QUANTITATIVE FINANCE RESEARCH CENTRE. The Economic Plausibility of Strict Local Martingales in Financial Modelling

Transcription

1 QUANTITATIVE FINANCE REEARCH CENTRE QUANTITATIVE F INANCE REEARCH CENTRE QUANTITATIVE FINANCE REEARCH CENTRE Research Paper 279 June 2010 The Economic Plausibiliy of ric Local Maringales in Financial Modelling Hardy Hulley IN

2 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING HARDY HULLEY Absrac. The conex for his aricle is a coninuous financial marke consising of a riskfree savings accoun and a single non-dividend-paying risky securiy. We presen wo concree models for his marke, in which sric local maringales play decisive roles. The firs admis an equivalen risk-neural probabiliy measure under which he discouned price of he risky securiy is a sric local maringale, while he second model does no even admi an equivalen risk-neural probabiliy measure, since he puaive densiy process for such a measure is iself a sric local maringale. We highligh a number of apparen anomalies associaed wih boh models ha may offend he sensibiliies of he classically-educaed reader. However, we also demonsrae ha hese issues are easily resolved if one hinks economically abou he models in he righ way. In paricular, we argue ha here is nohing inherenly objecionable abou eiher model. 1. Inroducion The seing for his aricle is a coninuous financial marke consising of a single non-dividendpaying risky securiy and a risk-free money-marke accoun. We consider wo models for he price of he risky securiy, boh of which are affeced by he presence of sric local maringales. The firs model admis an equivalen risk-neural probabiliy measure under which he discouned price of he risky securiy is a sric local maringale. In paricular, his model saisfies he srong no-arbirage requiremen of no free lunch wih vanishing risk (NFLVR) (see Delbaen and chachermayer [1994b]). The second model, on he oher hand, does no even admi an equivalen risk-neural probabiliy measure, since he densiy process associaed wih he puaive equivalen risk-neural probabiliy measure is a sric local maringale. Neverheless, we shall see ha his model does possess a numéraire porfolio, which ensures ha i saisfies he weaker no-arbirage requiremen of no unbounded profi wih bounded risk (NUPBR) (see Karazas and Kardaras [2007]). Following he erminology inroduced by Heson e al. [2007], we shall say ha he firs model is an example of a sock price bubble, while he second model is an example of a bond price bubble. ock price bubbles have recenly become a popular opic in he lieraure (see e.g. Cox and Hobson [2005], Eksröm and Tysk [2009], Heson e al. [2007], Jarrow e al. [2007a,b] and Pal and Proer [2008]). Alhough such models remain firmly wihin he ambi of he riskneural approach o coningen claim pricing, hey neverheless exhibi a number of anomalies ha have occupied he research aricles cied above. For example, he risk-neural forward price of he risky securiy is sricly less han is spo price grossed-up a he risk-free ineres rae. Furhermore, he risk-neural prices of European pus and calls on he risky securiy no longer obey he pu-call pariy relaionship. By conras, bond price bubbles have received comparaively lile aenion in he lieraure possibly due o a misguided concern abou heir arbirage properies. The only obvious excepion is he so-called benchmark approach of Plaen and Heah [2006], which develops a sysemaic approach o he pricing and hedging of coningen claims in financial markes ha Dae: April 6, Mahemaics ubjec Classificaion. Primary: 91B70; econdary: 60G35, 60G44, 91B28. Key words and phrases. ric local maringales; arbirage; Bessel processes; sock price bubbles; bond price bubbles; risk-neural pricing; real-world pricing; hedging porfolios; replicaing porfolios; pu-call pariy. 1

3 2 HARDY HULLEY possess a numéraire porfolio, wihou necessarily admiing an equivalen risk-neural probabiliy measure. The resuling pricing mehodology is referred o as real-world pricing, since i does no rely on any equivalen change of probabiliy measure. A firs blush, he real-world prices of coningen claims for he bond price bubble also appear o exhibi a number of anomalies. For example, he real-world price of a zero-coupon bond urns ou o be sricly less han he discouned value of is principal. I is also possible o generae a risk-free profi from no iniial invesmen, by consrucing a porfolio whose value is bounded from below. In his aricle we analyze he pricing and hedging of coningen claims wrien on he risky securiy, for boh he sock price bubble and he bond price bubble. Our main objecive is o demonsrae ha he anomalies associaed wih sric local maringales are acually no so srange afer all. In paricular, if one hinks economically abou he sric local maringales in our wo models, hen he apparenly srange behaviour of coningen claim prices begins o seem quie naural. The remainder of he paper is srucured as follows: Firs ecion 2 inroduces he wo models, boh of which are based on a Bessel process of dimension hree. ecion 3 hen derives pricing formulae and hedge raios for a number of European coningen claims wrien on he he risky securiy, for he case of he sock price bubble. By analyzing he behaviour of hese pricing formulae and hedge raios, we argue ha he anomalies associaed wih he claim prices for he sock price bubble are easily resolved. ecion 4 hen performs a similar analysis, for he case of he bond price bubble. 2. An Overview of he Two Models As saed in he inroducion, he subjec of his aricle is a financial marke consising of a single non-dividend-paying risky securiy and a risk-free money-marke accoun. We are concerned wih he following wo models for he price of he risky securiy: and d = 2 dβ (2.1a) d = 1 d + dβ, (2.1b) for all R +. Here β is a sandard Brownian moion residing on a filered probabiliy space (Ω, F, F, P), whose filraion F = (F ) R+ saisfies he usual condiions of compleeness and righ-coninuiy. We also adop he sandard convenion of assuming ha he risk-free ineres rae is zero, so ha he value of he money-marke accoun is given by B = 1. Observe ha he process specified by (2.1a) is by inspecion a local maringale. The firs of he wo models presened above hus clearly saisfies he NFLVR condiion, since P is iself he (unique) equivalen risk-neural probabiliy measure. In paricular, he densiy process for he equivalen risk-neural probabiliy measure for ha model is given by Z = 1. However, here is more o i han ha, since he process described by (2.1a) is in fac he inverse of a Bessel process of dimension hree, and is herefore a well-known example of a sric local maringale (see e.g. Revuz and Yor [1999], Exercise V.2.13, p. 194). This esablishes he firs model as an example of a sock price bubble. In he case of he second model, (2.1b) expresses he price of he risky securiy as a Bessel process of dimension hree. I hen follows ha is he numéraire porfolio for ha model (in he sense of Becherer [2001], Definiion 4.1), since B/ = 1/ and / = 1 are boh nonnegaive local maringales, and consequenly also supermaringales, by Faou s lemma. The model herefore saisfies he NUPBR condiion, by virue of Karazas and Kardaras [2007], Theorem However, i does no saisfy he NFLVR condiion, since he densiy process 1 Acually, his is no compleely rue, since Theorem 4.12 of Karazas and Kardaras [2007] requires he value of he numéraire porfolio o be a semimaringale up o infiniy. The Bessel process of dimension hree does

4 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING 3 for he puaive equivalen risk-neural probabiliy measure may be expressed in erms of he numéraire porfolio as follows: Z = ( 0 /B 0 )(B/) = 0 /, which is a sric local maringale. 2 This esablishes he second model as an example of a bond price bubble. We acknowledge ha he models presened by (2.1a) and (2.1b) have already been sudied in deail by Delbaen and chachermayer [1994a] and Delbaen and chachermayer [1995], respecively. However, whereas he above-menioned aricles focus on quesions of arbirage, we wish o address he pricing and hedging of coningen claims. Our paper may herefore be regarded as a companion for hose aricles. We begin our discussion of coningen claim pricing by inroducing a finie mauriy dae T > 0 and a payoff funcion h : (0, ) R +, which saisfy he following wo condiions: (i) h is coninuous and of polynomial growh; (ii) EZ T h( T )< ; and (iii) Ph( T ) > 0>0. We now consider a European coningen claim on he risky securiy, wih mauriy T and payoff h( T ). We shall denoe he value of his claim by he process V h = (V h ) [0,T ], which is deermined as follows: V h := 1 EZ T h( T )F, (2.2) Z for all [0, T ]. Observe ha condiion (ii) above ensures ha V h <, for all [0, T ], while condiion (iii) implies ha V h > 0, for all [0, T ). Also, noe ha since VT h = h( T ), i easily follows ha he process (Z V h ) [0,T ] is a maringale. We shall refer o (2.2) as he real-world price of he claim, since i does no require any ransformaion of probabiliy measure. 3 For he firs model i is easily seen ha (2.2) agrees wih he risk-neural price of he claim, since Z is a maringale. However, i is also clear ha (2.2) remains well-defined for he second model. In ha case here is no equivalen riskneural probabiliy measure, since Z is a sric local maringale, and hence risk-neural pricing is infeasible. In oher words, real-world pricing offers a proper exension of he risk-neural pricing concep. An imporan observaion for he wo models under consideraion is ha Z T if he price of he risky securiy is modelled by (2.1a); =(1 Z if he price of he risky securiy is modelled by (2.1b), T for all [0, T ]. We may herefore exploi he Markov propery of he price of he risky securiy, for boh models, by wriing V h = V h (, ), for all [0, T ], where he pricing funcion V h : [0, T ] (0, ) R + ha appears in his expression may be compued as no mee his requiremen, since is sample pahs diverge in he limi as ime goes o infiniy. Neverheless, his provides no pracical obsacle, since our applicaions only involve a finie ime-horizon. 2 ricly speaking, we are abusing erminology by calling Z a densiy process in his insance, since mos of he lieraure adops he convenion ha densiy processes are posiive maringales. In he case of he second model, i may herefore be more appropriae o describe Z as a pre-densiy process or a candidae densiy process. However, in he ineres of mainaining a uniform erminology for discussing boh models, we shall coninue o refer o Z simply as a densiy process, wih he undersanding ha he reader is sufficienly flexible on his poin. 3 The reader is direced o ecion 9.1 of Plaen and Heah [2006] for a more deailed accoun of he real-world approach o coningen claim pricing. Noe ha real-world prices are expressed here in erms of he numéraire porfolio, whereas (2.2) expresses hem in erms of he densiy process. These wo formulaions are equivalen, by din of he fac ha he densiy process is proporional o he numéraire-denominaed money-marke accoun.

5 4 HARDY HULLEY follows: V h (, ) :=8< ) :E,h( T if E T ),h( if T he price of he risky securiy is modelled by (2.1a); he price of he risky securiy is modelled by (2.1b), for all (, ) [0, T ] (0, ). 4 The following proposiion will allow us o obain convenien expressions for he pricing funcions of a number of sandard European coningen claims in ecions 3 and 4: Proposiion 2.1. A Bessel process ρ of dimension hree has he following (runcaed) momens: 1 E 1 x z Φx + z+φx x1 {ρ z} Φ ρ = x Φ x!; (2.4) E x1 {ρ>z} (2.3) E x1 1 Φx Φ x ρ =!; (2.5) x 1 1 Φx + z Φ!; x z (2.6) ρ = x P x (ρ z) = Φx + z Φx z + x φx + z φx!; z (2.7) and P x (ρ > z) = Φ x + z +Φx z x φx + z φx!, z (2.8) for all > 0, x, y (0, ) and z > 0. 5 In hese expressions Φ denoes he cumulaive disribuion funcion of a sandard normal random variable, while φ is he associaed densiy funcion. 6 Proof. To begin wih, Revuz and Yor [1999], Proposiion VI.3.1, p. 251 provides he following expression for he ransiion densiy of ρ: q(, x, y) := y P x(ρ y) = x 3 2 I 1 exp y x 2xy x2 + y!, 2 2 for all > 0 and x, y (0, ). The funcion I 1/2 ha appears above is he modified Bessel funcion of he firs kind wih index one half (see e.g. Abramowiz and egun [1972], Chaper 9). I saisfies he following ideniy: 1 I 1 (z) =r2 sinh z = (e z e z ), 2 πz 2πz 4 As usual, E, denoes he expeced value operaor wih respec o he probabiliy measure P,, under which he price of he risky securiy a ime R + is (0, ). Of course, we are abusing noaion slighly by using o denoe a paricular value for he price of he risky securiy, as well he price process for his asse. 5 In his case Ex should be undersood as he expeced value operaor wih respec o he probabiliy measure P x, under which ρ 0 = x, for all x (0, ). 6 In oher words, {z } 1 e ζ2 /2 2π φ(ζ) Φ(z) :=Zz dζ, for all z R. Also, recall ha he mapping R ζ 1 φ ζ μ σ σ is he probabiliy densiy funcion of a normal random variable wih mean μ R and sandard deviaion σ > 0.

6 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING 5 for all z R, according o Lebedev [1972], Equaion 5.8.5, p By combining hese wo expressions, we obain q(, x, y) = y 1 x φy 1 φy!, + x x for all > 0 and x, y (0, ). Now, fix > 0, x (0, ) and z > 0, and observe ha 1 1 E x1 {ρ z} ρ =Zz 0 y q(, x, y) dy = 1 Zz 1 x 1 + x φy dy Zz φy dy!. x 0 0 ince he final expression above involves he difference beween he runcaed zeroh momens of wo normal random variables, we obain (2.4) from Jawiz [2004], Table 1, for example. We may hen derive (2.5) from (2.4), by an applicaion of he monoone convergence heorem, while (2.6) follows direcly from (2.4) and (2.5). Nex, we see ha P x (ρ z) =Zz q(, x, y) dy = 1 Zz y x y + x φy dy Zz φy dy!. 0 x 0 0 In his case we recognize ha he final expression above involves he difference beween he runcaed firs momens of wo normal random variables. Once again, we may look hese up in Jawiz [2004], Table 1, o ge (2.7). Finally, (2.8) is obained direcly from (2.7). We urn our aenion now o he quesion of how o hedge he coningen claim inroduced above, for he wo models under consideraion. To begin wih, hroughou his aricle he phrase viable porfolio should be undersood o mean a self-financing porfolio consising of holdings in he risky securiy and he money-marke accoun, whose value is sricly posiive over [0, T ) and non-negaive a ime T. 7 uch a porfolio may be presened by a rading sraegy π = (π ) [0,T ), which specifies he fracion of is wealh invesed in he risky securiy. Given an iniial endowmen x > 0 and a rading sraegy π = (π ) [0,T ) for a viable porfolio, he associaed wealh process W x,π = (W x,π ) [0,T ] is easily seen o saisfy one of he following wo sochasic differenial equaions: dw x,π = π W x,π dβ (2.9a) or dw x,π = π 2 W x,π d + π W x,π dβ, (2.9b) for all [0, T ), depending on wheher he price of he risky securiy is modelled by (2.1a) or by (2.1b). 8 As an aside, observe ha (2.9) suggess he following sochasic exponenial represenaion for he wealh of a viable porfolio wih iniial endowmen x > 0 and rading sraegy π = (π ) [0,T ) : W x,π xez = 0 π s dx s, for all [0, T ]. The process X in he expression above is given by dx = dβ (2.10a) 7 ricly speaking, he las requiremen is redundan, since all self-financing porfolios in he financial marke inroduced above have coninuous sample pahs. 8 Of course, π should belong o an appropriae class of processes in order for he sochasic inegrals in (2.9) o be well-defined. For example, we could sipulae ha all rading sraegies should be progressively measurable.

7 6 HARDY HULLEY or dx = 1 2 d + 1 dβ, (2.10b) for all R +, depending on wheher he price of he risky securiy is deermined by (2.1a) or by (2.1b). The requiremen ha he porfolio wealh is sricly posiive over [0, T ) hen ranslaes ino he following concree condiion:z πs 2 d X s <, 0 for all [0, T ). Moreover, we obain he following correspondence beween evens: 0o=ZT = πs 2 d X s =, nw x,π T 0 by an applicaion of he law of large numbers for local maringales (see e.g. Revuz and Yor [1999], Exercise V.1.16, pp ). I will be useful for us o idenify wo noions of hedging, for he coningen claim inroduced earlier. The firs of hese corresponds o he siuaion when he value of a viable porfolio precisely maches he payoff of he claim a is mauriy, while he second describes he siuaion when he value of a viable porfolio precisely maches he real-world price of he claim a all imes up o is mauriy: Definiion 2.2. Consider a viable porfolio, which is deermined by an iniial endowmen x > 0 and a rading sraegy π = (π ) [0,T ). (i) The porfolio is said o hedge he claim if and only if W x,π T = h( T ). (ii) The porfolio is said o replicae he claim if and only if W x,π = V h, for all [0, T ]. I is immediaely eviden from he definiion above ha any viable porfolio ha replicaes he claim mus also hedge i. To make he relaionship beween hedging and replicaion more explici, suppose ha he iniial endowmen x hed > 0 and rading sraegy π hed = (π hed ) [0,T ) specify a viable porfolio ha hedges he claim. imilarly, le he iniial endowmen x rep > 0 and rading sraegy π rep = (π rep ) [0,T ) specify a viable porfolio ha replicaes he claim. We begin by observing ha he process Z W xrep,π rep is a maringale, since Z W xrep,π rep = Z V h Š [0,T ] = EZ T VTF h =EZ T W xrep,π rep T F, for all [0, T ], by virue of he already esablished fac ha (Z V h ) [0,T ] is a maringale. Nex, we noe ha he process Z W xhed,π hed is a local maringale. In he case of he Š [0,T ] sock price bubble, his is an easy consequence of (2.9a) and he fac ha Z = 1. In he case of he bond price bubble, he sochasic inegraion by pars formula yields dzw xhed,π hed 1 = 1 π hed ZW xhed,π hed dβ, for all [0, T ), wih he help (2.9b) and he fac ha Z = 0 /. We may herefore deduce ha Z W xhed,π hed is in fac a supermaringale, for boh models, by an applicaion of Š [0,T ] Faou s lemma. Finally, puing all of he above ogeher, we obain Z W xrep,π rep = EZ T W xrep,π rep T F =EZ T W xhed,π hed T F Z W xhed,π hed, for all [0, T ]. In paricular, since Z > 0 for boh he sock price bubble and he bond price bubble, we mus have W xrep,π rep for all [0, T ], irrespecive of which model is chosen. W xhed,π hed, (2.11)

8 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING 7 The imporance of (2.11) lies in he jusificaion i provides for he real-world pricing formula (2.2). In paricular, since he value of he replicaing porfolio if i exiss corresponds wih he real-world price of he claim, we see ha he real-world price represens he value of he cheapes possible hedging porfolio. This raises he following wo imporan quesions: Does he replicaing porfolio necessarily exis, and can we characerize i? These quesions are addressed by he nex proposiion, for which he crucial ingredien is a converse for he Feynman-Kac Theorem, due o Janson and Tysk [2006]: Proposiion :=8< 2.3. Define ) he funcions x rep : (0, ) (0, ) and π rep : [0, T ) (0, ) R + as follows: :E 0,h( T if he price of he risky securiy is modelled by (2.1a); x rep () (2.12) E T ) 0,h( if he price of he risky securiy is modelled by (2.1b), T for all (0, ), and π rep V h (, ) := V h (, ), (2.13) (, ) for all (, ) [0, T ) (0, ), respecively. In paricular, (2.13) is well-defined, since he pricing funcion (2.3) is boh sricly posiive and differeniable. The replicaing porfolio for he coningen claim under consideraion is hen deermined by he iniial endowmen x rep = x rep ( 0 ) and he rading sraegy π rep = (π rep ) [0,T ), wih π rep = π rep (, ), for all [0, T ). Proof. We have already observed ha by imposing condiion (iii) on he payoff funcion, we guaranee ha he pricing funcion is sricly posiive, for boh he sock price bubble and he bond price bubble. o o esablish ha he expression in (2.13) is well-defined, we need only demonsrae ha he pricing funcion of he claim is differeniable. We shall now analyze he he sock price bubble and he bond price bubble separaely: (i) uppose he price of he risky securiy is modelled by (2.1a). I hen follows from Janson and Tysk [2006], Theorem 6.1 ha he pricing funcion (2.3) is coninuously differeniable wih respec o he emporal variable and wice coninuously differeniable wih respec o he spaial variable, and ha i saisfies he following parial differenial equaion: V h (, ) V h (, ) = 0, (2.14) 2 for all (, ) [0, T ) (0, ). 9 The differeniabiliy of he pricing funcion ensures ha he rading sraegy (2.13) is well-defined, while Iô s formula combined wih (2.14) yields dv h (, ) = V h (, ) = 2 V h (, ) dβ = π rep (, ) V h (, ) dβ, 2 V h 2 (, )!d 2 V h (, ) dβ (2.15) for all [0, T ). Finally, by comparing (2.15) wih (2.9a), we see ha W xrep,π rep = V h (, ), for all [0, T ], by virue of he fac ha x rep ( 0 ) = V h (0, 0 ). (ii) uppose he price of he risky securiy is modelled by (2.1b), and consider he funcion ÒV h : [0, T ] (0, ) R +, given by ÒV h (, ) := V h (, ), (2.16) 9 Noe ha Theorem 6.1 of Janson and Tysk [2006] requires a coninuous payoff funcion of polynomial growh. This explains why we imposed condiion (i), when we inroduced he payoff funcion for he claim.

9 8 HARDY HULLEY for all (, ) [0, T ] (0, ). I hen follows from Janson and Tysk [2006], Theorem 6.1 and (2.3) ha he funcion defined by (2.16) is coninuously differeniable wih respec o he emporal variable and wice coninuously differeniable wih respec o he spaial variable, and ha i saisfies he following parial differenial equaion: ÒV h (, ) + 1 ÒV h (, ) + 1 2ÒV h (, ) = 0, (2.17) 2 2 for all (, ) [0, T ) (0, ). In paricular, he differeniabiliy of his funcion implies ha he pricing funcion is differeniable, from which i follows ha he rading sraegy (2.13) is well-defined. Nex, by combing (2.17) wih (2.16), we obain he following parial differenial equaion for he pricing funcion iself: V h (, ) + 1 V h (, ) V h 2 2 (, ) 1 V h (, ) = 0, (2.18) for all (, ) [0, T ) (0, ). By combining Iô s formula wih (2.18), we hen ge dv h (, ) = V h (, ) + 1 V h (, ) = 1 V h (, ) d + V h (, ) dβ 2 V h 2 (, )!d + V h (, ) dβ (2.19) = πrep (, ) 2 V h (, ) d + πrep (, ) V h (, ) dβ, for all [0, T ). As before, by comparing (2.19) wih (2.9b), we see ha W xrep,π rep = V h (, ), for all [0, T ], since x rep ( 0 ) = V h (0, 0 ). Togeher, (2.11) and Proposiion 2.3 esablish he canonical naure of he real-world pricing formula: Given any payoff funcion saisfying condiions (i) (iii) above, he real-world price of he corresponding claim provides he minimal cos of hedging i. Moreover, we see ha he rading sraegy (2.13) for he replicaing porfolio may be expressed in erms of he dela of he claim. The laer is in urn compleely deermined by a funcion Δ h : [0, T ) (0, ) R, which is defined by Δ h (, ) := V h (, ), for all (, ) [0, T ) (0, ). 3. The ock Price Bubble In his secion we focus on he siuaion when he price of he risky securiy is modelled by (2.1a). ince is price is hen an invered Bessel process of dimension hree, we may use Proposiion 2.1 o derive he pricing funcions and delas for a number of sandard European claims on he risky securiy, all of which share a common mauriy dae T > 0 and srike K > 0 (when appropriae). We begin by considering a zero-coupon bond wih a face-value of one dollar. 10 As one would expec, his insrumen is rivial in he case of he sock price bubble. In paricular, is dela is uniformly zero, so ha is replicaing porfolio consiss simply of an iniial endowmen equal o (he discouned value of) is principal invesed in he money-marke accoun: Example 3.1. The pricing funcion Z : [0, T ) (0, ) (0, ) for a zero-coupon bond is given by Z(, ) := E, (1) = 1, (3.1) 10 Here a dollar should be inerpreed as a generic uni of currency we assume ha he values of all insrumens are denominaed in dollars.

10 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING F, F, Figure 3.1. The pricing funcion and he dela of a prepaid forward on he risky securiy (T = 1). for all (, ) [0, T ) (0, ). The associaed replicaing porfolio is deermined by he dela Δ Z : [0, T ) (0, ) R of he conrac, which is given by for all (, ) [0, T ) (0, ). Δ Z (, ) := Z (, ) = 0, (3.2) Nex, we examine a prepaid forward conrac on he risky securiy. 11 Is pricing funcion and dela are presened below, and are ploed in Figure 3.1: Example 3.2. The pricing funcion F : [0, T ) (0, ) (0, ) for a prepaid forward on he risky securiy is given by F(, ) := E, ( T ) = 1 2Φ 1/!, (3.3) T for all (, ) [0, T ) (0, ), by an applicaion of (2.5). The associaed replicaing porfolio is deermined by he dela Δ F : [0, T ) (0, ) R of he conrac, which is given by Δ F (, ) := F (, ) = 1 2Φ 1/ for all (, ) [0, T ) (0, ). T 2 T φ 1/ T, (3.4) We immediaely noice from Figure 3.1 ha he value of he prepaid forward conrac is sricly less han he price of he risky securiy iself, a all imes prior o mauriy. In paricular, we have {z F (, ) = 2Φ 1/ } > 0, (3.5) T for all (, ) [0, T ) (0, ). The process γ(, inroduced above corresponds wih )Š R + wha Elworhy e al. [1999] refer o as he defaul of a sric local maringale (in his case he price of he risky securiy). 11 The principal difference beween prepaid forwards and convenional forward conracs is ha he laer are seled a mauriy, while he purchaser of a prepaid forward pays an up-fron premium for subsequen delivery of he underlying asse when he conrac maures. In oher words, a prepaid forward is simply a European call wih a srike price of zero. γ(,)

11 10 HARDY HULLEY Anoher ineresing observaion is ha he price of he prepaid forward is bounded from above a any fixed ime [0, T ), since (3.3) yields lim F(, ) = 2 È2π(T ). (3.6) This explains why one canno hope o exploi any arbirage opporuniy by simulaneously purchasing he prepaid forward and shor-selling he underlying risky securiy he value of he resuling porfolio is no bounded from below. I also explains why, for large values of he risky securiy price, he porfolio ha replicaes he prepaid forward is almos compleely invesed in he money-marke accoun, as evidenced by he surface plo of is dela in Figure 3.1. In paricular, (3.6) ensures ha he dela of he prepaid forward is zero in he limi, as he price of he risky securiy increases o infiniy. Le us now aemp o explain he inuiion behind he fac ha he prepaid forward is worh less han is underling risky securiy. To begin wih, consider for a momen a model ha does admi an equivalen risk-neural probabiliy measure, under which he (discouned) price of he risky securiy is in fac a maringale. In such a case he risk-neural value of he prepaid forward would exacly mach he marke price of he risky securiy. This phenomenon is ofen inerpreed o mean ha risk-neural valuaion allows one o rerieve he price he risky securiy iself, by compuing is discouned expeced fuure value. By his line of reasoning he sock price bubble seems incoheren, since i appears ha here are now wo prices for he risky securiy: a marke price and a model price. To resolve he conundrum above, we should firs poin ou ha he prices of he underlying primary securiies (i.e. he risky securiy and he money-marke accoun) are compleely exogenous in he modelling framework considered here hey are wha hey are, and hey are no subjec o any valuaion principle. The fac ha he risk-neural value of he prepaid forward agrees wih he marke price of he risky securiy, in he case when he (discouned) price of he risky securiy is a maringale under he equivalen risk-neural probabiliy measure, is simply a curiosiy induced by he properies of maringales. In ruh, he risky securiy and he prepaid forward conrac wrien on i are fundamenally differen insrumens. To make his poin clear, observe ha he prepaid forward may also be regarded as a European call on he risky securiy wih a srike price of zero. By he same oken, he risky securiy may be regarded as an American call on iself, also wih a srike price of zero. een in his ligh, i is no srange ha he value of he prepaid forward should be less han he price of he risky securiy he difference is simply an early exercise premium! Moreover, in he case of he sock price bubble when he price of he risky securiy is expeced o decrease over ime his early exercise premium becomes significan. In his way we would argue ha he inequaliy (3.5) is in fac quie naural. The nex insrumen we consider is a European call on he risky securiy. Is pricing funcion and dela are presened below, and are ploed in Figure 3.2: Example 3.3. The pricing funcion C : [0, T ) (0, ) (0, ) for a European call on he risky securiy is given by C(, ) := E,( T K) +=E,1 {T >K} T KP, ( T > K) 1/ 1/K 1/ + 1/K = ( K)Φ +( + K)Φ 2Φ 1/ T T T (3.7) + K T φ1/ 1/K T φ1/!, + 1/K T

12 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING 11 C, C, Figure 3.2. The pricing funcion and he dela of a call on he risky securiy (T = 1 and K = 1). for all (, ) [0, T ] (0, ), by an applicaion of (2.4) and (2.7). The associaed replicaing porfolio is deermined by he dela Δ C : [0, T ) (0, ) R of he call, which is given by Δ C (, ) := C 1/ 1/K 1/ + 1/K (, ) = Φ +Φ T T K + φ φ1/ 1/K + K 2 + φ1/ + 1/K T T 2 2Φ 1/ T T T 2 1/ K (3.8) T T K2 (T ) 1/K φ1/ 2 T T + K + K2 (T ) + 1/K φ1/ 2, T T for all (, ) [0, T ) (0, ). I is eviden from Figure 3.2 ha he call opion exhibis similar anomalies o he prepaid forward conrac. In paricular, for he price of he call we obain he following limi from (3.7): lim C(, ) = 2 2Φ È2π(T ) K 1/K (3.9) T 1!, for all [0, T ). Once again, his implies ha he call price is bounded from above a any fixed ime prior o mauriy. In addiion, i esablishes ha he pricing funcion for he call does no preserve he convexiy of is payoff funcion, wih respec o he price of he underlying risky securiy. This non-convexiy is easily observed in he surface plo of he call dela in Figure 3.2, where we see ha he second derivaive of he price of he call wih respec o he price of he risky securiy is negaive, for large enough prices of he risky securiy. Anoher easy consequence of (3.9) is ha he slope of he call price wih respec o he price of he risky securiy mus end o zero asympoically, a any fixed ime before mauriy, for large values of he risky securiy price. In oher words, he call dela should decrease o zero, when he price of he risky securiy becomes large, which implies ha he replicaing porfolio for he call should ulimaely become fully invesed in he money-marke accoun. Once again, his phenomenon is illusraed by Figure 3.2. The final claim considered in his secion is a European pu on he risky securiy. Expressions for is pricing funcion and dela are obained below, and are ploed in Figure 3.3:

13 12 HARDY HULLEY 2.0 P, 2.0 P, Figure 3.3. The pricing funcion and he dela of a pu on he risky securiy (T = 1 and K = 1). Example 3.4. The pricing funcion P : [0, T ) (0, ) (0, ) for a European pu on he risky securiy is given by T P(, ) := E,(K T ) +=KP, ( T K) E,1 {T K} = ( + K)Φ 1/ + 1/K T 1/K ( K)Φ1/ T + K T φ1/ 1/K T φ1/!, + 1/K T (3.10) for all (, ) [0, T ) (0, ), by an applicaion of (2.4) and (2.7). The associaed replicaing porfolio is deermined by he dela Δ P : [0, T ) (0, ) R of he pu, which is given by Δ P (, ) := P 1/ + 1/K Φ1/ 1/K (, ) = Φ T T + K + φ1/ + 1/K K 2 + φ1/ 1/K T T 2 (3.11) T T K K2 (T ) 2 T for all (, ) [0, T ) (0, ). φ1/ 1/K T + K + K2 (T ) 2 T + 1/K φ1/, T I is immediaely eviden from Figure 3.3 ha he behaviour of he pu is much more convenional han ha of he call. In paricular, we see ha he pricing funcion for he pu preserves he convexiy of is payoff funcion, wih respec he price of he risky securiy, a all imes prior o mauriy. Furhermore, he dela of he pu indicaes ha he replicaing sraegy for he claim converges o a shor posiion in one share of he risky securiy, if he price of his securiy is very low, bu has almos no exposure o he risky securiy, if is price is high. Once again, his corresponds wih he normal behaviour of European pus under he assumpions of he Black-choles model, for example. Finally, we urn our aenion o he quesion of pu-call pariy. I is already well-known ha pu-call pariy fails for he sock price bubble, and we may confirm his explicily, wih he help

14 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING 13 of (3.7) and (3.10): C(, ) P(, ) = 2Φ 1/ T 1! K < K, (3.12) for all (, ) [0, T ) (0, ). There are wo ways of inerpreing he above inequaliy: Eiher he model is pahological, or else he pu-call pariy relaionship is misspecified. We shall argue for he laer inerpreaion. To wi, noe ha pu-call pariy assers an equivalence beween a porfolio conaining a long European call and a shor European pu, on he one hand, and a long posiion in he risky securiy combined wih a shor posiion in he moneymarke accoun, on he oher hand. ince he laer porfolio offers invesors he possibiliy of aking advanage of favourable inerim movemens in he price of he risky securiy, while he porfolio of European opions does no, he inequaliy (3.12) simply expresses a ype of early exercise premium associaed wih holding he underlying securiies insead of he European opions. In he case of he sock price bubble, where he price of he risky securiy is expeced o decline over ime, he flexibiliy associaed wih holding he underlying securiies becomes valuable. A properly specified pu-call relaionship should only involve insrumens ha are European by naure, o avoid he ype of phenomenon described above. We herefore propose he following as he correc formulaion: C(, ) P(, ) = F(, ) KZ(, ), (3.13) for all (, ) [0, T ) (0, ). Noe ha his ideniy is easily verified for he sock price bubble, using he pricing funcions presened in his secion. Moreover, in ecion 4 we shall see ha (3.13) coninues o hold for he case of he bond price bubble. I is herefore, in some sense, he fundamenal saemen of pu-call pariy. 4. The Bond Price Bubble This secion examines he bond price bubble, when he price of he risky securiy is modelled by (2.1b). ince is price is a Bessel process of dimension hree, in his case, we may once again apply Proposiion 2.1 o derive he pricing funcions and delas for a number of European claims on he risky securiy, all of which share a common mauriy dae T > 0 and srike K > 0 (when appropriae). The firs insrumen we consider is a zero-coupon bond wih a face-value of one dollar, which we see is no longer rivial. In fac, we make he raher sarling discovery ha such a claim is acually an equiy derivaive, in he seing of he bond price bubble. Is pricing funcion and dela are presened below, and are ploed in Figure 4.1: Example 4.1. The pricing funcion Z : [0, T ) (0, ) (0, ) for a zero-coupon bond is given by Z(, ) := E,1 2Φ (4.1) T=1 T, for all (, ) [0, T ) (0, ), by an applicaion of (2.5). The associaed replicaing porfolio is deermined by he dela Δ Z : [0, T ) (0, ) R of he zero-coupon bond, which is given by for all (, ) [0, T ) (0, ). Δ Z (, ) := Z (, ) = 2 T φ T, (4.2) Figure 4.1 reveals ha he price of he bond is only significanly less han is (discouned) principal which would have been is price if an equivalen risk-neural probabiliy measure exised when he price of he risky securiy is low. The reason for his can be found in (2.1b),

15 14 HARDY HULLEY Z, 2.0 Z, Figure 4.1. The pricing funcion and he dela of a zero-coupon bond (T = 1). which reveals ha when he price of he risky securiy is small, is drif rae explodes, resuling in a srong repulsion from he origin. Under hese circumsances he risky securiy is an exremely aracive invesmen, since i generaes a posiive reurn over a shor period of ime, wih a high degree of cerainy. By comparison, he zero-coupon bond is relaively unaracive under hese condiions, unless i rades a a price subsanially lower han is discouned face-value. This explains why he price of he zero-coupon bond vanishes as he price of he risky securiy approaches zero. I is also ineresing o analyze he zero-coupon bond from he perspecive of is replicaing sraegy. The surface plo of he dela of his insrumen in Figure 4.1 indicaes ha, a any fixed ime before mauriy, is replicaing porfolio becomes progressively more heavily invesed in he risky securiy as he price of ha asse decreases. Once again, his simply akes advanage of he srong growh in he price of he risky securiy when i is low. In paricular, (4.2) yields lim Δ Z (, ) = 0 2 È2π(T ), for all [0, T ). An ineresing consequence of his is ha when mauriy is imminen and he price of he risky securiy is close o zero, he cerain principal paymen of he zero-coupon bond can be hedged by purchasing an arbirarily large number of unis of he risky securiy. In oher words, over shor periods he price of he risky securiy exhibis growh ha is almos deerminisic, if is iniial value is close o zero. This can be exploied o produce a non-random payoff a very low cos. We have already poined ou ha he bond price bubble does no admi an equivalen riskneural probabiliy measure, from which i follows ha his model does no saisfy he NFLVR condiion, by he resuls of Delbaen and chachermayer [1994b]. As we shall now demonsrae, Example 4.1 allows us o esablish he failure of NFLVR explicily. To do so, consider a porfolio comprising a long posiion in he zero-coupon bond, which is funded by borrowing Z(0, 0 ) from he money-marke accoun. The iniial value of his porfolio is obviously zero, bu is payoff a mauriy is Z(T, T ) Z(0, 0 ) = 2Φ 0 T>0, by (4.1). Moreover, since Z(, ) > 0, for all [0, T ], i follows ha he value of his porfolio is uniformly bounded from below by Z(0, 0 ). In oher words, he porfolio described above violaes wha Kabanov and ricker [2004] refer o as he rue no-arbirage propery, which is in fac a weaker condiion han NFLVR.

16 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING 15 C, C, Figure 4.2. The pricing funcion and he dela of a call on he risky securiy (T = 1 and K = 1). In ecion 3 we saw ha a prepaid forward conrac on he risky securiy is non-rivial, in he case of he sock price bubble. The opposie is rue for he bond price bubble, as demonsraed below: Example 4.2. The pricing funcion F : [0, T ) (0, ) (0, ) for a prepaid forward on he risky securiy is given by F(, ) := E, T T=, (4.3) for all (, ) [0, T ) (0, ). The associaed replicaing porfolio is deermined by he dela Δ F : [0, T ) (0, ) R of he conrac, which is given by for all (, ) [0, T ) (0, ). Δ F (, ) := F (, ) = 1, (4.4) The nex insrumen we consider is a European call on he risky securiy. Is pricing funcion and dela are presened below, and are ploed in Figure 4.2: Example 4.3. The pricing funcion C : [0, T ) (0, ) (0, ) for a European call on he risky securiy is given by ( T K) + C(, ) := E,!=P,( T > K) KE,1 1 {T >K} T T K + K = ( K)Φ + K)Φ (4.5) T +( T + T φ K T φ+ K T!, for all (, ) [0, T ) (0, ), by an applicaion of (2.6) and (2.8). The associaed replicaing porfolio is deermined by he dela Δ C : [0, T ) (0, ) R of he call, which is given by for all (, ) [0, T ) (0, ). Δ C (, ) := C (, ) = Φ K + K (4.6) T +Φ T,

17 16 HARDY HULLEY By inspecing Figure 4.2 we see ha he price of he call and is dela behave quie convenionally, when compared wih European calls in he Black-choles model, for example. In paricular, he call price preserves he convexiy of is payoff funcion, wih respec o he price of he underlying risky securiy, a all imes prior o mauriy. Also, conrary o wha we observed for he case of he sock price bubble, we see ha he call price is an unbounded funcion of he price of he risky securiy, a any fixed ime prior o mauriy. The dela of he call behaves as one would expec, for large values of risky securiy price, by converging o one. In oher words, he replicaing porfolio for he call is almos compleely invesed in he risky securiy when he conrac is deep in-he-money. However, he behaviour of he call dela is a lile unusual when he price of he risky securiy is very low. In paricular, we obain he following from (4.6): lim Δ C (, ) = 2Φ K 0 T, for all [0, T ). In oher words, a any fixed ime before o mauriy, he replicaing porfolio for he call always holds a leas a minimal number of unis of he risky securiy. Once again, his phenomenon is explained by he srong growh of he price of he risky securiy when i is small. By comparison, under he assumpions of he Black-choles model, for example, he price of he risky securiy does no exhibi such growh behaviour near he origin, and he dela of a European call converges o zero as he price of he underlying asse becomes very small. To complee his secion, we finally consider a European pu on he risky securiy. Is pricing funcion and dela are presened below, and are ploed in Figure 4.3: Example 4.4. The pricing funcion P : [0, T ) (0, ) (0, ) for a European pu on he risky securiy is given by (K T ) + 1 P(, ) := E,!=KE,1 {T K}, ( T K) T T P K K = ( K)Φ + K)Φ+ T ( T +2KΦ (4.7) T + T φ K T φ+ K T!, for all (, ) [0, T ) (0, ), by an applicaion of (2.6) and (2.8). The associaed replicaing porfolio is deermined by he dela Δ P : [0, T ) (0, ) R of he pu, which is given by Δ P (, ) := P (, ) = Φ K T Φ+ K T + 2K T φ for all (, ) [0, T ) (0, ). T, (4.8) Figure 4.3 reveals ha he pu opion shares some of he srange feaures of he zero-coupon bond, discussed earlier in his secion. In paricular, we observe ha he pu price converges o zero as he price of he risky securiy decreases, for any fixed ime before mauriy. Once again, here is a sound economic explanaion for his phenomenon, which akes ino accoun he behaviour of he risky securiy price near zero. In deail, since he drif erm in (2.1b) explodes near he origin, an almos cerain profi can be earned in a shor ime by invesing in he risky securiy when is price is very low. Due o is bounded payoff srucure, he pu herefore becomes unaracive relaive o an invesmen in he risky securiy, when he price of he laer is very low. Anoher feaure of he pu, which is clearly visible from he surface plo of is pricing funcion in Figure 4.3, is he fac ha is price does no preserve he convexiy of is payoff, wih respec

18 THE ECONOMIC PLAUIBILITY OF TRICT LOCAL MARTINGALE IN FINANCIAL MODELLING 17 P, P, Figure 4.3. The pricing funcion and he dela of a pu on he risky securiy (T = 1 and K = 1). o he underlying risky securiy price, a any fixed ime prior o mauriy. This non-convexiy is also observable in he surface plo of he pu dela, where we see ha he second derivaive of he pu price wih respec o he price of he risky securiy is negaive, for small enough prices of he risky securiy. In addiion o he above, he expression (4.8) for he dela of he pu reveals he following anomaly as well: lim Δ P K (, ) = 1 0 2Φ T + = 2Z K T 0 1 2π dζ 2Z +Z 2K È2π(T ) = K T φ(ζ) dζ > 0, 0 K T φ(ζ) dζ!+ 0 2K È2π(T ) for all [0, T ), by virue of he fac ha φ(ζ) φ(0) = 1 2π, for all ζ R. This ells us ha he replicaing porfolio for he pu opion conains a long posiion in he risky securiy, if he price of he laer is very low. In oher words, a long posiion in he risky securiy may be required o hedge a conrac whose payoff is negaively relaed o he price of he risky securiy a mauriy! In fac, as we see from he surface plo of he pu dela in Figure 4.3, when he price of he risky securiy is very low and mauriy is imminen, hen a very long posiion in he risky securiy may be required replicae he pu. The explanaion for his apparen conradicion is o be found, once more, in he srong growh of he risky securiy price near he origin. Finally, i is easily verified ha he pricing funcions derived in his secion are all relaed via he pu-call pariy relaionship (3.13) proposed in ecion 3. The fac ha his ideniy holds for he sock price bubble and he bond price bubble (as well as for models where all local maringales are maringales) suppors our claim ha i is he proper formulaion of pu-call pariy. References M. Abramowiz and I. A. egun, ediors. Handbook of Mahemaical Funcions. Dover, D. Becherer. The numeraire porfolio for unbounded semimaringales. Finance och., 5(3): , A. M. G. Cox and D. G. Hobson. Local maringales, bubbles and opion prices. Finance och., 9(4): , F. Delbaen and W. chachermayer. Arbirage and free lunch wih bounded risk for unbounded coninuous processes. Mah. Finance, 4(4): , 1994a.

19 18 HARDY HULLEY F. Delbaen and W. chachermayer. A general version of he fundamenal heorem of asse pricing. Mah. Ann., 300(3): , 1994b. F. Delbaen and W. chachermayer. Arbirage possibiliies in Bessel processes and heir relaions o local maringales. Probab. Theory Relaed Fields, 102(3): , E. Eksröm and J. Tysk. Bubbles, convexiy and he Black-choles equaion. Ann. Appl. Probab., 19(4): , K. D. Elworhy, X.-M. Li, and M. Yor. The imporance of sricly local maringales; applicaions o radial Ornsein-Uhlenbeck processes. Probab. Theory Relaed Fields, 115(3): , L. Heson, M. Loewensein, and G. A. Willard. Opions and bubbles. Rev. Finan. ud., 20 (2): , Janson and J. Tysk. Feynman-Kac formulas for Black-choles-ype operaors. Bull. Lond. Mah. oc., 38(2): , R. A. Jarrow, P. Proer, and K. himbo. Asse price bubbles in complee markes. In M. C. Fu, R. A. Jarrow, J.-Y. J. Yen, and R. J. Ellio, ediors, Advances in Mahemaical Finance, pages Birkhäuser, Boson, 2007a. R. A. Jarrow, P. Proer, and K. himbo. Asse price bubbles in incomplee markes. Johnson chool Research Paper eries No , Cornell Universiy, June 2007b. J. W. Jawiz. Momens of runcaed coninuous univariae disribuions. Adv. Waer Resour., 27(3): , Y. Kabanov and C. ricker. Remarks on he rue no-arbirage propery. In éminaire de Probabiliés XXXVIII, volume 1857 of Lecure Noes in Mahemaics, pages pringer, Berlin, I. Karazas and C. Kardaras. The numéraire porfolio in semimaringale financial models. Finance och., 11(4): , N. N. Lebedev. pecial Funcions and Their Applicaions. Dover, New York, Pal and P. Proer. ric local maringales, bubbles, and no early exercise E. Plaen and D. Heah. A Benchmark Approach o Quaniaive Finance. pringer, Berlin, D. Revuz and M. Yor. Coninuous Maringales and Brownian Moion. pringer, Berlin, hird ediion, Hardy Hulley, chool of Finance and Economics, Universiy of Technology, ydney, P.O. Box 123, Broadway, NW 2007, Ausralia address: hardy.hulley@us.edu.au