Quantitative Strategies Technical Notes

Transcription

1 Quaniaive Sraegies echnical Noes April 1997 Sochasic Implied rees: Arbirage Pricing Wih Sochasic erm and Srike Srucure of Volailiy Emanuel Derman Iraj Kani

2 QUANIAIVE SRAEGIES RESEARCH NOES Copyrigh 1997 Goldman, & Co. All righs reserved. his maerial is for your privae informaion, and we are no soliciing any acion based upon i. his repor is no o be consrued as an offer o sell or he soliciaion of an offer o buy any securiy in any jurisdicion where such an offer or soliciaion would be illegal. Cerain ransacions, including hose involving fuures, opions and high yield securiies, give rise o subsanial risk and are no suiable for all invesors. Opinions expressed are our presen opinions only. he maerial is based upon informaion ha we consider reliable, bu we do no represen ha i is accurae or complee, and i should no be relied upon as such. We, our affiliaes, or persons involved in he preparaion or issuance of his maerial, may from ime o ime have long or shor posiions and buy or sell securiies, fuures or opions idenical wih or relaed o hose menioned herein. his maerial has been issued by Goldman, & Co. and/or one of is affiliaes and has been approved by Goldman Inernaional, regulaed by he Securiies and Fuures Auhoriy, in connecion wih is disribuion in he Unied Kingdom and by Goldman Canada in connecion wih is disribuion in Canada. his maerial is disribued in Hong Kong by Goldman (Asia) L.L.C., and in Japan by Goldman (Japan) Ld. his maerial is no for disribuion o privae cusomers, as defined by he rules of he Securiies and Fuures Auhoriy in he Unied Kingdom, and any invesmens including any converible bonds or derivaives menioned in his maerial will no be made available by us o any such privae cusomer. Neiher Goldman, & Co. nor is represenaive in Seoul, Korea is licensed o engage in securiies business in he Republic of Korea. Goldman Inernaional or is affiliaes may have aced upon or used his research prior o or immediaely following is publicaion. Foreign currency denominaed securiies are subjec o flucuaions in exchange raes ha could have an adverse effec on he value or price of or income derived from he invesmen. Furher informaion on any of he securiies menioned in his maerial may be obained upon reques and for his purpose persons in Ialy should conac Goldman S.I.M. S.p.A. in Milan, or a is London branch office a 133 Flee Sree, and persons in Hong Kong should conac Goldman Asia L.L.C. a 3 Garden Road. Unless governing law permis oherwise, you mus conac a Goldman eniy in your home jurisdicion if you wan o use our services in effecing a ransacion in he securiies menioned in his maerial. Noe: Opions are no suiable for all invesors. Please ensure ha you have read and undersood he curren opions disclosure documen before enering ino any opions ransacions.

3 SUMMARY In his paper we presen an arbirage pricing framework for valuing and hedging coningen equiy index claims in he presence of a sochasic erm and srike srucure of volailiy. Our approach o sochasic volailiy is similar o he Heah-Jarrow-Moron (HJM) approach o sochasic ineres raes. Saring from an iniial se of index opions prices and heir associaed local volailiy surface, we show how o consruc a family of coninuous ime sochasic processes which define he arbirage-free evoluion of his local volailiy surface hrough ime. he no-arbirage condiions are similar o, bu more involved han, he HJM condiions for arbirage-free sochasic movemens of he ineres rae curve. hey guaranee ha even under a general sochasic volailiy evoluion he iniial opions prices, or heir equivalen Black-Scholes implied volailiies, remain fair. We inroduce sochasic implied rees as discree implemenaions of our family of coninuous ime models. he nodes of a sochasic implied ree remain fixed as ime passes. During each discree ime sep he index moves randomly from is iniial node o some node a he nex ime level, while he local ransiion probabiliies beween he nodes also vary. he change in ransiion probabiliies corresponds o a general (mulifacor) sochasic variaion of he local volailiy surface. Saring from any node, he fuure movemens of he index and he local volailiies mus be resriced so ha he ransiion probabiliies o all fuure nodes are simulaneously maringales. his guaranees ha iniial opions prices remain fair. On he ree, hese maringale condiions are effeced hrough appropriae choices of he drif parameers for he ransiion probabiliies a every fuure node, in such a way ha he subsequen evoluion of he index and of he local volailiy surface do no lead o riskless arbirage opporuniies among differen opion and forward conracs or heir underlying index. You can use sochasic implied rees o value complex index opions, or oher derivaive securiies wih payoffs ha depend on index volailiy, even when he volailiy surface is boh skewed and sochasic. he resuling securiy prices are consisen wih he curren marke prices of all sandard index opions and forwards, and wih he absence of fuure arbirage opporuniies in he framework. he calculaed opions values are independen of invesor preferences and he marke price of index or volailiy risk. Sochasic implied rees can also be used o calculae hedge raios for any coningen index securiy in erms of is underlying index and all sandard opions defined on ha index. We hank Indraji Bardhan, Peer Carr, Michael Kamal and Joseph Zou for helpful conversaions. We are also graeful o Barbara Dunn for her careful review of he manuscrip.

4 -1

5 ABLE OF CONENS INRODUCION... 1 LOCAL VOLAILIY SURFACE: HE EFFECIVE HEORY OF VOLAILIY... 4 HE EFFECIVE INERES RAE HEORY... 5 HE EFFECIVE VOLAILIY HEORY... 8 OWARDS A SOCHASIC HEORY OF VOLAILIY he Sochasic Ineres Rae heory he Sochasic Volailiy heory... 1 he HJM Condiions and he Sochasic heory of Ineres Raes he NO-ARBIRAGE CONDIIONS AND HE SOCHASIC HEORY OF VOLAILIY.. 16 SOCHASIC IMPLIED REES... 1 Our Noaion in Discree ime... 4 A Simple Example... 7 Pricing of Some Conracs wih Payoffs Based on Realized Volailiy HEDGING INDEX AND VOLAILIY RISKS IN SOCHASIC VOLAILIY MODELS MORE REALISIC SOCHASIC VOLAILIY MODELS SUMMARY APPENDIX A: EXPECAION DEFINIIONS OF LOCAL VOLAILIY APPENDIX B: MAHEMAICS OF EFFECIVE HEORIES APPENDIX C: LOCAL VOLAILIY VARIAIONAL FORMULAS IN EFFECIVE VOLAILIY HEORIES APPENDIX D: HE NO-ARBIRAGE CONDIIONS AND HE EXISENCE OF HE EQUIVALEN MARINGALE MEASURE IN SOCHASIC VOLAILIY HEORIES APPENDIX E: COMPUING DRIF PARAMEERS IN ARBIRAGE-FREE SOCHASIC VOLAILIY HEORIES

6 INRODUCION he Black-Scholes heory of opions pricing [Black 1973] assumes ha sock prices are sochasic and vary lognormally, bu ha fuure sock volailiies, ineres raes and dividend yields are known and deerminisic. he heory is based on he exclusion of arbirage: an opion s payoff can be replicaed by ha of a ime-varying porfolio of sock and riskless bonds, and mus herefore a any ime have he same value as he porfolio. he mos compelling consequence of his arbirage-free approach is ha opions values are preference-free: invesors of all risk preferences can agree on he unique fair value of an opion. his ranscenden qualiy of he heory has led o is grea pracical success, spawning more han wo decades of inensive research ha exended i o oher underlyers and relaxed is basic assumpions so as o beer mach he observed behavior of opions markes and underlyers. he curren generaion of models, even hough hey rea underlyers more realisically and can be calibraed o prevailing opions marke prices, are sill based on an arbirage-free approach, admiing no arbirage opporuniies in heir heoreical framework. he hisory of ineres rae opions pricing illusraes his developmen. Original models were simple adapaions he Black-Scholes formula wih bonds, raher han socks, as he underlyers. oday, mos ineres rae opions pricing models assume ineres raes hemselves are sochasic and mean-revering, allow for several sochasic facors, and can be calibraed o observed iniial bond prices (and heir volailiies), while consraining fuure ineres-rae evoluion o be arbirage-free. hese models fall ino wo basic families. Equilibrium models 1 consider ineres rae processes depending on one or more sae variables and are derived from general equilibrium argumens. he marke prices of risk are hen derived from associaed characerisics of he yield curve (such as level, slope, curvaure, ec.) or bond prices. In general hese models are no calibraed o all curren bond prices, and may herefore conain iniial arbirage violaions. Arbirage-free models, in conras, are calibraed o all iniial bond prices and also admi no fuure arbirage violaions. hey achieve his in wo differen ways. he firs class use sochasic ineres rae processes ha auomaically generae arbiragefree fuure scenarios, and equip he process wih enough parameers o be forcibly calibraed o he iniial raded bond prices. he second class 3, insead, sar wih exogenously specified sochasic process for bond prices or forward raes. hey hen derive consrains on he evoluion of bond prices or forward raes so ha no fuure arbirages occur. 1. See, for example, Cox, Ingersoll and Ross [1985].. See, for example, Vasicek [1977], Black, Derman and oy [1990]. 3. See, for example, Ho and Lee [1986], Heah-Jarrow and Moron [199]. 1

7 he hisory of sochasic volailiy modeling is shorer bu sill similar o he hisory of sochasic ineres raes. Exising sochasic volailiy models fall ino wo basic families. Complee-marke models 4 specify condiions under which he financial marke is complee in he presence of he volailiy risk. hey posi (if necessary) hypoheical raded volailiy insrumens ha can be used o hedge he volailiy risk and complee he marke. Coningen claim prices in hese models depend criically on he price dynamics of he volailiy insrumens and may also implicily depend on he marke price(s) of volailiy risk. Equilibrium models 5 end o assume (raher han derive) some parameric form for he sochasic evoluion of he index and is volailiy in equilibrium, and hen derive implici opions valuaion formulas which depend on he parameers of he process. he raded opions prices are hen invered for he unknown parameers. Complee-marke models can be somewha arbirary and someimes unnaural because of he specific assumpions hey make abou he hypoheical volailiy insrumens. he equilibrium volailiy models have he drawback ha he choice of he parameric form for he underlying sochasic processes remains largely arbirary. In addiion, i is usually difficul o inver complex and non-linear opions prices o obain he parameers. Finally, ad hoc specificaion of he marke prices of risk can lead o violaions of arbirage 6. In his paper we propose a new arbirage-based approach o coningen claims valuaion wih sochasic volailiy 7, similar o he Heah-Jarrow- Moron (HJM) mehodology for sochasic ineres raes 8. We begin wih a coninuous ime economy wih muliple facors. We work wih local (forward) volailiies, insead of implied volailiies (or opion prices), imposing an exogenous sochasic srucure on he local volailiy surface. he primacy of he local volailiy surface in our work is analogous o ha of he forward rae curve in he HJM framework. Our model akes as given he iniial local volailiy surface and posis a general muli-facor coninuous ime sochasic process for is evoluion across ime. o ensure ha his process is consisen wih an arbirage-free economy we characerize he condiions which guaranee absence of 4. See, for example, Meron [1973], Cox and Ross [1976], Johnson and Shanno [1987], Eisenberg and Jarrow [1994]. 5. See, for example, Wiggins [1977], Hull and Whie [1977], Sein and Sein [1991]. 6. See Cox, Ingersoll and Ross [1985], Heah, Jarrow and Moron [199]. 7. Presened in Risk Advanced Mahemaics for Derivaives Conference, New York, December For aemps in his direcion see, for example, Dupire [1993] and Bruno Dupire in he Proceedings of Risk Derivaives Conference, Brussels, February 1997.

8 explici arbirage opporuniies (a any fuure ime) among he various opion (and fuures) conracs defined and raded on he underlying index. Under hese condiions markes are complee and coningen claim valuaion is preference-free. Unforunaely, in conras o he HJM condiions, here he arbirage-free condiions are complex and non-linear (inegral) equaions, which are difficul o use in heir coninuous form. We hen inroduce Sochasic Implied rees as a discree-ime framework where he volailiy surface undergoes muli-facor (arbirage-free) sochasic variaions. Here we work wih rinomial sochasic implied rees 9. he locaion of he nodes in his kind of ree are fixed bu he ransiion probabiliies vary sochasically as ime changes and index level moves. As ime evolves, he index level moves randomly from node o node while local volailiies (and concurrenly he ransiion probabiliies) flucuae sochasically across he ree. Saring from any iniial node, he fuure movemens of he index and he local volailiy surface mus be resriced so ha oal ransiion probabiliies o all fuure nodes are simulaneously maringales. On he ree, hese maringale condiions can be saisfied by making an appropriae choice of he drif parameer for every fuure node. In he discree ime framework defined by he sochasic implied ree, his process sep-by-sep guaranees absence of arbirage opporuniies among differen opion (and forward) conracs and he underlying index. We draw exensively on he analogy beween ineres raes and volailiy hroughou his paper. We begin by reviewing he concep of he local (forward) volailiy surface and he effecive heory of volailiy which i defines. he local volailiy surface is he opions world analogue of he forward ineres rae curve. Sandard opion prices calculaed using oday s local volailiy surface mach heir marke prices, jus as he bond prices calculaed from oday s forward rae curve mach heir marke prices. he dynamics of sandard opion prices, as defined by oday s local volailiy surface, albei arbirage-free, is based on he assumpion of non-sochasic volailiy, as porrayed by he saic (non-random) naure of he local volailiy surface. his effecive dynamics of opion prices is analogous o he deerminisic, bu arbirage-free, bond price dynamics which resul from a saic forward rae curve. o allow sochasic dynamics we inroduce exogenous sochasic srucure on he effecive heory. his is o say ha we allow general (muli-facor) flucuaions of he local volailiy surface as ime and spo index level change. We impose dynamical condiions which explicily guaranee absence of arbirage among sandard opions, forwards and he underlying index. his process will augmen an effecive heory of volailiy o a full sochasic heory of volailiy in a manner which is he hallmark of he HJM approach o sochasic ineres raes. 9. See Derman, Kani and Chriss [1996], Kani, Derman and Kamal [1996]. 3

9 LOCAL VOLAILIY SURFACE: HE EFFECIVE HEORY OF VOLAILIY We can hink of local volailiy σ K, as he marke s consensus esimae of insananeous volailiy a he fuure marke level K and fuure ime. Local volailiies corresponding o differen fuure marke levels and imes ogeher comprise he local volailiy surface. he local volailiy surface indicaes he fair value of fuure index volailiy a fuure marke levels and imes as implied by he specrum of available sandard opion (and forward conrac) prices. he relaionship beween he local volailiies and opion prices (or implied volailiies) in he opions world is analogous o he relaionship beween he forward raes and bond prices (or yield-o-mauriies) in he fixed income world. We can calculae he forward ineres raes f corresponding o he fuure imes from he specrum of zero-coupon bond prices B wih differen mauriies, using a wellknown formula 1 f db = d B (EQ 1) Similarly, we can calculae he local volailiy σ K, corresponding o he fuure marke level K and ime from he specrum of sandard opion prices C K,, wih differen srikes K and mauriies, using he formula σ K, = C K, C K, + ( r δ)k + δc K K, C K K, K (EQ ) he riskfree discoun rae r and he dividend yield δ in Equaion are boh assumed o be consan. Also, he quaniies which we will discuss hroughou his paper are usually evaluaed a a specific imes or spo prices S, and conain oher explici or implici (deerminisic or sochasic) parameers which we may omi for breviy. For example, he quaniies in Equaions 1 and are evaluaed a he presen ime and spo price, hence f = f ( 0 ), σ K, = σ K, ( 0, S 0 ) ec. Equaions 1 ofen serves as a general definiion for forward raes, regardless of he specific naure of he ineres rae process. I can be shown 10 ha under very general assumpions, forward raes are risk-adjused expecaions of fuure shor raes f = E ( ) r [ ( )] (EQ 3) 10. See, for example, Jamshidian [1993]. 4

10 he expecaion E ( ) [ ] is performed a he presen ime and wih respec o a measure known as he -mauriy forward risk-adjused measure. he precise descripion of his measure is no necessary for our purposes here. he only hing o remember is ha Equaion 1 gives us a direc way for exracing hese expecaions of fuure shor raes from he raded bond prices. Similarly, i can be shown ha local volailiies are risk-adjused expecaions of fuure insananeous volailiies. More precisely, local variance σ K, is a risk-adjused expecaion of fuure insananeous variance σ () a ime as σ K, = E K, ( ) [ σ ( )] (EQ 4) Here he expecaion E ( K, ) [ ] is performed a he presen ime and marke level, and wih respec o a new measure which we call he K- srike and -mauriy forward risk-adjused measure, as described in Appendix A. Again he precise deails abou he measure are unimporan a his poin, only ha hese expecaions can be direcly exraced from he marke prices of sandard opions, as given by Equaion. A saic (non-random) local volailiy surface defines an effecive heory of volailiy in he same way as a saic forward rae curve defines an effecive heory for ineres raes. In an effecive heory, specific expecaions (or inegrals) of some or all of he underlying sochasic variables are exraced from he curren prices of he raded asses, and are subsequenly assumed o remain unchanged as ime evolves. he effecive dynamics which resuls is based on some of he sources of uncerainy being effecively inegraed ou of he full sochasic heory. Le us briefly review he ineres rae case firs. he Effecive Ineres Rae heory In he effecive ineres rae seing, he forward rae curve is evaluaed from he available bond prices a ime 0, and is assumed o remain unchanged hereafer as ime evolves, hus for all 0 : f () = f (EQ 5) As Figure 1 illusraes, his procedure inegraes all sources of ineres rae sochasiciy ou of he original heory, and herefore, he effecive dynamics of he raes in he effecive heory is compleely deerminisic. As physical ime elapses, he spo rae (or shor rae) r() rolls along he saic forward rae curve, coinciding wih he forward rae a ime : r () = f (EQ 6) 5

11 FIGURE 1. In an effecive heory defined by a saic forward rae curve, shor rae follows he insananeous forward raes. rae f r( 1 ) r( ) 0 1 ime he dynamics of zero-coupon bond prices is also deerminisic and is described by a simple backward equaion: d ---- f d B () = 0 (EQ 7) his equaion, wih he aid of Equaion 6, shows ha he asse price dynamics in he effecive heory is local and arbirage-free. Equaion 7 is also he dual of he forward equaion saisfied by he zero-coupon bond prices: d f d B () = 0 (EQ 8) he forward equaion is merely a resaemen of Equaion 1, and holds by he definiion of he forward raes regardless of specific assumpions concerning he behavior of ineres raes. he backward equaion describes propagaion forward in physical ime, for a fixed mauriy. More precisely, i relaes he prices of a - mauriy bond a differen ime poins, wih earlier imes in erms of he laer ones. his is bes undersood by inroducing he forward propagaor (or forward Green s funcion) p,', which relaes bond prices a imes and ', wih ', for any -mauriy bond, hrough a simple relaionship: B () = p ', B ( ' ) (EQ 9) he forward propagaor p,' describes bond price evoluion forward in physical ime, as illusraed by Figure (a). I saisfies he backward and forward differenial equaions wih boundary condiions p, = 1: d ---- f d p', = 0 ; d f (EQ 10) d' ' p', = 0 6

12 FIGURE. Forward propagaor describes he evoluion of bond prices forward in physical ime. Backward propagaor describes evoluion of bond prices backward in mauriy ime. (a) forward propagaor p,' : (b) backward propagaor φ,' : () () ' ' B () B (') B ' () B () and for any ', he composiion relaion: p' (, ) = p (, )p (, ' ) (EQ 11) Similarly, he forward equaion describes propagaion backward in mauriy ime, for a fixed physical ime. More precisely, i relaes he prices of bonds wih differen mauriies, bu a a fixed ime, wih longer mauriy bonds in erms of he shorer mauriy ones. he backward propagaor 11 φ,' relaes zero-coupon bond prices of mauriies and ', wih ', a any fixed ime, using he relaion B () = φ, ' B ' () (EQ 1) he backward propagaor φ,' describes bond price evoluion backward in mauriy ime, as depiced by Figure (b). I also saisfies he forward and backward equaions wih boundary condiions φ, = 1: d f d φ, ' = 0 ; d f (EQ 13) d' ' φ, ' = 0 and, for any ', he composiion relaion 11. he forward and backward propagaors for a saic yield curve are boh simply v equal o he discoun funcion i.e p uv, = φ uv, = exp f τ dτ. u 7

13 φ, ' = φ, φ, ' (EQ 14) he Effecive Volailiy heory In he effecive volailiy seing, he local volailiy surface is calculaed using he specrum of available opion prices (and fuures) a ime 0, and is assumed o remain unchanged hereafer as ime and index price S change: σ K, ( S, ) = σ K, (EQ 15) his procedure amouns o averaging ou all sources of sochasic volailiy, leaving he index price uncerainy as he only source of uncerainy lef wihin he heory. he resuling effecive dynamics only depends on he index price and ime and, as a funcion of hese variables, is deerminisic. As he physical ime elapses and index price S moves, he insananeous volailiy σ() follows along he local volailiy surface, as depiced in Figure 3, coinciding wih he local volailiy a ime and level S : σ() = σ S, (EQ 16) his is consisen wih an equilibrium (effecive) index price process described by he sochasic differenial equaion: ds = µ S d + σ S, dz (EQ 17) where µ is he index s expeced reurn and dz is he sandard Wiener measure a ime. In his process he insananeous volailiy is a known (deerminisic) funcion of ime and index price S. Implied ree models are he discree frameworks for implemening he (effecive) dynamics represened by Equaion 17. he dynamics of FIGURE 3. In an effecive heory represened by a saic local volailiy surface, insananeous volailiy σ() a ime follows he local volailiy a ime and index price S. level local vol 1 σ( 1 ) σ( ) ime level ( 1,S 1 ) (,S ) ime 8

14 sandard opion prices in he effecive heory is described by he backward equaion: 1 ( r δ)s --σ S S, S + + S r CK, ( S, ) = 0 (EQ 18) Since he only remaining source of uncerainy is he index price, he sandard opions are compleely hedgeable (using index as he hedge) wihin he effecive heory. Equaions 16 and 18 hen show ha he opion price dynamics in his heory is arbirage-free. Equaion 18 is also he dual of he forward equaion saisfied by he sandard opion prices: 1 ( r δ)k --σ K, K + + δ K K CK, ( S, ) = 0 (EQ 19) his forward equaion is he same as Equaion and holds by he definiion of local volailiy, regardless of any specific assumpions abou he behavior of volailiy. he forward propagaor p,s,',s' describes he relaionship beween he opion prices a he wo poins (, S) and (', S'), wih ', for any K-srike and -mauriy sandard opion, hrough he relaion C K, ( S, ) = p S',,, S' C K, ( ', S' ) ds' 0 (EQ 0) he forward propagaor p,s,',s' describes opion price evoluion forward in ime and index price, as illusraed by Figure 4(a). We can define he forward ransiion probabiliy densiy funcion p(,s,',s') in erms of he forward propagaor as p(,s,',s') = e r('-) p,s,',s'. I describes he oal probabiliy ha he index price will reach level S' a ime ', given ha he index price a ime is S. he mahemaical properies of p,s,',s' and p(,s,',s') are discussed in Appendix B. he backward propagaor Φ K,,K',' describes he relaionship beween prices of wo sandard opions corresponding o srike-mauriy pairs (K,) and (K','), wih ', a a fixed ime and index price S, as C K, ( S, ) = Φ K,, K', ' C K', ' ( S, ) dk' 0 (EQ 1) 9

15 FIGURE 4. Forward propagaor describes he evoluion of sandard prices in physical ime and index price. Backward propagaor describes he evoluion of opion prices in mauriy ime and srike price. (a) forward propagaor (b) backward propagaor p,s,',s' : Φ K,,K',' : (K,) (,S) (, S) (', S') (K',') (K, ) C K, (',S 1 ') C K1',' (,S) C (,S) K',' C K, (',S ') C K, (,S) C K, (,S) C K, (',S n ') C Kn',' (,S) As Figure 4(b) illusraes, We can also define he effecive heory backward ransiion probabiliy densiy funcion Φ(K,,K',') in erms of he backward propagaor as Φ(K,,K',') = e δ(-') Φ K,,K','. Appendix B discusses some of he mahemaical properies of Φ K,,K',' and Φ(K,,K','). We can use Equaion 17, eiher by performing simulaions or by using implied ree mehods, o price and hedge complex opions, wih he knowledge ha he sandard opions iniially used o derive he local volailiy surface will have model prices which mach heir marke values. In spie of his calibraion, if he volailiy has a subsanial sochasic behavior, he prices and hedge raios of mos opions wih pahdependen or volailiy-dependen payoffs will no be accuraely represened by he effecive heory resuls. he reason is simply ha effecive heory resuls are based on he assumpion ha local volailiies are saic or, equivalenly, ha he insananeous volailiy is subsanially a funcion of he marke level (and ime). his is a good assumpion in siuaions where he volailiy exhibis srong correlaion o he marke level and, hence, can be viewed predominanly as a funcion of i. For mos equiy index opion markes, for example, his more or less holds, specially for shorer daed opions. On he conrary, in he currency opions markes or in longer daed equiy (and mos oher) opions markes, he volailiy is predominanly sochasic and he effecive heory of saic local volailiies is no valid. We mus herefore move owards a full sochasic framework by allowing general muli-facor sochasic variaions of he volailiy surface. 10

16 OWARDS A SOCHASIC HEORY OF VOLAILIY o allow for sochasic dynamics we mus inroduce exogenous sochasic srucure on he effecive heory. In general, here are few resricions on he choice of his srucure. One imporan resricion, which is he cornersone of he arbirage framework, is he absence of any explici fuure arbirage opporuniies in he final sochasic heory. Anoher resricion is how close he number or he behavior of he sochasic facors are o wha is empirically observed. For now, we will consider very general (bu sufficienly regular) sochasic srucures and discuss he condiions which mus be imposed upon hem o guaranee he absence of arbirage. Le us briefly examine he sochasic ineres rae heory firs. he Sochasic Ineres Rae heory Figure 5 illusraes he dynamics of he forward raes in he sochasic framework. Here, he forward rae curve is allowed o flucuae sochasically wih several independen sochasic facors represened by Brownian moions W i, i = 1,...,n, wih facor volailiies ϑ i () generally depending on boh mauriy and ime, according o he sochasic differenial equaion: n i df () = α ()d + ϑ i = 1 ()dw i (EQ ) In he family of processes described by Equaion, he volailiy coefficiens ϑ i () reflec he sensiiviies of specific mauriy forward raes o he random shocks inroduced by he Brownian moions W i. hese coefficiens are lef unresriced, excep for mild measurabiliy and inegrabiliy condiions, and can depend on he pas hisories of FIGURE 5. In a sochasic ineres rae heory spo rae r() follows he insananeous forward rae f (). f ( 1 ) rae f ( 0 ) r( 1 ) r( 0 ) 0 1 ime 11

17 he Brownian moions W i. he drif coefficiens α () mus also saisfy mild measurabiliy and inegrabiliy condiions, bu mus be furher consrained by he no-arbirage requiremen. he spo rae a ime, r(), is he insananeous forward rae a ime, i.e, r () = f (). he sochasic inegral equaion saisfied by he spo rae is found by inegraing Equaion and evaluaing he resul a =. I is given by r () = f ( 0) + α ( u) du + 0 n i = 1 0 ϑ i ( u) dw i u (EQ 3) I has been argued by Heah, Jarrow and Moron, ha here will be no explici arbirage opporuniies in he heory defined by Equaion 3 if (and only if) he drif coefficiens are of he form: n i i α () = ϑ () ϑ u ()u d + λ i () i = 1 (EQ 4) Here λ i (), i = 1,..., n, denoe he marke prices of risk, which can no explicily depend on mauriy bu are oherwise arbirary. Under hese condiions, hey have shown ha markes are complee and coningen claims prices are independen of he marke prices of risk. he Sochasic Volailiy heory Our goal is o inroduce a similar sochasic srucure on he local volailiy surface. o do so, we allow he surface o undergo sochasic flucuaions wih several independen sochasic facors, W 0, W 1,...,W n, based on he following sochasic differenial equaion 1 : dσ K, ( S, ) = α K, ( S, )d + θ i K, ( S, )dw i n i = 0 (EQ 5) We include W 0 =Z, he index price s source of uncerainy, among he facors so ha he sochasic variaions of he local volailiy surface may depend on he prevailing marke level. he family of processes of Equaion 5 defines a muli-facor dynamics for he local volailiy surface, as illusraed by Figure 6. hese processes can be inegraed, 1. he variable S in he expression for local volailiy σ Κ,Τ (, S) is included for noaional purposes and does no imply dependence solely on he spo index level. In fac, local volailiies generally depend on he enire hisory of he index price and oher sochasic facors. Aside from ime and index price S, all oher variables have been explicily omied from expressions for local volailiies, drifs and facor volailiies. 1

18 FIGURE 6. In a sochasic volailiy heory insananeous volailiy σ() follows he local volailiy σ S, (,S ), a ime and index price S. local vol level σ( 0 ) σ( 1 ) ime 0 1 ime level ( 1,S 1 ) (,S ) saring from a fixed (non-random) iniial local volailiy surface σ K, (0,S 0 ) a ime = 0, as σ K, S, ( ) = σ K, ( 0, S 0 ) + α K, ( us, u ) du + 0 n θ i K, us, u 0 i = 0 ( ) dw i u (EQ 6) he facor volailiy θ i K, ( S, ) reflecs he sensiiviy of local volailiies σ K, (,S), across he whole surface, o he shock inroduced by he Brownian moion W i. Excep for mild measurabiliy and inegrabiliy condiions 13, he family of facor volailiies are unresriced, generally depending on ime and index price, and on he facors or heir pas hisories. However, for he sake of breviy we have omied explici references o all variables oher han ime and index price S from he expressions for facor volailiies, and we will do he same for oher quaniies such as drif coefficiens and local volailiies. he spo volailiy (or insananeous volailiy) a ime, σ(), is he insananeous local volailiy a ime and level S, i.e σ() = σ S (, ), S (EQ 7) I describes he variabiliy of index price reurn process, as given by he differenial equaion 13. he facor volailiy funcions θ i K, ( S, ) are assumed o be posiive, adaped and joinly measurable wih respec o he Borel σ-algebra resriced o 0, for some fixed maximum ime *. hey mus also saisfy ( θ i K, ) ( us, ) u u d <, i = 0,...,n, o 0 assure regulariy of spo volailiy process, and cerain addiional inegrabiliy condiions o assure regulariy of he sandard opion price processes. 13

19 ds = µ S d + σ()dw 0 (EQ 8) or is inegral form S = S 0 + µ u S u du σ( u)s u dw 0 u (EQ 9) where µ is he index s expeced reurn. Seing =and K=S in Equaion 6 we find he sochasic inegral equaion saisfied by he spo volailiy as σ () = σ S, ( 0, S 0 ) + α S, ( us, u ) du + 0 n θ i S, us, u 0 i = 0 ( ) dw i u (EQ 30) he drif coefficiens α K, ( S, ) mus also saisfy mild measurabiliy and inegrabiliy condiions, bu hey mus be furher resriced by he requiremen ha he sochasic heory described by Equaions 8 and 30 disallows explici arbirage opporuniies among he sandard opions, forwards and heir underlying index. his is similar o he HJM arbirage condiions on he spo rae process. Le us briefly examine (a variaion of) he HJM argumen below. he HJM Condiions and he Sochasic heory of Ineres Raes he bond price dynamics corresponding o he forward rae process of Equaion 85 is, by applying Io s lemma, described by he sochasic inegral equaion δb () db () = r ()B ()d d δ f u () f () du+ u 1 -- δ B () δ d f u ()δf u' () f () df u u' () dudu' (EQ 31) δ he symbol here denoes he variaional (or funcional) derivaive δ f u wih respec o he funcion f evaluaed a u. he firs erm in his equaion describes precisely he effecive heory bond price dynamics resriced o he fixed forward rae curve f () a ime. he nex wo erms describe he bond price dynamics resuling from he sochasic variaions of he effecive heory (defined by f ()) during he nex infiniesimal ime inerval d. I follows from he definiion of he forward raes (Equaion 1) ha he price of a -mauriy zero-coupon bond wih uni face, a ime, is given by 14

20 FIGURE 7. Sensiiviy of he forward and backward propagaors p,' and φ,' o he sudden changes of he forward rae f u. (a) forward propagaor (b) backward propagaor u u+du ' ' u u+du -1 u ' ' -1 u B () = exp f u ()u d (EQ 3) From his expression i is simple o see ha for any u ( u ): δb () = B δ f u () () (EQ 33) Anoher way of seeing his is by noicing how he forward and backward propagaors, p,' and φ,', corresponding o an oherwise fixed (non-random) forward rae curve, respond o sudden changes of a specific forward rae f u along he curve. I is simple o see ha p,' saisfies he following relaion, as depiced in Figure 7(a): δp( ', ) = p( u, )pu' (, ) = p' (, ) δ f u (EQ 34) and, as shown in Figure 7(b), ha φ,' saisfies he relaion: δφ, ' = φ δ f, u φ u' = u, φ, ' (EQ 35) hese relaions combined, respecively, wih Equaions 9 and 1, again lead o Equaion 33. he second order varia- Similarly, we can show ha for ional derivaives are given by: u u' 15

21 δ B () = δ f u ()δf u' () B () (EQ 36) he special f u -independen form of variaional relaions can be direcly aribued o he special form of he funcional relaionship beween he zero-coupon bond prices and he forward raes as described by Equaion 3. his feaure underlies he special simpliciy of no-arbirage condiions in he HJM framework. Using Equaions, 33 and 36 inside Equaion 31 we find n db () r ()d i ϑ B () u ()u d i = dw i = 0 n u i i α u () ϑ u () ϑ v ()v d du d i = 1 (EQ 37) If he drif coefficiens α () saisfy he no-arbirage condiions of Equaion 4 for some se of marke prices of risk λ i (), hen Equaion 37 shows ha in erms of he equivalen measure dw i = dw i + λ i d, defined by he Brownian moions W i W i = + λ i ( u) du, i = 1,..., n, he dynamics of zero-coupon bond 0 prices is: db () = r ()d B () n i = 1 i ϑ u ()u d dw i (EQ 38) herefore, {dw i ; i = 1,...,n} defines an equivalen maringale measure under which he rescaled bond prices B () exp ru ( ) du for all 0 mauriies are joinly maringale. Under his measure he ineres rae coningen claims prices are independen of he marke prices of risk and, hence, remain preference-free. HE NO-ARBIRAGE CONDIIONS AND HE SOCHASIC HEORY OF VOLAILIY he sandard opion prices C K, (,S) are funcionals of he local volailiies a ime and marke level S, jus as bond prices B () are funcionals of he forward raes a ime. As a resul, he dynamical variaions of he local volailiy surface induce correpsonding dynamical variaions of he sandard opion prices. During a ime inerval d, he index price moves and he local volailiies also change. We can hink of he local volailiy changes as comprised of wo componens. A predicable componen, due o movemens of ime and index price resriced o he saic local volailiy surface σ K, (,S) a ime and level S, and a non-predicable (sochasic) componen due o dynamic flucuaions away from his surface. I is 16

22 somewha simpler, bu enirely equivalen, o work wih he ransiion probabiliies, insead of opion prices. he ransiion probabiliy, P K, (,S), describes he oal probabiliy ha he index price will reach level K a ime, given ha he index price a ime is S, when boh he index price and volailiy are sochasic. I is relaed o he opion prices C K, (,S) hrough a general and well-known 14 formula: P K, ( S, ) = e r ( ) CK K, ( S, ) (EQ 39) he dynamical evoluion of ransiion probabiliies P K, (,S) based on he local volailiy process of Equaion 6 is given by he sochasic inegral equaion: P dp K K, (EQ 40), P K, 1 P µ ()S --σ S ()S K, + + S P d σ ()S K, = + dw 0 () + S ( S, ) δp K, 0 δσ K', ' dσ K', ' d K' d ' δ P K, dσ K', ' d 0 δσ K', ' δσ K'', '' σ K'', '' dk' dk'' d ' d'' All he probabiliy and local volailiy expressions in his equaion are evaluaed a (,S). he firs erm describes he effecive dynamics of he ransiion probabiliies P K, (,S) resriced o he fixed local volailiy surface σ K, (,S), prevailing a ime and level S. he bracke symbol, [ ] ( S, ), herefore, expresses he fac ha in his erm he fuure volailiy is a deerminisic funcion of he fuure ime and marke level K, given by σ K, (,S) viewed as funcion of hese wo variables. he nex wo erms describe he dynamical variaions of he ransiion probabiliies resuling from he sochasic flucuaions of he local volailiy surface during he nex insan of ime d. Conrary o Equaion 3, in general here are no explici expressions describing he funcional relaionship beween opion prices and local volailiies. herefore, we can no direcly compue he variaional 14. See Breeden and Lizenberger [1978]. 17

23 derivaives in Equaion 40. Insead, we can look a he variaions of he forward and backward ransiion probabiliies wih respec o he specific local volailiies. As shown in Appendix C and illusraed in Figure 8, he forward ransiion probabiliy p(,s,',s'), associaed wih he non-random local volailiy surface σ K, (,S) prevailing a ime and spo price S, has he following variaional derivaive wih respec o he local volailiy σ v,u (,S) on he surface, corresponding o fuure mauriy u and marke level v: δp( S',,, S' ) p( Suv,,, )v = puv' (,,, S' ) v δσ vu, (EQ 41) FIGURE 8. Sensiiviy of he forward and backward ransiion probabiliies p(,s,',s') and Φ(K,,K',') o he sudden changes of he local volailiy σ v,u. (a) forward (b) backward (, S) ( uv, ) ( ', S' ) ( K', ' ) ( vu, ) ( K, ) (, S) ( ', S' ) ( K', ' ) ( K, ) v v v v (, S) ( ', S' ) ( K', ' ) ( K, ) ( uv, ) ( vu, ) 18

24 his relaion holds for any u in he range u ', oherwise he variaional derivaive is equal o zero. Similarly, he backward ransiion probabiliy Φ(K,,K',') saisfies, for ' u, he relaion δφ( K,, K', ' ) δσ vu, = 1 --Φ ( K,, v, u)v Φ( vuk',,, ' ) v (EQ 4) and zero oherwise. Using Equaions 1 and 39, he sandard opion prices C K, (,S) and ransiion probabiliies P K, (,S) saisfy similar relaionships for u : δc K, δσ vu,, ( S) = 1 --Φ ( K,, v, u)v v Cvu, ( S, ) (EQ 43) and δp K, δσ vu,, ( S) = 1 --p( Suv,,, )v puvk (,,, ) v (EQ 44) in which he effecive ransiion probabiliies p ( ) and Φ ( ) correspond o he saic local volailiy surface σ K, (,S) prevailing a ime and marke level S. In arriving a Equaions 43 and 44 we have also used he following ideniies:, ( S, ) = psk (,,, ) P K psk (,,, ) = e r Φ( K,, S, ) = e δ ( ) CK K ( ) CK S, ( S, ), ( S, ) (EQ 45) (EQ 46) (EQ 47) As discussed in Appendix B, hese ideniies are all consequences of he fac ha he effecive heory associaed wih σ K, (,S) embodies all he informaion necessary for pricing sandard opions of all srikes and mauriies correcly. aking he variaional derivaives of boh sides of Equaions 41 and 4 wih respec o he local volailiy σ v',u' we find he second order variaional derivaives as δp( S',,, S' ) p( Suv,,, )v = puvu' (,,, v' )v' pu' (, v', ', S' ) δσ vu, δσ v', u' 4 v v' (EQ 48) 19

25 FIGURE 9. Second order variaional derivaives of he forward and backward ransiion probabiliies p(,s,',s') and Φ(K,,K',') wih respec o he local volailiies. (a) forward (b) backward (, S) v v v' v' ( ', S' ) ( K', ' ) v' v' v v ( K, ) ( u, v) ( u', v' ) ( v', u' ) ( v, u) for any u u' ', and δφ( K,, K', ' ) δσ vu, δσ v', u' 1 --Φ ( K,, v, u)v = Φ( vuv',,, u' )v' Φ( v', u', K', ' ) (EQ 49) 4 v v' for ' u' u. Figure 9 gives a graphical depicion of hese ideniies. he sandard opion prices C K, (,S) and ransiion probabiliies P K, (,S) saisfy similar relaionships for u u' : δc K, ( S, ) δσ vu, δσ v', u' 1 --p( Suv,,, )v = puvu' (,,, v' )v' 4 v v' Cv' u', ( S, ) (EQ 50) δp K, ( S, ) δσ vu, δσ v', u' 1 --p( Suv,,, )v = puvu' (,,, v' )v' pu' (, v',, K) 4 v v' (EQ 51) Using hese relaions, Appendix D proves ha Equaion 40 leads o n P K, dp K, σ()s dw 0 δp K, S δσ σ i = + K', ' θ K', ' dk' d' dw i 0 i = 0 K', ' (EQ 5) if and only if, for any S, K and, he drif funcions α K, (,S) saisfy he following no-arbirage condiions (EQ 53) n α K, ( S, ) θ i 1 K, ( S, ) psk (,,, ) θ i K', ' ( S, )ps' (,,, K' )K' p' (, K',, K) d K' d K' ' Π i = 0 i = 0 where Π 0 = 0 and Π i for i = 1,...,n are arbirary bu independen of K and, and where he equivalen measure { W i } is defined by dw 0 = dw 0 + ( d µ () r + δ) ; dw i = dw i + Π i d (EQ 54) σ() 0

26 he quaniies denoe he marke prices of risk associaed wih he volailiy risk facors W i, i = 1,..., n, while µ -(r-δ) is he marke price of risk associaed wih he index price risk facor W 0. Equaion 5 shows ha under he no-arbirage condiions he measure { dw i ; i= 1,...,n} isanequivalen maringale measure, wih respec o which he rescaled index price and rescaled opion prices for all srikes and mauriies are simulaneously maringales. Π i hese no-arbirage condiions in he presen case are significanly more involved han he HJM no-arbirage condiions described in he previous secion. he basic reason is ha local volailiies span a (wo-dimensional) surface on which (forward and backward) propagaion depends, in a raher complicaed and non-linear manner, on he srucure of local volailiies across he whole surface. his is eviden by he apparen complexiy of Equaions 44 and 51 as compared o he simpliciy of he corresponding Equaions 33 and 36 in he ineres rae framework. I is, herefore, raher difficul o use he no-arbirage condiions for sochasic volailiy in heir coninuous form direcly. In he nex secion we inroduce Sochasic Implied rees as a discree-ime framework for describing arbirage-free sochasic variaions of he local volailiy surface. SOCHASIC IMPLIED REES Figure 11 gives a schemaic illusraion of he dynamics in a sochasic volailiy heory. As he physical ime moves forward, he index price changes and, simulaneously, all local volailiies on he volailiy surface undergo muli-facor sochasic variaions. FIGURE 11. Schemaic illusraion of he dynamics of he index price and local volailiy surface in a sochasic volailiy heory. ime 1

27 o provide a more quaniaive descripion of his sochasic dynamics we choose o work wihin a discree-ime framework described by a Sochasic Implied ree. hese rees are exensions of he sandard (non-sochasic) implied rees, which are used o describe effecive volailiy models (see Derman, Kani and Chriss [1996]). Figure 1 shows an example of a 1-year, 5-period sandard implied rinomial ree which is calibraed o a marke where a-hemoney implied volailiy is 5% and here is an implied volailiy skew of 0.5% poin per 10 srike poins. In an implied rinomial ree FIGURE 1. Example of an Implied rinomial ree describing an effecive volailiy heory. sae space: forward diffusion up probabiliies: local volailiies: forward diffusion down probabiliies: backward diffusion up probabiliies: backward diffusion down probabiliies:

28 FIGURE 13. In a Sochasic Implied ree, as he index moves from node A o node B in a single ime sep, he local volailiies and ransiion probabiliies, for every node on he fuure subree beginning a node B, vary sochasically wih muliple sochasic facors. A B he locaion of he nodes, or he sae space, is more or less arbirarily. Once he sae space is fixed, however, he ransiion probabiliies a differen nodes are deermined from he requiremen ha sandard opions and forwards wih srike prices coinciding wih hose nodes and mauring a differen periods of he ree all have prices using he ree which mach heir marke prices. Since local volailiy a any node depends on he nodal levels and he ransiion probabiliies o he nearby nodes, he local volailiies a differen nodes are also deermined in his way. Sochasic implied rinomial rees are exensions of he implied rinomial ree in which he ransiion probabiliies are, in addiion, allowed o vary sochasically, wih several sochasic facors, as ime elapses and index level moves. he index level is allowed o move randomly from node o node, while he local volailiies, and simulaneously he ransiion probabiliies corresponding o he fuure nodes, all vary sochasically across he ree. his behavior is shown in Figure 13. Saring from any iniial node, he possible fuure movemens of he local volailiy surface mus be resriced o guaranee absence of any arbirage opporuniies in he discree heory represened by he sochasic implied ree. As discussed earlier, his is equivalen o he requiremen ha he oal ransiion probabilies o all fuure nodes be simulaneously maringales on he ree. his is also he same as 3

29 FIGURE 14. During a ime sep, he oal ransiion probabiliy P K, will move o one of M values P (i) K,, i = 1,...,M, as index price moves randomly o one of he nearby nodes and he local volailiy surface assumes one of N possible configuraions. + + w 1 P (1) K, w P K, w N P (M) K, he requiremen ha all rescaled sandard opion prices be simulaneously maringales on he ree. As Figure 14 shows, during he ime inerval, he spo price will move randomly (by amoun S) o one of he nearby nodes and, a he same ime, he local volailiy surface will assume one of is N possible configuraions, w 1,...,w N. As a resul, he oal ransiion probabiliy P K, (,S) o any given fuure node (K,) also moves o one of is several possible values P (i) K, (+, S+ S), i = 1,..., M, during his ime inerval. o guaranee no-arbirage, P K, mus be a maringale (fair game), ha is i mus equal he expecaion, under some (equivalen) measure, of is fuure values P (i) K, for all he fuure nodes (K,) on he ree. Our Noaion in Discree ime o make posiiviy manifes, i is more convenien o redefine he drif and volailiy funcions in Equaion 5 as α K, α K, σ K, and θ l K, θ l K, σ K,, l = 0,..., n, and begin by discreizing he following coninuous-ime differenial equaion: dσ K σ K , ( S, ) =, ( S, ) α K, ( S, )d + θ l K, ( S, )dwl n l = 0 (EQ 55) We le he ineger pair (i, label he node ( i,s j ) describing he curren locaion (i.e (,S)) of he index a he i h sep of he simulaion. 4

30 We also le he pair (n,m) label he fuure node ( n,s m ) corresponding he fuure ime and level (i.e (,K)). hen he discree form of Equaion 55 can be wrien as σ mn, ( i, σ l l = m, n( i, α mn, ( i, i + θ mn, ( i, W i n l = 0 (EQ 56) he vecor ( W i 0, W i 1,..., W i n ) is random and is drawn, a ime i, from he sample space of he incremens of n independen Brownian moions W l. l θ mn, he volailiy parameers ( i, are pre-specified bu he drif parameers α mn, ( i, mus be deermined from he no-arbirage requiremens ha he oal probabiliies P m, n ( i, of arriving a he fuure node (n,m) from he (fixed) iniial node (i, mus be joinly maringales for all fuure nodes (n,m). As we shall argue below, hese maringale condiions are precisely enough o compleely deermine all he drif parameers sep by sep during he simulaion process. A Sochasic Implied ree simulaion begins wih he consrucion of a rinomial implied ree calibraed o oday s prices of sandard opions and forwards. he simulaion begins a he node (0,0) of his ree. During he firs simulaion sep he drif parameers α mn, ( 00, ), for all fuure nodes (m,n), are deermined from he maringale condiions on he oal probabiliies P mn, ( 00, ). Figure 15 illusraes ha FIGURE 15. he drif parameer α 0,0 (0,0) in a Sochasic Implied ree is deermined from he maringale condiion on he oal ransiion probabiliy P 1, (0,0). (1,) (0,0) P 1, (0,0) is maringale (1,) a 0,0 (0,0) (0,0) (1,) (0,0) 5

31 he drif parameer α 00, ( 00, ) is deermined from he maringale condiion for P 1, ( 00, ) his also guaranees ha he ransiion probabiliies P 11, ( 00, ) and P 10, ( 00, ) are maringales. he reason is ha hese probabiliies are consrained by wo exra condiions which mus hold irrespecive of he specific behavior of he local volailiies: P 10, ( 00, ) + P 11, ( 00, ) + P 1, ( 00, ) = 1 (EQ 57) P 10, ( 00, )S 10, + P 11, ( 00, )S 11, + P 1, ( 00, )S 1, = S 00, e ( r δ) ( 1 0 ) he firs condiion is he normalizaion condiion, requiring ha he sum of he hree oal ransiion probabiliies a ime 1 mus be uniy. he second is he forward condiion, requiring ha he 1 - mauriy forward price a ime 0 mus mach is risk-neural value. In a similar way, he hree drif parameers α 1, ( 00, ), α 11, ( 00, ) and α 10, ( 00, ) are deermined from he maringale condiions of he hree oal ransiion probabiliies P 4, ( 00, ), P 3, ( 00, ) and P, ( 00, ). he remaining ransiion probabiliies P 1, ( 00, ) and P 0, ( 00, ) will hen also be maringales due o he normalizaion and forward condiions a ime. In his way all drif parameers α mn, ( 00, ) will be deermined during he firs simulaion sep. Finally, o complee his sep we draw a random vecor ( W 0 0, W 1 0,..., W n 0 ) from he sample space of he incremens of W i a ime 0, and use his vecor o simulaneously arrive a a (random) new locaion for he index price and new values for all fuure local volailiies. Equaion 56 is used direcly wih i=j=0o calculae he new local volailiy values from his choice of he random vecor. As for he index price, we use he random number W 0 0 o deermine which of he hree possible fuure nodes (i.e (1,), (1,1) or (1,0)) does he index price moves o during ime inerval. Figure 16 gives one simple possible mehod for FIGURE 16. Deermining which node he index price will go o during one simulaion sep using he renormalized random number W i 0. P u = P u (i, (i+1, j+) if W i 0 >= P m + P d (i, P m = P m (i, (i+1, j+1) if W i 0 >= P d and W i 0 < P m + P d P d = P d (i, (i+1, if W i 0 < P d 6