On Root's Barriers and Their Applications in Robust Pricing and Hedging of Variance Options

Transcription

1 On Roo's Barriers and heir Applicaions in Robus Pricing and Hedging of Variance Opions i Huang Mansfield College Universiy of Oxford isseraion for MSc in Mahemaical and Compuaional Finance riniy erm 1

2 Acknowledgemens I am hearily hankful o my supervisor, r Jan Obłój, whose grea encouragemen, guidance and suppor from he iniial o he final level enabled me o develop a deeper undersanding of his subjec. I believe, wihou him poining ou he righ direcion when I encouner difficulies, his hesis could no have been finished of his qualiy. I also offer my regards o all deparmen members who suppored me in any respec during he compleion of his hesis. i

3 Absrac he variance opion has recenly drawn grea aenion in financial research because i provides an invesor wih proecion on exposure o volailiy risk. In his paper I will reexamine he resul of Cox and Wang [3] which has made a remarkable conribuion o his area. hey explored he Roo s barrier in erms of seeking he soluion of a variaional inequaliy and consruced subhedging sraegies for variance opions hrough an alernaive proof of he opimal propery of Roo s barrier. In order o give a brief review of heir work I insaniae heir resuls by conducing boh numerical implemenaion and heoreical calculaion. o backup my resul, I backes my numerical soluion and compare he subhedging sraegies proposed in Cox and Wang [3] wih he subhedging sraegies separaely suggesed in Carr and Lee [17]. he comparison will well explain and demonsrae he generalizaion done in Cox and Wang [3]. Keywords: variance opion, Roo s barrier, Skorokhod embedding, obsacle problem, subhedging ii

4 Conens 1 Inroducion... 1 Mehodology and Resuls of Cox & Wang Alernaive of SEP on Solving Roo s Barrier he Subhedging Sraegy Implemenaion of he Obsacle Problem Finie ifference Mehod on he Obsacle Problem Brownian Moion Case Geomeric Brownian Moion Case Oher Examples Mone Carlo Backes Robus Subhedging Examples (Sub)hedging Sraegy for a Variance Swap Subhedging Sraegy for a Variance Call Conclusions... 3 Bibliography... 3 iii

5 Chaper 1 Inroducion In his paper, I mainly go hrough he resuls in Cox and Wang [3] on heir exension on he financial applicaion of Roo s barrier and he consrucion of subhedging sraegies. he moivaion comes from he recen increasing demand on he variance swaps and variance opions o conrol over exposure o volailiy risk. Since he marke for such producs is no well esablished, hedging and pricing of such producs is academically done in a model-independen fashion o avoid risks of model misspecificaion. However, model-independen approaches usually can no lead us o he exac price of a variance opion. Wha is done is o loosen he hedging problem o he problem of exploring he model-independen upper and lower bounds on prices by consrucing superhedging and subhedging sraegies. By narrowing he gap beween he upper bound and he lower bound for a variance opion, we can obain an close approximaion o he price of he variance opion. Cox and Wang [3] s work is mainly on he consrucion of an opimal lower bound for a variance opion, same wih hem, I will only focus on he lower bound and he subhedging sraegy of a variance call in his paper. Academically, he soluion of his problem is conneced o he well-known Skorokhod embedding problem (SEP) or someimes called he Skorokhod sopping problem. o furher illusrae his issue in a mahemaical manner, firs consider a simplified risk-neural environmen wih zero ineres rae and assume here is no arbirage in he marke. he sock price process S ( ) saisfies he following SE wih an unspecified : ds S dw (1.1) he log of sock price process hen saisfies: 1 d ln( S ) d dw (1.) By inegraing boh sides of (1.), we have: ln S d (1.3) I means ha he payoff of a variance swap or a variance opion can be seen as a funcion of he realized logarihmic quadraic variaion of he sock price process a mauriy. Afer wriing he payoff in form of F( ln S ), he variance opion wih srike K is hen a call or a pu wih F( x) ( x K) or F( x) ( K x) and he variance swap has F( x) x K. he nex sep is o adop a coninuous ime change o he sock price evoluion, which originaes from ubins and Schwarz [14]. By ubins-schwarz heorem, ake ln S and leing X S, we have: 1

6 d d (1.4) SE (1.4) means ha he law of ( S, ln S ) is jus equivalen o he law of ( X, ). herefore finding he lower bound of a variance call will be equivalen o finding an opimal o minimize E( K) subjec o he condiion ha he law of X is equal o he law of S. he law of S, demoed by, or he risk neural probabiliy densiy of S, can be easily obained hrough call prices. By aking he second derivaive wih respec o he srike price of call opions on S, we are able o ge a knowledge of he disribuion of S provided ha we have prices of call opions on all srikes. his problem ensembles SEP. he main issue of SEP is o find o minimal sopping ime wih a given underlying sochasic process X and a given probabiliy measure such ha X obeys he measure. Roo [9] has shown ha when X is a Brownian moion wih X and being a cenered law, he soluion of SEP is he firs hiing ime of a barrier B( ). his barrier B, which is called he Roo s barrier, is defined as a se ( x, ) in R Rsuch ha if ( x, ) B hen ( x, s) B for every s. Precisely, Roo s barrier is defined as follows: A closed subse B of [, ] [, ] is a barrier if 1) ( x, ) Bfor all x[, ]; ) (, ) Bfor all [, ]; 3) if ( x, ) B hen ( x, s) B whenever s. Roo [9] s resul is laer grealy exended by Ros [1] [11] o loosen he resricion on he underlying sochasic processx. Ros s resul only requires X being a Markov process and righ coninuous. Furhermore, he proved he exisence and uniqueness of a sopping ime of minimal residual expecaion o he Roo s barrier. By saying a sopping ime is of minimal residual expecaion we mean ha: A sopping ime is of minimal residual expecaion if for each following quaniy:, i minimizes he E( ) E ds P( s) ds over all { : X } where we assume X and X. he resul in Ros [1] [11] requires cerain resricions on boh and. Lierally, he condiion requires ha for all x he expecaion of - y x wih y is greaer han or equal o he expecaion of - y x wih y and boh of he expecaions mus be sricly less hen infiniy. Mahemaically, for all x : U ( x) y x ( y) dy y x ( y) dy U ( x) (1.5) Cox and Wang [3] makes furher conribuion o he soluion of SEP based on he work of Roo [9] and Ros [1] [11]. hey represened and proved an alernaive mehod o solve he

7 Roo s barrier problem wih a generalized underlying sochasic process. he alernaive is basically an obsacle approach bu hey furher applied he popular variaional inequaliies approach o generalize he basic obsacle problem. One of advanages of ransforming he SEP o an obsacle problem is ha he ransformed problem enables easier and sraighforward numerical implemenaion, especially wih a finie difference mehod. Anoher imporan resul of Cox and Wang [3] is heir consrucion of a dynamic subhedging sraegy based on he opimal characerizaion of Roo s barrier. hese ogeher inrigue me o make a brief review o he exension made in Cox and Wang [3] and illusrae some ypical examples by insaniaing heir resuls. o es he validiy of Cox and Wang [3], I will backes he subhedging sraegies hey proposed by applying he sraegies o a variance swap and a variance call and make a comparison wih he subhedging sraegies suggesed in Carr and Lee [17]. he bigges difference of Cox and Wang [3] and Carr and Lee [17] is ha hey look a he same problem from differen heighs. Generally, he relevan academic researches on variance opions are divided ino wo groups. he firs group invesigaes he usage of SEP and Roo s barrier o sudy model independen hedging sraegies while anoher group aims o consruc model independen super-replicaion or sub-replicaion direcly using European opions and oher financial producs. Cox and Wang [3] is in he firs group and he laes works on his aspec include Cox, Hobson and Obłój [], upire [7] and Obłój [13]. his angle develops quickly in recen imes because i enables subreplicaing he variance opion in very general seings. Carr and Lee [17] belongs o he second group. his group forms he basis for he consrucion of hedging sraegies of variance opions. heir work on his aspec mainly includes Neuberger [4] [5] and Carr and Lee [17]. However he work of he firs group is acually a developmen and re-formulizaion of he second group. By discovering and proposing he imporan link of he Skorokhod embedding problem and he model-independen in heir sudy by upire [6] and Carr and Lee [17], hey faciliae he research of he firs group. he res of his paper is organized as follows: Chaper review he mehodology and major resuls in Cox and Wang [3] relaed o Roo s barrier and he process of consrucing he subhedging sraegies. In Chaper 3, I perform a numerical implemenaion o solve he obsacle problem wih a Mone Carlo simulaion o backes he resuls. Chaper 4 is wo insaniaions of subhedging sraegies in Cox and Wang [3] and comparison wih resuls in Carr and Lee [17]. Chaper 5 concludes he paper. In Chaper, I will summarize he main deducing seps and resuls in Carr and Lee [17]. For he firs half of his chaper, he effor is on he alernaive o SEP and for he laer half of his chaper how o build he subhedging sraegy from Roo s barrier is explained. Chaper 3 is also composed by wo pars. Secion 3.1 shows he algorihm and resuls from numerically implemening he alernaive o SEP. he arge disribuion includes uniform disribuion, normal disribuion, log-normal disribuion, uniform disribuion wih aoms and exponenial disribuion. Secion 3. shows he resul by esing he barrier in Secion 3.1 in Mone Carlo simulaion. In he forh chaper of his paper, I demonsrae in deail ha he resul of Carr and Lee [17], a laes represenaive paper in subreplicaion of variance opions, is an insaniaion of Roo s barrier and Carr and Lee [17] s subhedging sraegy is included in he general sraegy provided in 3

8 Cox and Wang [3]. he approach I adop is o apply he hedging sraegies of Carr and Lee [17] and Cox and Wang [3] in a variance swap and a variance call. he comparison is made and he difference is well explained in he las par of Chaper 4. 4

9 Chaper Mehodology and Resuls of Cox & Wang he main conribuion of Cox and Wang [3] is wofold. Firs, hey proved a one-o-one correspondence of SEP and an obsacle problem (or sricly speaking a variaional inequaliy) and secondly, hey consruced subhedging sraegies for variance opions wih he opimal propery of Roo s barrier. herefore in his chaper, I will summaries heir resuls in separae wo secions..1 Alernaive of SEP on Solving Roo s Barrier Before moving forward o summarize he alernaive mehod of SEP on solving Roo s barrier, I resae he Roo s barrier problem formally. Generally, he sochasic process X under sudy can be wrien as: dx ( X ) dw (.1) wihw being a Brownian moion and X following a cerain law. Cox and Wang [3] makes cerain resricions on : : For some posiive consan K, ( x) ( y) K x y ( x) K(1 x ) is smooh And we require for some sricly posiive consan,. As in a financial conex, we are mos ineresed in wo major applicaions of :, namely ( x) and ( x) x, which sand for he Brownian moion and geomeric Brownian moion. As for ( x), i is naurally included as he mos rivial siuaion. For he case ( x) x, hough excluded by he above resricion, Cox and Wang [3] (Secion 4.6) employed a simple ransformaion by leing v( x, ) u( e z, ) o ensure ha all conclusions discussed in he res of his paper can cover he case ( x) x. Wih he above resricions on :, he Roo s Skorokhod embedding problem is a problem as follows (quoed from Cox and Wang [3]): 5

10 SEP (,, ) : Find a lower semi-coninuous funcion Rx ( ) such ha he domain defined by {( x, ) : R( x)} has X, and ( X ) is a uniformly inegrable process, where is he iniial law of X and he diffusion coefficien. where Rxis ( ) a lower semi-coninuous funcion R : [, ] suggesed by Loynes [19] o describe he Roo s barrier by rewriing B as B {( x, ) : R( x)}. As discussed before, Ros [11] and Loynes [19] have shown he exisence and uniqueness of he Roo s barrier o SEP. However, aemp o solve he SEP direcly is ime-consuming, complex and only resuls in an explici expression of he domain {( x, ): R( x)} in a few simple cases. herefore, we hope ha we could find an alernaive mehod exacly equivalen o SEP bu is more feasible and can be easily implemened in a numerical way. he alernaive would be even beer if i could provide a soluion ha is one-o-one corresponding o he domain {( x, ): R( x)} and has a very general seup. he alernaive suggesed by Cox and Wang [3] mees all requiremen menioned above. I is basically an obsacle problem and hey have shown ha for every soluion of he following obsacle problem, we can consruc a domain ha solves he associaed SEP. he descripion of he obsacle problem is from Cox and Wang [3]: 1,1 OBS(,, ) : Find a funcionu( x, ) ( ) such ha U ( x) u( x,) (.) U ( x) u( x, ) (.3) u 1 u x ( x, ) ( x) ( x, ) (.4) u 1 u x ( ( x, ) ( x) ( x, ))( U ( x) u( x, )) (.5) where U ( x) and U ( x) is he funcions defined in (1.5). Condiion (.4) can be undersood as requiring he following in a probabilisic manner: u 1 u x ( ( x) ( x, ) ( x) ( x, ) ( x)) dx (.6) And condiion (.5) indicaes ha whenever U ( x) u( x, ) : u 1 u x ( x, ) ( x) ( x, ) In oher words, he funcion u( x, ) will evolve wih ime according o he PE (.7) and will sop immediaely whenever i his he corresponding boundary specified by U ( x) for all x. hen (.7) 6

11 we can define he domain as {( x, ) : U ( x) u( x, ), } based on heorem3.1 in Cox and Wang [3]: Suppose is a soluion o SEP(,, ) and is such ha: henu solves OBS (,, ). u x E X x (.8) 1,1 (, ) ( ) However here are wo main drawbacks of his obsacle problem. Firs i 1,1 resrics u( x, ) ( ) o make sure ha he second derivaive of u( x, ) is well defined. his obviously excludes many cases. In realiy, we usually do no expec a very smooh law of S since i is unlikely o obain prices of European calls for all srikes K. Even if we have access o his knowledge, he disribuion of S derived from he prices of Europeans calls will no performs perfecly due o he unavoidable imperfecion in he marke. herefore may conain aoms which makes u( x, ) no firs order coninuous in x. Furhermore, heorem3.1 in Cox and Wang [3] only ensures a one-way projecion from he soluion of SEP o OBS. I neiher ensures he exisence nor uniqueness of he soluion of OBS. I prevens us from consrucing a one-o-one correspondence beween SEP and OBS. he drawback of he obsacle problem leads Cox and Wang o reformulae he problem and furher generalize he seup. hey reformulae he obsacle problem hrough he popular approach called variaional inequaliies where exisence and uniqueness of he soluion are boh ensured inherenly. By coordinaing he seup wih SEP, he variaional inequaliies problem is described as follows (briefly quoed from Cox and Wang [3]): VI (,, ) : Find a funcion v: [, ] saisfying: 1, v v L (, ; H, ) and L (, ; H ) (.9) v 1, (, v) a( ; v, v), H such ha () a.e. (, ) (.1) v(, ) ( ) a.e. (, ) (.11) v(,) v (.1) m, where H and m L (, ; H, ) are Hilber spaces wih naural inner producs. And all he coefficiens are given by he following definiions: x v v (.13) x x a ( ; v, ) e [ a( x, ) b( x, ) ] dx 7

12 where a( x, ) ( x) andb( x, ) ( x) ( x) ( x)sgn( x) ( x) U ( x) (.14) v U ( x) (.15) Noe ha in order o ensure he exisence and uniqueness of soluion o VI (,, ), some similar resricions as hose in obsacle problem are added o :. Bu as menioned earlier, he resuls can cover ( x) x wih ransformaion of v( x, ) u( e z, ). he formulaion of VI (,, ) hen leads o he main resul of Cox and Wang [3] heorem 4., which ensures a one-o-one correspondence of SEP and VI: Le and v be he soluions o SEP (,, ) and VI(,, ) respecively, define: And v u( x, ) E X x (.16) {( x, ) [, ]: v( x, ) ( x)} (.17) hen we have [, ] and for all( x, ) [, ], u( x, ) v( x, ) (.18) he above heorem gives a perfec link beween SEP and VI, which shifs he aemps o solve SEP o aemps o solve he sysem wih VI. If is no oo weird, implemenaion of VI (,, ) or OBS(,, ) will usually give ou a nice approximaion of Roo s barrier. Noe ha our purpose of solving he SEP is o find he lower bound on he price of variance opions. Hence here wih he approximaion of Roo s barrier we could be able o do i. he connecion of soluion of SEP o he lower bound of variance call is described by he minimal residual expecaion characerizaion of Roo s barrier. Alhough Ros [11] has given a complee proof on he minimal residual expecaion characerizaion, Cox and Wang [3] explored an alernaive way o prove i in which hey also found a submaringale G and a funcion H( x) o form a subhedging sraegy for variance calls.. he Subhedging Sraegy Suppose is a given cenered disribuion, X evolves as he process in (.1) and X, he minimal residual expecaion characerizaion of Roo s barrier is equivalen o minimizing EF [ ( )] subjec o X where is a sopping ime and funcion F is convex, increasing wih F(). enoe is righ derivaive f. he subhedging sraegy is based on 8

13 definiion of four funcions, namely M ( x, ), Zx ( ), G( x, ) and H( x ). M ( x, ), Zx ( ), G( x, ) and H( x) are defined wih a given domain ha solves a corresponding SEP: ( x, ) M ( x, ) E f ( ) (.19) Z( x) x y M( z,) dzdy () z (.) G G( X, ) and G( x, ) M ( x, s) ds Z( x) (.1) R( x) H( x) ( f ( s) M ( x, s)) dx Z( x) (.) where is he firs hiing ime of he given domain and Rxis ( ) he funcion used o describe he barrier. here are hree main resuls here given by Cox and Wang [3]: (1) For all ( x, ), we have G( x, ) H( x) F( ). () G( X, ) is a maringale and G( X, ) is a sub-maringale if: [ ( s) ( s) ] E Z X X ds and EZ [ ] (3) E[ F( )] E[ F( )] where X Proposiion (1) and () ogeher show ha G( x, ) plays he role of a dynamic subreplicaion of he variance opion wih payoff funcion F. If is Roo s barrier, his subreplicaing sraegy will provide an opimal lower bound for he price of he variance opion. his lower bound is given by EF [ ( )] in Proposiion (3). And Proposiion (3) shows ha i is indeed he lower bound and is opimal when equaliy in (3) holds. Wih definiions in (.19)-(.), he specific subhedging sraegy is consruced hrough he following process assuming now we have a sock price pah following an arbirary sochasic process wih iniial price S.he process can be similarly expressed as ha in (.1): ds ( S ) dw (.3) And I assume ha I have obained he Roo s barrier from he disribuion of S implied by he call prices of all available srikes. enoe he corresponding domain of his Roo s barrier as. As discussed before, he subhedging sraegy in Cox and Wang [3] consiss of finding a submaringale denoed by G( S, ) and a funcion denoed by HS ( ) o saisfy he inequaliy: G( S, ) H( S ) F( ) (.4) 9

14 where he equaliy holds when. Here denoes he firs hiing ime of he obained Roo s barrier. Afer obaining G( S, ) and HS ( ), i is able o build up a sub-replicaion porfolio separaely subhedges G( S, ) and fully hedges HS ( ). he process begins wih a ime change which is similar o ha in Secion Inroducion bu is more general: X ln( S ) S which enables us o rewrie (.3) as: dx X dw (.5) wherew is a Brownian moion under he new filraion. his ime change is based on heorem of ubins and Schwarz [14]. By Io s lemma, we have: d ln( S ) ds d S ds d ln( S) S S S (.3) indicaes ha S is a coninuous local maringale hence so is 1 1 Y ds ln( S ) ln( S) s S s By ubins-schwarz heorem, here exiss a Brownian moionw wih: and hence we have: his leads o SE (.5). W ln( S ) Y 1 ln( S ) W ln( S) ln( S ) Wih his ime change, inequaliy (.4) becomes: Y where: G( X, ) H( X ) F( ) F( ln( S) ) (.6) A where A is defined hrough S A X. Firs consider he subhedging sraegy for G( X, ). he submaringale par of G( X, ) can be subhedged hrough he Maringale Represenaion heorem. By he Maringale Represenaion heorem, here exiss a process saisfying he following inequaliy: G( X, ) G( X,) dx s s (.7) Since G( X, ) is a maringale, he equaliy in (.7) holds when and moreover: G G( X, ) G( X,) ( X, ) dx (.8) s x herefore wih some cerain process, i is able o sub-replicae G( S, ) hrough: 1

15 G( X, ) G( X,) dx G( S,) ds s s s s (.9) or: G( S, ) G( S,) ds s s (.3) wihs s. Hence he subreplicaing porfolio can be consruced as holding s unis of sock and he iniial invesmen equalsg( S,) S. As for HS ( ), Cox and Wang [3] fully replicaes i hrough he following decomposiion: H( S ) H( S ) H ( S )( S S ) ( S K) H ( dk) ( K S ) H ( dk) ( S, ) (, S] (.31) (.31) indicaes ha HS ( ) can be replicaed by puing H( S) H ( S) S in he bank accoun, holding H ( S) unies of shares, holding H ( dk ) unies of calls wih srike K for K greaer han S and holding H( dk) unies of pus wih srike K for K smaller han or equal o S. Based on he sub-replicaion or replicaion of G( S, ) and HS ( ), he lower bound for he price of he variance opion wih payoff F( ln S ) is hen: G( S,) H( S ) C( K) H ( dk) P( K) H ( dk) ( S, ) (, S] (.3) 11

16 Chaper 3 Implemenaion of he Obsacle Problem In his secion, my aim is wofold. Firs, I will perform a numerical implemenaion o solve he Roo s barrier in erms of an obsacle problem as described in Secion. here are wo reasons ha I choose o perform under he sysem of OBS (,, ) insead of VI (,, ). he firs reason is ha I only solve he Roo s barrier wih simple assumpions of (,, ) and hey essenially mee he resricions pu on OBS(,, ). herefore here is no need o generalize from OBS (,, ) o VI (,, ). he second reason is ha variaional inequaliies are jus an exension of OBS (,, ) and do no change is basic algorihm. Solving VI (,, ) and C solving OBS (,, ) is essenially he same, which boh provide us wih a barrier B wih respec o a specified group of (,, ). My second objec in his secion is o verify he conclusion of Cox and Wang [3] ha he barrier obained from OBS (,, ) is indeed he soluion o he corresponding SEP(,, ). In order o do his, I apply Mone Carlo simulaion o execue a backes. he idea is based on he following fac: If we use Mone Carlo mehod o simulae a sufficienly large number of pahs of he underlying sochasic process X, sop hem immediaely when hey hi he barrier and make record of he posiions when hey sop, we can ge a close approximaion of he disribuion of X. If he barrier is acually he Roo s barrier ha solves he SEP, his approximaion of he disribuion of X will be an approximaion of he given. 3.1 Finie ifference Mehod on he Obsacle Problem In order o solve he OBS(,, ) described in Secion numerically, I employ a forward explici finie difference scheme. he advanage of a finie difference scheme is ha we can sop whenever we wan during he evoluion. Since in finance we are mos ineresed in cases where ( x) and ( x) x, I only solve he OBS(,, ) in his wo represenaive cases. he solving process generally consiss of he following wo seps. he firs sep is o build up he arge disribuion of X denoed by. Here sands for he sopping ime when U ( x) u( x, ) and he whole evoluion sops. o explain his, noe ha we have an explici expression of u( x, ) given by (.8) as: When ( x), u x E X x 1,1 (, ) ( ) X is a Brownian moion and when ( x) x, X is a geomeric 1

17 Brownian moion. hese wo processes boh ensure ha E X x is non-decreasing in. herefore u( x, ) is obviously a non-increasing funcion in given x is fixed. Furhermore we have U( x) u( x,) U( x) as a necessary assumpion. Hence from he beginning u( x, ) (if sricly greaer han U ( x) ) will evolve according o PE (.7) and as he ime goes forward, u( x, ) will ge closer and closer o U ( x). A some momen in ime, u( x, ) will evenually be equal o U ( x). From his poin, u( x, ) will no longer evolve wih ime according o PE (.7) bu equal o U ( x) for all greaer han his momen. he momen when u( x, ) his U ( x) is jus and he collecion of values of x saisfying u( x, ) U ( x) is hen he barrier we ge. o illusrae he above argumen, I plo he figure of ux (,), U ( x) and he corresponding evoluion process in Figure under he assumpion of ( x) and being he uniform disribuion. Figure 3.1.1: Evoluion of u( x, ) wih ime and U ( x) he black line on he op in Figure represens ux (,) and he red parabola below is he arge U ( x). We can see ha wih ime increasing, he shape of u( x, ) is pulled down smoohly and ges closer and closer o U ( x). Before he green line ouches he red line, i will be coninually pulled down by PE (.7). Once i ouches he red line, i remains in he same place of he red line. Obviously, seing up he arge disribuion of X is equivalen o building up U ( x). From (1.5) we know ha U ( x) is he expecaion of yx wih y.for simple like uniform, normal and log-normal disribuion, U ( x) can be expressed explicily by calculaing ou he inegraion of y x ( y) dy. For cases where he inegraion can no be calculaed ou explicily, we can adop Mone Carlo simulaion o ge an approximaion of he expecaion for each x. he second sep in numerical implemenaion is o solve he sysem in OBS (,, ). As discussed above, before u( x, ) his U ( x), u( x, ) saisfies PE (.7): u 1 u x ( x, ) ( x) ( x, ) herefore he algorihm for solving he OBS (,, ) goes like his: for every ime sep 13

18 from, I check wheher u( x, ) is smaller han or equal o U ( x). If i is, hen make u( x, ) evolves according o PE (.7) for he nex ime sep. If i is no, hen se u( x, ) equal o U ( x) for he res of he ime seps and ge record of he posiion of x immediaely. he forward explici finie difference scheme o implemen he above algorihm is formulaed as follows. Assume we have a given mauriy, a large enough value be considered and a small enough value Xmin X max as he ceiling for x o as he floor for x o be considered. We hen equally divide he space [ X min, X max ] [, ] ino a M N grid wih M 1 poins in x-grid and N 1 poins in -grid. For 1 i N1and 1 j M 1, denoe U( j, i) as he value of u( x j, i), where x j is he j-h poin in x-grid and i is he i-h grid in -grid. efine N and x ( X max X min ) M. hen for every ime-sep, PE (.7) can be approximaed by he following equaion: U ( j, i 1) U ( j, i ) 1 ( 1, ) (, ) ( 1, ) ( x j ) U j i U j i U j i ( x) for 1iN and j M. Hence he value of U( j, i 1) can be derived from values of U( j 1, i), U( j, i) and U( j 1, i) which are already known. Precisely: (3.1) ( x j ) ( x j ) ( x j ) U( j, i 1) U( j 1, i) (1 ) U( j, i) U( j 1, i) ( x) ( x) ( x) (3.) for1inand j M. o sum up, he second sep of he forward finie difference scheme sars by seing up he value for U(1, j) for all j. hen by increasing he value of i one by one while calculaing he value of U( i, j) for all j, we are able o simulae he evoluion process. he boundary condiion for he scheme is formulaed according o he explici expression of u( x, ) given by (.8). For values a, u( x,) E X x. Hence: U( j,1) u( x,) X x for all1 j M 1 (3.) j and for values a x Xmax and x Xmin : i min i U(1, i) u( X, ) i max i U( M 1, i) u( X, ) E X X if min E X X oherwise min E X X if max E X X oherwise max i i (3.3) (3.4) Here we have obained all necessary seing for implemening OBS (,, ) numerically wih a forward finie difference scheme. In he nex secion, I will represen he barriers I obained from solving he obsacle problem. Firs I will represen he barrier by solving he OBS (,, ) under ( x) for being he uniform disribuion and he normal disribuion. hen I urn o he case where ( x) xand explore he barrier wih being he log-normal disribuion. he reason why I choose hese seings for is ha we have already known wha he 14

19 shape of he barrier should be, which is very nea and regular. If he barrier I obained is differen from he shape i supposed o be, hen I can quickly find ha here mus be somehing wrong eiher wih my implemenaion or wih he resuls given by Cox and Wang [3]. Afer his saniy check, I will implemen wo groups of(,, ) where he explici expression of Roo s barrier can no be obained immediaely from heoreical calculaion and see wha he shape of barrier looks like. For simpliciy, assume is a disribuion ha wih probabiliy one equal o a consan X in he res of his secion Brownian Moion Case here is no doub ha he simples and mos commonly used funcion for is ha being a consan. In his case, he underlying sochasic process X evolves as a Brownian moion: dx dw (3.5) I has been shown ha if is a uniform disribuion cenered a X, we should ge a very nice barrier symmeric and concave o X. And if is se o be he normal disribuion cenered a X, we shall obain a verical sraigh line a some poin in x-grid. his is a naural conclusion. If we sop he Brownian moion in (3.5) a any ime, X is equal o X W, which is normally disribued wih mean X and variance. Figure 3.1. shows he barriers I obained from numerical implemenaion. Obviously, excep for some implemenaion errors near boundaries, hey are boh good approximaions of he arge Roo s barriers and herefore give an iniial suppors for he resul in Cox and Wang [3] Geomeric Brownian Moion Case In he financial conex, he underlying sochasic process under invesigaion is usually a geomeric Brownian moion. In his case, ( x) x. Now for, he log-normal disribuion becomes a rivial case. Because if we sop a geomeric Brownian moion a ime, X is hen log-normal disribued. enoe Z ln( X), we have: 1 Z N(ln( X ), ) Bu as for X being a geomeric Brownian moion, he finie difference scheme in equaion (3.) migh encouner some problems when calculaing ( x j ) ( x) when x j is very close o. In order o avoid his undesirable feaure, I make a ransformaion o he issue by z leing u( x, ) u( e, ) v( z, ). hen PE (.7) changes o: v 1 1 v v z z (3.6) In his way an M z-grid replaces he original M x-grid and he forward explici finie difference scheme (3.1) changes o: V ( j, i 1) V ( j, i ) 1 ( 1, ) (, ) ( 1, ) 1 ( 1, ) ( 1, ) V j i V j i V j i V j i V j i ( x) x (3.7) 15

20 for1inand jm and where V ( j, i) sands for he value of v( z j, i). Similarly z j is he j-h poin in z-grid and i is sill he i-h grid in -grid, According o (3.7), value of V ( j, i 1) can also be derived from values of V ( j 1, i), V ( j, i) and V ( j 1, i) which are already known. enoe: a ( z) and b z Wih he above denoaion, (3.) is ransformed ino: 1 1 V ( j, i 1) ( a b) V ( j 1, i) (1 a) V ( j, i) ( a b) V ( j 1, i) (3.8) Figure represens he barrier obained in his case. I is a sraigh line and again accords wih our predicion. I suppors he idea ha he resul of Cox and Wang [3] can indeed be exended o he geomeric Brownian moion. Figure 3.1.: Barriers under uniform disribuion (above) and normal disribuion (below) 16

21 Figure 3.1.3: Barriers under log-normal disribuion Oher Examples Here I pick wo disribuions for ha are no commonly seen and invesigae wha he barrier would look like. he aim o do his is o give more generalized examples since under mos circumsances he barrier is eiher unsmooh or irregular. iscussion and represenaion in Secion and Secion 3.1. only shows some very simple and rivial Roo s barriers. here barriers usually serve as benchmarks as a saniy check or used as an inheren assumpion in he consrucion of hedging sraegies. One example is given by Carr and Lee [17]. heir resul urns ou o be a subhedging sraegy based on a consan barrier. he lower graph in Figure 3.1. and Figure are wo mos imporan consan barriers. However, in pracice, he shape of Roo s barrier will no be so perfec due o he deviaion of he sochasic sock process from eiher a sandard geomeric Brownian moion or a Brownian moion. he firs disribuion is a uniform disribuion wih an aom a X. I means ha he law of has probabiliy p o be a uniform disribuion beween and X while i has probabiliy 1-p o be a he poin X. Here I se p=.8 and ge he barrier depiced in he upper figure in Figure I can be easily deduced ha if is jus an aom a X, he barrier will be nohing bu a horizonal line beginning from X o infiniy. And i has already been shown in he upper figure in Figure 3.1. ha he barrier of a uniform disribuion under Brownian moion is a parabolic curve symmerically concave o X. herefore he barrier in he upper graph of Figure can be viewed as a combinaion of he above wo barriers. I is a parabolic curve concave o X wih a spike poining o X. If we keep increasing he number of seps in finie difference evoluion, he spike will evenually become a horizonal line. I is he shape of a spike due o he resricion on he obsacle problem. he barrier from solving he obsacle problem will be firs order coninuous in x and herefore we can only approximae he horizonal componen smoohly by a coninuous spike. Also, he shape of his barrier is conneced o he value of p. If he value of p decreases, he lengh of his spike will be greaer. I accords wih inuiion since if we se p=, he barrier will become he one in he pure aom disribuion case. 17

22 We can argue ha he acual shape of Roo s barrier will be he combinaion of a parabola and Figure 3.1.4: Barriers under uniform disribuion wih an aom a X (above) and kinks in he arge disribuion (below) a horizonal line poining o he aom hrough he evoluion process shown in he lower graph of Figure he graph in he lef below represens he siuaion where here exiss only one aom a X in he uniform disribuion. he graph in he righ below shows he siuaion where here exis wo aoms, one a X and anoher a.3x. We can see ha he number of aoms in a uniform disribuion is shown by he number of kinks in he arge disribuion (red line in he graph). When he funcion u( x, ) evolves wih ime and pulled down by he corresponding PE, he poins in x-grid near hese kinks will be he firs o hi he arge red line. herefore here will form a horizonal line in he Roo s barrier which will be approximaed by a spike. he second disribuion I choose is a arge exponenial disribuion under geomeric Brownian moion. he drawback of my numerical implemenaion here is ha we abandoned z values ha are smaller han he minimal value se on z-grid o avoid he negaive infiniy of log zero. By doing so, we direcly alered he probabiliy of hiing he barrier. Especially in his case, for an exponenial disribuion, he probabiliy assigned o a smaller value of he underlying random variable is much greaer han he probabiliy assigned o hose larger values. herefore by cuing off smaller z values, he underlying sochasic process X will be more likely ending up hiing he floor se o z-grid as ha shown in Figure However we can sill obain a nice 18

23 barrier. Figure 3.1.5: Barriers under exponenial disribuion wih z-grid 3. Mone Carlo Backes Alhough he shape of Roo s barrier of occasions wih uniform, normal and log-normal disribuions have already been proved heoreically, I implemen a numerical backes using Mone Carlo simulaion o beer illusrae he connecion beween SEP and OBS. he heoreical Roo s barrier is a resul of naural and menal inducion of properies of he underlying sochasic process. However a Mone Carlo simulaion dynamically reproduces he whole process of a SEP in which we are able o ge a very close approximaion of he disribuion of X. I adop he simple Euler mehod o simulae he pah of X. When X evolves as a Brownian moion in (3.5), he pah is approximaed by: X j 1 X j Bj (3.9) for j M. Here B is a sandard normally disribued variable chosen independenly for j every sep in (3.9). When X evolves as a geomeric Brownian moion, he pah is approximaed by: X j 1 X j X j Bj (3.1) for j M. Bu here we adop he z-grid and he Euler mehod for approxima8ing he pah of Z is hen: 1 Z j 1 Z j Bj (3.11) And in he Mone Carlo simulaion, I keep he ime sep he same lengh as ha in implemenaion of barriers in Secion 3.1. he hollow recangles in he following hisograms represen he disribuion of X wih being he firs hiing ime of he barrier obained in Secion 3.1. he broken red line shows he heoreical probabiliy densiy funcion of X wih Figure 3..3 as an excepion. In Figure 3..3 he broken red line represens he probabiliy densiy of a pure uniform disribuion since he probabiliy densiy of a uniform disribuion wih an aom is jus he probabiliy densiy of a pure uniform disribuion wih a jump a he aom. 19

24 he upper graph in Figure 3..1 shows he disribuion of X wih being he firs hiing ime of he barrier represened in he upper figure of Figure By comparing he disribuion of recangles and he broken red line we can see ha i indeed approximaes a uniform disribuion beween and X under limied number of simulaed pahs. Also, he lower graph in Figure 3.1. shows a very close approximaion of a normal disribuion cenered a X wih limied number of simulaion pahs. Figure 3..1: isribuion of X under uniform disribuion (above) and normal disribuion (below) Figure 3.. shows he disribuion of X wih being he firs hiing ime of he barrier in Figure I is he shape of a log-normal disribuion wih he normal componen cenered a Z which equalsln( X ).

25 Figure 3..: isribuion of X under log-normal disribuion Figure 3..3 represens he Mone Carlo resul associaed wih he barrier in he upper graph of Figure We can see ha he disribuion shown in Figure 3..3 is acually a combinaion of a uniform disribuion and an aom a X. his suppors ha he resul of Cox and Wang [3] works on barriers ha is no smooh. Figure 3..3: isribuion of X under uniform disribuion wih an aom a X Figure 3..4 shows he disribuion of X wih being he firs hiing ime of he barrier consruced from he exponenial disribuion. he resul in Figure 3..4 is obained applying he z-grid. As menioned in Secion 3.1.3, he limi z-grid of numerical implemenaion wiss he disribuion of X especially wih small value of X. I makes X end up wih value closer o wih greaer probabiliy. Hence he resul of Mone Carlo should be a wised exponenial disribuion wih more probabiliy on values close o. However we can see ha wih limied grids in hisogram his disorion is no obviously represened. Figure 3..4 also provides srong suppor for Cox and Wang [3]. 1

26 Figure 3..4: isribuion of X under exponenial disribuion wih z-grid

27 Chaper 4 Robus Subhedging Examples In his secion I re-examine he subhedging sraegy given in Secion 5 and Secion 6 of Cox and Wang [3] and compare he resul wih ha in Carr and Lee [17]. he sraegy consruced in Cox and Wang [3] is generalized o any variance derivaives wih final payoff saisfying he funcion F( ln S ) provided ha F is a convex, increasing funcion wih F(). Is righ derivaive is denoed by f and is required o be bounded. Bu he subhedging sraegy in Carr and Lee [17] (Proposiion.6 and Proposiion 3.1) only limis o variance swaps and variance calls. In fac, he sraegy of Carr and Lee [17] (Proposiion 3.1) can be viewed as an insaniaion of he resul in Cox and Wang [3] wih a chosen consan barrier. I will illusrae his in deail by applying he sraegy summarized in Secion. in wo specific examples. he firs example is a variance swap wih a simplified payoff ln S and he second example is a variance call wih final payoff ( ln S K). he firs example in variance swap is acually used as a saniy check which demonsraes he idea of he subhedging sraegy in Cox and Wang [3]. he deailed comparison will be represened in he second example. 4.1 (Sub)Hedging Sraegy for a Variance Swap Wih a variance swap we have F() for, which is perfecly linear in he variable. Hence we have a rivial case where f ( ) 1no maer wha he barrier looks like. According o definiion: where M ( x, ) is surely bounded on ( x, ) M( x, ) E f ( ) 1 (4.1). Zx ( ) is hen defined as: M( x,) Zx ( ) ( x) x (4.) Equaion (4.) is equivalen o equaion (.). So: ln( x) Z( x) (4.3) Noe here I choose a paricular normalizaion o ( ) Zx o simplify he calculaion. his normalizaion makes no difference o he final subhedging sraegy because any wo soluions of (4.3) only differ by a consan. Hence he final subhedging sraegies derived from any wo soluions of (4.3) are essenially he same. G( x, ) and H( x) are hen calculaed hrough he definiion given in (.1) and (.): 3

28 ln( x) G( x, ) (4.4) ln( x) H( x) (4.5) G( x, ) H( x) F( ) (4.6) he ideniy in (4.6) means ha we can perfecly replicaes he payoff of he variance swap. In order o consruc such a sraegy, consider subsiue he corresponding ime-changed process X (.5) ino he expression of G( x, ). he aim here is o find he admissible dynamic sraegy o saisfy: s s ln( X ) G( X, ) G( X,) dx ln( X ) dx (4.7) s s s s SinceX follows a uni-variance geomeric Brownian moion, ln( X ) can be explicily expressed hroughln( X) W. herefore i can be easily deduced ha saisfies he inequaliy in (.6). G( X, ) X X S (4.8) his resul accords wih he radiional sub-replicaing sraegy for a variance swap. Neuberger [4] and Carr and Lee [17] have boh deduced he same replicaing sraegy for a variance swap paying ln( S) a some specific ime. Neuberger [4] argued by wriing ln( S) as: ds ln( S) ( ) S and consider applying Io s lemma on ln( S ) : Compare (4.9) and (4.1), we have: (4.9) ds ds ln( S ) ln( S ) ( ) 1 S (4.1) S ln( S) ln( S ) ln( S) ds (4.11) S Based on (4.11) and assume ha call opions wih all srikes are publicly raded, Neuberger [4] suggesed ha he variance conrac wih final payoff ln( S) has a model independen iniial price equal o (ln( S) ln( S )). By dynamically holding S unis of he underlying sock we are able o replicae he final payoff. We can see ha. Carr and Lee [17] buil up he replicaion sraegy by defining a difference of convex funcions : which saisfies: yy ( y) ( y) (4.1) y where is he second derivaive of and : [, ) is a weigh funcion. Proposiion.16 yy 4

29 in Carr and Lee [17] saes ha (I quoe from Carr and Lee [17] Proposiion.16 wih some minor adjusmen in heir noaion): Le be a sopping ime. If claims on ( S ) and ( S ) are radable, hen he sraegy of holding a each ime (, ] : and holding a each ime (, ] : 1 claim on ( S ) I claim on ( ) S 1 claim on ( S ) ( S ) shares y ( S ) ( S ) ds S ( S ) bonds y s s y subreplicaes he forward-saring weighed variance ofln( S ), defined by: S ( S ) ln( ), s d S s If he equaliy in (4.1) holds, hen he sraegy replicaes S, exacly. If we se ( y) 1and, we reproduce he variance swap wih payoff ln( S) a. And moreover, if we se he equaliy in (4.1) and solve he OE, we have ( y) ln( y). Hence he exac replicaing sraegy consiss of holding ( S ) S unis of shares, which is again he same wih he hedging sraegy from Cox and Wang [3]. y Noe here he reason why we obain he same rading sraegy in boh Cox and Wang [3] and Carr and Lee [17] or Neuberger [4] is ha he hedging sraegy in Cox and Wang [3] has nohing o do wih he chosen barrier. he special form of he payoff funcion of a variance swap makes funcion M ( x, ) in (4.1) independen from, which resuls in a universal hedging sraegy for a variance swap. his conceals he significance of he subhedging sraegy in Cox and Wang [3]. If he firs derivae of he payoff funcion is no independen of, we may obain differen subhedging sraegies when we replace by anoher barrier. Cox and Wang [3] shows ha he subhedging sraegy obained by choosing Roo s barrier is opimal among oher sraegies obained using oher barriers. In he nex secion, I move forward o consider a variance call wih payoff ( ln( S) K) and furher discuss his poin. 4. Subhedging Sraegy for a Variance Call As for a variance call, funcion f() is dependen on he barrier we chose. herefore in his 5

30 case, we need o assume a barrier o calculae ou M ( x, ), Zx ( ), G( x, ) and H( x ). As menioned a he end of Secion 4.1, Cox and Wang [3] shows ha hough we can apply any barrier o consruc he subhedging sraegy, we can only obain he opimal one by choosing Roo s barrier. herefore I will choose a Roo s barrier and anoher differen barrier o see wha he hedging sraegies will be like and discuss he resuls I obain. he Roo s barrier I choose is a consan one wih inf{ : R( x) K}. I accords wih he siuaion where S follows a geomeric Brownian moion wih volailiy. Figure has represened ha if S is log-normal disribued wih mean ln( S ) and variance, he Roo s barrier will be a verical line. Firs, from F( ) ( K) we have: 1 f() if K if K (4.13) ( x, ) According o (.19), we have M( x, ) E f ( ) 1 for all. I is because when R( x) K, and when K, K. herefore Zx ( ) can be calculaed hrough (.): ln x Z( x) (4.14) which is he same as he Zx ( ) for a variance swap afer normalizaion. G( x, ) hen follows as; ln( x) G( x, ) M ( x, s) ds Z( x) (4.15) which is also he same as he G( x, ) for a variance swap. Finally, H( x) is calculaed hough (.): herefore we have: K ln( x) H( x) ( 1) dx Z( x) K (4.16) G( x, ) H( x) K ( K) F( ) (4.17) which confirms he argumen of Cox and Wang [3] Proposiion 5.1. Follow a similar deducion in variance swap case, subsiue he corresponding ime-changed process X ino he expression of G( x, ), we aim o find he admissible dynamic sraegy which saisfies: s s ln( X ) G( X, ) G( X,) dx ln( X ) dx (4.18) s s s s Here (4.18) is he same as (4.7) and hence choosing S forms a subhedging sraegy for he variance call. Noe he subhedging sraegy for a variance call is exacly he same as he replicaing sraegy for a variance swap. he idea is based on he fac ha he payoff of a variance call is always greaer or equal o he payoff of a variance swap wih a same srike. herefore by replicaing he variance swap we ge a naural subhedging for he corresponding variance call. 6

31 Now I subsiue anoher barrier o see wha he subhedging sraegy will look like. Choose inf{ : R( x) Q} wihq K. Alhough he expression of f() will no be changed, M ( x, ) will no longer equal one for all. When R( x) Q, Q and hence f ( ). When R( x) Q, and here are wo cases we need o consider. If Q K, f ( ) bu if K, f ( ) 1. herefore M ( x, ) now becomes: if K M ( x, ) 1 if K (4.19) By subsiuing (4.19) ino definiion of Zx ( ), G( x, ) and H( x ), we have Zx ( ), Hx ( ) and: Hence we have: if K G( x, ) K if K if K F( ) G( x, ) H( x) K if K (4.) (4.1) Obviously, expression of curren G( x, ) and H( x) implies a differen subhedging sraegy from he one using Roo s barrier. Here he independence of G( x, ) and x means ha we choose o hold zero unis of shares o subhedge a variance call. However, here we have he equaliy E[ F( )] E[ F( K)] E[ F( Q)], which means ha boh subhedging sraegies are opimal. his resuls from he special payoff form of a variance call. he realized volailiy of a variance call is always a consan so we will obain an opimal subhedging sraegy if he barrier we choose is a consan. Noe here if we resric our choice of Rxin ( ) o be a consan, hen he funcion in Carr and Lee [17] acually corresponds o G( x, ). he subhedging sraegy of Carr and Lee [17] consiss of he following componens (parly quoed from Proposiion 3.1 from Carr and Lee [17] wih minor adjusmen in noaion): A each ime hold 1 claim on ( S ) N shares BS( S, K; ) NsdSs NS bonds where N is defined as: 7

32 N BS y( S, K ln( S) ; ) if K y( S) if K (4.) And he funcion BS( y, v; f ) is defined as: BS( y, v; f ) f( y) Z 1 v f ( ye ) exp( ( z ) v) dz if v v (4.3) if v here are many choices for funcion. However we can see ha he subhedging sraegy obained by using Roo s barrier corresponds o choosing ( y) ln( y), which gives equaliy in (4.1). he subhedging sraegy obained by using R( x) Q K corresponds o. o see his, when ( y) ln( y), (4.3) becomes: BS( y, v; f ) if v ln( y) 1 v (ln( y) z) exp( ( z ) v) dz if v v (4.4) and (4.) becomes simply N S for all, which is exacly he in he subhedging sraegy of applying Roo s barrier in Cox and Wang [3]. And choosing will make N meaning ha we hold zero unis of shares o subhedge. Generally, by solving he equaliy in (4.1) we are acually solving he equaliy in (.8). I means ha holding N shares where N can be view as an opimal soluion o: BS y( S, K ln( S) ; ) if K y( S) if K G( X, ) G( X,) dx s s where is he subhedging sraegy in Cox and Wang [3]. We can jus subsiue above inequaliy and check ha i does saisfy he condiion. N ino he As menioned in Remark 5.4 in Cox and Wang [3], by accuraely choosing Roo s barrier for he sock process wih respec o each variance opion, we are able o obain he opimal subhedging sraegy which gives he opimal lower bound for he price of he variance opion. Bu he lower bound given by Carr and Lee [17] is merely by fixing he barrier as a consan. In general he subhedging sraegy in Carr and Lee [17] will be subopimal compared o ha from Cox and Wang [3] s consrucion. he reason why here choosing he opimal funcion of in Carr and Lee [17] s consrucion corresponds o choosing Roo s barrier in Cox and Wang [3] is he same as menioned earlier. he special form of a variance call covers he fac ha fixing he barrier as a consan will resul in a subopimal sraegy. If in anoher case where he Roo s barrier is no a consan, Carr and Lee [17] can no provide an opimal subreplicaing sraegy. Generally speaking, he generalizaion of Cox and Wang [3] comes from wo aspecs. One is 8

33 ha hey provide more choices for he shape of he barrier and anoher is ha hey enable us o apply he subhedging sraegy on variance opions differen from a variance swap and a variance call. Carr and Lee [17] s resul mainly works on variance swaps and variance calls bu Cox and Wang [3] s resul can work on more forms of he volailiy derivaives. Supplemenary o he firs poin, Example 5.6 in Cox and Wang [3] demonsraes a barrier ha is no a consan. hey choose {( x, ) : R( x)} wih R( x) ( x )( x )1 (, ). Here we require ha,,. Given he underlying process is a Brownian moion and payoff funcion as F( ), G( x, ) and H( x) car calculaed as: G( x, ) [ ( x )( x )] Z( x) 1 if R( x) R ( x) ( 1) Z( x) if R( x) We can see ha: R ( x) H( x) Z( x) ( 1) G( x, ) H( x) F( ) for all. his sraegy obviously can no be deduced from resuls in Carr and Lee [17]. In realiy, he barrier is mos likely o be non-consan and some non-consan barrier examples are ploed in Secion 3.1 Figure 3.1. (above), Figure and Figure

34 Chaper 5 Conclusions I conclude his paper firs by summarizing he seps I have aken o give a brief review of Cox and Wang [3]. Cox and Wang [3] s work is based on he connecion of Roo s barrier and he Skorokhod embedding problem and heir aim is o find a model-independen subhedging for variance opions. heir work mainly consiss of wo pars. he firs par is ha hey proved a one-o-one correspondence of Roo s barrier and an obsacle problem or sricly, a variaional inequaliy problem. And he second par is ha hey provided an alernaive proof of he opimaliy propery of Roo s barrier leading o a consrucion of subhedging sraegies for variance opions. As for he firs par of Cox and Wang [3] s work, I provide a brief review by solving he obsacle problem numerically and checking wheher i gives he righ resul of he corresponding Skorokhod embedding problem. he obsacle problem is solved hrough an explici forward finie difference scheme and he resul is checked by performing a Mone Carlo simulaion. he resul from numerically solving he obsacle problem represens us a barrier and I have shown in Secion 3.1 ha he barrier is a close approximaion of he suggesed Roo s barrier from Skorokhod embedding. Hence my numerical implemenaion gives an iniial suppor o Cox and Wang [3] s work. As for back checking he barrier, I simulaed a sufficienly large number of pahs o reproduce he sopping process. By reconsrucing he disribuion of he underlying sochasic process a he sopping ime, I furher represened ha solving he obsacle problem acually renders us he same Roo s barrier as solving he Skorokhod embedding problem. o reexamine he second par of Cox and Wang [3], I subsiue he subhedging sraegy of Cox and Wang [3] ino a variance swap and a variance call and compare he sraegies wih hose given by Carr and Lee [17]. For a variance swap, no maer wha barrier we choose in Cox and Wang [3] s consrucion, we will end up in he unique replicaing sraegy since is payoff funcion is perfecly linear in he variable. Hence we obain he same hedging sraegy from Carr and Lee [17] and Cox and Wang [3]. For a variance call, I also obained he same subhedging sraegy as ha in Carr and Lee [17] by using a consan Roo s barrier in Cox and Wang [3]. However i is because he realized volailiy of a variance call is always a consan and his conceals he fac ha Cox and Wang [3] s sraegy is opimal while Carr and Lee [17] is only subopimal. he radeoff beween Cox and Wang [3] and Carr and Lee [17] is ha even hough boh papers work for any coninuous posiive maringale, Cox and Wang [3] s mehod is less explici han Carr and Lee [17] bu is opimal as opposed o sub-opimal. ue o he limi of ime I am unable o provide an example fully shows he opimaliy of Cox and Wang [3]. his will be aken as a furher sudy in his subjec. For he second par of conclusion I would like o provide some oher furher seps ha migh be aken o give a more profound undersanding of he subjec. he firs exension ha comes o mind is o develop more Roo s barriers by numerically implemening he obsacle problems. Wih barely some simple examples in his paper, I can no fully represen he srong connecion of he 3