Conditional Density Models for Asset Pricing

Transcription

1 Condiional Densiy Models for Asse Pricing Damir Filipović, Lane P. Hughson & Andrea Macrina arxiv: v2 [q-fin.p 11 Nov 211 Swiss Finance Insiue, Ecole Polyechnique Fédérale de Lausanne, Swizerland Deparmen of Mahemaics, Imperial College London, London SW7 2AZ, UK Deparmen of Mahemaics, King s College London, London WC2 2LS, UK Insiue of Economic esearch, Kyoo Universiy, Kyoo , Japan Absrac We model he dynamics of asse prices and associaed derivaives by consideraion of he dynamics of he condiional probabiliy densiy process for he value of an asse a some specified ime in he fuure. In he case where he price process is driven by Brownian moion, an associaed maser equaion for he dynamics of he condiional probabiliy densiy is derived and expressed in inegral form. By a model for he condiional densiy process we mean a soluion o he maser equaion along wih he specificaion of (a) he iniial densiy, and (b) he volailiy srucure of he densiy. The volailiy srucure is assumed a any ime and for each value of he argumen of he densiy o be a funcional of he hisory of he densiy up o ha ime. In pracice one specifies he funcional modulo sufficien parameric freedom o allow for he inpu of addiional opion daa apar from ha implici in he iniial densiy. The scheme is sufficienly flexible o allow for he inpu of various ypes of daa depending on he naure of he opions marke and he class of valuaion problem being underaken. Various examples are sudied in deail, wih exac soluions provided in some cases. 9 November 211 Classificaion: C6, C63, G12, G13. Key words: volailiy surface, opion pricing; implied volailiy; Bachelier model; informaion-based asse pricing; nonlinear filering; Breeden-Lizenberger equaion. 1 Inroducion This paper is concerned wih modelling he dynamics of he volailiy surface. The problem is of grea pracical ineres o raders, and as a consequence has an exensive mahemaical lieraure associaed wih i. In his brief repor, we shall no aemp o review earlier work in he area, bu refer he reader, for example, o Schönbucher (1999), Gaheral (26), Schweizer & Wissel (28a,b), Carmona & Nadochiy (29, 211), and references herein. 1

2 Puinformally, hegeneralidea ofhepaperisasfollows. Wefixanumeraire, andwrie{a } for he value process of some radable financial asse expressed in unis of ha numeraire. We fix a ime T and assume ha no dividends are paid from ime up o T. Leing Q denoe he maringale measure associaed wih he chosen numeraire, we have for T, and A = E Q [A T F, (1.1) C T (K) = E Q[ (A T K) + F, (1.2) where C T (K) denoes he price a ime of a T-mauriy call opion wih srike K. The associaed condiional densiy process {f (x)} for he random variable A T is defined by y Then for he asse price we have f (x)dx = E Q [1{y > A T } F. (1.3) A = xf (x)dx, (1.4) and he corresponding opion prices are given by C T (K) = (x K) + f (x)dx. (1.5) Insead of modelling {A } and hen deermining {C T (K)}, our sraegy is o model he condiional densiy process. Then he underlying asse price process and he associaed opion prices are deermined by (1.4) and (1.5). oughly speaking, he idea is o model {f (x)} in such a way ha i conains some parameric freedom ha can be calibraed o a specified range of iniial opion prices. Models for condiional densiies have been considered in various conexs in finance. These include for example applicaions o ineres raes(brody & Hughson 21a,b, 22, Filipović e al. 21), and o credi risk (El Karoui e al. 21). Alhough mosly differen from wha has previously appeared in he lieraure, our approach o modelling he volailiy surface is similar in spiri in some respecs o ha of Davis (24). Le us consider in more deail he class of asses ha will form he basis of our invesigaion. We inroduce a probabiliy space (Ω,F,P) wih filraion {F }, where P is he physical measure and {F } is he marke filraion. We assume ha price processes are adaped o {F }. We assume he absence of arbirage, and he exisence of an esablished pricing kernel {π } associaed wih some choice of base currency as numeraire. We work in he seing of a muli-asse marke, and do no assume ha he marke is complee. We wrie {A i } i=,1,...,n for he price processes of a collecion of non-dividend-paying radable financial asses. Prices are expressed in unis of he base currency. We refer o asse i as A i. We model he {A i } as Io processes, and for each Ai we require ha π s A i s = E P[ π A i F s (1.6) for s. Such an asse is characerized by is value A i T a some erminal dae T. In some siuaions i is useful o regard he asse as offering a single paymen a T. In ha case 2

3 {A i } <T represens he price process of he asse ha offers such a paymen. In oher siuaions we can consider A T as being a snapsho of he value of he asse a ime T. Usually he conex will make i clear which meaning is inended. Le {A } be a money-marke accoun in he base currency, iniialised o uniy. If we se ρ = π A for, i follows ha {ρ } is a P-maringale. A sandard argumen shows ha {ρ } can be used o make a change of measure. The resuling measure Q is he riskneural measure associaed wih he base currency, and has he propery ha if he price of any non-dividend-paying asse is expressed in unis of he money-marke accoun, hen he resuling process is a Q -maringale. Thus, for each i we have [ A A i i s = A s EQ A F s. (1.7) A similar siuaion arises wih oher choices of numeraire. Specifically, for any nondividend-paying asse A i of limied liabiliy (A i > ), wih price {Ai }, here is an associaed measure Q i wih he propery ha if he price of any non-dividend-paying asse is expressed in unis of A i hen he resul is a Q i -maringale. Thus for s and for all i,j for which he price of A i is sricly posiive we have: [ A j s = A j Ai s EQi A i F s. (1.8) Bearing hese poins in mind, we observe ha he opion pricing problem can be formulaed in he following conex. We consider European-syle opions of a Margrabe ype, for a pair of non-dividend-paying asses A i and A j, where he opion-holder has he righ a ime o exchange K unis of A i for one uni of A j. The payoff H ij of such an opion, in unis of he base currency, is of he form H ij (K) = ( A j KA i ) +. (1.9) The value of he opion a s, expressed in unis of he base currency, is given by Cs(K) ij = 1 [ ( E P π A j ) KA i + Fs. (1.1) π s If A i is of limied liabiliy, hen he opion value, expressed in unis of A i, is a Q i -maringale: C ij s(k) A i s = E Qi [( A j A i + K) F s. (1.11) This relaion can be expressed more compacly as follows. Wrie A s for he price a ime s of a generic asse expressed in unis of a generic numeraire, and C s (K) for he price, in unis of he chosen numeraire, a ime s, of a -mauriy K-srike opion. Then he opion payoff is given (in numeraire unis) by H (K) = (A K) + (1.12) 3

4 The value of he opion a ime s is C s (K) = E Q[ (A K) + F s, (1.13) where Q is he maringale measure associaed wih he numeraire. By generic we mean any choice of a non-dividend-paying asses A i and A j such ha A i is of limied liabiliy. Sandard opions are no included in he caegory discussed above. A sandard call has he payoff H = (A K) + where A is he price of he underlying a ime in currency unis, and K is a fixed srike in currency unis. This is no an opion o exchange K unis of a non-dividend-paying asse for one uni of anoher non-dividend-paying asse. One should hink of he fixed srike as being K unis of a uni floaing rae noe. The only asse ha mainains a consan value (in base-currency unis) is a floaing rae noe and such an asse pays a dividend. The dividend is he ineres rae. In a given currency, by a floaing rae noe we mean an idealised noe ha pays ineres coninuously (raher han in lumps). The associaed dividend is he shor rae. One migh argue ha he srike on a sandard opion is cash, and ha cash is a non-dividend-paying asse: his poin of view leads o paradoxes. In he sandard heory we regard cash as paying an implici dividend, a convenience yield, in he form of a liquidiy benefi equivalen o he ineres rae. In summary, a sandard opion is an opion o exchange he asse wih a floaing-rae noe, which pays a dividend. Thus a sandard opion is a complicaed eniy i is an opion o exchange a cerain number of unis of a dividend-paying asse for one uni of a non-dividend-paying asse. I is more logical firs o examine an opion based on a pair of non-dividend-paying asses. In he lieraure his approach is implicily adoped hrough he device of seing he ineres rae equal o zero. In ha siuaion he floaing rae noe is non-dividend-paying; hus, he seing we operae wihin includes he zero-ineres case. One would like o ackle he general problem of an opion o exchange K unis of one dividend-paying asse for one uni of anoher dividend-paying asse (a sandard foreign exchange opion falls ino ha caegory); bu, unless he dividend (or he ineres rae) sysems are deerminisic, his is a more difficul problem han he one we propose o consider here. The srucure of he paper is as follows. In Secion 2 we derive a dynamical equaion for he condiional densiy, which we call he maser equaion, given by (2.13). An inegral form of he equaion is presened in (2.14), which forms he basis of he soluions presened in laer secions. In Secion 3 we specify he general form we require he volailiy srucure of he condiional densiy o ake, and give a characerizaion of wha we mean by a condiional densiy model for asse pricing. In Secion 4 we consider in deail he class of models for which he volailiy srucure of he condiional densiy is a deerminisic funcion of wo variables. This family of models admis a complee soluion by use of a filering echnique. The resuling asse prices exhibi a sochasic volailiy ha is adaped o he marke filraion bu ha is no in general of he local-volailiy ype. In Secion 5 we consider he case when he volailiy srucure is linear in he erminal value of he asse. In ha case he resuling models are Markovian, and can be calibraed o an arbirary iniial densiy. In Secion 6 an alernaive represenaion of he semi-linear case is presened using a Brownian-bridge echnique. In Secion 7 we show ha he Bachelier model and he geomeric Brownian moion model arise as special cases of he semilinear models, for paricular choices of he iniial densiy. We conclude in Secion 8 wih he calculaion of opion prices. 4

5 2 Condiional densiy processes The marke is undersood as having he seup described in he previous secion. We have a probabiliy space (Ω,F,Q) wih filraion {F }. A non-dividend-paying limied-liabiliy asse is chosen as numeraire, and all prices are expressed in unis of ha numeraire. The measure Q has he propery ha he price process of a non-dividend-paying asse, when expressed in unis of he numeraire, is a maringale. We refer o Q as he maringale measure associaed wih his numeraire. We fix T >, and assume for < T he exisence of an F -condiional Q-densiy f (x) fora T. In ourapplicaions wehave inmind hecases x andx +, bu i isuseful o be flexible as regards he choice of he domain of he densiy funcion. In wha follows we rea he case x, and leave i o he reader o supply he necessary adjusmens for oher domains. Thus we assume he exisence of a densiy process {f (x)}, x, < T, such ha (1.3) holds. I follows ha for any bounded, measurable funcion g(x), x, we have g(x)f (x)dx = E[g(A T ) F. (2.1) We ask ha {f (x)} should have hose properies ha follow heurisically as consequence of he formula obained by formally differeniaing (1.3), namely: f (x) = E[δ(x A T ) F. (2.2) In paricular, we require he following: (a) ha for each x he process {f (x)} <T is an {F }-maringale, and hence f s (x) = E[f (x) F s for s < T; (b) ha he price of he asse can be expressed in erms of he densiy by (1.4) for < T; and (c) ha lim T xf (x)dx = A T. (2.3) We also assume, where required, ha expressions analogous o (1.4) can be wrien for claims based on A T. For example, if C T (K) denoes he price a of a T-mauriy, K-srike European call opion, hen we assume ha (1.5) holds. In he applicaions ha follow, we inroduce a Q-Brownian moion {W }, which we ake o be adaped o {F }, and specialize o he case for which he dynamical equaion of {f (x)} is of he form f (x) = f (x)+ σ f s (x)f s(x)dw s, (2.4) for some process {σ(x)}, f x, < T, represening he volailiy of he densiy. I follows ha A = A + σ A s dw s, (2.5) where A = xf (x)dx, and where he volailiy of A is given by σ A = xσ(x)f f (x)dx. (2.6) For simpliciy, we consider in his paper he case where {W } is a one-dimensional Brownian moion. The exension o he muli-dimensional siuaion is unproblemaic. 5

6 Lemma 2.1. The normalisaion condiion f (x)dx = 1 (2.7) holds for all [,T) if and only if here exiss a process {σ (x)} such ha σ(x) f = σ (x) f (y)dy σ (y)f (y)dy (2.8) for almos all [,T), and we have he iniial condiion f (x)dx = 1. (2.9) Proof. Firs we show ha (2.8) and (2.9) imply (2.7). Saring wih (2.4), we inegrae wih respec o x o obain f (x)dx = f (x)dx+ σs f (x)f s(x)dxdw s. (2.1) Insering (2.8) and using (2.9) we obain (2.7). Conversely, if we assume (2.7) hen (2.9) holds as well, and hence σs f (x)f s(x)dxdw s =, (2.11) from which i follows ha σ f (x)f (x)dx = (2.12) for almos all [,T), and hus ha (2.8) holds for almos all [,T) for some σ (x). Thus, once we specify {σ (x)} and {f (x)}, he dynamical equaion for he densiy he so-called maser equaion akes he form [ f (x) = f (x)+ σ s (x) σ s (y)f s (y)dy f s (x)dw s. (2.13) Lemma 2.2. The condiional desiy process {f (x)} saisfies he maser equaion (2.13) wih iniial densiy {f (x)} and volailiy srucure {σ (x)} if and only if ( f (x)exp σ ) s(x)dz s 1 σ 2 s 2 (x)ds f (x) = ( f (y)exp σ ) s(y)dz s 1, (2.14) σ 2 s 2 (y)ds dy where Z = W + σ s (y)f s (y)dyds. (2.15) 6

7 Proof. Wriing (2.13) in differenial form, we have df (x) = f (x)(σ (x) σ )dw, (2.16) where for convenience we wrie σ = σ (x)f (x)dx (2.17) for he condiional mean of he volailiy. We inegrae (2.16) o obain [ f (x) = f (x)exp (σ s (x) σ s )dw s 1 2 Expanding he exponen, we have [ exp σ s(x)(dw s + σ s ds) 1 2 f (x) = f (x) exp Then we inroduce a process {Z } by wriing [ σ s (dw s + σ s ds) 1 2 (σ s (x) σ s ) 2 ds. (2.18) σ s 2 (x)ds. (2.19) σ s 2 ds and i follows ha Z = W + [ exp f (x) = f (x) exp σ s(x)dz s 1 2 [ σ s dz s 1 2 σ s ds, (2.2) σ s 2 (x)ds. (2.21) σ s 2 ds Applying he normalizaion condiion (2.7) o equaion (2.21) above, we see ha ( ) exp σ s dz s 1 σ 2 s 2 ds ( ) = f (x)exp σ s (x)dz s 1 σ 2 2 s (x)ds dx. (2.22) I follows ha (2.21) reduces o (2.14). Conversely, if f (x) is given by (2.14), hen i is a sraighforward exercise in Io calculus o check ha he maser equaion is saisfied. 3 Condiional densiy models We are in a posiion now o say more precisely wha we mean by a condiional densiy model. In doing so, we are moivaed in par by advances in he sudy of infinie-dimensional sochasic differenial equaions. By a model for he densiy process we undersand he following. We consider soluions of he maser equaion (2.13) saisfying he normalizaion 7

8 condiion (2.7), in conjuncion wih he specificaion of: (a) he iniial densiy f (x); and (b) he volailiy srucure {σ (x)} in he form of a funcional σ (x) = Φ[f ( ),,x. (3.1) For each x and he volailiy σ (x) depends on f (y) for all y. Hence (2.13), hus specified, deermines he dynamics of an infinie-dimensional Markov process. The iniial densiy f (x) can be deermined if one supplies iniial opion price daa for he mauriy dae T and for all srikes K. In paricular, we have C T (K) = E[(A T K) + = (x K) + f (x)dx. (3.2) By use of he idea of Breeden & Lizenberger (1978) we see, in he presen conex, ha for each value of x one has f (x) = 2 C T (x) x 2. (3.3) Here f (x) is no generally he risk-neural densiy, bu raher he Q-densiy of he value a ime T of he asse in unis of he chosen numeraire. For simple pracical applicaions, we ake in common wih much of he lieraure he numeraire o he be he money-marke accoun, and se he ineres rae o zero. Then f (x) is he risk-neural densiy. Once f (x) has been supplied, he choice of he funcional Φ deermines he model for he condiional densiy: we give some examples laer in he paper. In pracice, one would like o specify Φ modulo sufficien parameric freedom o allow he inpu of addiional opion price daa. Wha form his addiional daa migh ake depends on he naure of he marke and he class of valuaion problems being pursued. For example, a sandard problem would be o look a a limied-liabiliy asse and consider addiional daa in he form of iniial opion prices for all srikes in + and all mauriies in he srip < T. We require ha Φ should be specified in such a way ha once he daa are provided, hen Φ is deermined and he maser equaion provides an evoluion of he condiional densiy. Once we have he condiional densiy process, we can work ou he evoluion of he opion price sysem for he specified srip, and hence he evoluion of he associaed implied volailiy surface. The daa do no have o be presened exacly in he way specified in he previous paragraph here may be siuaions where more daa are available (e.g., in he form of barrier opion prices or oher derivaive prices) or where less daa are available (less welldeveloped markes). One should hink of he parameric form of Φ as being adaped in a flexible way o he naure of a specific problem. The philosophy is ha here are many differen markes for opions, and one needs a mehodology ha can accommodae hese wih reasonable generaliy. I should be eviden ha an arbirary soluion o he maser equaion need no be a densiy process for an F -measurable random variable addiional assumpions are required concerning he naure of he volailiy srucure in order o ensure ha f (x) converges in an appropriae sense o a suiable Dirac disribuion. In he examples given, we indicae how his can be achieved, in various siuaions, by he choice of he volailiy srucure. 8

9 4 Models wih deerminisic volailiy srucures We proceed o presen a raher general class of condiional densiy models characerized by a deerminisic volailiy srucure. These are models for which {σ (x)} is of he form σ (x) = v(,x) (4.1) for some deerminisic funcion v(,x) defined for appropriae values of and x. We are able o give a more or less complee consrucion of such models in he form of a weak soluion of he maser equaion. By a weak soluion we mean ha on a probabiliy space (Ω,F,Q) we consruc a filraion {F }, a Brownian moion {W }, and a condiional densiy process {f (x)} saisfying he maser equaion and having he desired properies. More specifically, le T (, ) be fixed, and le f (x) : + be a prescribed iniial densiy. Le v(,x) be a funcion on [,T) saisfying v(s,x) 2 ds < (4.2) for < T and x, and le γ() be a funcion on [,T) such ha limγ() =, lim T γ() v(s,x)ds = g(x), (4.3) T for some inverible funcion g(x) on. We have he following: Proposiion 4.1. Le X have densiy f (x), and le {B } be an independen Brownian moion. Le {F } be he filraion generaed by he informaion process {I } defined by Le {f (x)} be defined by f (x) = and define {W } by seing I = B + [ f (x)exp v(s,x)di s 1 2 f (y)exp[ v(s,y)di s 1 2 W = I v(s, X)ds. (4.4) v2 (s,x)ds v2 (s,y)ds, (4.5) dy E Q [v(s,x) F s ds. (4.6) Then: (a) he random variable X is F T -measurable, and f (x) is he associaed condiional densiy; (b) he process {W } is an {F }-adaped Brownian moion; and (c) for [,T), he densiy process {f (x)} saisfies he maser equaion [ f (x) = f (x)+ f s (x) v(s,x) v(s,y)f s (y)dy dw s. (4.7) 9

10 Proof. I follows from (4.3) ha g(x) = lim T γ()i is F T -measurable. Since g(x) is inverible, we conclude ha X is F T -measurable. Now le he filraion {G } be defined by G = σ({b s } s,x). (4.8) Clearly F G. The random variable X is G -measurable, and {B } is a ({G },Q)-Brownian moion. We inroduce a ({G },Q)-maringale {M } by seing [ M = exp v(s,x)db s 1 2 v 2 (s,x)ds, (4.9) for [,T), and we le he probabiliy measure B be defined by db dq = M. (4.1) G We observe ha di = db +v(,x)d. By Girsanov s heorem, {I } is a ({G },B)-Brownian moion. We noe ha {M 1 } is a ({G },B)-maringale. Le H be a bounded measurable funcion on. Since H(X) is G -measurable, we have he generalised Bayes formula E Q [H(X) F = EB[ M 1 H(X) F E B[ M 1. (4.11) F Le us work ou he righ-hand side of his equaion. To his end we show ha X and I are B-independen for all. In paricular, we show ha he generaing funcion facorises. We have: where E B [exp(yi +zx) (4.12) E B [exp(yi +zx) = E Q [M exp(yi +zx), (4.13) ( = E [exp Q v(s,x)db s 1 v 2 (s,x)ds 2 [ ) +y B + v(s, X)ds exp(zx), (4.14) = E Q[ m exp ( 1 2 y2 ) exp(zx), (4.15) ( m = exp [ v(s,x)+ydb s 1 2 ) [ v(s,x)+y 2 ds. (4.16) By use of he ower propery and he independence of {B } and X under Q, we have E B [exp(yi +zx) = E Q[ m exp ( 1 2 y2 ) exp(zx), (4.17) = exp ( 1 2 y2 ) E Q[ E Q [m exp(zx) X, (4.18) = exp ( 1 2 y2 ) E Q[ E Q [m XE Q [exp(zx). (4.19) 1

11 One observes ha E Q [m X = 1. Thus we obain he desired facorizaion: E B [exp(yi +zx) = exp ( 1 2 y2 ) E Q [exp(zx). (4.2) Now ha we have shown ha X is B-independen of I (and hus of F ), we can work ou he righ-hand side of (4.11). We have: E Q [H(X) F = EB[ M 1 H(X) F E B[ M 1, (4.21) F ( E [H(X)exp B v(s,x)di ) s 1 2 v2 (s,x)ds F = ( E [exp B v(s,x)di ) s 1, (4.22) 2 v2 (s,x)ds F f (x)h(x)exp( v(s,x)di ) s 1 2 v2 (s,x)ds dx = f (y)exp( v(s,y)di ) s 1. (4.23) 2 v2 (s,y)ds dy In paricular, seing H(X) = 1(X x), we deduce ha f (x) is he F -condiional densiy of X, as required. Tha proves he firs par of he proposiion. Nex we show ha {W } is an ({F },Q)-Brownian moion. We need o show (1) ha (dw ) 2 = d, and (2) ha E Q [W u F = W for u. The firs condiion is evidenly saisfied. The second condiion can be shown o be saisfied as follows. For simpliciy, we suppress he superscrip Q. We have [ u E[W u F = E[I u F E E[v(s,X) F s ds F. (4.24) Firs we work ou E[I u F. Since {B } is a ({G },Q)-Brownian moion, we have [ u E[I u F = E B u + v(s,x)ds F (4.25) [ u = E[B u F +E v(s,x)ds F (4.26) = E [ E [ [ u B u G F +E v(s,x)ds F (4.27) = E [ [ u B F +E v(s,x)ds F. (4.28) We inser his inermediae resul in (4.24) o obain [ u E[W u F = E[I u F E E[v(s,X) F s ds F, (4.29) = E [ [ u B F +E v(s,x)ds [ u F E E[v(s,X) F s ds F. (4.3) 11

12 Nex we spli he inegrals in he las wo expecaions by wriing, E[W u F = E [ [ B F +E v(s,x)ds [ u F +E v(s,x)ds F [ E E[v(s,X) F s ds [ u F E E[v(s,X) F s ds F. (4.31) Observing ha and ha E [ [ B F +E v(s,x)ds F = I, (4.32) [ u E v(s,x)ds [ u F = E E[v(s,X) F s ds F, (4.33) we see ha he expecaion E[W u F reduces o [ E[W u F = I E E[v(s,X) F s ds F = I E[v(s,X) F s ds = W. (4.34) Tha shows ha {W } is an {F }-Brownian moion. An applicaion of Io calculus shows ha he densiy process (4.5) saisfies he SDE (4.7). emark 4.1. For simpliciy we have presened Proposiion 4.1 for he case of a onedimensional sae space. I should be clear from he proof how he resuls carry over o condiional densiies on higher-dimensional sae spaces. emark 4.2. For simulaions of he dynamics of he condiional densiy process, he following alernaive represenaion for (4.5) may prove useful: ( f (x)exp [v(s,x) v(s,x)db ) s 1 2 [v(s,x) v(s,x)2 ds f (x) = ( f (y)exp [v(s,y) v(s,x)db ) s 1. (4.35) 2 [v(s,y) v(s,x)2 ds dy The unnormalised densiy he numeraor in (4.35) is for all x condiionally lognormal given X. The simulaion of he densiy requires only he numerical implemenaion of he sandard Brownian moion and of he random variable X. emark 4.3. The densiy models wih deerminisic volailiy srucure presened in Proposiion 4.1 can be exended o a class of models ha saisfy he following sysem: Le f (x) : + be a densiy funcion. A filered probabiliy space (Ω,F,{F },Q) can be consruced along wih (i) an F -measurable random variable X wih densiy f (x), (ii) an {F }-adaped densiy process {f (x)}, and (iii) an {F }-adaped Brownian moion {W }, such ha (a) for some funcion γ() on [, ), (b) for some funcion g(x) ha is inverible ono, and (c) for some suiably inegrable funcion v(,x) on [, ) wih he properies lim γ() =, lim γ() v(s,x)ds = g(x), (4.36) 12

13 he following relaions hold for all [, ). We have Q[X dx F = f (x)dx and [ f (x) = f (x)+ f s (x) v(s,x) v(s,y)f s (y)dy dw s. (4.37) emark 4.4. The consrucion of condiional densiy models admis an inerpreaion as a kind of a filering problem. The process (4.4) plays he role of an observaion process, and (4.6) has he inerpreaion of being an innovaion process. See [1, [15, [16. emark 4.5. I is reasonable on a heurisic basis o expec ha he general deerminisic volailiy srucure model can be calibraed o he specificaion of an essenially arbirary volailiy surface. In paricular, he parameric freedom implici in a deerminisic volailiy srucure coincides wih ha of a volailiy surface. The siuaion is raher similar o ha of he relaion arising in he Dupire (1994) model beween he local volailiy (which is deermined by a deerminisic funcion of wo variables, one wih he dimensionaliy of price and he oher wih ha of ime) and he iniial volailiy surface (which represens a wo-parameer family of opion prices, labelled by srike and mauriy). The precise characerisaion of such relaions consiues a non-rivial and imporan inverse problem. emark 4.6. In he general deerminisic volailiy srucure model, he dynamics of he underlying asse price {A } are of he form da = V dw, wih V = xv(,x)f (x)dx xf (x)dx v(,x)f (x)dx, (4.38) where f (x) is given by (4.5). Thus he absolue volailiy V a ime akes he form of a condiionalcovariancebeween X andv(,x). Ingeneral, {V }isan{f }-adapedsochasic volailiy process ha canno be expressed in he form Σ(,A ) for some funcion Σ(,x), and he dynamics do no consiue a simple diffusion of he Dupire (local volailiy) ype. In he semilinear case v(, x) = σt x/(t ), however, he associaed informaion process is Markovian, and V can indeed be expressed in he form Σ(,A ). 5 Semilinear volailiy srucure We consider in his secion he case where A T has a prescribed uncondiional densiy f T (x), and consruc a family of condiional densiy processes {f T } ha solve he maser equaion (2.13) over he ime inerval [,T). The filraion wih respec o which {f T } is defined will be consruced as follows. We inroduce a process {ξ T } T given by ξ T = σa T +β T, whereσ isaconsan and{β T } T isasandardbrownianbridge, akenobeindependen from A T. We assume ha {F } is given by F = σ ( {ξ st } s ). (5.1) Clearly {ξ T } is {F }-adaped, and A T is F T -measurable. I is shown in Brody e al. (27, 28) ha {ξ T } is an {F }-Markov process (see also ukowski & Yu 27). 13

14 Proposiion 5.1. Leheiniialdensiy f T (x)beprescribed, andlehevolailiysrucure be of he semilinear form σ T (x) = σ T x, (5.2) T for < T, and le {F } be defined by (5.1). Then he process {W } <T defined by W = ξ T 1 T s (σt E[A T ξ st ξ st ) ds (5.3) is an {F }-Brownian moion, and he process {f T (x)}, given by f T (x)exp [ ( T σξt x 1 T 2 f T (x) = σ2 x 2 ) f T(y)exp [ ( T σξt y 1 2 σ2 y 2 ) dy, (5.4) saisfies he maser equaion (2.13) wih he given iniial condiion. T Proof. The fac ha {W } <T is an {F }-Brownian moion is shown in Brody e al. (27, 28). We work ou E[A T ξ T by use of he Bayes formula, E[A T ξ T = xf T (x)dx, (5.5) where f T (x) = f T (x)ρ(ξ T A T = x) f T(y)ρ(ξ T A T = y)dy. (5.6) Here ρ(ξ T A T = x) is he condiional densiy of ξ T given he value of A T. We observe ha ξ T is condiionally Gaussian: [ T ρ(ξ T A T = x) = 2π(T ) exp 1 T 2(T ) (ξ T σx) 2. (5.7) Thus he densiy process is given by [ f T (x)exp 1 2 f T (x) = f T(y)exp[ 1 2 T (ξ (T ) T σx) 2 T (T ) (ξ T σy) 2 dy. (5.8) The las expression can be simplified afer some rearrangemen so as o ake he form f T (x)exp [ ( T σξt x 1 T 2 f T (x) = σ2 x 2 ) f T(y)exp [ ( T σξt y 1 2 σ2 y 2 ) dy. (5.9) T Wih his resul a hand, we can wrie he process {W } <T in he form ( 1 W = ξ T σt xf st (x)dx ξ st )ds. (5.1) T s 14

15 We recall ha he maser equaion (2.13) can be wrien as (2.14). We shall prove ha (5.4) saisfies (2.13) by showing ha (2.14) reduces o (5.4) if we inser (5.3) in (2.2) and choose he volailiy srucure o be (5.2). For he process {Z } in (2.2) we obain Z = ξ T + The nex sep is o inser {Z } in he exponen σ st (x)dz s 1 2 ξ st ds. (5.11) T s σ st (x) 2 ds (5.12) appearing in equaion (2.14). Expression (5.12) can be simplified by use of (5.2) o give ( 1 σtx dξ st + ξ st ) T s T s ds 12 1 (σtx)2 (T s) ds = T ( σxξt T σ2 x 2 ). (5.13) To derive his resul, we make use of he relaion dξ st T s = ξ T T ξ st ds. (5.14) (T s) 2 Wih equaion (5.13) a hand, we see ha (2.14) reduces o (5.4) if (5.2) holds. 6 Semilinear volailiy: Brownian moion approach We proceed o show how he models consruced in Secion 5 are relaed o he densiy models wih deerminisic volailiy srucure reaed in Secion 4. In paricular, we consider a deerminisic semilinear volailiy funcion of he form v(,x) = σ T x, (6.1) T where < T. For his volailiy funcion he process {I } has he dynamics We are hus able o work ou he exponen di = σ T T X d+db. (6.2) v(s,x)di s 1 2 in equaion (4.5) making use of (6.1) and (6.2). We have: v(s,x)di s 1 2 = T [ T σx (T ) v 2 (s,x)ds db s +T(T )σx T s v 2 (s,x)ds (6.3) ds 12 (T s) 2 T2 σ 2 x 2 ds (T s) 2. (6.4) 15

16 The firs inegral gives rise o a ({G },Q)-Brownian bridge {β T } over he inerval [,T. More specifically, we have db s β T = (T ) T s. (6.5) The deerminisic inegral in (6.4) gives ds (T s) = 2 T(T ). (6.6) Armed wih hese resuls, one can wrie (6.3) as follows: v(s,x)di s 1 2 v 2 (s,x)ds = T T σx(σx +β T) 1 2 T T σ2 x 2. (6.7) Le {ξ T } be defined for [,T by ξ T = σx+β T. Then for (6.7) we obain v(s,x)di s 1 2 v 2 (s,x)ds = T ( σxξt 1 2 T σ2 x 2 ). (6.8) We conclude ha he condiional densiy process {f (x)} in (4.5) reduces o he following expression in he case for which he volailiy srucure is given by (6.1): f (x)exp [ ( T σxξt 1 T 2 f (x) = σ2 x 2 ) f (y)exp [ ( T σyξt 1 2 σ2 y 2 ) dy. (6.9) T Fromequaion(6.8)we seeha {ξ T }akes heroleof heinformaionprocess hageneraes {F }. Since {β T } vanishes for = T, he random variable X is revealed a T. Thus X is F T -measurable, and {ξ T } is he process generaing he informaion-based models of Brody e al. (27, 28). This conclusion is suppored by he following consrucion. We consider he measure B defined in (4.1). Under B he process {I } is a Brownian moion over he inerval [,T). We consruc a B-Brownian bridge by use of he B-Brownian moion {I } as follows. On [,T) we se ξ T = (T ) 1 T s di s. (6.1) Nex we recall definiion (4.4) and inser his in he expression above. The resul is ξ T = (T ) db s +(T ) T s 1 v(s,x)ds. (6.11) T s Thefirsinegral definesa({g },Q)-Brownianbridgeover heinerval [,T)whichwedenoe {β T }. For he volailiy funcion we se This leads o and hence o (6.8). ξ T = σx(t )T v(,x) = σ T x. (6.12) T 16 1 (T s) 2 ds+β T, (6.13)

17 7 Bachelier model The Bachelier model is obained by seing A = γw where γ is a consan. We shall show ha he class of models defined by Proposiion 5.1 conains he Bachelier model. We consider a random variable A T associaed wih a fixed dae T. We assume ha A T N[,1/(Tσ 2 ), where N[m,v is he class of Gaussian random variables wih mean m and variance v. In he noaion of Secion 5, we have f T (x) = σ T 2π exp ( 1 2 σ2 Tx 2). (7.1) We recall he process {ξ T } T defined by ξ T = σa T +β T. If A T N(,1/(Tσ 2 )), hen {ξ T } is an {F }-Brownian moion over [,T. This is because {ξ T } is a coninuous Gaussian process wih ξ T = and Cov[ξ st,ξ T = s for s T. We recall he definiion of he Brownian moion {W } associaed wih {ξ T }, given by (5.3). Since for A T N(,1/Tσ 2 ) he process {ξ T } is a Brownian moion, i follows ha E[A T ξ st = 1 σt E[ξ TT ξ st = 1 σt ξ st. (7.2) Thus we see ha W = ξ T. As a consequence we have A = E[A T ξ T = 1 σt E[W T W = 1 σt W. (7.3) Hence o mach he Bachelier model wih an elemen in he class of models consruced in Secion 5, i suffices o se σ = 1/(γT). Proposiion 7.1. The condiional densiy process {ft B (x)} of he Bachelier price process, defined over he inerval [,T), given by [ exp 1 1 (x γw ft B (x) = 2 γ 2 (T ) ) 2 [, (7.4) exp 1 1 (y γw 2 γ 2 (T ) ) dy 2 is a special case of he family of he models of Proposiion 5.1, and is obained by seing f T (x) = σ T exp ( 1 2π 2 σ2 Tx 2), σ T (x) = σ T x, and σ = 1/(γT). (7.5) T Proof. We inser (7.5) in (5.4). Compleion of squares gives f T (x)exp [ ( T σξt x 1 T 2 f T (x) = σ2 x 2 ) f T(y)exp [ ( T σξt y 1 2 σ2 y 2 ) dy T = [ exp 1 2 σ 2 T 2 T [ exp 1 σ 2 T 2 2 T ( x 1 ξ ) 2 σt T ( y 1 σt ξ T ) 2. (7.6) dy ecalling ha ξ T = W, and seing σ = 1/(γT), we obain he desired resul. 17

18 emark 7.1. Le he iniial densiy f T (x), he volailiy srucure σ T (x), and he parameer σ be given as in (7.5). Then he Bachelier condiional densiy {ft B (x)} saisfies he maser equaion (2.13), and {W } coincides wih {ξ T }. emark 7.2. Suppose we chose an asse price model wih a cerain law. Then we know ha we can derive he corresponding condiional densiy process where he relaed volailiy srucure and iniial densiy are specified. We may hen wonder how he condiional densiy ransforms, and wha he new volailiy srucure looks like, if we consider a new law for he asse price model. For insance, we may begin wih he Bachelier model and ask wha is he condiional densiy and volailiy srucure associaed wih a log-normal model. We presen a ransformaion formula for he condiional densiy. This resul allows for he consrucion of a variey of condiional densiy processes from a given one. Le {f (x)} solve (2.13), and le ψ : be a C 1 -bijecion. Then i is known ha if X has condiional densiy f (x) hen Z = ψ(x) has condiional densiy g (z) given by g (z) = f (ψ 1 (z)) ψ (ψ 1 (z)), (7.7) where ψ 1 (z) is he inverse funcion and ψ (x) is he derivaive of ψ(x). The condiional densiy {g (z)} saisfies dg (z) = g (z)[ν(,z) ν dw, (7.8) where ν(,z) = v(,ψ 1 (z)) and ν = v(,y)g (y)dy. (7.9) We see ha he volailiy srucure v(,x) associaed wih f (x) becomes he volailiy srucure ν(,z) associaed wih g (z). For example, consider he Bachelier model where A T N(,1/(Tσ 2 )). The associaed iniial densiy and volailiy srucure are given in (7.5). Now suppose ha Z = exp(a T ), so ψ(x) = exp(x). The price process {A } is hen given by he log-normal model A = exp(γw ), (7.1) where W = ξ T. I follows by (7.7) ha he condiional densiy process {g T (x)} associaed wih he log-normal price process (7.1) is [ exp 1 1 (ln(z) γw 2 γ g T (z) = 2 (T ) ) 2 z [ exp 1 1 (ln(y) γw 2 γ 2 (T ) ) 2, (7.11) dy for z >. Indeed we see ha g T (z) is he log-normal condiional densiy. The associaed volailiy srucure is ν(,z) = v(,ln(z)) = σ T (z) = σ T T ln(z). (7.12) 18

19 8 Opion prices We consider a European-syle call opion wih mauriy, srike K, and price as deermined by equaion (1.13). The price process {A } < of he underlying asse is given by xf A = E Q (x)exp( v(s,x)di ) s 1 2 v2 (s,x)ds dx [X F = f (y)exp( v(s,y)di ), (8.1) s 1 2 v2 (s,y)ds dy where {F } s < is generaed by (4.4). We recall ha {B } is a ({G },Q)-Brownian moion, and ha here exiss an ({F },Q)-Brownian moion {W } such ha di = dw + v d, (8.2) wherehebracke noaionisdefinedby(2.17). Weinroduceaposiive({F },Q)-maringale ( ) Λ = exp v s dw s + 1 v 2 s 2 ds, (8.3) which induces a change of measure from Q o measure Q given by dq dq = Λ. (8.4) F The Q -measure is characerised by he fac ha {I } is an ({F },Q )-Brownian moion. We observe ha by he relaionship ( ) f (x)exp v(s,x)di s 1 v 2 (s,x)ds dx 2 ( ) = exp v s di s 1 v 2 s 2 ds (8.5) = Λ, (8.6) we can use he denominaor in (8.1) o wrie he opion price C s in erms of a condiional expecaion aken wih respec o Q under which {I } is a Brownian moion. Equaion (8.6) is obained by applying he relaionship (8.2). We hen have C s = E Q[ (A K) + F s (8.7) F s [ = E Q Λ 1 (N KΛ ) + F s = E Q [ (N Λ 1 K ) + (8.8) (8.9) where N = Λ = = Λ 1 s E Q [ (N KΛ ) + F s, (8.1) ( xf (x)exp v(s,x)di s 1 2 ( f (x)exp v(s,x)di s ) v 2 (s,x)ds dx, (8.11) ) v 2 (s,x)ds dx. (8.12)

20 Since {I } is an ({F },Q )-Brownian moion, he condiional expecaion simplifies o he calculaion of a Gaussian inegral provided he zero of he max funcion can be compued. Wih hese formulae in hand, we observe ha a closed-form expression for he condiional expecaion can be worked ou in he case of a binary iniial densiy funcion of he form f (x) = q 1 δ(x x 1 )+q 2 δ(x x 2 ). (8.13) Here δ(x) is he Dirac disribuion and q i = Q[X = x i for i = 1,2. I follows ha N = 2 x i q i E (x i ), Λ = where we inroduce he process [ E (x i ) = exp v(s,x i )di s 1 2 i=1 2 q i E (x i ), (8.14) i=1 v 2 (s,x i )ds. (8.15) In he case where he random variable X akes he values x 1 and x 2, he opion price is ( 2 + C s = Λ 1 s E Q s (x i K)q i E (x i )) F s. (8.16) i=1 We observe ha {E(x i )} is posiive and has he propery E Q [E (x i ) = 1. In paricular, we can use {E (x 1 )} o define a change of measure from Q o a new measure Q by seing d Q dq = E (x 1 ), (8.17) F ogeher wih di = dĩ+v(,x 1 )d. Nex we pull E (x 1 ) ou o he fron of he max funcion in equaion (8.16) o obain ( C s = Λ 1 s E [E Q (x 1 ) q 1 (x 1 K)+q 2 (x 2 K) E ) + (x 2 ) F s. (8.18) E (x 1 ) By use of he Bayes formula we express he opion price in erms of Q: C s = Λ 1 s E s (x 1 )E Q [ ( q 1 (x 1 K)+q 2 (x 2 K) E ) + (x 2 ) F s. (8.19) E (x 1 ) For he sake of a simplified noaion we define = E (x 2 )/E (x 1 ), and we observe ha he process { } is an exponenial ({F }, Q)-maringale: ( ) = exp [v(s,x 2 ) v(s,x 1 )dĩs 1 [v(s,x 2 2 ) v(s,x 1 ) 2 ds. (8.2) We wrie = s s so ha C s = Λ 1 s E s (x 1 )E Q [ (q 1 (x 1 K)+q 2 (x 2 K) s s ) + Fs, (8.21) 2

21 and we noe ha s is F s -measurable. One is hus lef wih he ask of finding he range of values of s for which he max funcion vanishes. Then we calculae he Gaussian inegral arising from he condiional expecaion. ecalling ha {Ĩ} is an ({F }, Q)-Brownian moion, we noe ha he logarihm of s is Gaussian. Le Y be a sandard Gaussian variable. Then we can wrie ln( s ) = [v(u,x 2 ) v(u,x 1 ) 2 du Y 1 [v(u,x 2 2 ) v(u,x 1 ) 2 du. (8.22) s By solving for he logarihm of s in he argumen of he max funcion in (8.21), we deduce ha he max funcion is zero for all values y ha Y may ake for which s y ln [ q1 (K x 1 ) q 2 (K x 2 ) s + 1 [v(u,x 2 s 2) v(u,x 1 ) 2 du. (8.23) [v(u,x s 2) v(u,x 1 ) 2 du I follows herefore ha he opion price can be wrien in he form [ C s = Λ 1 1 s E s (x 1 ) q 1 (x 1 K) exp ( 1 y2) dy 2π 2 y 1 +q 2 (x 2 K) s exp ( 1 2π 2 η2 (y) ) dy, (8.24) y where η(y) = y [v(u,x 2 ) v(u,x 1 ) 2 du. (8.25) s The wo Gaussian inegrals can be wrien in erms of he normal disribuion funcion N(x). To highligh he similariy wih he Black-Scholes opion price formula, we define d s = y, d + s = d s + [v(u,x 2 ) v(u,x 1 ) 2 du, (8.26) s so ha one can wrie C s = Λ 1 s E s (x 1 ) [ q 1 (x 1 K)N(d s)+q 2 (x 2 K) s N(d + s). (8.27) We can simplify his expression furher by use of (8.14). Finally, we conclude ha he price of he call opion is given by he following compac formula: C s = (x 1 K) q 1 N(d q 2 q 1 +q 2 s)+(x 2 K) s q 1 s 1 N(d + +q s). (8.28) 2 21

22 Acknowledgemens. The auhors hank T. Björk, Y. Kabanov, M. Monoiyos, M. Schweizer, J. Sekine, J. Zubelli, and paricipans a he Sixh World Congress of he Bachelier Finance Sociey for useful discussions. Par of his research has been carried ou as par of he projec Dynamic Asse Pricing, Naional Cenre of Compeence in esearch Financial Valuaion and isk Managemen (NCC FINISK), a research insrumen of he Swiss Naional Science Foundaion. LPH and AM are graeful o he Ludwig-Maximilians-Universiä München and he Fields Insiue, Torono, for hospialiy. LPH acknowledges suppor from Lloyds TSB Bank Plc, Kyoo Universiy, Shell UK Ld, and he Aspen Cener for Physics. AM acknowledges suppor by ESF hrough AMaMeF gran 2863, and hanks he Vienna Insiue of Finance and EPFL for he simulaing work environmen. Par of his research was carried ou while AM was a member of he Deparmen of Mahemaics, ETH Zürich. eferences. 1. Bensoussan, A. (1992) Sochasic Conrol of Parially Observable Sysems. Cambridge Universiy Press. 2. Breeden, D. T. & Lizenberger,.H. (1978) Prices of sae-coningen claims implici in opion prices. Journal of Business 51, Brody, D. C. & Hughson, L. P. (21a) Ineres raes and informaion geomery. Proceedings of he oyal Sociey A 457, Brody, D. C. & Hughson, L. P. (21b) Applicaions of informaion geomery o ineres rae heory. Disordered and Complex Sysems, eds. P. Sollich, A. C. C. Coolen, L. P. Hughson &. F. Sreaer (New York: AIP). 5. Brody, D. C. & Hughson, L. P. (22) Enropy and informaion in he ineres rae erm srucure. Quaniaive Finance 2, Brody, D. C., Hughson, L. P. & Macrina, A. (27) Beyond hazard raes: a new approach o credi risk modelling. In Advances in Mahemaical Finance, Fesschrif volume in honour of Dilip Madan.. Ellio, M. Fu,. Jarrow & Ju-Yi Yen eds., Birkhäuser. 7. Brody, D. C., Hughson, L. P. & Macrina, A. (28) Informaion-based asse pricing. Inernaional Journal of Theoreical and Applied Finance 11, No. 1, Carmona,. & Nadochiy, S. (29) Local volailiy dynamic models. Finance and Sochasics 13, Carmona,. & Nadochiy, S. (211) Tangen models as a mahemaical framework for dynamic calibraion. Inernaional Journal of Theoreical and Applied Finance 14, No. 1, Davis, M. (24) Complee-marke models of sochasic volailiy. Proceedings of he oyal Sociey London A 46, Dupire, B. (1994) Pricing wih a smile. isk 7,

23 12. El Karoui, N., Jeanblanc, M. & Jiao, Y. (21) Wha happens afer a defaul: he condiional densiy approach. Sochasic Processes and heir Applicaions 12, No. 7, Filipović, D., Tappe, S. & Teichmann, J. (21) Term srucure models driven by Wiener process and Poisson measures: exisence and posiiviy. SIAM Journal on Financial Mahemaics 1, Gaheral, J. (26) The Implied Volailiy Surface: a Praciioner s Guide. Wiley. 15. Krylov, N. V. (198) Conrolled Diffusion Processes. Springer. 16. Lipser,. S. & Shiryaev, A. N. (21) Saisics of andom Processes, I. General Theory, and II. Applicaions. 2nd ediion, Springer. 17. ukowski, M.& Yu, N.(27) An exension of he Brody-Hughson-Macrina approach o modeling of defaulable bonds. Inernaional Journal of Theoreical and Applied Finance 1, No. 3, Schönbucher, P. J. (1999) A marke model for sochasic implied volailiy. Philosophical Transacions of he oyal Sociey A 357, Schweizer, M. & Wissel, J. (28a) Term srucure of implied volailiies: absence of arbirage and exisence resuls. Mahemaical Finance 18, No. 1, Schweizer, M. & Wissel, J. (28b) Arbirage-free marke models for opion prices: he muli-srike case. Finance and Sochasics 12,