CUING EDGE. OPION PRICING Calibraing and pricing wih embedded local volailiy models Consisenly fiing vanilla opion surfaces when pricing volailiy derivaives such as Vix opions or ineres rae/ equiy hybrids is an imporan issue. Here, Yong Ren, Dilip Madan and Michael Qian Qian show how his can be accomplished, using a sochasic local volailiy model as he main example. hey also give, for he firs ime, quano correcions in local volailiy models Local volailiy models inroduced by Dupire (1994) and Derman & Kani (1994) are now widely used o price and manage he risks of srucured producs. he dimensionaliy of risks o be simulaneously managed coninues o expand wih he demand for hybrid producs and he growh of markes direcly rading volailiy. he formulaion and implemenaion of local volailiy models in hese higher-dimensional Markov conexs is now becoming an imporan issue. Of paricular ineres o he financial indusry are he accommodaion of sochasic volailiy, sochasic ineres raes, and he pricing of opions on foreign socks, quanos and baskes, in he presence of volailiy skew. he general recipe for models based on a Brownian filraion was provided by Gyöngy (1986), who showed how o associae wih a general Iô process a Markov process wih he same marginal disribuions. We illusrae he required compuaions wih paricular emphasis on he presence of sochasic volailiy as he addiional dimension. Sochasic volailiy in a local volailiy conex permis he exac calibraion of vanilla opions while a he same ime addressing he exposure of financial conracs o he rae of mean reversion in volailiy and he volailiy of volailiy. We implemen he algorihm developed for opions on realised variance and opions on he Vix index. Gyöngy and he maching of one-dimensional marginals Firs, we briefly summarise he imporan resul in Gyöngy (1986). Consider a general n dimensional Iô process of he form: d, d,dw his process gives rise o marginal disribuions of he random variables () for each. Gyöngy hen shows ha here is a Markov process x() wih he same marginal disribuions. he explici consrucion is given by: where: dx b, x d, x dw, x E,, b, x E, x x In he res of his aricle, we will repeaedly find his resul useful in idenifying he local volailiy funcion and he local drif funcion b of he one-dimensional process wih he same marginal disribuions as he rue high-dimensional dynamics. he sochasic local volailiy model Here, we consider an exension of he Dupire (1994) local volailiy model ha incorporaes an independen sochasic componen o volailiy. We develop dynamics under a risk-neural measure where he sock price process (S(), ) has a drif equal o he risk-free ineres rae r less he dividend yield q and in our case an average volailiy given by a deerminisic funcion of he sock price and calendar ime, (S, ). he independen sochasic componen of volailiy is modelled by a sochasic process (Z(), ) ha sars by assumpion a Z() = 1. Hence he evoluion of he sock price is given by: ds r q Sd S, Z SdW S (1) where (W S (), ) is a sandard Brownian moion. We illusrae using a mean-revering lognormal model for Z(), he sochasic componen of volailiy. We suppose ha: d ln Z ln Z d dw Z () where is he rae of mean reversion and is he volailiy of volailiy. here is a long-erm deerminisic drif given by ((), ). Wih a view o inerpreing (S, ) as he average local variance, we force he uncondiional expecaion of Z() o be uniy by requiring ha: 1 e he dependence of volailiy on he sock price is already capured in he leverage funcion (S, ), so we assume ha he Brownian moion driving he sochasic componen of volailiy W Z is uncorrelaed wih he Brownian moion driving he sock price W S. We could exend he model o a non-zero correlaion, which ideally we can observe from he marke, hen some of he skew will come from he correlaion and some from he leverage funcion (noe ha if has no dependence on S, he model is very Heson-like). Here we will only describe he simples model o implemen and calibrae. Oher approaches a making local volailiy sochasic include Derman & Kani (1998) and Dupire (4). he fac ha volailiies are sochasic and capable of rising wihou a movemen in spo prices is now widely recognised and pracical consideraions of risk managemen require an assessmen of he magniude of his exposure for a variey of sruc- 138 Risk Sepember 7
ured producs. his necessiaes he inroducion of an independen sochasic volailiy unrelaed o he sochasiciy inroduced by leverage funcions and permis he assessmen of exposure o he volailiy of volailiy ha is now independenly parameerised here by he coefficien. Oher sochasic volailiy models based on he Heson model and is generalisaions also permi such an assessmen bu do no capure he surface of implied volailiies as precisely as a local volailiy model. his is why here is ineres in inroducing sochasic volailiy ino a local volailiy conex. One-dimensional Markov process for he sock price marginals From Gyöngy (1986), we see ha he one-dimensional Markov process wih he same marginal disribuions as hose of our model is given by: where we mus have ha: LV ds r qsd LV S, Sd W % S (3) K, K, EZ S K For he one-dimensional Markov process (3), he Dupire (1994) and Derman & Kani (1994) relaionship holds beween call opion prices of srike K and mauriy, C(K, ), and he onedimensional leverage funcion LV (S, ). In paricular, we have: LV K, C r q C K qc K C KK Hence, we may now recover our leverage funcion from opion prices provided we have synhesised he funcion (K, ) = E[Z() S() = K] by: K, LV K, K, We now address he issue of he simulaneous soluion of (K, ) and (K, ) saisfying equaion (4). For his, we inroduce he forward join ransiion densiy p(x, y, ) for he logarihm of he sock price X, o reach he level x and he logarihm of he sochasic componen of volailiy Y = ln(z) o reach he level y a ime, saring from S = S() and Z = 1 respecively a ime zero. his densiy elemen saisfies he Kolmogorov forward equaion: p r q e y e x, x p x e y e x, p y y p y p subjec o he boundary condiion ha a ime zero we sar a X = ln(s()) and Y = lnz() = wih probabiliy one. We hen recover he funcion as: K, (4) (5) e y p e K, y, n dy Y p p e K j e y n Y j / p j (6), y, dy j1 j1 he sraegy is o solve he forward equaion in he densiy elemen (5) forward one sep a a ime, recognising ha a he sar we have (S(), ) from he local volailiy equaion (4) since (S(), ) = 1. We hen use (6) a he nex ime sep o deermine he funcion a his poin and ha allows us o infer from (4) he funcion (K, ) a his ime sep. We hen proceed in his fashion hrough ime o recover simulaneously boh he funcions (K, ) and (K, ). Along he way we have also solved for all he densiy elemens p(x, y, ) for all ime poins : K, 1, S, S, px, y, S, px, y, S, px, y,3 S,3... Our paricular example uses alernaing direcion implici mehods o solve for he probabiliy elemen. Wih p denoing p(x, y, ) for ij i j poins (x i, y j, ) on he space ime grid, and using operaor spliing wih a fully implici mehod in boh direcions, we ge: p 1/ ij r q ey j e X i 1, X p 1/ i1, j r q ey j e X i 1, p 1/ i1, j e Y j e X X i 1, 1/ p i1, j e Y j e X i, p 1/ ij e Y j e X i 1, 1/ p i1, j p ij p ij Y Y Y i, j1 p i, j1 p ij p i, j1 p i, j1 1/ p ij Y i, j1 p i, j1 1/ One firs solves for p ij from p ij using a ridiagonal solver, hen p can be solved from p 1/. ij ij he advanage of such an approach is ha he probabiliies solved for in his way are very smooh and sable (jus like he vanilla opion values one calculaes by parial differenial equaion (PDE) mehods). For a similar bu less rigorous numerical approach in he conex of foreign exchange markes and a squareroo volailiy process solved on rinomial rees, we refer he reader o Jex, Henderson & Wang (1999). Once (K, ) has been calculaed, hey can be used in eiher Mone Carlo or PDEs o price securiies ha are sensiive o sochasic volailiy. I is naural o price opions on realised variance using he Mone Carlo approach where he realised variance can be summed up along simulaed pahs. Because he opion on he risk.ne 139
CUING EDGE. OPION PRICING 1 (K, ) calibraed o vanilla opion surface E[Z S = K] 5. 4.5 4. 3.5 3..5. 1.5 1. 1yr 3yr 5yr 1yr.5 5 7 9 1,1 1,3 1,5 1,7 1,9 S Calibraion of SLV o SPX marke implied volailiies Implied volailiy.35.3.5..15.1.5 3mo SLV 3mo mk 1yr SLV 1yr mk 3yr SLV 3yr mk 5yr SLV 5yr mk 1yr SLV 1yr mk 5 7 9 1,1 1,3 1,5 1,7 1,9,1 Srike Vix index is wrien on he sum of expeced fuure variances, he pricing can be easily implemened on he PDE grid. One may evaluae he value of he forward Vix level a ime on he grid using ransiion probabiliies. his is given by: 1 Vix = E h +h Z u σ ( S( u),u)du o do so, we noice ha we have already calculaed (K, ), and Z is simply e Y, one of he dimensions of our PDE grid. We sar a ime + h, and pu he value of Z (S, ) on he grid. We hen use a PDE solver o propagae backwards on a space ime grid, a condiional expecaion a ime + h o he grid poins a ime + h. As we propagae backwards on he grid, he value on he grid becomes he expeced fuure value of he variance a + h (care mus be aken ha he value is no auomaically discouned). We add o his quaniy he value of Z (S, ) a he curren grid poin, and propagae his oal value back o + h, and so on. When we reach ime, we have on he grid he value for he +h expeced forward variance, E [ Z u (S(u), u)du]. Or expressed as a formula, he 3-day expeced variance is calculaed as: 1 +h Var 3 ()= E Z u σ ( u)du h 3 1 = E 3 Z +iσ +i i=1 = 1 3 E Z +1 = 1 3 E Z +1 σ +1 σ +1 3 + E +1 Z +i i= + E +1 Z + σ +i σ + + L + E +9 Z + 3 σ + 3 All ha remains is o define he payou for a call opion on Vix of srike k as (Vix k) + and propagae wih discouning o he valuaion ime. Since we have calibraed our model o all plain vanilla opions, Vix conracs ha can be regarded as he underlying for opions on he Vix index are priced correcly as well, according o he argumen in Carr & Madan (1998). hus our model is able o price volailiy derivaives such as opions on he Vix index consisenly wih vanilla opions and volailiy conracs (variance swaps, Vix fuures). his is done in a conex ha incorporaes leverage effecs on volailiy along wih independen shocks o volailiy. Sochasic local volailiy resuls We consider a lognormal sochasic volailiy wih mean reversion =.5 and a volailiy of volailiy of =.5. We recognise ha calibraion of he vanilla opion surface alone ypically poorly idenifies he rae of mean reversion and he volailiy of volailiy in previous sochasic volailiy models. In our model, he local volailiy funcion may be used o calibrae he enire surface of implied volailiies, a any choice for he mean-reversion rae and he volailiy of volailiy. However, he mean-reversion rae and he volailiy of volailiy do joinly affec he erm srucure of volailiy opions, which we discuss a he end of his secion. Here, we focus aenion on he sensiiviy of volailiy producs, such as opions on realised variance and opions on he Vix index, o hese parameers. We ake vanilla opions a each of 13 mauriies from wo weeks o 1 years, and solve simulaneously for he leverage funcion (K, ) and: K, EZ S K he vanilla opions used are on he S&P 5 index as a Ocober 3, 5. Since he leverage funcion is jus he raio of he usual local volailiy funcion divided by, we presen a graph in figure 1 of he funcion as a funcion of K a he one-, hree-, five- and 1- year poin. We observe ha hese expecaions are convex in K near he a-he-money poin, where hey dip below uniy and hey rise above uniy in boh ails. Addiionally, in figure we presen he marke-implied volailiies and he calibraed implied volailiies from he sochasic local volailiy model. he calibraion speed on a presen-day Linux box was abou seconds for mauriies up o hree years. For mauriies up o 1 years, he corresponding speed was 6 seconds. he use of sochasic local volailiy models arises in assessing he naure of produc exposure o he presence of an independen sochasic volailiy. he model formulaed here inroduces wo addiional parameers conneced wih sochasic volailiy he rae of mean reversion in volailiy and he volailiy of volailiy. As an example, we presen he exposure of a-he-money one-year sraddles on realised variance o hese componens of sochasic volailiy. For he calibraion dae of December 1, 6, we show he resuling valuaions in able A. We firs observe from able A ha he sochasic volailiy model reduces o local volailiy a a zero volailiy of volailiy. 14 Risk Sepember 7
We furher observe ha he a-he-money one-year realised variance sraddle is convex in he volailiy of volailiy and one may herefore be exposed o an undervaluaion if in fac he volailiy of volailiy is iself sochasic. Wih respec o he rae of mean reversion, we have concaviy and an overvaluaion if here is sochasiciy in he rae of mean reversion. In able B, we also presen a sample of six-monh a-he-money sraddles on Vix for a range of and values and he calibraion dae of December 1, 6. We observe ha opions on he Vix rise in value wih an increase in volailiy and are convex in he volailiy. hey fall wih an increase in he rae of mean reversion and are concave in his direcion. Sochasic volailiy wih a zero volailiy of volailiy again reduces o he local volailiy model. As we previously noed, as realised variance opions are wrien on an average of pas squared reurns, hey are calculaed using Mone Carlo from he calibraed model. he opions on he Vix index are based on averages of fuure squared reurns and hese may be calculaed as explained easily by backward propagaion on he PDE grid. While we have given here he qualiaive effec of our model parameers on a seleced se of volailiy derivaives, we have no aemped a horough sudy of simulaneously calibraing for he examples of he Vix surface and he equiy opion surface. A his sage, we have a fairly advanced equiy opion model embedded ino an elemenary, almos Black-Scholes-ype volailiy model. We anicipae ha he join calibraion of boh surfaces will evenually involve muli-facor models for he volailiy and may go as far as incorporaing a local volailiy-of-volailiy surface. We leave hese maers for fuure research. However, i is insrucive o observe ha in our model, as: Z exp 1 e e and he realised variance ill mauriy is given by: 1 Z S, d dw he variance of realised variance quickly decays in he presence of a large mean-reversion rae. In such cases, he innovaions become independen and he variance of he realised variance will be proporional o 1/ as he mauriy of he opion ges longer. On he oher hand, for a negligible mean-reversion rae, one is averaging sums and his does no diminish as fas. hese are simple observaions on he erm srucure of he variance of realised variance. Models wih more facors may exhibi richer erm dynamics. he erm srucure of he variance of realised variance is embedded in he prices of opions on realised variance and hese may be used o calibrae he dynamics of volailiy. Oher exensions of local volailiy Apar from sochasic volailiy exensions of local volailiy models, here are oher exensions well undersood in he Black-Meron-Scholes conex of a consan volailiy ha are somewha more involved when we come o a local volailiy formulaion. hese include he pricing of foreign sock opions in he domesic currency ha is inclusive of boh he sock and exchange rae risk. he pricing of quanos ha shaves ou he exchange rae risk is also well known for consan volailiy, and we show here he precise adjusmens needed in he presence of local volailiy models A. Valuaions of one-year a-he-money sraddles on realised variance Local volailiy Volailiy of volailiy % 5% 1%.5.5.5.99 3.94 1..5.5 3.16 4.3..5.5 3.3 4.45 4..5.5 3.13 4.3 B. Valuaion of half-year a-he-money sraddles on he Vix Volailiy of volailiy % 5% 1%.5 3.67 4.81 6.7 1. 3.67 4.58 6.4. 3.67 4.6 5.53 4. 3.67 3.96 4.71 for boh he sock and he exchange rae. Nex we consider opions on baskes and, finally, he case of sochasic ineres raes. hese opics are aken up in separae subsecions. Foreign sock. Consider a wo-dimensional Markov process for he foreign price of sock and he exchange rae, where boh models are of he local volailiy form: ds r f qsd s S, SdW S dx r r f Xd x X, XdW X dw S dw X S, X,d where r f is he foreign ineres rae, r is he domesic ineres rae and q is dividend yield, while s, x are he wo local volailiy funcions, and (W S (), W X (), ) are sandard Brownian moions wih insananeous correlaion. Le Y() be he domesic price of foreign sock or: Y()= S()X () For he domesic currency of he dollar, his is a dollar-denominaed asse, so i has a risk-neural evoluion on his filraion given by he maringale represenaion heorem, and we may also wrie Y() as an Iô process wih he represenaion: dy r qyd S YdW S X YdW X By Gyöngy s resul, here is a one-dimensional Markov process wih he same one-dimensional marginals and he evoluion: dy r qyd Y, YdW Y where: y y, E s S, x X, S, X, s S, x X, (7) Y y When he volailiies are srike-independen, we recover he wellknown Black-Scholes resul. Similar relaion also holds for he cross exchange rae and he wo corresponding currency exchange raes. he expression (7) is paricularly useful in a join sock and exchange rae local volailiy conex because i relaes he skew of he sock price process and he exchange rae process o ha of he risk.ne 141
CUING EDGE. OPION PRICING foreign sock process. In he absence of such an expression, we are lef wih he need o price a call opion on Y() and use he Dupire equaion o infer he required local volailiy funcion. Equaion (7) provides an alernaive way o direcly infer he local volailiy from he erminal join densiy, which can be available as copula funcions or calculaed by oher means. Quanoed sock. We now noe ha he quaniy S() on which we may wrie quanoed opions is given by: ()= Y() X() S We have he risk-neural law wih respec o he dollar numeraire of X() and Y(). Specifically, we wrie: dy r qyd s S, YdW S x X, YdW X dx r r f Xd x X, XdW X We hen wrie he law for S from an Iô analysis of he raio as: ds s S, SdW S r q r r f s S, x X, S, X, d Applying Gyöngy, we obain a one-dimensional Markov process wih he dynamics: ds r q r r f s S, E x X, S, X, S d (8) s S, SdW S Here, we have an example of how he local volailiy in he exchange rae ransfers ino a localised drif in he one-dimensional law for he quanoed sock. Again, we see if x and have no dependence on X or S, his revers o he well-known quano adjusmen. his resul is non-rivial as here is no more direc way of calculaing he adjusmen. Opions on baskes. We now consider he case of a weighed baske of socks S i, each of which has a local volailiy model. he Brownian moions are correlaed wih insananeous correlaion (S i, S j, ). he sock dynamics are: ds i r q i S i d i S i, S i dw i Now define he baske by he sum wih S() = i w i S i () and we may develop he expression: ds rs w i q i S i i d w i i S i, S i dw i i An applicaion of Gyöngy s resul yields ha: where: K ds r q S, Kq( K, )= E w i q i S i i K, Sd S, SdW ()S()= K (9) w i i S i, S i i i, j w i w j E S i,s j, S i S j i S i, j S j, S K (1) Again, he expression (1) relaes he skews in he individual asses o he baske skew. his is an imporan consideraion in pricing opions on baskes. We also noice ha by seing he weighs o be negaive, he resul also applies o ouperformance or Margrabe opions. For he implemenaion of an approximaion o equaion (1), he reader is referred o Avellaneda e al (). he resuls conained in his secion on he pricing of foreign sock, quano and baske opions are easily derived under he Black-Scholes assumpion, bu are no so obvious in a local volailiy seing. hey no only elucidae he role of skew in he dynamics, bu are also of pracical ineres. Firs, approximaions can be made in hese formulas for pracical use; second, expecaions in hese formulas can be evaluaed using eiher known densiies, or by Mone Carlo and PDE mehods. Once he drif and local volailiies are known, he dimension of he dynamics of he asse is effecively reduced o one. While he dimensionaliy reducion is less necessary in he pricing of simple opions on hese asses, i is crucial in more complex srucures where, for example, PDE mehods become impracical wih he higher dimensionaliy of he problem. Dimensional reducion will also help in many risk managemen applicaions where speed is crucial. No surprisingly, hese are he siuaions where he resuls derived in Black-Scholes seings are ofen used (and abused). Sochasic ineres raes. Our las example involves a sochasic ineres rae economy wih local volailiy sock dynamics. We noice aemps along a similar line have been made by Alan (6). Our approach follows from he same calibraion mehod we used for he sochasic local volailiy case: ds S r q d S, dw S (11) Applying Gyöngy direcly o equaion (11), we observe ha he Markov dynamics embedded in he marginal sock reurn disribuions are given by: ds S E r S S q d S, dw S and hence he effecs of sochasic ineres raes on equiy opions are fully capured on deermining he condiional expecaion of he spo rae given ha he sock prices reach he level S a ime. For calibraion of he local volailiy funcion, we analyse he call prices. In his case, we may wrie he opion prices as: CK, E exp rudu S K Elemenary manipulaions show, on applying he Meyer-anaka formula o he discouned payou and simplifying, ha: K, C qc qkc K KE exp K C KK rvdv r 1 S We herefore need, in addiion o opion prices, he price of he conrac ha pays r()1 S() > K. Using he forward measure, we may wrie his price as: w P, E % r 1 S K K 14 Risk Sepember 7
Hence, we need o develop he join law of r(), S() under he forward measure. By way of an example, we work using a Heah-Jarrow-Moron model for he fixed-income dynamics wih he pure discoun bond prices P(, ) wih dynamics given by: dp(, ) P(, ) = r ( ) dw α α ()d + σ ( α,u)du for a se of correlaed Brownian moions wih correlaions. We now shif o he forward measure denoed E ~ ( ), wih forward dae where is he opion mauriy. he dynamics of he sock price under his measure wih correlaion dw S dz = a are given by: ds S r q,udu S, d (1) S, dz S where Z S is a sandard Brownian moion under he forward measure. he dynamics of he spo rae under he forward measure may be calculaed o be: r f, s, s,w dw ds s, dz a s () (13) where Z are he new forward moion Brownian moions. We may now apply Gyöngy o he join equaions (13) and he soluion o (1) o consruc he wo-dimensional Markov process in (r(), S()) wih he same marginals as hese Iô processes: dr r,r,sd r,r,sdz ds S r f,,r,sd S, dz S,,wdw E s, s, dw ds s, s,wdw ds r E, r r,s S S S r q E,udu r r,s r r,s S S S, o calibrae he local volailiy in he presence of sochasic ineres raes, he Heah-Jarrow-Moron ineres volailiies are firs calibraed o ineres rae markes. We are hen lef wih he sock price and spo rae dimensions wih he dynamics of he spo rae fully deermined. his is very similar o our sochasic local volailiy case. We hen calculae, using he Kolmogorov forward equaion, he densiy disribuion (S, r, ), which is hen used in calculaing he condiional drif of he sock price r()1 S() > K ha goes ino he local volailiy calculaion. We repea his procedure going forward in ime unil all values of local volailiy (S, ) are known. hen he original hybrid model can be implemened using eiher Mone Carlo or a PDE. Conclusion We exploi Gyöngy s (1986) resul o represen he marginal laws of Iô processes by Markov processes in a local volailiy conex. his represenaion leads o generalisaions of he Dupire (1994) and Derman & Kani (1994) equaions for deerminaion of local volailiy or leverage funcions joinly wih cerain condiional expecaions of he Markov process. hese may be solved by PDE mehods ha exrac he join densiy using a soluion of he Green s funcion. he mehod is applied in paricular o a sochasic volailiy model in a local volailiy conex ha permis an exac calibraion of vanilla opions while a he same ime addressing quesions on conrac exposure o he volailiy of volailiy. Calibraion and repricing speeds are observed o be around seconds for mauriies exending up o hree years. For mauriies up o 1 years, he speed was 6 seconds. hese are reasonable speeds, making he mehod boh racable and useful for he invesigaion of high-dimensional exensions o he base local volailiy model. We illusrae he required calculaions for opions on realised variance and opions on he Vix index. We furher give he resuls for pricing opions on foreign socks, quanos and baskes in a local volailiy seing. Finally, we address he problem of calibraing hybrid models wih sochasic ineres raes. Yong Ren was execuive direcor in he equiy analyical modelling group of Morgan Sanley in New York and is currenly a direcor a Saba Principal Sraegies, Deusche Bank in New York. Dilip Madan is professor of finance a he Smih School of Business and serves as a consulan o Morgan Sanley. Michael Qian Qian is execuive direcor in he equiy analyical modelling group of Morgan Sanley in New York. Email: yren@alumni.princeon.edu, dbm@rhsmih.umd.edu, michael.qian@morgansanley.com References Alan M, 6 Localizing volailiies Working paper, Laboraoire de Probabiliés, Universié Pierre e Marie Curie Avellaneda M, D Boyer-Olson, J Busca and P Friz, Reconsrucing volailiy Risk Ocober, pages 87 91 Carr P and D Madan, 1998 oward a heory of volailiy rading In Volailiy, edied by Rober Jarrow, Risk Publicaions Derman E and I Kani, 1994 Riding on a smile Risk February, pages 3 39 Derman E and I Kani, 1998 Sochasic implied rees: arbirage pricing wih sochasic erm and srike srucure Inernaional Journal of heoreical and Applied Finance 1, pages 61 11 Dupire B, 1994 Pricing wih a smile Risk January, pages 18 Dupire B, 4 A unified heory of volailiy In Derivaives Pricing: he Classic Collecion, edied by Peer Carr, Risk Publicaions Gyöngy I, 1986 Mimicking he one-dimensional marginal disribuions of processes having an Io differenial Probabiliy heory and Relaed Fields 71, pages 51 516 Jex M, R Henderson and D Wang, 1999 Pricing exoics under he smile Risk November, pages 7 75 risk.ne 143