he mahemaical machinery ha has been developed in relaion wih probabiliy measures. We will see ha his is useful a a laer sage. For now, as long as we remember o reverse he arificial compounding wih a subsequen equivalen discouning, we will always find our way back o sae prices, he rue deerminans of each arbirage-free price. In order o calculae he arbirage-free price of a redundan securiy a ime, we can simply muliply he payoff in each sae wih he corresponding risk-neural probabiliy, sum up all hese producs, and hen discoun his sum back o ime. 5 1 θ = X ( i) q( i) (7) 1+ r i= 1 12
(7) is equivalen o equaion (1). Any price diverging from his calculaion would enable arbirage, and marke forces would push hem o be in line again. In mahemaical erms, we have done nohing else han discouning he payoff expecaion under he risk-neural measure. Formulae (8) and (7) are equivalen. 1 θ = E [ X ] (8) 1 + r We do now have an inuiive undersanding of wha risk-neural probabiliies are; hey have a direc economic inerpreaion as compounded sae prices. Any facor ha is relevan for deermining he supply and demand of an Arrow securiy, and hence is price, has herefore a direc influence on he corresponding risk-neural probabiliy. Any risk-premia implied by marke prices are herefore incorporaed in he risk-neural probabiliies. The frequenly encounered saemen ha he risk-neural probabiliy is independen of he real probabiliy is false. The real probabiliy affecs he sae price, and is hence relevan for he risk-neural probabiliy. Of course, if we already know he sae price, hen here is no need o esimae he real probabiliy for he sake of calculaing he risk-neural probabiliy; as he esimae of he real probabiliy is already incorporaed in he sae price. 13 Thus far we have no explained why we rely on risk-neural probabiliies a all. Why should we calculae an arbirage-free price via risk-neural probabiliies if we already know he sae prices? The answer is ha i migh no be so sraighforward o ge o know he sae prices; in realiy, here is a coninuous range of saes., and herefore an infinie number of Arrow securiies. Hence, we will laer assume ha sae prices are unknown. We will use he connecion beween risk-neural probabiliies and sae prices he oher way round i.e. we will obain sae prices from risk-neural probabiliies for he purpose of arbirage-free pricing. We will be able o use heorems from probabiliy heory in order o obain riskneural probabiliies hrough a differen roue. This is he reason why we consruced an arificial probabiliy measure. Wha follows in he nex half of his paper is a brief overview of how we can calculae risk-neural probabiliies wihou using sae prices. Before deriving an arbirage-free pricing equaion, however, we firs have o gradually move owards a more realisic marke seing, and inroduce some furher elemens. 13 A similar misake is usually made when claiming ha a forward FX rae is independen of he expeced inflaion in he wo currencies, since he arbirage-free forward FX rae is fully deermined by he curren FX rae and he wo ineres raes. Correcly, one would have o say ha he curren ineres raes are affeced by expeced inflaion. Hence, expeced inflaion is relevan for he arbirage-free forward FX rae. However, we do no need o esimae expeced inflaion for he purpose of calculaing he arbirage-free forward FX rae, since his expecaion is already refleced in he currencies ineres raes. 13
2.7 Trading sraegy and dynamic compleeness So far, we have looked a a one period model only, wih ime and 1. Concepually, we can inroduce a furher elemen when moving o a muli period model. This furher elemen is he rading sraegy. In a one period model, all rading is a one-off decision a ime, which canno be changed before mauriy of he securiy a ime 1. Le us assume a simple muli-period model wih imes, 1 and 2, and wo sae branches a each ime kno. Sae #5 Sae #4 Sae #3 Sae #3 Sae #2 Sae #1. Time Time 1 Time 2 Figure 6 Muli-period sae space In his model we are allowed o make rading decisions a ime and a ime 1. A rading sraegy can help in consrucing a paricular Arrow securiy. 14 If we can ener ino a rade a ime, adap he rade a ime 1 wihou incurring any addiional cos or benefi, and be sure of he payoff a ime 2 for each possible sae, hen he iniial price of his fuure sochasic payoff is deermined by he upfron cos of he iniial rade. Any deviaion from his price would lead o arbirage aciviies. Basically, as soon as here is a replicaion sraegy o perfecly mach he payoff of a securiy a ime 2, one can always hedge he posiion, hereby making sure ha here is no exposure lef a ime 2. Price differences can be locked in a ime as a risk-free profi. The possibiliy of inermediae rading does no make maers more complicaed, quie he opposie; i makes i easier o consruc Arrow securiies. However, i is imporan o noe ha he rading sraegy mus be self-financing, i.e. here is no 14 By rading sraegy we are no referring o a sraegy designed o make a profi, bu o a replicaion sraegy where we simply ry o mach a payoff. 14
money coming in or ou before mauriy. I is only abou rebalancing he rading porfolio beween is componens a ime 1 in a value-neural way. 15 If he rading sraegy was no self-financing, hen we could no be sure of he iniial cos o aain he final sochasic oucome, 16 and i would herefore no serve as a poenial price enforcing arbirage vehicle. In he one-period model we required he availabiliy of an Arrow securiy for each final oucome, which in urn requires a leas an equal number of linearly independen securiies in case we need o consruc hese Arrow securiies firs. 17 If we can adap our rade hrough ime, we can poenially consruc such Arrow securiies wih fewer primary securiies. A marke, in which every Arrow securiy is available due o replicaion-enabling rading sraegies, is called dynamically complee. A dynamically complee marke deermines he sae prices and he riskneural measure, i.e. every redundan securiy has a unique arbirage-free price. For example, he Black-Scholes equaion for calculaing he arbirage-free price of a European call opion relies on a rading sraegy whereby he replicaing posiions in he primary asses are coninuously adjused according o he dela hedging rule. Illusraions of such dynamic replicaion sraegies can be looked-up in every book lised in he bibliography. Taleb (1997) should be consuled in order o become aware of he difficulies of dynamic replicaion. For he concep of arbirage-free pricing via risk-neural probabiliies, we acually do no rely on any knowledge abou he exac replicaion sraegy iself. We only rely on he fac ha a replicaion sraegy does exis. Of course, a replicaion sraegy migh no exis as ofen as we would like o assume in pracice. All we wan o keep in mind a his poin is ha replicaion migh involve some acive rading of he primary asses before mauriy of he redundan asse, and no all payoffs can be reached by simply buying and holding a mix of primary securiies from sar unil mauriy. This rading should be self-financing, and he rading rule needs o be clear wih he informaion available a he ime of applicaion. We canno use he acive replicaion sraegy for arbirage reasoning wihou hese condiions. 2.8 Discouned asse prices as maringales In a muli-period model wih imes, 1 and 2 we can wrie an asse price wih a ime subscrip. θ is he price of he asse a ime. θ follows a sochasic process, adoping differen values depending on he sae of he economy a ime. 15 Think of your own sock porfolio. You may decide o sell some amoun of one sock o ge ino anoher one. If you are fully invesed before and afer your rade wihou aking any money ou or injecing any more, hen your rade is self-financing. 16 We do no necessarily know a ime wha he value of he securiy and he value of he replicaion porfolio are a ime 2 (sochasic / uncerain). All we know is ha hey will be of equal value. 17 Linear algebra: o solve for a number of unknowns, we need a leas an equal number of equaions. 15
Obviously, he price a ime 2 is equal o he final payoff: θ 2 = X. I is imporan o disinguish beween an acual price pah and is expeced pah a ime. In realiy, a price will follow a cerain pah as uncerainy wih respec o he payoff-relevan facors is resolved over ime; his is he acual price observed over ime represened by θ. However, we are here no ineresed in he acual pah of he sochasic price process, because we simply canno know wha i will look like beforehand. The only hing we can do is aking expecaions abou he fuure a ime. To perform an expecaion under a probabiliy measure, he measure needs o be specified. We do no claim o be able o fully specify he real probabiliy measure, i.e. he real probabiliy of he price aking a cerain pah. However, le us assume ha he risk-neural probabiliy measure is known. Therefore, we wan o see how he price pah behaves under he risk-neural expecaion operaor, simply in order o discover is mahemaical properies. We can only ake expecaions a ime, because all he uncerainy is sill ahead of us. This can be represened by: E [ ], θ (9) This is simply a case of aking an expecaion of an expecaion, because he price iself is an expecaion [as we saw in equaion (8)]. The quesion is: wha is our expecaion a ime abou our expecaion a some poin in he fuure? Le us sar wih he expeced price a ime : E 1 [ θ ] = E E [ X ] (1) 2 ( 1 + r) We simply replaced he price wihin he expecaion operaor wih equaion (8). Since we are now in a wo period model, we are discouning for wo periods. The discoun facor is sae-independen and can be aken ou of he expecaion operaor. Finally, aking an expecaion of he same expecaion is he same as aking i only once. We herefore obain: E 1 θ = E (11) [ ] ( 1 + r) 2 [ X ] This is acually our arbirage-free pricing equaion (8). We could have wrien his equaion direcly. We have now shown wha happens when aking he same expecaion wice in a row, i is he same as aking i only once. We are now moving o he expeced price a ime 1, where we ge: 16
E 1 θ 1 1 (12) 1+ r [ ] = E E [ X ] Since he price a ime 1 is only one period before he payoff, we discoun for one period only. Again, we can ake he discoun facor ou of he expecaion. Our expecaion a ime of he final payoff, and our expecaion a ime of our expecaion a ime 1 of he final payoff, mus be he same. We do no know a ime wha will happen o he sae of he economy beween ime and 1. This relaionship is known as he law of ieraed expecaions. 18 We herefore obain: E 1 θ 1 E ] (13) 1+ r [ ] = [X Finally, we expec he price a ime 2 o be: E [ ] [ ] E E [ X ] θ (14) 2 = 2 A ime 2, however, we will know wha he oucome of he payoff is. The expecaion of X aken a ime 2 mus be he payoff iself, we can herefore wrie: E [ ] E [ X ] θ (15) 2 = We can clearly recognise ha he expeced price grows a he risk-free rae r wih each period. This is no a surprising paern. In fac, i is enforced by he arbiragefree price of he risk-free zero-coupon bond (or bank accoun erms), which is raded for all mauriies and grows a he risk-free rae r, and he way we defined risk-neural probabiliies. Relaive o a sae price, each individual risk-neural probabiliy is inflaed unil mauriy of he payoff by he risk-free rae, simply because his growh rae is incorporaed in is definiion [see equaion (5)]. Any expecaion under he risk-neural measure herefore grows a he risk-free rae. Therefore, he expeced price of any asse under he risk-neural measure grows a he risk free rae. Tha is, no he acual price will grow a he risk free rae, bu he expecaion of he price hrough ime, where he expecaion is calculaed a a fixed ime (ime in our example). This arificial drif was necessary o disguise 18 Imagine ha you have an expecaion of he final score of a fooball mach before he game sars. A he same ime you can have an expecaion abou wha your expecaion of he final score will be afer he firs half of he game. When aking hese expecaions a he same ime before he mach, hey coincide. I is only wih passing ime, as he game progresses and uncerainy is resolved, ha hese expecaions can sar o diverge, because hen hey are aken wih a differen se of informaion. The ime subscrip under he expecaion operaor effecively refers o he amoun of informaion ha we have available. When aking several expecaions, we always have o work wih he expecaion ha is based on he mos resricive se of informaion, which is obviously always he amoun of informaion a an earlier poin in ime. 17
he sae prices as a probabiliy measure. We emphasize ha we are no saying anyhing abou how asse prices are acually developing over ime as uncerainy is being resolved; all we know are he prices a ime. We have now discovered an imporan mahemaical propery of our arificial probabiliy measure. We can ge rid off his arificial expeced drif only by going hrough he reverse operaion. By doing so, every discouned expeced fuure price 19 becomes equal o he price a ime. We have: 1 ( 1+ r) E [ ] = θ, θ (16) We call his expression he normalised expeced price. By going hrough his normalisaion (i.e. discouning), we have achieved ha he whole expression exhibis no more drif. By removing he drif, we have shown ha he normalised expeced asse price process is a maringale. A maringale is a sochasic process wihou expeced drif. Since we can form a maringale ou of every normalised expeced asse price pah wih he help of risk-neural probabiliies, he risk-neural probabiliy measure is also called a maringale measure. The reason for ransforming he expeced price pah ino a maringale is o open he door for maringale heory. Any mahemaical machinery and heories ha have been developed for maringales can now be applied o arbirage-free asse pricing. The maringale propery is very useful for our purpose of calculaing arbirage-free asse prices. We can simply urn equaion (16) around o calculae he arbiragefree price a ime. This is he muli-period equivalen of equaion (8): 1 θ = E (17) ( 1 + r) [ θ ], In he nex secion, we will move o a framework where asse prices follow a coninuous process. I can be shown mahemaically ha any coninuous squareinegrable maringale can be represened by a Brownian moion unfolding a a cerain speed [Björk, 24]. A Brownian moion is a sochasic process where each incremen sems from a normal disribuion. Hence, he ransformaion of asse prices ino maringales wih help of he risk-neural measure effecively enables us o work wih he normal disribuion; bu only under he resricive assumpion of finie variance. 2 19 From here on, whenever we alk of an expecaion, we mean he expecaion calculaed wih risk-neural probabiliies, i.e. aking a sum by weighing wih compounded sae prices. 2 Square-inegrable means ha he variance of he process is finie. However, his assumpion migh no be accurae in realiy, which would lead o a breakdown of our pricing framework. 18
2.9 Coninuous-ime model We now move on o a seing where he price of a securiy is coninuous in ime and adops values on a coninuous range from o +. Securiy prices now follow coninuous sochasic processes. Under he one-period or muli-period model we were assuming a discree probabiliy disribuion for he price and final payoff. All ha we are doing now is move o coninuous probabiliy disribuions of he price and payoff. The probabiliy of a price being a a cerain level a a cerain ime can be characerised by a probabiliy densiy funcion. The mahemaics in a coninuous seing are fairly advanced, bu he key conceps developed so far remain all he same. In realiy, price processes, such as sock prices over ime, are no coninuous in any dimension, i.e. hey are disconinuous in ime and disconinuous in price. Wha do we mean? Transacions ake place a a specific price, quaniy and ime. A ransacion can herefore be represened by a poin in a hree dimensional space, characerised by price, quaniy and ime. The real price process is a sequence of ransacion poins. There is no connecion beween hese poins. The lines we usually draw hrough all he ransacion poins on a price char are arificial. In realiy, even if here was guaraneed coninuous bid/offer quoing, a marke maker migh sill no be able o execue a he quoed levels. I is herefore advisable o consider only ransacion levels as price realisaions, no quoed levels. We noe ha a coninuous price process happens only in our model seing, no in he real world. How can we specify he probabiliy of an asse price o be a a cerain level a a cerain ime, assuming ha he process is now coninuous? We choose he classic example of he geomeric Brownian moion as a sock price model. The geomeric Brownian moion is governed by he following sochasic differenial equaion (SDE): ds = μ S d + σs dw (18) P P dw sands for he incremen in a Brownian moion under a cerain probabiliy measure P. The incremens of a Brownian moion follow a normal disribuion, and his disribuion is fully specified by is mean E P [ dw ] = E P 2 ( dw ) = [ ] d, and is variance. The parameer σ scales he random shocks from he Brownian moion, μ specifies he deerminisic drif hrough ime. The soluion o he SDE in (18) is given by: 19
S S 1 2 exp μ σ + σw (19) 2 = This can be shown via an applicaion of Io s Lemma. We will here no ge ino he opic of solving SDEs, however, and simply presen he resul. Equaion (19) S specifies our probabiliy disribuion for, as we know ha follows a normal N( W ~, ) Z disribuion:. Hence, he whole exponen follows a normal disribuion, since he oher facors are only shifing and scaling he disribuion in a non-random way: W Z 1 2 = μ σ + σ 2 P W (2) 2 σ ~ N, σ 2 μ (21) Z 2 We will now ry o give an arbirage-free price o a redundan securiy defined by a X payoff funcion ha depends solely on S (a cerain sock price a ime T). As a firs approach, le us ry o obain he payoff expecaion of he redundan securiy under he real probabiliy measure. In order o do ha, we have o sum up (over all possible scenarios) he payoff under each scenario wih is corresponding probabiliy densiy. As we know, normally disribued random variables ake on values on he whole range from o +, our sum is herefore an inegral. We ge he following payoff expecaion: T ZT [ X ( S )] X ( S e ) p( ) + P E T = ω dω (22) where p ( ω) is he normal probabiliy densiy funcion of Z T. Hence, we can calculae he real expecaion of he securiy s payoff a mauriy T, IF we have reliable esimaes of μ and σ. We do know S and T, and here are no furher parameers. However, μ and σ are very difficul o esimae objecively in realiy, especially since we are ineresed in he fuure drif and volailiy. How can we measure somehing ha lies ahead of ime? Moreover, wha we are really ineresed in is he arbirage-free price θ, no he expeced payoff a ime T. Such a payoff expecaion under he real probabiliy measure has no been adjused for any sor of risk premia such as hose incorporaed in he marke prices of he 2
primary securiies. We have no shown ha he above calculaion leads o an arbirage-free price. If we could somehow find a way of ransforming he expecaion from he real probabiliy measure o he risk-neural measure, hen we could simply discoun wih he risk-free rae, and be sure ha he obained price is arbirage-free. Such a probabiliy measure ransformaion is based on he Girsanov Theorem. 21 2.1 Obaining he risk-neural probabiliy measure We have learn so far ha risk-neural probabiliies are obained by compounding Arrow securiy prices. We have in our coninuous-ime example no even made any aemp o replicae he redundan securiy s payoff, hereby finding ou he required combinaion and amouns of Arrow securiies and he required self-financing rebalancing sraegy, which would hen give us he arbirage-free price. Is i possible o find ou he arbirage-free price in any oher way? The answer is yes. The Girsanov Theorem offers an alernaive roue. I shows how we can move a sochasic process driven by a Brownian moion from one probabiliy measure o anoher, equivalen probabiliy measure. Such a probabiliy measure ransformaion is unique, and herefore, if we manage o obain he dynamics under he riskneural measure, we can conver his o a sae price disribuion by simple risk-free discouning. The move from one probabiliy measure o an equivalen measure 22 is done via he W ω, where ω represens a single scenario and Ω Radon-Nikodym derivaive ( ) represens he se of all possible scenarios. ( ) ( ω) ( ω) q ω =, ω Ω p W (23) The densiy q ( ω) is he risk-neural probabiliy densiy wih respec o he sae ω, and p ( ω) is he real probabiliy densiy. The Radon-Nikodym derivaive is he riskneural probabiliy densiy wih respec o he real probabiliy densiy. We can now do he following ransformaion: E ZT [ X ( S )] X ( S e ) p( ω) ( ω) ( ω) + q = dω p T (24) 21 See for example Björk (24) 22 We have seen in secion 2.4 ha risk-neural probabiliies are equivalen o real probabiliies under he absence of arbirage possibiliies. 21
Noe ha, as we change from one probabiliy measure o anoher, he superscrip above he expecaion has changed from P o. This is he risk-neural payoff expecaion a ime. Forunaely, we can easily move from he risk-neural probabiliy densiy o sae price densiy ς ( ω) via simple discouning, jus like in he discree marke seing. In a coninuous seing, raher han hinking in erms of discree Arrow securiy prices, i makes more sense o use sae price densiy, since we would oherwise have o hink in erms of an infinie number of Arrow securiies disribued over a coninuous range. The exisence of a unique riskneural probabiliy measure implies he exisence of a unique sae price measure. 23 ς ( ω) e rt q( ω) = (25) The sae price densiy is he marke s fundamenal pricing funcion wih respec o all ω. In a complee marke, he arbirage-free price of every asse mus be consisen wih he sae price densiy, i.e. we have for every asse: Ω ( ω) ς ( ω) θ = X dω (26) This is he coninuous-range equivalen of equaion (1). The sae price densiy ς ( ω) is simply he equivalen of he Arrow securiy price in a coninuous seing. The Radon-Nikodym derivaive allows us, herefore, o obain he arbirage-free price of an asse. In our example, where he underlying asse follows a geomeric Brownian moion, we have: q( ω) ( ω) [ ( )] + rt rt Z e T θ = e E X ST = X ( S e ) p( ω) dω (27) p So far, we know how o proceed concepually, bu how does q ( ω) in his paricular case look like? The Girsanov Theorem saes ha a Brownian moion under he real probabiliy measure convers o a Brownian moion under an equivalen probabiliy measure PLUS a drif componen. We have: dw P = dw + ϕ d (28) We can plug his expression ino he SDE in (18), and obain he sock price dynamics under he new probabiliy measure: 23 This is based on he Fundamenal Theorem of Asse Pricing. 22
ds ( μ + σϕ ) d + σs dw = S (29) However, we sill know more han his. We know ha he relaive change in S is composed of an expeced drif of r under he risk-neural measure. The deerminisic drif componen arising from he measure change herefore precisely cancels he risk premium in μ. The risk-neural dynamics of he underlying asse are now fully specified wih: ds = S rd + σs dw (3) We can direcly move o hese dynamics wihou ever knowing he specific risk premium. We know ha he risk-premium mus be cancelled, because we know he ransformed drif already beforehand. Afer equaion (27), we are lef wih he final arbirage-free pricing equaion for a redundan asse which is based on an underlying asse following a geomeric Brownian moion: rt + ZT ( S e ) q( ω) θ = e X dω (31) where Z is normally disribued according o he normal densiy q ( ω) : T 2 σ ~ N r T, σ T 2 Z T 2 (32) 2 2 1 1 σ q( ω ) = exp ω r T 2 (33) σ 2πT 2σ T 2 The nice aspec of his pricing approach is ha we no longer need o esimae he parameer μ. This is he whole meri of risk-neural pricing. The qualiy of our arbirage-free pricing now depends on our abiliy o esimae σ. 24 How can we esimae his parameer? The risk-neural probabiliy approach has helped us wofold in his respec. Firsly, we go rid of he drif componen which leaves only one parameer o be esimaed. Secondly, if we have access o he marke prices of oher securiies which depend only on he same volailiy parameer, hanks o our risk-neural pricing formula, 24 Do no forge ha we have inroduced a new parameer which also needs o be esimaed, and his is he risk-free rae r. I is no necessarily clear in pracice wheher such a rae acually exiss and which one i is. Nobody can rule ou he possibiliy of a governmen defaul. 23
hen we can calculae he implied volailiy in hese marke prices, since i is he only unknown parameer. By calibraing our pricing formula wih he implied volailiy from oher marke prices we aim o ge a marke consisen, arbirage-free price. Finally, wheher his can be achieved depends largely on he validiy of he assumpions inheren in our pricing framework. 3. Conclusion 3.1 Puing he pieces ogeher We have gone hrough many seps before finally arriving a an example of an arbirage-free pricing equaion. Here, we wan o briefly summarise he chain of argumens again. 1. If a marke is complee and arbirage-free, hen we have a unique sysem of sae prices. The arbirage-free price of any sochasic payoff is he sum (over all scenarios) of he scenario-payoff weighed wih is sae price. Based on he sae prices, we can define a unique risk-neural probabiliy measure. For his purpose, we need o compound he sae prices wih he risk-free rae. This inroduces a drif (a he risk-free rae) ino any expecaions under he risk-neural measure aken a a fixed poin in ime. The arbirage-free price of any sochasic payoff is herefore equal o he expeced payoff under he risk-neural measure, discouned a he riskfree rae. This firs chain of argumens has effecively linked arbirage-free pricing wih a probabiliy concep. We can use his chain as a bridge, from arbirage-free prices o equivalen probabiliies, and vice versa. Mos imporanly, we can now use any ools and mehods known in probabiliy heory, and use his bridge o ge back o arbirage-free prices. 2. A redundan securiy s payoff is defined in erms of he price of an underlying securiy. If he uncerainy in he underlying securiy is driven by he innovaion of a Brownian moion, hen he Girsanov Theorem ells us how he price dynamics look like under he risk-neural probabiliy measure. This effecively allows us o obain he risk-neural probabiliy disribuion of he payoff ha we are rying o price. Wih he help of he second chain of argumens we have obained he risk-neural probabiliy disribuion of he redundan securiy s payoff. The firs chain of argumens can now be used as a bridge o conver his probabiliy disribuion back o an arbirage-free price. 24
3.2 Final houghs The main difficuly in inuiively grasping he risk-neural pricing concep arises from he fac ha he probabilisic mehods and ools used were no primarily developed from a financial pricing perspecive. This resuls in he use of wo differen languages, one from economics and he oher from mahemaics, which can easily be confusing. We have shown in his paper ha risk-neural probabiliies are, from he economis s sandpoin, sae prices compounded wih he risk-free rae. One can always use his ranslaion in order o make economic sense when using he ools from probabiliy heory. I should be menioned ha he risk-neural probabiliy concep is only useful for arbirage-free pricing. An arbirage-free price is no necessarily a fair price, or he correc price; i is only a marke consisen price. We have wo general conclusions: 1) if a marke paricipan was buying (selling) a redundan asse above (below) is arbirage-free price, hen we can say ha here would be a more efficien way for his marke paricipan o express his view, namely via he replicaion sraegy; 2) if a marke paricipan was buying (selling) he underlying asse of a redundan securiy, where he redundan securiy rades below (above) is arbirage-free price, hen here would be a more efficien way for his marke paricipan o express his view, again via he replicaion sraegy. Beyond hese wo saemens, a unique arbirage-free price only serves as a rading crierion if he marke paricipan is ready o engage in arbirage aciviies, rying o lock in price differenials via replicaion. Oherwise, one needs o ake ino accoun ha he arbirage-free price of a redundan asse is NOT independen of risk premia. The risk-neural valuaion approach implicily uses he risk premia incorporaed in he marke price of he underlying asse. As we have seen, he price of he underlying asse is sill par of he arbirage-free pricing formula, herefore, he marke risk premia sill make heir way ino he equaion. The marke paricipan, however, migh have differen views on adequae risk premia han he marke, and his needs o be aken ino accoun when rading he redundan asse on a sand-alone basis. Finally, we noe ha our pricing equaion was no model-independen. Wha does his mean? The pricing equaion in our example was sill relying on he fac ha a geomeric Brownian moion was he correc model for he descripion of he underlying asse s dynamics. Pleny of simplifying assumpions were made. Wha he risk-neural probabiliy concep essenially helps us wih is he specificaion of a model, by removing he need o specify he drif parameer. However, if he form of he model is no correc in he firs place, hen he specificaion hereof migh no be of much use alogeher. If he form of he model was differen, hen he so-called 25