Bruno Dupire. Banque Paribas Swaps and Options Research Team 33 Wigmore Street London W1H 0BN United Kingdom

Transcription

1 ARBIRAGE PRICING WIH SOCHASIC VOLAILIY Bruno Dupire Banque Paribas Swaps and Opions Research eam 33 Wigmore Sree London W1H 0BN Unied Kingdom Firs version: March 199 his version: May 1993 Absrac: We address he problem of pricing coningen claims in he presence of sochasic volailiy. Former works claim ha, as volailiy iself is no a raded asse, no riskless hedge can be esablished, so equilibrium argumens have o be invoked and risk premia specified. We show ha if insead of rying o find he prices of sandard opions we ake hese prices as exogenous, we can derive arbirage prices of more complicaed claims indexed on he Spo (and possibly on he volailiy iself). his paper is an expansion of a former version presened a he AFFI conference in Paris, June 199. I am graeful o Nicole El Karoui, Darrell Duffie, John Hull, and my colleagues from SOR (Swaps and Opions Research eam) a Paribas for enriching conversaions. All errors are indeed mine.

2 Arbirage Pricing wih Sochasic Volailiy 1. Inroducion he celebraed Black-Scholes formula is he universal benchmark for opion pricing. I relies on a major assumpion ha he volailiy of he Spo is consan (Black & Scholes [1]), or a mos a known funcion of ime (Meron [15]). Paradoxically, his assumpion is permanenly violaed, which calls for a heory of opion pricing where he volailiy iself is non-deerminisic. Work on sochasic volailiy has flourished in he las few years. A series of papers in 1987 (Hull & Whie [1], Sco [17], Johnson & Shanno [13], Wiggins [18]) brough ino ligh he miigaed resul ha opion prices could be compued 1, bu a he expense of a sringen assumpion: defining a risk premium for he volailiy. I ells us wha prices would be in an economy where agens have some ypes of preferences, bu offers no way o lock hese prices, i.e. o make sure profis if a price deviaed from he heory. Our goal is o expose in a simple fashion he ingrediens allowing for arbirage pricing wih sochasic volailiy, wih no need for any volailiy risk premium o be specified. he curren siuaion is in many ways similar o ha of he ineres rae heory before is recen developmens (Ho & Lee [11], Heah, Jarrow & Moron [10], El Karoui, Myneni & Viswanahan [7]), which mainly consised of a shif from equilibrium consideraions o arbirage argumens. he aim is no o explain he yield curve, bu, aking i for graned and along wih evoluion assumpions, o obain indispuable arbirage prices for derivaive securiies. Our approach sraighforwardly mimics his pah in ha we do no ry o explain sandard opion prices observed in he marke: we merely use he fac ha hey are raded asses, offering (once again along wih sochasic assumpions) he basis for shaping higher order asses, such as forward sar opions or any pah-dependen claim. More precisely, he problem we address here is: finding he fair price for claims coningen on boh he Spo price and is volailiy (or only one of hem), hedging hem, i.e. exhibiing a rading sraegy ha ensures a perfec replicaion. he ouline of his paper is as follows: In Secion, we obain he prices of logarihmic profiles from he sandard Call prices. In Secion 3, we inroduce he forward variance markes and compue heir arbirage free values from he prices of Secion. Secion 4 inroduces sochasic assumpions on he forward variance process, and esablishes he risk neural volailiy and Spo processes. Secion 5 presens opion pricing and hedging, 1 Moreover, hey can be expressed as weighed average of deerminisic volailiy opion prices if he volailiy and he Spo are independen.

3 Arbirage Pricing wih Sochasic Volailiy 3 which involves numerical echniques addressed in Secion 6. Secion 7 explores some exensions, including sochasic correlaion and Secion 8 concludes.. Prices of logarihmic profiles from Call prices ` For he sake of simpliciy, he ineres rae is assumed o be zero a all imes. his hypohesis grealy eases he presenaion, and can be relaxed wih no serious damage o he heory (see Secion 7). We hus consider a fricionless marke wih a riskless asse B (bond) bearing no ineres, and a risky securiy S, he price of which a ime is denoed S (Spo process). We associae o a real valued funcion f and o a ime he financial insrumen f delivering f(s ) a ime. Is value a ime < is denoed f (). A Call opion of mauriy and exercise price (Srike) K is a claim ha promises C K (S ) = Max(S K,0) a ime. I will hereafer be denoed C K,. We assume ha he coninuum of all (C K, ) K, are raded and ha heir prices a ime 0 (i.e. oday) (C K, (0)) K, are consisen wih no arbirage 3. For all, he knowledge of (C K, (0)) K allows us o obain φ, he risk neural disribuion of S, which in urn deermines he price of any European coningen claim (defined hrough he end of period payoff). he firs sep is accomplished by considering a simple porfolio of Calls (known as buerfly in financial markes): C C + C K ε, K, K+ ε, ε his is no so far from realiy, on O..C. Foreign Exchange markes, for example. Marke makers are supposed o give wo-way prices for all srikes and mauriies. 3 his hypohesis can be proved o be equivalen o he following condiions: - For all, C K, ( 0 ) is convex decreasing owards 0 in K, A K =, C ( ) = S 0 and K, For all K, CK, 0 = Max ( S0 K, 0) - For all 1 <, C, C, K K 1 C K, K More generally, his laer condiion mus hold for any convex funcion of S. 1

4 Arbirage Pricing wih Sochasic Volailiy 4 As ε ends owards zero, is profile converges o a Dirac funcion a poin K (Arrow- Debreu securiy associaed wih sae K), and is price converges o φ (K) if finie (if no, here is a lump disribuion on K). We hus ge a simple descripion of φ from (C K, ) K : C φ ( K ) = K K, he equaliy is o be aken in a disribuional sense (C is a convex funcion of K, hence admis lef and righ firs derivaives). We refer o Breeden & Lizenberger [3] for more deails. Denoing P he associaed probabiliy (which densiy funcion is φ ), we can compue he price a ime 0 of any profile f in L 1 (P ) hrough: P f = f ( S) φ( S) ds E [ f ] 0 In paricular, if L, a claim delivering he logarihm of S a ime, belongs o L 1 (P ), P L ( 0 ) = E [ln S ] We hus obain, under appropriae inegrabiliy condiions, he exisence of (L (0)), which is acually he only piece of informaion we need for he sequel. he family (C K, (0)) K, was inroduced because of is daily use by he praciioners, bu only served our purpose as a device in view of obaining (L (0)), which will accoun of he erm srucure of volailiy. A word of cauion is in order: he knowledge of (C K, ) K, is equivalen o he knowledge of (φ ). However, we can go no furher, i.e. deducing he risk neural diffusion process 4, as wo differen processes may generae he same disribuions a all imes. herefore, given only (C K, ), we canno price pah-dependen or American opions, nor can we compue he dynamic hedging parameers. Wihou any addiional assumpion, we can merely price by arbirage (possibly infinie) combinaions of Calls. 3. Arbirage free value of forward variance We now add srucure o he picure by classically inroducing a filered probabiliy space (Ω,,, P), where ( ) is a righ coninuous filraion conaining all P-null ses, 0 being 4 However, recen work shows ha if we resric o one dimensional Io diffusions wih a specified riskneural drif, a unique diffusion process is obained (Dupire [6]).

5 Arbirage Pricing wih Sochasic Volailiy 5 rivial, and a Spo price process S governed by he following sochasic differenial equaion: (3.1) ds S = µ d + σ dw1,, where W 1 is a P-Brownian moion adaped o ( ), and µ and σ may in urn be sochasic processes measurable and adaped o 5. By Io s lemma, ds d ln S = σ S d, which, inegraed beween 1 and, yields which we rewrie as ds 1 ln S ln S = d, 1 1 S σ 1 ds (3.) σ d = (ln S ln S ). he sochasic inegral 1 1 S 1 ds can be inerpreed as he wealh a ime of a sraegy 1 S consising of permanenly keeping one uni of he riskless bond B invesed in he risky asse S beween ime 1 and ime (hrough he possession of 1 S sraegy is self financing, for he ineres rae is zero. unis of S). his he lef hand side of (3.) is he cumulaive insananeous variance of he Spo reurn beween 1 and. I can be reproduced by he porfolio (L L ), associaed wih a 1 dynamic self-financing sraegy consising in keeping unis of he riskless asse B permanenly invesed in he risky asse S beween 1 and. If a conrac delivering σ 1 1 d a is raded a ime, i herefore has a unique possible value, equal o ( L ( ) L ( ) ), which is deermined by arbirage. In oher words, if is marke value were differen, definie profis could hen be generaed. his means ha even if here is no such forward marke, we can neverheless synhesize i; we will herefore assume i exiss. 5 For he sake of simpliciy, inegrabiliy requiremens will no be sressed and we assume a erminal ime for he economy. We refer o Karazas & Shreve [14] for a clear presenaion of sochasic calculus.

6 Arbirage Pricing wih Sochasic Volailiy 6 Le V be a forward conrac on he insananeous variance o be observed a ime. From he above relaion, we ge by differeniaion he value of he conrac V a any ime < : V L () () = As L is a convex funcion of S, V () is indeed posiive (see foonoe 3). From he values (L (0)) of Secion, we are now able o deduce he iniial insananeous forward variance curve: V L = 4. Sochasic assumpions on he forward variance and risk neural processes A his poin, more sochasic assumpions are needed. While i may seem naural o apply hem on he insananeous volailiy, his unforunaely leads o he undesirable necessiy of specifying risk premia. I is simpler, along he lines of Heah, Jarrow & Moron [10], o direcly model he forward variance, which auomaically ensures compaibiliy wih (L (0)). We make, among many oher possible choices, he assumpion ha V () is Lognormal. hus: (4.1) dv () V () = ad + bdw, where a and b are consan 6 and W is anoher Brownian moion adaped o (F ), possibly correlaed wih W 1. We now look for he risk-neural process of several quaniies. By risk-neural process, we undersand expressing he dynamics wih a Brownian Moion ha can be "raded" in a self financing way Risk neural process for he forward variance Defining dw (4.) dw +,, dv () V () a, he equaion (4.1) can be rewrien as b d = bdw, 6 In fac, b can be a deerminisic funcion of ime, which would allow for mean reversion on he insananeous volailiy (see Secion 7.4).

7 Arbirage Pricing wih Sochasic Volailiy 7 where W is a Brownian moion under Q, he P-equivalen probabiliy classically obained by Girsanov s heorem. Applying Io s lemma o (4.): dln V () = b Inegraing beween 0 and leads o d + b dw, (4.3) ln V () = ln V (0) b + b W Indeed, under Q, V is a maringale: Q E [ V ( ) V ] = V, 4.. Risk neural processes for he insananeous variance and volailiy Le us define v V() = σ, he insananeous variance a ime. From (4.3), we obain ln v which can be differeniaed ino (4.4) dln v b = ln V + bw, and finally gives, by Io s lemma (4.5) dv v ln V b = d + b dw ln V = d + b dw, his las expression is he risk neural process of he insananeous variance (he real process is no needed), and provides a way o esimae he parameer b from he single Spo process. he drif erm ln V ensures compaibiliy wih he iniial volailiy erm srucure given by (L ). We can also derive he risk neural process for he insananeous volailiy, which is Lognormal as he insananeous and forward variances. From relaion (4.4),, 1 ln V b b d lnσ = d + dw,

8 Arbirage Pricing wih Sochasic Volailiy 8 and hanks o Io s lemma once again, (4.6) dσ 1 ln V b b = d + dw σ 8, A his poin, we can price any claim coningen on he volailiy. he hedge will be accomplished hrough a dynamic rading of (L ). his allows us o rade W Risk neural process for he Spo he Spo process follows he following sochasic equaion (3.1): ds S = µ d + σ dw1, he ineres rae being zero, he risk neural process for he Spo, which acually governs prices, is (as i will be made clear in Secion 5): (4.7) ds S = σ dw1, wih σ saisfying (4.6) and dw1, dw1, + µ d σ A his sage, a mos imporan poin has o be sressed: o obain he risk neural process for he Spo. Cancelling is drif is no enough, as he insananeous volailiy also has o be replaced by he risk neural one esablished earlier in his secion. 5. Opion pricing and hedging under sochasic volailiy 5.1.Pricing We now make he addiional assumpion ha ( ) is he naural augmened filraion associaed wih W 1 and W. he filraion associaed wih W and W conveys he same informaion and herefore is he same. We define Q as he P-equivalen measure, defined 1

9 Arbirage Pricing wih Sochasic Volailiy 9 hrough Girsanov s heorem 7, under which W and W are Q-Brownian moions. he processes S and V are Q-maringales: we recall from Secion 4 ha wih and ds S = σ dw1, dσ 1 ln V b b = d + dw σ 8 dv () V () = bdw We can hen express dw and dw in erms of ds and dv : (5.1) dw 1, 1 ds = σ S dv (5.) dw, = () bv (), Le A be a coningen claim ha delivers in a payoff dependen on he pahs followed by he Spo and variance, i.e. a square inegrable F -measurable random variable on Ω. We hen define h by: Q h () E [ AF ] Clearly, h is a Q-maringale adaped o (F ). I can herefore be represened (Karazas & Shreve [14]) by: h () = h() 0 + αu dw1, u βu dw, u 0 he sochasic inegrals can be expressed in erms of ds and dv hanks o (5.1) and (5.): α u h () = h() 0 + S ds u + σ 0 0 u u βu dv bv ( u) hey can herefore be inerpreed as a self financing rading sraegy on he Spo and V. he iniial wealh h(0), associaed wih his dynamic rading sraegy, gives exacly he coningen claim A a ime. herefore:, ( u) 7 If W1 and W are correlaed, some care is needed, for insance performing he ransformaion on 1 orhogonalized Brownian moions. However, W and W will exhibi he same correlaion as W1 and W.

10 Arbirage Pricing wih Sochasic Volailiy he price of A a ime 0 is h(0) = E Q [ A F0 ].. here exiss a dynamic hedging sraegy. In he erminology of Harrison & Kreps [8], here exiss a self-financing porfolio which ransforms h(0) ino A a ime. he value of he claim a ime is given by Q (5.3) h() = E [ A F]. where he expecaion is aken over all pahs for S and σ, generaed by he risk neural processes of Secion 4. From he knowledge of (L (0)), we derived he values of (V (0)). Given some sochasic assumpions on V, we were able o compue he value of any pah-dependen opion, and, in paricular, values for sandard European Calls. However, if (L (0)) were obained from he marke Call prices, one should no expec o recover he laer exacly hrough his compuaion 8, as he -dimensional informaion (C K, (0)) K, was compaced ino a 1-dimensional emporal one, (L (0)), in he process 9. he ransversal informaion (C K, (0)) K, i.e. he deviaion from he Black-Scholes prices, may provide a way complemenary o (4.5) o esimae b, he volailiy of he variance. 5..Hedging As we deal wih a one facor model, all forward variances are perfecly correlaed and heoreically any one of hem could be used o hedge volailiy risk. Moreover, any opion would do hough i enails an undesired associaed Spo posiion ha mus be filered ou. his one facor assumpion is blaanly unrealisic for praciioners daily observe a wide variey of wiss in he volailiy curve, as given by he L. I is herefore imporan o elaborae a mauriy by mauriy hedge. his can be performed by locally alering he volailiy curve and compuing he associaed opion price change. We hus obain a heoreical equivalen posiion in L which, when expressed in erm of ( C K, ) K, would lead o a porfolio comprising an infinie number of calls. Indeed in pracice, only a few of hem may be used, even hough leading o a high hedging cos due o imperfec liquidiy. However from a global porfolio risk-managemen viewpoin, his mauriy analysis of parially offseing posiions allows 8 We acually ge Hull & Whie prices for European Calls. 9 his should no be a poin of worry, for i parallels he fac ha he yield curve is compued from Bond prices ha canno be exacly recovered from ha curve. However, i is possible o rerieve he exac price even wih a deerminisic volailiy model! (see Dupire [6])

11 Arbirage Pricing wih Sochasic Volailiy 11 o precisely decompose he volailiy risk hrough he imescale and o decide wheher acion should be aken or no. 6. Numerical mehods 6.1.Mone Carlo In he general case, (5.3) has o be evaluaed numerically, by Mone-Carlo simulaions (Boyle []), which, in is simples form, runs as follows: (a) generaion of a pah for σ (hrough a pah of W ), (b) generaion of a pah for S (hrough a pah of W ), depending on (a) and he correlaion beween he wo Brownian Moions. (c) compuaion of he erminal payoff for hese sample pahs, (d) ieraing housands of imes he firs hree seps and averaging he values obained from (c). Indeed, if A merely depends on σ, sep (b) can be omied. Hedging parameers can be compued a he same ime hrough a small shif of he same pahs Discreisaion he risk-neural process for S and σ can be used o wrie he wo dimensional (in S and σ) parial differenial equaion and discreised i implicily, which is somewha cumbersome, even wih ADI or Hopscoch mehods. We prefer o expose an explici discreisaion of he diffusion process followed by S and σ. wo main feaures are required: (1) he discree scheme mus exhibi he same mean and covariance marix as he coninuous model. () he scheme mus be recombining, prevening an inracable exponenial explosion. o discreise a process, he mos sraighforward approach is o generae from one node (moher node) oher nodes (daugher nodes) as o verify condiion (1). For insance, boh he logarihm of he Spo and he volailiy can be discreised binomially (mean plus or minus one sandard deviaion), leading o a recangular scheme wih a paern of

12 Arbirage Pricing wih Sochasic Volailiy 1 probabiliy reflecing he correlaion ( 1 + ρ 1 ρ on he firs diagonal, on he second 4 4 one). Unforunaely, if on wo following seps he volailiy increases hen decreases, he pah corresponding o an up move of he Spo followed by a down move will no recombine wih he pah obained by he reverse order. On he second sep, he volailiy is higher and he wo pahs will herefore cross, violaing condiion (). here are several ways o overcome his difficuly. Probably he simples is he following: We discreise he insananeous volailiy binomially bu he Spo rinomially. A rinomial scheme has he grea advanage of being able o caer for a wide range of variance. his means ha we can define a recangular grid a each ime sep, wih he Spo discreisaion sep calibraed on he average volailiy level. For exreme values of volailiy, he one ime sep variance of he coninuous ime process canno fi ino he discree scheme bu his can be safely ignored by crysallising he volailiy above he hreshold, for probabiliies in hese porions of he grid are very low. Once he grid is buil and he connecions from a moher o is six daughers are defined, we proceed as usual o value European or American claims. We firs compue is value a he las ime sep, hen proceed backwards in ime o obain a each node he (discouned) expeced value (bounded by he inrinsic value, in he case of American opions), unil we reach he roo node where he desired premium is obained. 6.3.Analyical approximaion In he case of no correlaion, we can consider ha he volailiy rajecory is drawn firs and ge he value of he claim condiioned on ha rajecory now seen as deerminisic. For European Calls, he condiioned price merely depends on he cumulaive variance (sum of he insananeous variance hroughou he life of he opion), so we can express he full price as an expecaion of deerminisic prices wih differen variance parameers. We hen ge he price as a one dimensional inegral: C0 = C BS ( V) ψ ( V) dv 0 where ψis he densiy of V, he cumulaive variance and C BS is he Black-Scholes price wih he associaed consan volailiy. his has been clearly exposed by Hull&Whie [1] who proposed an analyic approximaion based on he aylor expansion up o he second order of he Call price as a funcion of he cumulaive variance. Acually, he same accuracy can be achieved by insead approximaing he densiy by a sum of wo Dirac masses of.5 locaed one sandard deviaion away on each par of he mean. his means

13 Arbirage Pricing wih Sochasic Volailiy 13 ha we obain a quie accurae approximaion by aking he average of wo Black-Scholes prices, which praciioners easily adop. his represenaion of he sochasic volailiy price as a weighed average of deerminisic volailiy prices applies indeed o any European claim (i.e. pay-off coningen on he final value of he spo) bu also exends o slighly more complicaed insrumens, for insance forward sar opions which appear in he popular "clique" or rache srucure. A forward sar opion grans a ime he amoun Max( S S, ). Is sochasic volailiy price 0 1 is he average over values of he cumulaive variance beween 1 and of he deerminisic prices and can be as well approximaed by an average of wo deerminisic prices. However some care should be aken in he pah dependen case in general. For insance, i is emping bu wrong o wrie he sochasic price of a barrier opion as an average of he deerminisic prices in he case of non-zero ineres raes. In effec, wo volailiy rajecories ha exhibi he same cumulaive variance will no necessarily yield he same opion value, for he associaed ime change may desroy cerain spo rajecories. 7.Exensions 7.1.Non-zero ineres raes Secion 3 showed ha he forward variance could be synhesized hrough - he iniial purchase of a porfolio - a dynamic sraegy of holding he asse In he case of sochasic ineres raes, his sraegy will consis in invesing a ime B(, ), value a ime of a zero-coupon bond of mauriy, in he risky asse. he sraegy is no self financing any more bu is cos can be exacly assessed independenly of he model. Unforunaely, he ineress a ime of he proceeds of he porfolio a ime 1, depend boh on he value a ime 1 of he spo and of he ineres raes. We hen need a model of ineres rae, ogeher wih he correlaion beween spo and raes. Once he risk neural processes for he spo, he ineres raes and he volailiy are obained, we make use of he maringale represenaion of Secion 5, now wih hree Brownian moions, o deduce coningen claim prices ogeher wih he replicaing hedge. In he case of deerminisic ineres raes, compuaions run smoohly wih no major changes wih respec o he zero rae case.

14 Arbirage Pricing wih Sochasic Volailiy Sochasic correlaion Mulifacor models now proliferae due o wo reasons. Firsly, more and more absruse cross-marke insrumens are proposed and deal by invesmen banks, noedly "quano" srucures which pay in one currency an inrinsic value expressed in anoher, "bes of" and spread opions. Secondly, o finely une he pricing and hedging of sandard opions, i is imporan o infuse some sochasiciy on parameers oherwise assumed o be consan or deerminisic funcions of ime. When a modelling enails several Brownian Moions, correlaions indeed come ino play and may hemselves be sochasic. We consider he case of wo underlying asses X and Y, raded on he marke as well as European Call opions wrien on hem. he analysis of he preceding secions ells us we can price and hedge pah dependen or American opions on each of he underlying asses in an arbirage free fashion. We now pay aenion o coningen claims wrien on boh asses, in which case prices are affeced by he correlaion as well. We address he problem of pricing and hedging under sochasic correlaion. I should be emphasised ha his correlaion is he one beween he wo Brownian Moions associaed wih X and Y and no he one beween he Brownian Moions of he Spo and of he variance. Praciioners daily manipulae volailiy and increasingly correlaion. However, from a mahemaical sand-poin, he more naural noions are variance and covariance: whenever a correlaion eners a formula, i is wih he produc of he associaed volailiies. I is unforunaely somewha consraining o make on he covariance a sochasic assumpion which is compaible wih he wo volailiies (inferior o heir produc). We herefore prefer o make an assumpion on he sochasic evoluion of correlaion, or more precisely of he forward correlaion. o keep he correlaion beween -1 and 1, we express i as he cosine of a sochasic angle. We assume ha European Calls are raded on boh X and Y, X Y which gives us he forward variances V and V as in Secion 3. Moreover, we assume ha cross-markes insrumens as spread opions or quanos are raded, which gives us he forward covariance V. We define RO, he forward correlaion as he forward covariance divided by he squared roo of he produc of he wo forward variances: RO V X V V Y I can be inerpreed as he insananeous forward correlaion bu, hough compued from arbirage values, i is no in iself an arbirage value! We make, along wih sochasic assumpions on V X Y andv homologous o (4.1), sochasic assumpions on R, or more precisely on he angle of which i is he cosine. We hence deduce he sochasic differenial equaion for V by applying Ios lemma o he relaionship:

15 Arbirage Pricing wih Sochasic Volailiy 15 V = R O X V V Y X Y As V, V and V are maringales under he risk neural probabiliy, we obain heir risk neural dynamics by simply cancelling he drifs. he nex sep consiss in deriving he risk neural dynamics of he insananeous (no forward) quaniies. We saw how o do i for he variance in Secion 4. For he covariance, an analogous compuaion leads from: dv = cdw V o dv v ln V = d + c dw and we are now in a posiion o derive he risk neural process of he insananeous correlaion by anoher applicaion of Ios lemma o he relaion: ρ = v X Y v v he dynamics of he join process (X,Y) is hen fully defined and we can make use of he maringale represenaion of Secion 5 wih now five Brownian Moions, associaed o X, Y, σ, σ and ρ. X Y 7.3.Diffusion-jumps models In a precursor work, Meron [16] considered jumps of random ampliude occurring a random Poisson poins in ime, in addiion o a diffusion Black-Scholes model. he model is no complee and demands he assessmen of risk premia. Once done, sochasic prices are compued as he expecaion of deerminisic prices. Our assumpions do no allow us o make Meron model complee. However, in he simplified case where he ampliude of he jump is deerminisic, compleeness is achieved. 7.4.Mean revering volailiy We now allow volailiy of V o be ime dependen insead of consan as in Secion 4. he risk neural dynamics of V are now:

16 Arbirage Pricing wih Sochasic Volailiy 16 dv () V () = b (, ) dw, Along he lines of Secion 4, we ge α 0 l n v () () bs (,) dw, s = + 1 where α( ) ln V b (,) s ds 0 b and dln v() = ( α() + (,) s dw, s) d + b(,) dw 0, s he presence of a sochasic inegral in he drif of d ln v is roublesome, inroducing in general pah dependency, excep in he case where i can be expressed as a funcion of ln v ( ). As boh he inegral and ln v ( ) are Gaussian, he relaionship has hen o be affine and here is necessarily a funcion of ime λ such ha: b (,) s dw, s = λ() bs (,) dw 0 0, s and by equaing he erms in dw, s and inegraing, we ge: b (, ) = be (,) λ ( s) ds and he sochasic differenial equaion for ln v can be resaed as d ln v( ) = α ( ) λ( )(ln v( ) α( )) d + b(, ) dw, In oher words, he logarihm of he insananeous variance follows an Ornsein- Uhlenbeck process wih a pull-back force of λ. 8. Conclusion In his paper, we sared by assuming ha European Calls of all Srikes and mauriies were raded, and ha heir marke prices were consisen wih no arbirage. From hese prices, we deduced he arbirage price L of coningen claims ha promised ln S a dae. hese prices L were hen shown o be relaed o he insananeous variance of he Spo reurn process and furhermore permied o uniquely se he value of a forward marke on hese insananeous variances, a any mauriy. In oher words, i was shown ha such forward markes could be synhesized from he mere knowledge of he (L ).

17 Arbirage Pricing wih Sochasic Volailiy 17 he use of hese synhesized markes was he key ha could open he door of volailiy hedging and pricing. A his poin, we needed o se assumpions on he variance process o go furher. We focussed on a one facor model (in which all forward markes and curren variance are perfecly correlaed) and derived he risk neural process for boh he insananeous variance and he Spo iself. We obained arbirage free prices ha do no depend on any risk premia nor on a volailiy drif. hey do depend on he erm srucure of volailiy, on he correlaion beween Spo and volailiy, and on he volailiy of he laer. his was achieved by maringale mehods, hanks o he abiliy o - perform he inegral represenaion, hrough he choice of he appropriae filraion, - inerpre i as a self financing sraegy, due o he possibiliy of rading he wo Brownian moions involved. Any claim measurable wih respec o he associaed σ-field can be dynamically spanned and herefore fairly priced. In a more general way, o achieve pricing and hedging, we need he risk neural dynamics of all he sochasic variables. o ensure compaibiliy wih curren known parameers, i is easies o make an assumpion on he forward value of he variable and hen proceed on wih wo seps: firsly obain he risk neural dynamics of he forward value and hen ge he risk neural dynamics of he insananeous value. References [1] Black, F. and M. Scholes (1973). he Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy. 81, pp [] Boyle, P. (1977). Opions: A Mone Carlo Approach, Journal of Financial Economics. 4, pp [3] Breeden, D. and R. Lizenberger (1978). Prices of Sae-Coningen Claims Implici in Opion Prices, Journal of Business, 51, pp [4] Cox, J.C., J.E. Ingersoll, S.A. Ross (1985). A heory of he erm Srucure of Ineres Raes, Economerica, 53, p [5] Duffie, D. (1988). Securiy Markes, Sochasic Models. San Diego: Academic Press. [6] Dupire, B., (1993). Pricing and Hedging wih Smiles, working paper.

18 Arbirage Pricing wih Sochasic Volailiy 18 [7] El Karoui, N., R. Myneni, R. Viswanahan (199). Arbirage Pricing and Hedging of Ineres Claims wih Sae Variables, working paper. [8] Harrison, J.M. and D. Kreps (1979). Maringales and Arbirage in Muliperiod Securiies Markes, Journal of Economic heory. 0, pp [9] Harrison, J.M. and D. Pliska (1981). Maringales and Sochasic Inegrals in he heory of Coninuous rading, Sochasic Processes and heir Applicaions. 11, pp [10] Heah, D., A. Jarrow, A. Moron (1987). Bond Pricing and he erm Srucure of Ineres Raes: A New Mehodology, preprin., Cornell Universiy. [11] Ho,.S., S.B. Lee (1986). erm Srucure Movemens and Pricing Ineres Rae Coningen Claims, Journal of Finance, 41, p [1] Hull, J. and A. Whie (1987). he Pricing of Opions on Asses wih Sochasic Volailiies, he Journal of Finance. 3, pp [13] Johnson, H. and D. Shanno (1987). Opion Pricing when he Variance is Changing, Journal of Financial and Quaniaive Analysis., pp [14] Karazas, I., Shreve S.E. (1988). Brownian Moion and Sochasic Calculus, Springer-Verlag, New-York. [15] Meron, R. (1973). he heory of Raional Opion Pricing, Bell Journal of Economics and Managemen Science. 4, pp [16] Meron, R. (1976). Opion Pricing when Underlying Sock Reurns are Disconinuous, Journal of Financial Economics. 3, pp [17] Sco, L. (1987). Opion Pricing when he Variance Changes Randomly: heory, Esimaion, and an Applicaion, Journal of Financial and Quaniaive Analysis., pp [18] Wiggins, J. (1987). Opions Values under Sochasic Volailiy, Journal of Financial Economics. 19, pp