Article from: Risks & Rewards. February 2010 Issue 55

Transcription

1 rticle from Rks & Rewrds Februry 2010 Issue 55

2 negtive secury s Osbne negtive 1959) secury pprently s unwre Osbne 1959) Bchelier s pprently wk unwre Bchelier s wk proposed geomric Brownin proposed moti GBM) geomric s Brownin moti sck s GBM) s prt cuse sck s prt cuse GBM cn negtive Tht GBM cn ws negtive ecomics Tht from ws 1960s bly ecomics from 1960s bly vlue s Blck Scholes vlue 1973) s lso Blck GBM Scholes ir 1973) rky lso ss GBM sce ir rky ss sce n Osbne s GBM n sck Osbne s s GBM hs ten en clled sck s Blck-Scholes hs ten en clled Blck-Scholes In sequel will st In strd sequel Brownin will st moti t strd Brownin hs dependent moti t hs dependent crements ctuous trjecries crements ctuous trjecries hs n N dtrii hs n N dtrii rhmic Brownin moti rhmic BM) Brownin moti BM) b re cstnts b re cstnts STOCHSTIC VOLTILITY ND OPTION PRICING usul fncil ti GBM usul goes fncil bck ti Blck Scholes GBM goes ir bck Blck Scholes ir By predecesss Dniel fmilir predecesss schstic dferentil fmilir equti schstic dferentil equti Th mens exctly sme Th g mens s exctly sme g s ltter shows me clerly t ltter log-s shows m me n clerly BM t log-s m n BM Schstic voltily s re now routely dcre-time vestments equivlent rhmic pricg dcre-time Brownin brief equivlent troducti moti rhmic rom Brownin wlk moti rom wlk s n mhod pricg s descrid iid Here sts log- iid Here sts In =dference cses log- vrince log-s) In per cses time period vrince cstnt =squred voltily) rurns =squred =dference voltily) log-s) rurns per =dference time period re cstnt log-s) voltily per time s period Observe t rn- cstnt schstic re cstnt t lter voltily cstnt cme s re Observe cstnt s Brownin t voltily rom s wlk Observe my t hve ny rom dtrii wlk while my hve S GBM moti ny dtrii ppered while Bchelier ny GBM dtrii n 1900) s while necessrily GBM n hs nml necessrily hs nml dtrii Schstic secury dtrii Voltily s NB Bchelier Opti nme imged dtrii rom Pricg wlk secury NB ten hs nme wider s rom meng u s wlk ten e fnce hs wider nme meng rom wlk fnce ten hs wider meng mmticl fnce defi lerture here) I m sttg clssicl m- rhmic lerture Brownin I m sttg moti lerture clssicl defed mmticl low) I m sttg defi hs clssicl here) By Dniel shtcomg llowg negtive secury s Osbne emticl defi here) Schstic Wher Voltily log-s ctully Opti do Wher follow Pricg log-s n rhmic ctully Brownin do follow moti n ws rhmic chllenged Brownin moti ws chllenged Schstic voltily 1959) pprently unwre Bchelier s wk proposed geo- Schstic so s fter re Voltily t now routely ws multed so Opti fter vestments Melbrot t Pricg ws 1963) multed pricg n Fm Melbrot 1965)1963) n Fm 1965) By brief Dniel troducti mric suggested Brownin s stble moti Lévy GBM) suggested s s tter n stble mhod sck Lévy s log-s pricg s s Lévy tter Wher es log-s re log-s ctully Lévy do follow es n rhmic re Brownin descrid prt By ctuous-time cuse Dniel GBM es cn negtive ctuous-time dependent Tht crements es ws t re dependent moti necessrily ws crements chllenged nmlly t re so necessrily fter t nmlly ws multed Stble Melbrot es nml crements 1963) my Stble n Fm es 1965) my suggested Schstic voltily ecomics dtried s from BM re now 1960s routely ly bly dtried Lévy vlue vestments BM s nml Blck ly crements Lévy pricg brief schstic troducti t lter cme s Brownin moti ppered Scholes Schstic 1973) voltily s lso s GBM re now ir routely n rky mhod ss vestments pricg sce s stble Lévy pricg s tter log-s Lévy es pricg descrid Bchelier 1900) s secury s Bchelier imged secury s n rhmic n brief Brownin Osbne s troducti moti GBM defed s low) sck hs s hs shtcomg ten n en mhod llowg re ctuous-time s es dependent cre- descrid negtive secury clled s Blck-Scholes Osbne 1959) pprently unwre Bchelier s wk ments t re necessrily nmlly dtried BM schstic t lter cme s Brownin moti ppered Bchelier proposed geomric 1900) schstic Brownin moti secury t GBM) s lter cme s Bchelier imged s sck s Brownin s secury moti prt ly cuse Lévy s ppered nml crements Stble es n GBM rhmic cn In Brownin Bchelier negtive sequel moti 1900) Tht ws will defed s st ws low) secury strd ecomics hs s Brownin shtcomg from Bchelier moti 1960s imged llowg bly my secury good f s s rurn series y we hve negtive vlue s secury n t n Blck rhmic s hs Scholes d dependent co 1973) lso rd Osbne Brownin 1959) moti pprently crements defed GBM t unwre low) h ctuous ir Bchelier s rky hs ss tr- shtcomg wk n sce undesirble llowg property t expectti expentil proposed n Osbne s geomric GBM jecries negtive nd Brownin c secury moti s sck GBM) s Osbne hs ten en clled Blck-Scholes hs n s 1959) N pprently dtrii sck unwre s prt Bchelier s cuse vrible wk stble dtrii hs fe expectti GBM cn proposed negtive n geomric rhmic Tht BM) Brownin Brownin ws pr moti moti ecomics GBM) BM) s s from 1960s sck bly s ftl prt cuse pricg Stble es re s vlue s GBM Blck cn Scholes negtive 1973) Tht lso b GBM ws re hs cstnts ir N ecomics rky ss dtrii from sce 1960s bly n In Osbne s sequel will st strd Brownin moti t hs dependent s crements rhmic vlue GBM s ctuous Brownin Blck trjecries moti sck s Scholes BM) hs 1973) ten en lso clled hs n GBM N Blck-Scholes ir rky dtrii ss good log-s f sce rurn series pricg y we s hve Lévy n undesirb n Osbne s GBM sck good s f hs ten rurn en series clled es y Blck-Scholes expectti we hve me hve n expentil undesirble en property proposed vrible t t lest stble dtrii rhmic b usul re cstnts Brownin fncil moti ti BM) expectti GBM goes bck expentil Blck cdemic expectti vrible lerture) stble ftl dtrii rurns hs pricg fe lso Stble es s re In sequel Scholes b will re st cstnts ir predecesss strd Brownin moti fmilir t schstic dependent Note log-s t Lévy es pricg hve dependent s Lévy crements es so hve me usul fncil ti GBM goes expectti bck good Blck f ftl Scholes rurn series ir pricg y we Stble es hve n re undesirble s property s t crements In dferentil In ctuous sequel equti trjecries will st strd Brownin hs N moti dtrii t proposed hs dependent t lest cdemic lerture) rurns lso predecesss fmilir schstic log-s dferentil expectti equti pricg expentil vrible s dependence Lévy es stble dtrii rurns hve me hs time fe ws en ctested usul fncil rhmic crements ti Brownin ctuous GBM moti goes trjecries bck BM) Blck Scholes hs ir n n N Note t dtrii Lévy es hve dependent crements so depende expectti proposed t lest ftl cdemic pricg lerture) Melbrot Stble es Fm rurns re Interestgly lso s s Blck s predecesss fmilir schstic dferentil equti time ws ctested Melbrot Scholes Fm Interestgly 1972) Bl Th mens exctly b re cstnts rhmic Brownin log-s Note moti t Lévy BM) es pricg hve dependent s sme g s lso crements questied Lévy es so wher hve dependence voltily me rurns cstnt en 1972) lso questied wher voltily cstnt time time Th mens exctly b re sme cstnts proposed time g s t ws lest ctested cdemic Melbrot lerture) Fm rurns Interestgly lso Blck s Scholes Th usul mens fncil exctly ti sme g GBM s goes Note 1972) bck t lso Lévy questied Blck es wher Scholes hve ir dependent voltily crements cstnt so time dependence rurns Voltily wht we deed GBM bove cse predecesss ltter shows usul me fmilir fncil clerly schstic ti t log-s dferentil GBM time ws m goes equti n bck ctested BM Blck Scholes Melbrot ir Fm Interestgly Blck Scholes log-s strd devi Ecomric mod ltter predecesss shows me clerly t fmilir 1972) Voltily log-s schstic lso questied wht we deed m dferentil wher n BM equti voltily cse GBM cstnt bove rom time wlk cse log-s rom wlk strd dcre-time equivlent ltter rhmic shows me Brownin clerly moti t log-s log-s rom m wlk strd devi 1970s 1980s Ecomric t refed s rom ppered wlk 1970s mkg Th mens exctly sme g s dcre-time BM Th equivlent mens exctly rhmic sme Voltily Brownin g 1970s s s wht 1980s we deed t refed rom moti rom wlk GBM schstic 1980s bove wlk t refed ledg mkg cse rom well- rom vrince wlk wlk RCH mkg es iid Here sts log- schstic log-s ledg In cses strd well- devi vrince RCH es Ecomric schstic s ledg ppered well- ltter shows me clerly t log-s m n BM =squred voltily) iid dcre-time Here rurns sts equivlent =dference log- rhmic 1970s log-s) Brownin 1980s per t In time moti refed period cses n rom Opti pricg ltter shows me clerly t log-s m n BM vrince RCH wlk es mkg vrince Not ly GBM f sck s well lso unstf cstnt re cstnt voltily s Observe t rom wlk my hve =squred dcre-time voltily) equivlent rom wlk rurns rhmic =dference schstic Opti pricg Brownin log-s) ledg moti per rom time period well- wlk RCH es Not ly GBM f sck pricg s ledg well lso well- unstfcry smile effect ny cstnt dtrii re while dcre-time cstnt voltily GBM equivlent n s rhmic Observe t Brownin necessrily rom moti hs wlk nml rom my OPTION hve wlk PRICING dtrii ny dtrii NB iid while Here nme rom sts GBM wlk Opti pricg n ten log- hs pricg ledg well- smile effect wider necessrily meng In cses hs fnce nml vrince Not RCH ly es GBM hve en f suggested sck s pricg well s lso n cl lerture I m sttg clssicl mmticl Not ly defi GBM here) f sck s well lso unstfcry =squred dtrii voltily) NB nme rurns iid rom Here =dference wlk RCH ten sts hs es log-s) wider log- hve meng per time en period suggested In fnce In unstfcry found cses pricg me s frequently vrince n pricg lerture clss ledg nmely s well- voltily pricg ledg well- smile effect cstnt lerture In I =squred m cstnt sttg cses voltily) clssicl vrince s mmticl =squred rurns found Observe =dference me voltily) defi t frequently rom here) log-s) rurns wlk lerture per my time smile time nmely hve period effect voltily ctuoustime s Melbrot es 1963) hve necessrily Observe t Wher log-s ctully do follow n rhmic Brownin moti ws chllenged ny dtrii cstnt so fter t while ws re GBM cstnt multed n RCH voltily en n suggested hs Fm nml rom wlk my hve 1965) pricg s n clss s Wher log-s ctully do follow n rhmic Brownin moti ws chllenged suggested dtrii stble NB ny Lévy dtrii nme rom while s tter wlk found ten GBM me hs n log-s frequently wider meng Lévy es lerture necessrily fnce re nmely hs nml voltily ctuoustime rom crements wlk defi t ten re hs here) necessrily wider meng nmlly fnce so fter t ws multed Melbrot 1963) n Fm 1965) ctuous-time lerture I dtrii m es sttg NB clssicl dependent nme mmticl suggested stble Lévy s tter log-s Lévy es re Here rk-free rte terest sce multed un dtried BM lerture ly I I Lévy m sttg clssicl nml mmticl crements defi Stble es here) my Wher ctuous-time log-s es ctully do follow dependent Here n rhmic crements Brownin t rk-free re moti rte necessrily terest ws chllenged nmlly sce neutrl mesure) multed re Brownin rkneutrl motis possibly crelted) dtried so fter BM Wher t log-s ly ws Lévy multed ctully do do Melbrot follow nml mesure) rhmic 1963) crements Brownin n Stble re Fm Brownin es moti schstic 1965) ws motis my chllenged es possibly t crelted) my depend In re suggested stble Lévy so fter t s tter ws Here schstic multed es log-s rk-free Melbrot t rte Lévy my 1963) es terest depend sce re en n Fm replced In 1965) multed schstic GBM rkneutrl en crements s s replced mesure) tter t re schstic log-s necessrily re Brownin Lévy nmlly Severl motis es s possibly re squred crelted) voltily hve en proposed re y re two problems hs squred voltily 28 RISKS ctuous-time ND REWRDS suggested es FEBRURY stble dependent Lévy 2010 dtried BM ctuous-time ly Lévy es schstic Severl dependent nml s es crements t my Stble hve t depend es re en proposed necessrily my In comti y nmlly re two If problems GBM estimti hs s n logic dtried BM ly Lévy comti nml If crements should Stble f es s my n s logiclly me precely wht sought d en replced schstic squred voltily

3 FEBRURY 2010 RISKS ND REWRDS 29 RCH es hve en suggested pricg s n clss s found me frequently lerture nmely voltily ctuous-time e terest sce m Here l rk-free rte terest sce multed rk-neutrl mesure) -free rte o es t m re Brownin motis possibly crelted) M mode re schstic es t my depend In GBM hs en replced schstic In oposed t squred voltily Severl s hve en proposed y re two problems estimti comti If s n logiclly should f s me precely wht sought dtrii S rk-neutrl mesure Th esy estimte cuse numr s usully que smll Only cse GBM t problem cuse qudrtic vri logs t time uns t cse sme cstnt re Brownin motis possibly crelted) re schstic es t my depend In GBM hs en replced schstic squred voltily Severl s hve en proposed y re two problems estimti comti If s n logiclly should f s me precely wht sought dtrii rk-neutrl mesure Th esy estimte cuse numr s usully que smll Only cse GBM t problem cuse qudrtic vri log-s t time uns t cse sme cstnt edir s e ****should σ 2 t***) physicl mesure rk-neutrl mesure ree GBM sck dtrii physicl mesure implies GBM sck dtrii rk-neutrl mesure sme mesures hence e ly needs estimte from pst sck s pplies ly GBM/Blck-Scholes schstic voltily hs en estimted re usully rems n problem t comg s g dtrii ten turns out eir complicted just un Mte Crlo simulti possibily e so relizes t simulti schstic dferentil 2 t physicl mesure rk-neutrl mesure ree GBM sck dtrii physicl mesure implies GBM sck dtrii rk-neutrl mesure od f rurn series y we hve n undesirble property t ectti expentil vrible stble dtrii hs fe ti ftl pricg Stble es re s s ces pricg s Lévy es hve me en ed t lest cdemic lerture) rurns lso s t Lévy es hve dependent crements so dependence rurns e ws ctested Melbrot Fm Interestgly Blck Scholes lso questied wher voltily cstnt time y wht we deed GBM bove cse rom wlk log-s strd devi Ecomric s ppered 0s 1980s t refed rom wlk mkg vrince tic ledg well- RCH es pricg ly GBM f sck s well lso unstfcry ledg well- smile effect es hve en suggested pricg s n clss s e frequently lerture nmely voltily ctuousocess rk-free rte terest sce multed rkmesure) re Brownin motis possibly crelted) re tic es t my depend In GBM hs plced schstic squred voltily l s hve en proposed y re two problems estimti comti If s n logiclly f s me precely wht sought dtrii rk-neutrl mesure Th esy estimte cuse r s usully que smll Only cse GBM t problem cuse qudrtic vri log-s t s t cse sme cstnt edir s e ****should ) physicl mesure rk-neutrl mesure ree ck dtrii physicl mesure implies GBM sck ti rk-neutrl mesure sme mesures e ly needs estimte from pst sck s pplies ly lck-scholes he schstic voltily hs en estimted re usully rems n t comg s g dtrii n turns out eir complicted just un Mte Crlo simulti sibily e so relizes t simulti schstic dferentil sme mesures hence e ly needs estimte good f rurn series y we hve n undesirble property t expectti expentil vrible stble dtrii hs fe expectti ftl pricg Stble es re s s log-s pricg s Lévy es hve me en proposed t lest cdemic lerture) rurns lso s Note t Lévy es hve dependent crements so dependence rurns time ws ctested Melbrot Fm Interestgly Blck Scholes 1972) lso questied wher voltily cstnt time Voltily wht we deed GBM bove cse rom wlk log-s strd devi Ecomric s ppered 1970s 1980s t refed rom wlk mkg vrince schstic ledg well- RCH es Opti pricg Not ly GBM f sck s well lso unstfcry pricg ledg well- smile effect RCH es hve en suggested pricg s n clss s found me frequently lerture nmely voltily ctuoustime Here rk-free rte terest sce multed rkneutrl mesure) re Brownin motis possibly crelted) re schstic es t my depend In GBM hs en replced schstic squred voltily Severl s hve en proposed y re two problems estimti comti If s n logiclly should f s me precely wht sought dtrii rk-neutrl mesure Th esy estimte cuse numr s usully que smll Only cse GBM t problem cuse qudrtic vri log-s t time uns t cse sme cstnt edir s e ****should σ 2 t***) physicl mesure rk-neutrl mesure ree GBM sck dtrii physicl mesure implies GBM sck dtrii rk-neutrl mesure sme mesures hence e ly needs estimte from pst sck s pplies ly GBM/Blck-Scholes schstic voltily hs en estimted re usully rems n problem t comg s g dtrii ten turns out eir complicted just un Mte Crlo simulti possibily e so relizes t simulti schstic dferentil from pst sck s pplies ly GBM/Blck-Scholes schstic voltily hs en estimted re usully rems n problem t comg s g dtrii S ten turns out eir complicted just un Mte Crlo simulti possibily e so relizes t simulti schstic dferentil equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily equtis fr from trivil e must del errs rg fr smple dcrizti dferentil equti T pplictis mke mtters wse erly us o schstic voltily s hve suggested mhods especilly questible vlue cv empr) Perhps st- schstic voltily e wh stfy so-clled squre-root re cstnts Th ws Willim 1940s ws s terest rtes squred vo n-negtive ii) mny s re explicly cse 2001 dils) fdg dtrii res t solut Wht mtters fdg dtrii t teg ltter rrely e swchg schstic voltily prmer chnges vlue ccdg M come que populr t lest cdemic lerture voltily csts ltg Mrkov ch t tk kd lmost lwys hve sm sttes regimes ) sy 3 Th ppe ppernce grphs voltilies trctble t defe cstnt voltily s ble re re ly vlues dtrii hs en lerture while Ch Du dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) sh versi mul my Europen s clls s t wy ws new clssicl prob chrctertic g trnsm densy probbily densy o mul R rel le i imgry un ) so Prsevl s em similr expressi expectti stfy so-clled squre-root equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily stfy so-clled squre-root re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help fdg dtrii res t soluti Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e swchg schstic voltily prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg Mrkov ch t tkes vlues s kd lmost lwys hve smll numr sttes regimes ) sy 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly vlues dtrii explicly hs en lerture while Ch 2009) dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) t versi mul my Europen s clls ide expressg s t wy ws new clssicl probbily y t chrctertic g trnsm probbily densy probbily densy versi mul R rel le i imgry un ) cdis Prsevl s em similr expressi expectti bc re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily stfy so-clled squre-root re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help fdg dtrii res t soluti Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e swchg schstic voltily prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg Mrkov ch t tkes vlues s kd lmost lwys hve smll numr sttes regimes ) sy 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly vlues dtrii explicly hs en lerture while Ch 2009) dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) t versi mul my Europen s clls ide expressg s t wy ws new clssicl probbily y t chrctertic g trnsm probbily densy probbily densy versi mul R rel le i imgry un ) cdis Prsevl s em similr expressi expectti cse 2001 dils) help fdg dtrii equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily stfy so-clled squre-root re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help fdg dtrii res t soluti Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e swchg schstic voltily prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg Mrkov ch t tkes vlues s kd lmost lwys hve smll numr sttes regimes ) sy 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly vlues dtrii explicly hs en lerture while Ch 2009) dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) t versi mul my Europen s clls ide expressg s t wy ws new clssicl probbily y t chrctertic g trnsm probbily densy probbily densy versi mul R rel le i imgry un ) cdis Prsevl s em similr expressi expectti res t soluti equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily stfy so-clled squre-root re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help fdg dtrii res t soluti Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e swchg schstic voltily prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg Mrkov ch t tkes vlues s kd lmost lwys hve smll numr sttes regimes ) sy 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly vlues dtrii explicly hs en lerture while Ch 2009) dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) t versi mul my Europen s clls ide expressg s t wy ws new clssicl probbily y t chrctertic g trnsm probbily densy probbily densy versi mul R rel le i imgry un ) cdis Prsevl s em similr expressi expectti equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily stfy so-clled squre-root re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help fdg dtrii res t soluti Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e swchg schstic voltily prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg Mrkov ch t tkes vlues s kd lmost lwys hve smll numr sttes regimes ) sy 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly vlues dtrii explicly hs en lerture while Ch 2009) dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) t versi mul my Europen s clls ide expressg s t wy ws new clssicl probbily y t chrctertic g trnsm probbily densy probbily densy versi mul R rel le i imgry un ) cdis Prsevl s em similr expressi expectti Wht mtters fdg dtrii equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily stfy so-clled squre-root re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help fdg dtrii res t soluti Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e swchg schstic voltily prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg Mrkov ch t tkes vlues s kd lmost lwys hve smll numr sttes regimes ) sy 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly vlues dtrii explicly hs en lerture while Ch 2009) dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) t versi mul my Europen s clls ide expressg s t wy ws new clssicl probbily y t chrctertic g trnsm probbily densy probbily densy versi mul R rel le i imgry un ) cdis Prsevl s em similr expressi expectti t tegrted squred voltily equtis fr from trivil e must del errs rg fr smple dcrizti dferentil equti T pplictis mke mtters wse erly us o schstic voltily s hve suggested mhods especilly questible vlue cv empr) Perhps st- schstic voltily e wh stfy so-clled squre-root re cstnts Th ws Willim 1940s ws s terest rtes squred vo n-negtive ii) mny s re explicly cse 2001 dils) fdg dtrii res t solut Wht mtters fdg dtrii t teg ltter rrely e swchg schstic voltily prmer chnges vlue ccdg M come que populr t lest cdemic lerture voltily csts ltg Mrkov ch t tk kd lmost lwys hve sm sttes regimes ) sy 3 Th ppe ppernce grphs voltilies trctble t defe cstnt voltily s ble re re ly vlues dtrii hs en lerture while Ch Du dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) sh versi mul my Europen s clls s t wy ws new clssicl prob chrctertic g trnsm densy probbily densy o mul R rel le i imgry un ) so Prsevl s em similr expressi expectti ltter rrely e REGIME SWITCHING STOCHSTIC VOL- TILITY MODEL prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily stfy so-clled squre-root re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help fdg dtrii res t soluti Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e swchg schstic voltily prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg Mrkov ch t tkes vlues s kd lmost lwys hve smll numr sttes regimes ) sy 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly vlues dtrii explicly hs en lerture while Ch 2009) dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) t versi mul my Europen s clls ide expressg s t wy ws new clssicl probbily y t chrctertic g trnsm probbily densy probbily densy versi mul R rel le i imgry un ) cdis Prsevl s em similr expressi expectti Mrkov ch t tkes vlues s equtis fr from trivil e must del errs rg from romness smple dcrizti dferentil equti Th rel problem pplictis mke mtters wse erly us pricg schstic voltily s hve suggested mhods especilly series t re questible vlue cv empr) Perhps st- schstic voltily e squred voltily stfy so-clled squre-root re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help fdg dtrii res t soluti Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e swchg schstic voltily prmer chnges vlue ccdg Mrkov ch hve come que populr t lest cdemic lerture schstic voltily csts ltg Mrkov ch t tkes vlues s kd lmost lwys hve smll numr sttes regimes ) sy 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly vlues dtrii explicly hs en lerture while Ch 2009) dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) t versi mul my Europen s clls ide expressg s t wy ws new clssicl probbily y t chrctertic g trnsm probbily densy probbily densy versi mul R rel le i imgry un ) cdis kd lmost lwys hve smll numr sttes regimes ) sy equtis fr from trivil e must del errs rg fro smple dcrizti dferentil equti Th pplictis mke mtters wse erly us schstic voltily s hve suggested mhods especilly s questible vlue cv empr) Perhps st- schstic voltily e whe stfy so-clled squre-root re cstnts Th ws Willim 1940s ws s terest rtes squred vol n-negtive ii) mny s re explicly cse 2001 dils) i fdg dtrii res t solutio Wht mtters fdg dtrii t tegr ltter rrely e swchg schstic voltily prmer chnges vlue ccdg M come que populr t lest cdemic lerture voltily csts ltg Mrkov ch t tke kd lmost lwys hve smll sttes regimes ) sy 3 Th ppe ppernce grphs voltilies trctble defe cstnt voltily s ble re re ly vlues dtrii hs en lerture while Ch Du dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) sho versi mul my Europen s clls T s t wy ws new clssicl prob chrctertic g trnsm densy probbily densy mul R rel le i imgry un ) som 3 Th pper mtch vul ppernce grphs voltilies trctble cert defe cstnt voltily s ble dvntge t re re ly equtis fr from trivil e must del errs rg smple dcrizti dferentil equti pplictis mke mtters wse erly us schstic voltily s hve suggested mhods especill questible vlue cv empr) Perhps st- schstic voltily e w stfy so-clled squre-root re cstnts Th ws Willi 1940s ws s terest rtes squred n-negtive ii) mny s re explici cse 2001 dils) fdg dtrii res t sol Wht mtters fdg dtrii t te ltter rrely e swchg schstic voltily prmer chnges vlue ccdg come que populr t lest cdemic lertu voltily csts ltg Mrkov ch t t kd lmost lwys hve sm sttes regimes ) sy 3 Th pp ppernce grphs voltilies trctble defe cstnt voltily s b re re ly vlues dtrii hs en lerture while Ch dils) Pricg s tegrls In cse squre-root voltily Hesn 1993) s versi mul my Europen s cll s t wy ws new clssicl pro chrctertic g trnsfo densy probbily densy CONTINUED ON PGE 30 MODELS UNDER WHICH PRMETER CHNGES VLUE CCORDING TO MRKOV CHIN HVE RECENTLY BECOME QUITE POPULR T LEST IN THE CDEMIC LITERTURE

4 questible vlue cv empr)s hve suggested mhods especilly series t re tible vluequestible cv empr) schstic voltily vlue cv empr) mke mtters wsere erly us pricg Th ws Willim s Perhps Perhps st- schstic voltily e squred voltily cstnts st- schstic voltily e squred voltily questible vlue cv empr) ily s hve suggested mhods especilly series t re voltily Perhps ws svoltily terest rtes squred cuse voltily stfy so-clled squre-root stfy squre-root so-clled st- schstic voltily e i) squred ps 1940s st- schstic e squred voltily Perhps st- schstic voltily e squred lue cv empr) n-negtive s re explicly) dtrii voltily stfyii) mny so-clled squre-root tfy so-clledstfy squre-root Perhps st- schstic voltily e STOCHSTIC VOLTILITY FROM PGE 29 squred voltily so-clled squre-root cse 2001 dils) re cstnts Th wshelp Willim stfy squre-root re cstnts Th ws Willim s s voltily t-fdg schstic eso-clled squred voltily dtrii res t soluti 1940s ws s terest rtes squred cuse i) re cstnts Th ws Willim s voltily 1940s ws s terest rtes squred voltily cuse i) re cstnts Th ws Willim s re o-clled squre-root re cstnts Th ws Willim s mny n-negtive ii) mny cuse s re explicly) dtrii n-negtive ii) s re explicly) dtrii 1940s ws s terest rtes squred voltily cuse i) 0s ws s terest rtes squred voltily i) re cstnts Th ws Willim s 1940s ws s terest rtes squred voltily cuse i) n-negtive ii) mny s re explicly) dtrii cse 2001 dils) help Th ii) mny s re explicly) n-negtive cse 2001 dils) helpvoltily 1940s ws s dtrii terest ii) mny s re explicly) squred dtrii cuse i) enegtive cstnts ws Willim s rtes fdg dtrii res t soluti fdg dtrii res t soluti cse 2001 dils) help own cse 2001 dils) explicly) n-negtive ii) mny dils) s dtrii cse 2001 help hosen s terest rtes squred voltily cuse i) helpre fdg dtrii res t soluti help ng mny dtrii res t soluti cse 2001 dils) fdg dtrii res t soluti d ii) s re explicly) dtrii Wht mtters fdg dtrii t Th tegrted squred voltily soluti dil Sce pyfs Th ldil l Sce pyfs fdg dtrii res t cse 2001 dils) Th dil l Sce h help ltter rrely resurnce ctrcts re smeth s re cll s ie ) pyfs resurnce ctrcts smes cll s ie Th dil Sce l dil l pyo ii res t soluti resurnce ctrcts rectrcts smere sresurnce cll s ie ctrcts re sme s cllpo resurnce sme s cll s ie sme ides pply resurnce mul t pper smesce ides pply resurnce pricg mul t Th dil l pyfs pricg Thsme dil l Sce pyfst ides pply resurnce pricg mul pp swchg voltily Wht Wht mtters fdg t tegrted squred voltily dtrii sme ides pply resurnce p resurnce ctrcts re cll s ie ) sme ides pricg mul prove sme t s tegrted squred voltily mttersschstic fdg dtrii Th pply dil ls Sce pyfs resurnce ctrcts re sme sresurnce cll ie prmer chnges vlue ccdg Mrkov ch hve Wht mtters fdg dtrii t tegrted squred voltily ltter rrely e sme ides pply resurnce pricg mul t pper t e tegrted squred voltily Th t mtters fdg dtrii resurnce ctrcts reresurnce dil smes cll s ie ltter rrely lpricg Sce pyfs Wht mtters populr fdg dtrii t tegrted squred voltily sme ides pply mul t pp come que t lest cdemic lerture schstic Th dil l Sce py rrely ltter e ch sme ides pply resurnce pricg mul t pp voltily ltter rrely e mtters fdg dtrii vlues t tegrted squred voltily resurnce ctrcts re sme s cll s ie csts ltg Mrkov t tkes s ltterwht rrely e resurnce ctrcts re sme s cll s ie Th dil ll Sce Th dil Sce p Th dil l Sce pyfs swchg schstic voltily swchg schstic voltily Th rel prt s ides ly ssumptis rom vrible posive cll rel prt re ly ssumptis re rom vrible posi ltter rrely e sme pply resurnce pricg mul t pp kd lmost lwys hve smll numr dil l Sce pyfs fdg dtrii t tegrted squred voltily resurnce ctrcts re sme s cll s ie resurnce ctrcts re sme s cll s ie sme ides pply resurnce pricg mul prov ) resurnce ctrcts re sme s ie Th dil l Sce pyfs rel prt ly ssumptis re rom vrible posiv prmer chnges vlue ccdg Mrkov ch hve schstic prmer chnges vlue ccdg men Mrkov ch hve swchg voltily schstic voltily ly rel prt l ly ssumptis rel prt similr ssumptis re rom ) resurnce ctrcts re sme s cll s ie hs fe posive) re re muls re hs fe men cstnt posive) re similr muls vrible Th Th dil dil l Sce Sce py p sttes regimes ) sy schstic resurnce 3 Th pper mtch vul swchg voltily erme swchg rrely e sme ides pply resurnce pricg p sme ides pply resurnce pricg mul ctrcts re lerture s cstnt cll s ie ) sme ides pply resurnce pricg mul t pper come que populr tsme lest cdemic lerture schstic hs fe men cstnt posive) re re similr muls mul come que populr trel lest cdemic schstic chnges vlue prmer chnges vlue ccdg Mrkov ch hve els prmer ccdg Mrkov ch hve T hs fe men cstnt posive) Th prt ly ssumptis re rom vrible posive swchg schstic voltily hs fe men cstnt posive) re re similr mul resurnce resurnce ctrcts ctrcts re re sme sme s s cll cll s s ie ie ppernce grphs voltilies trctble cert comtil dvntge mul t necessry comtil dvntge mul t necess Th prmer chnges vlue ccdg sme Mrkov ch hve ides pply resurnce pricg mul vrible t pper posiv dil t l Sce pyfs sme ides pply resurnce mul pper rel prt ssumptis re rom voltily ltg Mrkov ch t tkes vlues ly s t voltily csts ltg csts Mrkov ch tkes vlues pricg come que populr t lest cdemic lerture schstic Mrkov comtil dvntge mul dvntge t necess ntly come que populr t lest cdemic lerture schstic s prmer chnges vlue ccdg s ch hve defe cstnt voltily ble dvntge t come que populr tmen lest cdemic re schstic comtil typ hs fe cstnt lerture posive) re similr muls ides comtil dvntge mul t pro sme sme ides pply pply resurnce resurnce re pricg pricg mul mul prov rel prt ly ssumptis re rom vrible posiv fd dtrii e comg sp-loss fd dtrii e comg sp-loss resurnce ctrcts re sme s cll s ie ) g schstic voltily kd lmost lwys hve smll numr hs fe men cstnt posive) re similr muls kd lmost lwys hve smll numr voltily csts ltg Mrkov ch t tkes vlues s tily csts ltg Mrkov ch t tkes vlues s e comg sp-loss fd dtrii comg pyf lso fster n Mte come que populr t lest cdemic lerture schstic re re ly vlues dtrii voltily csts ltg Mrkov ch t tkes vlues s vlues explicly e comg fd dtrii comtil dvntge mul t necessry prmer chnges vlue ccdg Mrkov ch hve e comg sp fd dtrii n sgle tegrl Europen n sgle tegrl Europen hs men cstnt posive) re re similr muls sme ides pply resurnce pricg mul dvntge t pper fe rel prt ly ssumptis re rom posiv lmost sttes kd lwys hve smll numr regimes ) sy 3 Th pper mtch vul vlues prt ly ssumptis re rom vrib comtil mul vrible t necess kd lwys hve lmost smll numr 3 Th pper mtch vul sttes en regimes ) sy voltily csts ltg Mrkov ch t tkes srel n sgle tegrl kd lmost lwys hve smll2009) numr Crlo simulti hs while Ch hs en lerture while Ch e que populr t lest cdemic lerture schstic tegr n Europen sgle tegr lerture fd dtrii e comg sp-loss n re sgle tegrl v Eu comtil dvntge mul t necess n comg cll Th n fdg dtrii cll Th n fdg dtrii n comg expecte ppernce grphs voltilies trctble cert hs fe men cstnt posive) re similr muls ppernce grphs voltilies trctble cert sttes regimes ) sy 3 Th pper mtch vul rel prt ly ssumptis re rom vri rel prt ly ssumptis re rom hs fe men cstnt posive) re re similr mu rel prt ly ssumptis re rom vrible posive fd dtrii e comg sp-loss 3 Th pper s mtch pper vul scsts regimes ) sy regimes ) Th kd lmost lwys hve smll numrn cll Th fdg dtrii n comg dils) 3 mtch vul sttes sy ltg Mrkov ch t tkes vlues 2009) dils) pyf lso fster n Mte Crlo simulti pyf lso fster n Mte Crlo simulti cll Th n fdg dtrii n sgle tegrl Europen cll Th n fdg dtrii n comg defe cstnt voltily s ble dvntge t rel prt ly ssumptis re rom vrible posive fd dtrii e comg sp-loss comtil dvntge cstnt mul re t re defe cstnt voltily s ble dvntge pper t ppernce voltilies grphs voltilies 3 Th trctble cert fster rnce grphs fe trctble cert posive) similr m hs fe men cstnt fe men posive) re re similr hs rel prt ly ssumptis re rom vrible ide posive comtil dvntge necessr t fo hs n sgle tegrl mul Europen posive) re similr muls men cstnt sttes regimes ) sy re mtch vul pyf lso n Mte Crlo simulti ppernce grphs voltilies trctble cert odels kd lmost lwys hve re smll numr wy pply Mte descrid Ch pyf lso fster n Mte Crlo simult pyf lso fster n Crlo simulti vlues dtrii explicly re ly n comg cll Th n fdg dtrii rel rel prt prt ly ly ssumptis ssumptis re re rom rom vribl vrib vlues dtrii explicly re re ly defe cstnt voltily s ble dvntge t n sgle tegrl Europen hs fe men cstnt posive) re re similr muls e comg comg sp-loss fd dtrii muls ne cstnt voltily ble dvntge t comtil dvntge n mul t dvntge mul ppernce pper grphs mtch voltilies trctble pply cert comtil hs fe men cstnt dvntge posive) re similr fd dtrii e st s comtil Ch t necessry Pricgdefe s tegrls cll Th n fdg dtrii comg cstnt voltily s dvntge ble re t 3re Th vul mes ) sy wy pply ide descrid wy rmul ide descrid Ch V dependent Brownin moti pyf OPTIONS lso fster n Mte Crlo simulti vlues dtrii explicly re ly hs en lerture while Ch 2009) PRICING WITH FOURIER INTE vlues dtrii comtil explicly e re ly hs hs fe fe men men cstnt cstnt posive) posive) re re re re similr similr mul mu hs en lerture prt while Ch 2009) comtil dvntge Ch mul t necessry n comg cll Th n fdg dtrii rel ly ssumptis re rom vrible posive n sgle tegrl Europen defe cstnt voltily s ble dvntge t wy pply ide descrid In cse squre-root voltily Hesn 1993) t pyf lso fster n Mte Crlo simulti re re ly vlues dtrii explicly fd dtrii e comg e comg oe fd dtrii dvntge mul t necessry n sgle tegrl fd dtrii e comg sp-loss rphs voltilies dils) trctble cert Brownin wy pply ide descrid Ch wy pply ide bove) descrid Ch chrctertic Supp deed bove) t dependent moti drivg deed bove) t dependent Brownin moti drivg drivg Sexplicly t dils) hsgrls en lerture while Ch 2009) pyf lso fster n Mte Crlo simulti deed comtil comtil dvntge dvntge mul mul t tio hs en lerture while Ch 2009) re re ly vlues dtrii e comg sp-loss dtrii fd posive) re re similr muls hs fe men cstnt versi mul my Europen s clls ide expressg n comg cll Th n fdg dtrii n dependent Brownin moti drivg deed bove) t n sgle tegrl hs en lerture while Ch 2009) sgle tegrl fd dtrii e comg sp-loss ement cstnt voltily s ble dvntge t cll Th n fdg dtrii n com n sgle tegrl Europen de dependent drivg n Brownin drivg wy pply ide chrctertic descrid Ch dependent tegrted Brownin deed bove) squred voltily tegrted squred moti chrctertic tegrted voltily moti n e dils) ls) dils) s ws new squre-root clssicl probbily y 1993) t wy ide lso descrid Ch pyf pply lso fster n Crlo simulti fd fd dtrii dtrii n e e comg comg s s In cse voltily Ch n sgle tegrl Esimulti e Europen hs lerture while 2009) en comtil dvntge mul cll t voltily necessry vluest wy dtrii explicly pyf fster n Mte Crlo n compu Th n fdg dtrii n comp cll Th fdg dtrii n Hesn sgle tegrl Mte Europen chrctertic tegrted squred voltily n e cll Th fdg dtrii n comg Pricg n tegrls dependent s Brownin moti drivg deed bove) t Pricg s tegrls wy pply ide descrid Ch chrctertic tegrted squred voltil n e my cdi V cll chrctertic tegrted squred voltily cll my cdi my cdi cll dils) chrctertic g trnsm probbily dependent Brownin moti drivg deed bove) t n n sgle sgle tegrl tegrl E t versi mul my comg cll Th n fdg dtrii In fd dtrii e comg sp-loss pyf lso n Mte Crlo simulti pyf lso fster n Mte Crlo simulti en lerture s 2009) pyf lso fster n Mte Crlo simulti n comg fster cll Th n fdg dtrii my cdi cll cse squre-root voltily Hesn 1993) t In cse while squre-root voltily Hesn 1993) t wy Pricg Ch tegrls g s tegrls deed bove) t dependent Brownin moti drivg my cdi cll chrctertic tegrted squred voltily n n e my cdi dtrii cll pply descrid e densy probbily densy pyf versi Pricg s tegrls lso fster Mte Crlo simulti wy pply ide Ch descrid Ch Sup n n comg com cll cll Th Th n n fdg fdg dtrii Europen s clls expressg s ide n sgle tegrl ide Europen chrctertic tegrted squred voltily n pyf lso fster n Mte Crlo simulti versi mul my Europen s clls ide expressg versi mul my Europen s clls ide expressg In cse squre-root voltily Hesn 1993) t e cse squre-root Hesn 1993) t Pricg s tegrls mul cll my cdi deed bove) t dependent Brownin moti drivg In cse voltily squre-root voltily Hesn 1993) t chrctertic tegrted squred voltily n e Th mens cdig ) Now vlue Th mens cdig ) Now vlue os pyf pyf lso lso fster fster n n Mte Mte Crlo Crlo simulti simulti wy pply ide descrid Ch wy pply ide descrid Ch 2009) dependent Brownin moti drivg deed bove) n wy pply ide descrid Ch t wy ws new clssicl probbily y s t wy ws new clssicl probbily y t n comg cll Th n fdg dtrii gen cll t wy ws new clls clssicl probbily y tcdig versi mul my Europen s clls my idecdi expressg Th mens ) Now vlue rsi mul my s Europen s ide expressg Inmy cse squre-root voltily Hesn 1993) t mul dependent Europen s clls ide expressg wy pply ide descrid Ch Th mens cdig Th mens cdig ) Now Th mens cdig cll my cdi re ssumed schstic tegrl re ssumed schstic tegrl s versi tegrls chrctertic tegrted squred voltily n e dependent Brownin moti drivg deed bove) dependent Brownin moti drivg deed bove wy pply ide descrid Ch pyf lso fster n Mte Crlo simulti Brownin moti drivg deed bove) t t chrctertic g chrctertic trnsm probbily chrctertic tegrted squred voltily know s t wy ws new clssicl probbily y t chrctertic g trnsm probbily s t wy ws clssicl probbily y t re ssumed schstic tegrl versi mul my Europen s clls ide expressg s new t wy ws new clssicl probbily t wy wy drivg pply pply ide ide descrid descrid ssumed Ch Ch Sup Su dependent y Brownin moti deed bove) voltily t sch e squre-root voltily Hesn 1993) t re ssumed sch Th mens cdig )deed Now vlue re ssumed tegrl my cdi chrctertic cll Brownin moti bove) t probbily densy versi densy ws tegrted squred kno chrctertic tegrted squred voltily k trnsm dependent probbily densy drivg probbily densy versi densy g trnsm probbily chrctertic chrctertic re cll my cdi tegrted squred voltily n e Th mens cdig )schstic Now vlue ctertic g trnsm probbily s t wy new clssicl probbily y t chrctertic g trnsm probbily l my Europen s clls ide expressg dependent dependent Brownin Brownin moti moti drivg drivg deed deed bove) bove) wy pply ide descrid Ch re ssumed schstic tegrl schstic tegrl chrctertic tegrted squred voltily n e ) Now vlue Th mens cdig mul R rel le i imgry un )densy cdis mul densy probbily versi cll my cdi my cdi cll chrctertic tegrted squred voltily n e probbily densy versi mul y cll my cdi reprobbily ssumed schstic tegrl t g trnsm chrctertic y densy similr probbily t wy ws new clssicl expressi probbily my Prsevl s em densy expectti cdi versi dependent Brownin moti drivg deed bove) t chrctertic chrctertic tegrted tegrted squred squred voltily vlue know cll re ssumed schstic tegrl mul Th )voltily Now ) Now ul my cdi probbily cll Th mens cdig densy cdig versi hs nml dtrii men vrince hs mens nml dtrii men vrince ncti mul g densy trnsm probbily hs nml dtrii men vrince my my cdi cdi cll cll chrctertic tegrted squred voltily re n e hs ssumed schstic tegrl mul Th mens cdig ) Now hs nml dtrii men vri Th mens cdig ) Now give nml dtrii men vrince re ssumed schstic tegrl Th mens cdig ) Now vlue probbily densy versi n Th mens cdig Now schstic vlue tegrl tegrl cdi le cll R Th dtrii rel i un ) cdis ssumed hsmy men vrince re ssumed schstic R rel le inml imgry un )imgry cdis mens cdig ) Now re vlue ) re ssumed schstic tegrl hs nml dtrii men vrince hs nml dtrii men vrince Th Th mens mens cdig cdig ) ) Now Now Prsevl s em similr expressi expectti re ssumed schstic tegrl expressi ssumed expectti tegrl Ri em rel le isimilr imgry un ) cdis s rel le Prsevl s imgry ) cdis re schstic hs nml dtrii R rel le un i imgry un ) cdis Hence cll Hence cll men vrince re re ssumed ssumed schstic schstic tegrl tegrl Prsevl s em similr expressi expectti Th mens cdig expectti ) Hence vlue hs evl s emprsevl s similr expressi expectti R le i i un R rel le imgry imgry un ) cdis em rel similr expressi Now nml dtrii men vrince Hence cll Hence cll cll hs nml dtrii men vrince em Prsevl s re ssumed schstic tegrl Prsevl s similr expressi expectti cdis em similr expressi cll ne i imgry un Hence ) cdis hs nml dtrii men vrince hs nml dtrii men vrince hs strd nml dtrii dependent t pot e hs strd nml dtrii dependent t pot o hs nml dtrii men Hence vrince cll dtrii dependent strd nml t hs em similr expressi expectti expectti nml dtrii vrince Hence hs men cll hs strd nml dtrii hs strd nml dtrii dependent pot t hs nml dtrii men vrince no-rbrge cll dry Blck-Scholes no-rbrge cll dry Blck-Scholes Hence cll no-rbrge cll dry Blck-Scholes hs hs nml nml dtrii dtrii men men vrince vrince no-rbrge cll hs strd nml dtrii dependent t pot e no-rbrge cll dry Blck-Sch Hence cll replced Dee cllcll dependent t t replced Dee cll dtrii hs Hence strd nml pot dry replced Dee cll t hs nml dtrii men vrince no-rbrge cll Blck-Scholes replced Dee su replced Dee cll hs strd nml dtrii dependent t pot Hence cll Hence cll Hence cll cll s OW schstic voltily cll schstic voltily sme s no-rbrge sme cll dry Blck-Scholes cll schstic sme s On cll replced Dee Hence cll t no-rbrge voltily cll dry strd nml voltily dependent s pot Hence cll Th dil l Sce pyfs dtrii cll schstic mod schstic depent sme hs cll nml Blck-Scholes dependent strd nml dtrii replced strd Dee cll voltily trnsm hs found Prsevl's em my n fd trnsm dtrii found Prsevl's em my n fd t Th dil l Hence Hence cll cll trnsm found Prsevl's em my n fd replced Dee cll resurnce ctrcts re sme s cll s ie ) no-rbrge cll dry Blck-Scholes cll schstic voltily sme s hs strd nml dtrii dependent hs strd nml dtrii dependent found Prsevl's Tt trnsm no-rbrge cll dry Blck-Sc dent pot e no-rbrge hs strd nml dtrii dependent t pot e trnsm found Prsevl's em my n use followg mul Ch 2009 ems 21 22) followg mul Ch 2009 ems 21 22) cll schstic voltily sme s On Sce pyfs resurnce ctrcts re sme s echnique dil l pyfs Hence hs Sce cll followg mul Ch 2009 ems 21 22) hs strd nml dtrii dependent ems t e sme ides pply resurnce pricg mul t pper replced Dee cll Blck t no-rbrge Blck-Scholes cll pot dry no-rbrge cll dry Blc strd nml dtrii dependent mul t pot e followg mul Ch 2009 o trnsm found Prsevl's em my n fd On cll schstic voltily sme s cll dry no-rbrge cll dry Blck-Scholes followg Ch ) replced Dee cll cll ie sme rnce ctrcts re sme s clls s ie ) found Prsevl's em my n fd t no-rbrge trnsm Th Th dil dil dil Th l l Sce Sce pyfs pyfs dil dry ldtrii Sce Th hs hs strd strd nml nml dtrii pyfs dependent dependent cll Blck-Scholes l Sce pyfs no-rbrge cll dry Blck-Scholes replced Dee cll replced Dee cll followg mul Ch 2009 ems 21 22) replced Dee em n ny em n ny replced Dee cll t mul cll schstic voltily 21 sme s trnsm found Prsevl's em my n fd cll schstic voltily sme s resurnce resurnce ctrcts ctrcts re re sme sme s s cll cll s s ie ie )) Th pper dil lno-rbrge Sce pyfs ides pply pricg resurnce pricg ) On resurnce ctrcts re sme s cll s ie me ides pply resurnce ie t followg Ch 2009 ems 22) ) resurnce ctrcts re sme s mul cll s em n ny Blck-Sc no-rbrge cll cll n dry dry Blck-S hs strd nml dtrii mul dependent t pot e replced Dee cll t em em ny pply replced Dee s t cll pper t cll followg mul Ch 2009 ems 21 22) trnsm found Prsevl's em my n fd ) resurnce ctrcts re sme cll s ie cll schstic voltily sme s cll schstic voltily sme s cll schstic voltily sme s sme sme ides ides pply resurnce resurnce pricg pricg mul mul t t pper t pper trnsm found Prsevl's em my n sme ides pply resurnce pricg mul t pper sme ides pply resurnce pricg mul dry t pper Blck-Scholes replced replced Dee cll u no-rbrge cll Dee n ny pricg cll schstic sme s cll followg mul Ch 2009 ems 21 22) em sme ides pply resurnce mul Prsevl's t pper cll schstic voltily sme svoltily trnsm found em my n trnsm found Prsevl's em my n n ny em followg mul Ch 2009 ems 21 22) trnsm found Prsevl's em my n fd rel prt ly ssumptis re rom vrible posive replced cll t cll cll schstic schstic voltily voltily sme sme s g) found Prsevl s em my em n n ny s22) trnsm found Prsevl's em my 2009 fd Dee mul Ch 2009 ems 21 22) followg mul Ch ems 21 trnsm found Prsevl's em my ntrnsm fd followg mul Ch 2009 ems 21followg 22) hs fe men cstnt posive) re re similr muls cll schstic voltily sme s n fd followg mul Ch trnsm trnsm found found Prsevl's Prsevl's em em my my n n usu followg mul Ch 2009 ems 22) em 21 n ny n ny followg Ch 2009 ems 21 22) em rel prt ly ssumptis re mul rom vrible posive comtil dvntge mul t necessry rel prt ly ssumptis re rom vrible posive 2009 ems 21 22) followg followg mul mul Ch Ch ems ems ) 22) ssumptis found Prsevl's em my n fd ny em em vrible n n ny posive) rel reltrnsm prt muls ly re re rom rom vrible vrible posive posive rel n em rel prt ly ssumptis rn prt ny ly ssumptis re rom posive s fe men re similr cstnt fd hs dtrii re e comg sp-loss Ch posive) re re similr muls fe men cstnt em n re nysimilr followg mul 2009 ems 21 22) hs rel prt re similr ly ssumptis re rom vrible posive em n ny hs hs fe fe men men cstnt cstnt posive) posive) re re re similr muls muls dom vrible S posive fe men cstnt K comtil dvntge mul t necessry hs fe men cstnt posive) re muls dvntge mul t necessry comtil n sgle tegrl Europen em ny em n ny em posive) re resimilr muls hs fe men cstnt dtrii comtil dvntge dvntge sp-loss mul mul t t necessry necessry comtil dvntge mul tn necessry posive) re re similr muls d dtriicll Th comg sp-loss fd comtil e comg e n fdg dtrii n comg ny em n ny dvntge mul t necessry fd fd dtrii dtrii e e comg comg sp-loss sp-loss comtil dvntge mul t dtrii um) pyf n sgle tegrl Europen fd e comg ipti sp-loss n sgle tegrl comtil Europen lso fster n Mte Crlo simulti fd dtrii e sp-loss n comg cll Th n fdg dtrii dtrii n n sgle sgle tegrl tegrl Europen Europen comg n sgle tegrl Europen necessry fd S e comg h n fdg dtrii n comg pyf lso fster Mte Crlo simulti n n sgle tegrl Europen lso fster Mte Crlo simulti cll cll Th Th n n fdg fdg dtrii n n comg comg n wy pply ide n descrid Ch dtrii n comg cll Th dtrii sp-loss fdg pyf pyf lso lso fster fster n n Mte Mte Crlo Crlo simulti simulti n comg cll Th n fdg dtrii pyf lso fster n Mte Crlo simulti dependent Brownin moti drivg deed bove) t n sgle tegrl Europen wy pply ide descrid Ch wy pply ide descrid Ch pyf lso n Mte Crlo simulti cll n fdg dtrii n dependent Th Brownin moti drivg deed bove) t chrctertic tegrted squred voltily fster S n e wy wy pply pply ide idebove) descrid descrid Ch Ch endent Brownin moti drivg deed t wy pply ide descrid Ch chrctertic tegrted n e my cdi squred cllvoltily wy pply ide descrid bove) dependent dependent Brownin Brownin moti moti drivg drivg deed deed bove) bove) Ch t t dependent Brownin moti drivg deed t tertic tegrted squred voltily n e my cdi cll dependent Brownin moti drivg deed bove) t chrctertic chrctertic tegrted tegrted squred squred voltily voltily n n e e chrctertic tegrted squred voltily n e di Th cdig cll similr mens ) Now vlue mul obted s regime swchg schstic voltily ws test mul Crlo simulti lso explic mul chrctertic tegrted voltily n e Th mens ) Now cdi vlue squred my mycdig cdi cdi cll cll gst cllmte my re ssumed schstic tegrl cse voltily tkes vlues) dtrii re ssumed schstic tegrl cll my cdi mens cdig ) Now vlue tegrl ts lterntives esily comg time s well s codg eft Th mens mens cdig cdig Th mens ))Now Now vlue vlue em t vlue tril ly downside t re free prmer ) cdig Now re ssumed Th schstic tegrl 30 RISKS ND REWRDS FEBRURY 2010 Th mens cdig err required fd good)rnge Th comm fure Now vlue tegrls re ressumed ssumed schstic schstic tegrl re ssumed schstic tegrl tegrl t volve oscilltg s hs nml dtrii men vrince hs nml dtrii men vrince re ssumed schstic tegrl Dniel FS PhD Centre cturil Studies Universy Melbourne cn