Hedging Variance Options on Continuous Semimartingales

Transcription

1 Hedging Variance Opions on Coninuous Semimaringales Peer Carr and Roger Lee This version : December 21, 2008 Absrac We find robus model-free hedges and price bounds for opions on he realized variance of [he reurns on] an underlying price process. Assuming only ha he underlying process is a posiive coninuous semimaringale, we superreplicae and subreplicae variance opions and forward-saring variance opions, by dynamically rading he underlying asse, and saically holding European opions. We hereby derive upper and lower bounds on values of variance opions, in erms of Europeans. 1 Inroducion Variance swaps, which pay he realized variance of [he reurns on] an underlying price process, have become a leading ool for managing exposure o volailiy risk. As repored in he Financial Times, [19], Volailiy is becoming an asse class in is own righ. A range of srucured derivaive producs, paricularly hose known as variance swaps, are now he preferred roue for many hedge fund managers and proprieary raders o make bes on marke volailiy. Dealers have me he demand for variance swaps wih he help of he model-free log conrac mehodology which replicaes realized variance, and which became in 2003 he basis for he CBOE s calculaion of he VIX index. Exending ha mehodology, we replicae he forward-saring weighed variance of general funcions of coninuous semimaringales bu mainly we focus on variance opions. Variance opions calls and pus on realized variance allow porfolio managers greaer conrol over volailiy risk exposure, offering hem he abiliy o go long or shor variance while limiing he Bloomberg LP and Couran Insiue, NYU. pcarr@nyc.rr.com Universiy of Chicago. RL@mah.uchicago.edu We d like o acknowledge exremely valuable conversaions wih Bruno Dupire, David Hobson, and Greg Pels. We d also like o hank he edior, Marin Schweizer, he anonymous referees, and seminar paricipans a Columbia, DRW Trading, he Fields Insiue, Illinois Insiue of Technology, IMPA, Minnesoa, Princeon, Purdue, Sanford, he Sevanovich Cener, and UBS. 1

2 downside o he premium paid for he opion. However, hey presen greaer hedging difficulies o he dealer. According o one praciioner [1] in 2007, The indusry is aking a big risk wriing such producs [opions on variance] and a some poin ha will be a risk ha you can assess. This indusry has o fulfill invesors needs, bu a he same ime I don wan o wrie a icking bomb. We ake a robus model-free approach o his hedging problem. Assuming only ha he underlying process is a posiive coninuous semimaringale, we superreplicae and subreplicae variance opions and forward-saring variance opions, by dynamically rading he underlying asse, and saically holding European opions. We hereby derive upper and lower bounds on he values of variance opions, in erms of Europeans. 1.1 Relaed Work In [7], Carr-Geman-Madan-Yor priced opions on realized variance, assuming reurns follow pure jump dynamics wih independen incremens; whereas we work wih arbirary coninuous dynamics wihou assuming independen incremens. They did no address hedging, whereas we develop boh subhedges and superhedges. They found pricing formulas in erms of he characerisics/parameers of he underlying process; whereas we derive bounds direcly in erms of Europeansyle opion prices wihou imposing a model on he underlying dynamics, hence wihou bearing he risks of misspecificaion and miscalibraion associaed wih any specific model. Indeed, we regard our resuls as par of a broad program which aims o use European opions which pay funcions of he ime-t underlying Y T o exrac informaion model-independenly abou risks dependen on he enire pah of Y, and o hedge or replicae hose risks robusly. Three prominen examples of such pah-dependen risks are: firs, he maximum of a price process, robusly hedged in Hobson [20] by holding a call opion o subreplicae, and by gradually selling off a porfolio of calls o superreplicae (also see Hobson-Pedersen [21] for subreplicaion of a forwardsaring digial on he maximum); second, barrier-coningen call and pu payoffs, robusly hedged in Brown-Hobson-Rogers [6] using European opions ogeher wih a ransacion in he underlying a he barrier passage ime; and hird, he variance swap payoff, robusly replicaed in Neuberger [23], Dupire [16], Carr-Madan [9], Derman e al [14], and Brien-Jones/Neuberger [5], using a log conrac ogeher wih dynamic rading of he underlying. This paper includes exension and unificaion of he replicaion sraegies for he various flavors of variance swaps (including gamma swaps, corridor variance, and variance of ransformed prices), bu our main conribuion o his program is o exend he managemen of pah-dependen risks o include sub/superreplicaion of variance opions. In [8], Carr-Lee ook a model-free approach o he exac pricing and replicaion of general funcions of realized variance, bu ha paper made an independence assumpion on he volailiy process (while carefully immunizing is pricing and rading mehodology, o firs order, agains 2

3 violaions of he independence assumpion). Here we do no assume independence; insead we work in he very general seing of an arbirary coninuous semimaringale price process. Such minimal assumpions do no deermine uniquely he prices of variance opions; bu we will show ha hey do imply bounds on hose prices, enforceable by superreplicaing and subreplicaing porfolio sraegies. In [17], Dupire found lower bounds and subhedges for spo-saring variance opions. Here we exend hose bounds and subhedges o forward-saring opions. Moreover, we find upper bounds and superhedges for spo-saring and forward-saring opions. 1.2 Assumpions Le Y denoe he price of a share of he underlying asse, ogeher wih all reinvesed dividends. Assume ha Y is a posiive coninuous semimaringale on a filered probabiliy space (Ω, F, {F }, P ) saisfying he usual condiions. We inerpre P as physical probabiliy measure, so Y is no necessarily a local P -maringale. Excep in Secion 5, our proofs will have no need of risk-neural measure. Fix some T > 0. If we say ha [a claim on] some F T -measurable payoff A is radeable, we mean ha a imes T, i may be bough and sold fricionlessly a some finie price, denoed by V A. Assume he absence of arbirage in he class of predicable self-financing semi-saic sraegies in he radeable payoffs. By semi-saic we mean sraegies which rade a mos once in (0, T ). We assume he exisence of he following radeables: he underlying asse, wih payoff Y T and price V Y T = Y ; and a bond, wih payoff 1 and price V 1 = 1. Depending on he conex, we may add oher radeables. We view he radeables as he basic asses from which we will synhesize conracs on realized variance. Le X := ϕ(y ) where ϕ is he difference of convex funcions, for example ϕ(y) = log y. Le (X) and (Y ) denoe he quadraic variaion of X and Y respecively, wih he convenion ha quadraic variaion a ime 0 is zero. We will model-independenly (super/sub)-replicae claims wrien on (X) including opions on forward-saring variance (X) T (X) θ for any consan θ [0, T ) using predicable self-financing sraegies which dynamically rade Y and saically hold European-syle claims on Y T and Y θ. Superreplicaion implies upper bounds on variance opion prices, by he sandard logic ha shoring an opion bid above he upper bound, and going long he superreplicaing sraegy, produces an arbirage. However, our noion of a superreplicaing sraegy does no promise any noion of ameness or admissibiliy, so o be careful and complee, we show moreover ha our sraegies saisfy naural margin consrains a all imes [0, T ]. We do likewise for he subreplicaion sraegies which give lower bounds. To summarize, we impose consisency among he prices of he radeable basic asses by assuming he absence of semi-saic arbirage among hem. We impose consisency beween each variance opion and is super/sub-replicaing porfolios of radeable basic asses (including Y, which we rade 3

4 fully dynamically), by assuming, moreover, he absence of dynamic arbirage saisfying naural margin consrains. Remark 1.1. The consan bond price assumpion does no resric us o zero ineres raes, because we regard all prices in his paper (excep Y ' and Z ' in his Remark) o be denominaed in unis of he bond. If in pracice we wish o use a differen uni of denominaion le us say he dollar hen we have he following conversions. Leing Y ' denoe he dollar-denominaed share price and Z ' denoe he dollar-denominaed bond price (for a bond ha pays 1 dollar a mauriy T ), we have he bond-denominaed share price Y = Y ' /Z ', and he bond-denominaed bond price Z = Z ' /Z ' = 1. In pracice, variance conracs are wrien on dollar-denominaed logarihmic variance, no on bond-denominaed logarihmic variance; bu under arbirary deerminisic (including non-consan) ineres raes given by a shor rae process r, he wo noions of variance are idenical. Indeed, we have Z ' = exp( quadraic variaion T r d), hence he bond-denominaed share price Y = Y ' /Z ' has logarihmic log Y = log(y ' /Z ' ) = log Y ' + r d = log Y ' (1.1) because he d inegral has finie variaion. Therefore a T -mauriy conrac on any funcion of (log Y ' ) T is idenical o a T -mauriy conrac on ha funcion of (log Y ) T. This holds rue even if he former conrac pays in dollars while he laer conrac pays in bonds, because 1 ime-t dollar equals 1 ime-t bond. In conclusion, our consan bond price assumpion enails no loss of generaliy relaive o arbirary non-random ineres raes. Noe ha he irrelevance of ineres raes shown in (1.1) conrass o he cases of lookback and barrier opions [6, 20] where more care was required, because max (Y ' /Z ' ) does no equal max (Y ' ). T 2 Model-free replicaion By Meyer-Iô we have 1 dx = ϕ y (Y )dy + L a ϕ yy (da), (2.1) 2 R where ϕ y denoes he lef-hand derivaive of ϕ, and ϕ yy denoes he second derivaive in he sense of disribuions, and L a denoes he local ime of X a a. Since ϕ yy is he difference of wo posiive measures and L a is increasing, he local ime erm has finie variaion. Therefore Le h(y, q) be C 2,1. Then for all, by Iô s rule, d(x) = ϕ 2 y(y )d(y ). (2.2) 1 h(y, (X) ) = h(y 0, 0) + h y dy s + h yy d(y ) s + h q d(x) 2 = h(y 0, 0) + h y dy s + h + ϕ h d(y ) yy y q s s 4

5 where subscrips on h denoe parial differeniaion. More generally, we will need Proposiions 2.1 and 2.2 which are sligh exensions of Bick [3] o a larger class of sopping imes. Proposiion 2.1. Le U be an open se wih (Y 0, 0) U R 2. Le h be C 2,1 on U and coninuous on U. Then for all T and all sopping imes τ inf{ : (Y, (X) ) / U}, T τ T τ 1 h(y T τ, (X) T τ ) = h(y 0, 0) + h y dy s + h yy + ϕ y 2 h q d(y ) s. (2.3) If moreover ϕ y (y) > 0 for all y hen T τ T τ 1 h yy h(y T τ, (X) T τ ) = h(y 0, 0) + h y dy s + + h q d(x) s. (2.4) ϕ 2 y In he inegrands, he he h y, h yy, and h q are evaluaed a (Y s, (X) s ), and ϕ y is evaluaed a Y s. Proof. Le τ n := inf{ : here exis (y, q) (R R + ) \ U such ha Y y + (X) q < 1/n}. Iô s rule applies o he sopped process (Y τn, X τn ), so for all T T τ n T τ n 1 h(y T τn, (X) T τn ) = h(y 0, 0) + h y dy s + h yy + ϕ y 2 h q d(y ) s Now le n. By coninuiy of Y and h, we have (2.3). By (2.2) we have (2.4). 2.1 Vanishing (X) inegral To proceed from (2.4), we can choose h o make he d(x) inegral vanish. Then he (X)-dependen LHS can be creaed, using he rading sraegy in bonds and shares given by he wo remaining erms on he RHS. Proposiion 2.2. Under he hypoheses of Proposiion 2.1, we assume, moreover, ha 1 h yy + hq = 0 for (y, q) U. (2.5) 2 ϕ2 y Then for any T he payoff h(y T τ, (X) T τ ) can be replicaed by holding a each ime T τ h y (Y, (X) ) h(y, (X) ) Y h y (Y, (X) ) The replicaing porfolio has ime-0 value h(y 0, 0). shares bonds. (2.6) Proof. Wih Z := 1 denoing he bond price, he porfolio s value a any ime τ T is V := (h(y, (X) ) Y h y (Y, (X) )) Z + h y (Y, (X) ) Y = h(y, (X) ) = h(y 0, 0) + h y (Y s, (X) s )dy s + 0, 0 5

6 by (2.4) and (2.5). Therefore dv = h y (Y, (X) )dy + (h(y, (X) ) Y h y (Y, (X) ))dz which is by definiion he self-financing condiion. Remark 2.3. Equaion (2.5) is a backward Kolmogorov PDE, wih quadraic variaion playing he role of ime. We reurn o his poin in Remark We will need he following sligh exension of Bick [3], who has he case ha f is a pu payoff. Le R + = (0, ) denoe he posiive reals. Proposiion 2.4 (Claims on price when variance reaches a barrier). Le X = log(y /Y 0 ). Le τ be he firs passage ime of (X) o level Q. For any y > 0, any v 0, and any coninuous f : R + R such ha f(e z ) F (e z ) for some polynomial F and all z R, le z ) 1 (z+v/2) 2 f(ye exp 2v dz if v > 0 2πv BS(y, v; f) := (2.7) f(y) if v = 0 and le BS y denoe is y-derivaive. Then he sraegy of holding a each ime T τ replicaes he ime-(t τ) payoff BS y (Y, Q (X) ; f) BS(Y, Q (X) ; f) Y BS y (Y, Q (X) ; f) shares bonds (2.8) The replicaing porfolio has ime-0 value BS(Y 0, Q; f). f(y τ )I τ T + BS(Y T, Q (X) T ; f)i τ>t (2.9) Proof. Le h(y, q) := BS(y, Q q; f). Direcly verify ha 1 y 2 h yy + h q = 0 on U = R + (, Q) 2 and h is coninuous on U; hen apply Proposiion 2.2. Remark 2.5. No longer purely heoreical, similar conracs, of perpeual ype, have been raded by Sociéé Générale [2], and described as imer opions. Remark 2.6. Inuiively he BS(y, v; f) funcion gives he value of he payoff f(y τ ), which is compued by he Black-Scholes formula wih dimensionless volailiy parameer Q q. We say dimensionless o emphasize ha his parameer represens a oal unannualized variance unil expiraion, no variance per uni ime. Proposiion 2.4 shows ha saring wih bonds and shares of oal value BS(Y 0, 0), and a each ime dela-hedging a dimensionless BS volailiy Q (X) will produce f(y τ ) if and when (X) reaches Q. 6

7 Corollary 2.7 (How o make profi/loss if realized volailiy / implied BS volailiy). Under he condiions of Proposiion 2.4, furher assume convexiy of f, which herefore has a lef derivaive f '. Then sraegy (2.8), exended o imes > τ by holding a all (T τ, T ] he saic porfolio f ' (Y τ ) shares f(y ) f ' τ (Y τ )Y τ bonds (2.10) subreplicaes (resp. superreplicaes) f(y T ) if τ T (resp. τ T ). Proof. If τ T, hen by (2.9) he porfolio has ime-t value BS(Y T, Q (X) T ; f) f(y T ) by convexiy. If τ T, hen he porfolio has ime-t value f(y τ ) + f ' (Y τ )(Y T Y τ ) f(y T ). Remark 2.8. Suppose a conrac paying f(y T ) has ime-0 value BS(Y 0, Q, f); for example, his holds if f is a call, and Q is is BS implied volailiy. Then Corollary 2.7 implies immediaely ha going long he f(y T ) conrac and shor he porfolio (2.8,2.10) is a zero-iniial-cos sraegy whose ime-t value is nonnegaive if τ T, nonposiive if τ T. In addiion o variance-barrier conracs, we also replicae price-barrier conracs. Proposiion 2.9 (Claims on variance unil price reaches a down-barrier). Le X = log(y /Y 0 ). Le τ be he firs passage ime of Y o a barrier b (0, Y 0 ). For any coninuous g : R R such ha g G for some polynomial G, le g( q + z) log(b/y) exp (log(b/y)+z/2)2 0 2z dz if y = b BP (y, q; b, g) := 2πz 3 (2.11) g(q) if y = b and le BP y denoe is y-derivaive. Then he sraegy of holding a each ime T τ replicaes he ime-(t τ) payoff BP y (Y, (X) ; b, g) BP (Y, (X) ; b, g) Y BP y (Y, (X) ; b, g) shares bonds (2.12) g((x) τ )I τ T + BP (Y T, (X) T ; b, g)i τ>t. (2.13) If g is monoonically increasing, hen he sraegy superreplicaes g((x) τ T ). The replicaing porfolio has ime-0 value BP (Y 0, 0; b, g). Proof. Le h(y, q) := BP (y, q; b, g). Then direcly verify ha 1 y 2 h yy + h q = 0 on U = (b, ) R, 2 and ha h is coninuous on Ū. Proposiion 2.2 implies replicaion of (2.13), and superreplicaion of g((x) τ T ) follows because BP (y, q; b, g) g(q) for increasing g. We chose he noaion BP for Brownian passage, o be explained in Remark 2.11; bu firs we give he analogue of Proposiion 2.9 for a double barrier more precisely, for claims on he variance from ime 0 unil he price s exi ime from a finie inerval. 7

8 Proposiion 2.10 (Claims on variance o an exi ime). Le X = log(y /Y 0 ). Le 0 < b d < Y 0 < b u, and le τ be he exi ime of Y from he inerval (b d, b u ). For any coninuous g : R R such ha g G for some polynomial G, le 0 g(q + z)p(log(y/b d ), log(y/b u ), z)dz if b d < y < b BP (y, q; b d, b u, g) := g(q) oherwise u (2.14) p( β ) := z/8 [ β d/2 d, β u, z e e ψ( β u, β u β d, z) + e βu/2 ψ( β d, β u β d, z)] (2.15) o R r + 2kR 2 ψ( r, R, z) := e (R r+2kr) /(2z). (2.16) k= 2πz 3/2 Then he sraegy of holding a each ime T τ replicaes he ime-(t τ) payoff BP y (Y, (X) ; b d, b u, g) BP (Y, (X) ; b d, b u, g) Y BP y (Y, (X) ; b d, b u, g) shares bonds (2.17) g((x) τ )I τ T + BP (Y T, (X) T ; b d, b u, g)i τ>t. (2.18) If g is monoonically increasing, hen he sraegy superreplicaes g((x) τ T ). The replicaing porfolio has ime-0 value BP (Y 0, 0; b d, b u, g). Proof. Le h(y, q) := BP (y, q; b u, b d, g). Then direcly verify ha 1 y 2 h yy +h q = 0 on U = (b d, b u ) 2 R, and ha h is coninuous on U. Proposiion 2.2 implies replicaion of (2.18), and superreplicaion of g((x) τ T ) follows because BP (y, q; b, g) g(q) for increasing g. Remark By Borodin-Salminen [4] Formula , he p funcion is he densiy of he exi ime of drif 1/2 Brownian moion from he inerval (β d, β u ). Inuiively, he BP funcion gives he expeced value of g a his Brownian Passage ime. Remark The formulas (2.7) and (2.11) and (2.14)-(2.16) can be undersood via ime change. We have dx = dy d(y ) = dy d(x). Y 2Y 2 Y 2 Under risk-neural measure he underlying Y is a coninuous local maringale, hence so is M where 1 1 M := dy s = X + (X). 0 Y s 2 By Dambis/Dubins-Schwarz ([12, 15]; henceforh DDS), here exiss (on an enlarged probabiliy space if needed) a Brownian moion W wih W (X) = M for all T. So X = W (X) 1 (X) and 2 hence Y = G (X) where G u := Y 0 exp(w u u/2). Therefore, wih respec o business ime (X), he underlying Y is drifless geomeric Brownian moion. So, even in our compleely general coninuous 8

9 semimaringale seing, Black-Scholes prevails under he sochasic clock which idenifies ime wih quadraic variaion. Forde [18] independenly noes he relevance of DDS o pricing variance-o-a-barrier claims. Dupire [17] uses DDS o cas volailiy derivaives ino he framework of he Skorokhod embedding problem. Our hedging proofs do no rely on DDS indeed hey do no even rely on he exisence of a risk-neural measure bu he ime change perspecive adds insigh. In he case of a call payoff, we find an easily compuable formula for BP. Proposiion 2.13 (Fourier represenaion for calls on variance unil an exi ime). For a call payoff g(q) = (q Q) +, he funcion BP of Proposiion 2.10 for y (b d, b u ) has he represenaion BP = αi y/b u sinh(log(b d /y) 1/4 2iz) y/b d sinh(log(b u /y) 1/4 2iz) dz αi 2πz 2 e i(q q)z sinh(log(b u /b d ) 1/4 2iz) where α > 0; any such α gives he same value for he inegral. Remark Abusing noaion, we will wrie BP (y, q; b d, b u, Q) o mean BP (y, q; b d, b u, g), where g(q) := (q Q) +. Proof. Combine Borodin-Salminen [4] Formula , which gives he Laplace ransform of p, wih Lee [22] Theorem 5.1 (for he G 2 payoff), which obains BP from ha ransform. Proposiion 2.15 (Properies of BP for a call). For any q 0, Q 0, y > 0, For q = Q = 0 and b d < y < b u, BP (y, q; b d, b u, Q) BP (y, 0; b d, b u, Q) (q Q) +. (2.19) log(b u /b d ) BP (y, 0; b d, b u, 0) = 2 log(y/b u ) 2 (y b u ). (2.20) b u b d Proof. For y / (b d, b u ), inequaliy (2.19) clearly holds. For y (b d, b u ), BP (y, q; b d, b u, Q) = (q + z Q) + p(log(y/b d ), log(y/b u ), z) dz 0 0 [(q Q) + + (z Q) + ]p(log(y/b d ), log(y/b u ), z) dz = (q Q) + + BP (y, 0; b d, b u, Q), and (2.19) again holds. Equaion (2.20) holds because each side equals he expecaion of he exi ime of drif 1/2 Brownian moion from he inerval (log(b d /y), log(b u /y)): he LHS by definiion of BP and Remark 2.11, and he RHS by he usual mehod of exracing an expecaion from he known Laplace ransform of he exi ime densiy. 9

10 2.2 Nonvanishing (X) inegral An alernaive way o proceed from (2.4) is o generae he quadraic variaion dependence in he d(x) s inegral, insead of in h(y T, (X) T ). In paricular, by making h(x, q) depend on x alone, he quadraic-variaion-dependen inegral on he RHS of (2.3) can be creaed from he LHS (which has hereby become simply a European claim) minus he bonds and shares erms on he RHS. Proposiion 2.16 ((Sub)replicaion of forward-saring weighed variance of ϕ(y )). Le he weigh w : R + [0, ) be a Borel funcion and le τ be a sopping ime. Le λ : R + R be a difference of convex funcions, le λ y denoe is lef-hand derivaive, and assume ha is second derivaive in he disribuional sense has a (signed) densiy, denoed λ yy, which saisfies for all y R + λ yy (y) 2ϕ 2 (y)w(y). (2.21) y If claims on λ(y T ) and λ(y τ T ) are radeable, hen he sraegy of holding a each ime (0, τ T ] and holding a each ime (τ T, T ] 1 claim on λ(y T ) 1 claim on λ(y τ T ) (2.22a) 1 claim on λ(y T ) λ(y τ T ) λ y (Y s )dy s + Y λ y (Y ) bonds, τ T λ y (Y ) shares (2.22b) subreplicaes he forward-saring weighed variance of X = ϕ(y ), defined by T (X) w w(y s ) d(x) s. τ,t := The subreplicaing porfolio has ime-0 value V 0 λ(y T ) V 0 λ(y τ T ). τ T If equaliy holds in (2.21) hen he sraegy replicaes (X) w τ,t exacly. Proof. The sraegy clearly self-finances and has he claimed ime-0 value. By Meyer-Iô T T 1 λ(y T ) = λ(y 0 ) + λ y (Y s )dy s + λ yy (Y s ) d(y ) s and, by Meyer-Iô applied o he sopped process Y τ, τ T τ T 1 λ(y τ T ) = λ(y 0 ) + λ y (Y s )dy s + λ yy (Y s ) d(y ) s

11 Taking he difference, hence which proves subreplicaion of which proves exac replicaion. T T 1 λ(y T ) = λ(y τ T ) + λ y (Y s )dy s + λ 2 yy (Y s ) d(y ) s (2.23) τ T T τ T T λ(y τ T ) + λ y (Y s )dy s + ϕ 2 y(y s )w(y s ) d(y ) s (2.24) τ T τ T T T = λ(y τ T ) + λ y (Y s )dy s + w(y s ) d(x) s, (2.25) τ T τ T λ(y T ) λ(y τ T ) T τ T λ (Y )dy (X) w y s s τ,t, (2.26) w (X) τ,t. If equaliy holds in (2.21), hen i holds in (2.24) and (2.26), Remark The sraegy (2.22) can be described as dela-hedging he λ claim a zero vol, because is share holding λ y (Y ) is idenical o BS y(y, v; λ). v=0 Proposiion 2.16 includes as special cases he classical resuls on replicaion of various flavors of variance swaps. Example 2.18 (Replicaion of forward-saring variance of log Y ). Consider he weigh funcion w(y) := 1. If λ(y) = A 1 y + A 0 2 log y (2.27) where A 0, A 1 are arbirary consans, hen (2.21) holds wih equaliy, so if claims on λ(y T ) and λ(y τ T ) are radeable, hen he sraegy (2.22) replicaes (X) T (X) τ T, where X = log Y. This recovers he known sraegy (Neuberger [23], Dupire [16], Carr-Madan [9], Derman e al [14]) of using a log conrac o replicae logarihmic quadraic variaion. Example 2.19 (Replicaion of forward-saring corridor variance of ϕ(y )). Le he corridor C be a Borel se and le he weigh funcion be he indicaor w(y) := I(y C). If λ is convex and λ yy = 2ϕ2 y in C and λ yy = 0 ouside of C, hen (2.22) replicaes corridor variance [9] T τ T I(Y s C) d(x) s. The replicaing porfolio has ime-0 value V 0 λ(y T ) V 0 λ(y τ T ). Taking C = R + produces he full forward-saring variance of ϕ(y ). Example 2.20 (Replicaion of forward-saring gamma swap on ϕ(y )). Le he weigh funcion be w(y) := ay where a is a consan, ypically a = 1/Y 0. If λ is convex and λ yy (y) = 2aϕ 2 (y)y hen y (2.22) replicaes he gamma swap payou T τ T ay s d(x) s. 11

12 In paricular, for he usual logarihmic case ϕ(y) = log(y), he ODE λ yy (y) = 2a/y is solved by λ(y) = ay log y + A 1 y + A 0 for arbirary consans A 0 and A 1. The replicaing porfolio has ime-0 value V 0 λ(y T ) V 0 λ(y τ T ). The final example we designae as a Corollary, due o is relevance o one of our main goals subreplicaing a forward-saring variance call. Corollary 2.21 (Subreplicaion of forward-saring variance of log Y ). Le λ : R + R be a difference of convex funcions, le λ y denoe is lef-hand derivaive, and assume ha is second derivaive in he disribuional sense has a densiy, denoed λ yy, which saisfies for all y R + λ yy (y) 2/y 2. (2.28) Le τ be a sopping ime. If claims on λ(y τ T ) and λ(y T ) are radeable, hen T λ(y T ) λ(y τ T ) λ y (Y s )dy s (X) τ,t = (X) T (X) τ T (2.29) τ T and he sraegy (2.22) subreplicaes (X) T (X) τ T, where X = log Y. Proof. Take w = 1 and ϕ(y) = log y in Proposiion Variance call: Lower bound In his secion le X := log(y /Y 0 ). Le Q 0 and T > Spo-saring variance call: Dupire s subreplicaion Consider a variance call wih srike Q and expiry T. Dupire s [17] subreplicaion sraegy has he following inuiion. Le λ be convex and saisfy he hypoheses of Corollary If and when (X) his Q prior o ime T, we need o subreplicae a variance swap, so we wan o have a claim on λ(y T ) plus a claim on λ(y τq ). The former is a European claim, and he laer is synhesized by a bond-and-shares sraegy, according o Proposiion 2.4. If (X) does no hi Q prior o ime T, hen our ime-t porfolio is λ(y T ) minus a claim on λ(y τq ). By convexiy of λ, he laer has greaer value han he former. So he porfolio value is negaive, as desired. 12

13 Proposiion 3.1 (Dupire [17]). Consider a variance call which pays ((X) T Q) +. Assume λ is convex and saisfies he hypoheses of Corollary Define BSy (Y, Q (X) ; λ) if τ Q N := λ y (Y ) if > τ Q. Then for any T he following sraegy subreplicaes he variance call: a each ime < T hold 1 claim on λ(y T ) BS(Y 0, Q; λ) + N s dy s N Y bonds. 0 The subreplicaing porfolio has ime-0 value BS(Y 0, Q; λ) + V 0 λ(y T ). Proof. The sraegy clearly self-finances and has he claimed ime-0 value. If τ Q T, hen he ime-t porfolio value is N shares (3.1) τ Q T T BS(Y 0, Q; λ) + N s dy s + N s dy s + λ(y T ) = λ(y τq ) + N s dy s + λ(y T ) 0 τ Q τ Q (X) T (X) τq = ((X) T Q) + by Proposiion 2.4 and Corollary If τ Q > T, hen he ime-t porfolio value is T BS(Y 0, Q; λ) + N s dy s + λ(y T ) = BS(Y T, Q (X) T ; λ) + BS(Y T, 0; λ) (3.2) 0 0 = ((X) T Q) +. (3.3) Equaliy (3.2) holds by Proposiion 2.4. Inequaliy (3.3) holds because he convexiy of λ implies ha BS is increasing in is second argumen. Remark 3.2. Dupire chooses λ o maximize he lower bound, as follows. Le ( y K) + if K Y 0 van K (y) := (K y) + if K < Y 0 (3.4) denoe he payoff funcion of he OTM vanilla opion a srike K, and assume radeabiliy of van K (Y T ) for all K. Define he ime-0 dimensionless Black-Scholes implied volailiy for an underlying Y, srike K, and expiry T, o be he unique I 0 (K, T ) such ha BS(Y 0, I 0 (K, T ); van K ) = V 0 van K (Y T ). (3.5) 13

14 Then we may rewrie he lower bound as V 0 λ(y T ) BS(Y 0, Q; λ) = λ yy (K)[V 0 van K BS(Y 0, Q; van K )]dk 0 (3.6) = λ yy (K)[BS(Y 0, I 0 (K, T ); van K ) BS(Y 0, Q; van K )]dk. 0 Under he consrain 0 y 2 λ yy (y) 2, he opimal λ consiss of 2/K 2 dk OTM vanilla payoffs a all K where he dimensionless BS implied volailiy I 0 (K, T ) exceeds Q: λ(y) = 2 van K (y) dk. {K:I 0 (K,T )>Q} K 2 If a variance call is offered below is lower bound, hen shor he λ(y T ) claim and borrow BS(Y 0, Q; λ), he λ claim s Black-Scholes valuaion using dimensionless volailiy Q. Use he proceeds o buy he variance call, for a ne credi. Then dynamically rade shares o lock in his credi. 3.2 Forward-saring variance call: Subreplicaion Le he forward-sar dae be a consan θ [0, T ). Proposiion 3.3. Consider a forward-saring variance call which pays ((X) T (X) θ Q) +. Assume ha λ is convex and saisfies he hypoheses of Corollary Le τ Q := inf{ : (X) (X) θ Q} and BSy (Y, Q ((X) (X) θ ); λ) if θ τ Q N := λy (Y ) if > τ Q. If claims on BS(Y θ, Q; λ) and λ(y T ) are radeable, hen he following sraegy subreplicaes he forward-saring variance call. A each ime [0, θ] hold 1 claim on λ(y T ) 1 claim on BS(Y θ, Q; λ). (3.7a) A each ime (θ, T ) hold 1 claim on λ(y T ) BS(Y θ, Q; λ) + N s dy s N Y bonds. θ The subreplicaing porfolio has ime-0 value V 0 λ(y T ) V 0 BS(Y θ, Q; λ). N shares (3.7b) 14

15 Proof. The sraegy clearly self-finances and has he claimed ime-0 value. If τ Q T, hen he ime-t porfolio value is τ Q T T BS(Y θ, Q; λ) + N s dy s + N s dy s + λ(y T ) = λ(y τq ) + N s dy s + λ(y T ) θ τ Q τ Q (X) T (X) τq = ((X) T Q) + by Proposiion 2.4 and Corollary If τ Q > T, hen he ime-t porfolio value is T BS(Y θ, Q; λ) + N s dy s + λ(y T ) = BS(Y T, Q ((X) T (X) θ ); λ) + BS(Y T, 0; λ) (3.8) θ 0 = ((X) T Q) +. (3.9) Equaliy (3.8) holds by Proposiion 2.4. Inequaliy (3.9) holds because he convexiy of λ implies ha BS is increasing in is second argumen. 3.3 Subreplicaion under a margin consrain The value V sub of he subreplicaing porfolio is a lower bound on he price of he variance call, in he sense ha if he variance call is offered a a price below V sub, hen buying he variance call and shoring he porfolio produces an arbirage ha is well-behaved in he following way. We prove ha he subreplicaion sraegy (3.7) saisfies a naural margin consrain on [0, T ]. Specifically, we show ha V sub is a all imes T dominaed by he marke s ime- expecaion of (X) T (X) ( θ) τq, by which we mean he RHS of (3.10). This consrain prevens he magniude of our shor posiion in he subreplicaing porfolio from becoming oo large, relaive o he collaeral ha we own, having gone long he variance call. Definiion 3.4 (Call buyer s margin consrain). Assume ha claims on log(y T ) and log(y θ ) are radeable. We say ha a self-financing rading sraegy wih ime- value V saisfies he call buyer s margin consrain if for all [0, T ], V (X) (X) τq 2V log(y T /Y θ ). (3.10) Proposiion 3.5 (Subreplicaing sraegy saisfies he margin consrain). Assume ha claims on log(y T ) and log(y θ ) are radeable. Then, under he hypoheses of Proposiion 3.3, he subreplicaing sraegy (3.7) saisfies he call buyer s margin consrain. Therefore he (3.7) sraegy s value V sub is a lower bound on he buyer s price of he variance call, where buyer s price is defined as he supremum of he prices of all subreplicaing sraegies saisfying he call buyer s margin consrain. Proof. For all τ Q, Corollary 2.21 implies V sub = λ(y τq ) + N s dy s + V λ(y T ) = λ(y τq ) + N s dy s + λ(y ) λ(y ) + V λ(y T ) τ Q τ Q (X) (X) τq + 2 log(y ) 2V log(y T ). 15

16 For all (θ, τ Q ), Proposiion 2.4 and he convexiy of λ imply V sub = BS(Y, Q ((X) (X) θ ); λ) + V λ(y T ) BS(Y, 0; λ) + V λ(y T ) = λ(y ) + V λ(y T ) 2 log(y ) 2V log(y T ). For all θ, he convexiy of λ implies V sub = V BS(Y θ, Q; λ) + V λ(y T ) V λ(y θ ) + V λ(y T ) V (2 log(y θ ) 2 log(y T )), as claimed. 3.4 Forward-saring variance call: Lower bounds For any λ saisfying he hypoheses of Corollary 2.21, we have esablished he lower bound V 0 λ(y T ) V 0 BS(Y θ, Q; λ) on he ime-0 value of he variance call. Exending Dupire o he forward-saring case, we choose λ o maximize he lower bound, as follows. Define van K by (3.4), and assume radeabiliy of van K (Y θ ) and van K (Y T ) for all K. Define he ime-0 dimensionless Black-Scholes forward implied volailiy for an underlying Y, a srike K, and a ime inerval [θ, T ] o be he unique I 0 (K, [θ, T ]) such ha Then we may rewrie he lower bound as V 0 BS(Y θ, I 0 (K, [θ, T ]); van K ) = V 0 van K (Y T ). (3.11) V 0 λ(y T ) V 0 BS(Y θ, Q; λ) = λ yy (K)[V 0 van K (Y T ) V 0 BS(Y θ, Q; van K )]dk 0 = λ yy (K)[V 0 BS(Y θ, I 0 (K, [θ, T ]); van K ) V 0 BS(Y θ, Q; van K )]dk. 0 Under he consrain 0 y 2 λ yy (y) 2, he opimal λ is λ consising of 2/K 2 dk OTM vanilla payoffs a all K for which he dimensionless BS forward implied volailiy exceeds Q: 2 λ (y) = K 2 van K (y) dk {K:I 0 (K,[θ,T ])>Q} where forward implied volailiy on [θ, T ] is defined by (3.11). Noe ha we have shown ha in his conex he appropriae noion of forward implied volailiy I 0 (K, [θ, T ]) involves he enire marke-implied disribuion of Y θ ; saring from his disribuion (no necessarily lognormal) a ime θ, run a geomeric Brownian moion wih dimensionless volailiy Q on [θ, T ]; he unique Q which recovers he ime-0 price of he K-srike T -expiry opion is wha we mean by forward implied volailiy. The opimized lower bound is V SUB := V 0 λ (Y T ) V 0 BS(Y θ, Q; λ ). (3.12) 16

17 If variance call is offered below his lower bound, hen shor he λ (Y T ) claim and go long a claim on BS(Y θ, Q; λ ), which is he λ claim s Black-Scholes ime-θ valuaion using dimensionless volailiy Q; his fuure value is compleely deermined by Y θ, so i can be synhesized a ime 0 using θ expiry Europeans. Use he proceeds o buy he variance call, for a ne credi. Saring a ime θ, dynamically rade shares o lock in his credi. Remark 3.6. Inuiively, his lower bound says ha a variance call dominaes a τ Q -saring corridor variance swap, where he corridor can be arbirarily chosen (and need no be coniguous). In urn, he τ Q -saring corridor variance swap value a ime 0 dominaes he sum, over all K in he corridor, of (2/K 2 )dk OTM T -expiry vanillas less hose vanillas ime-0 Black-Scholes valuaion using dimensionless volailiy Q on [θ, T ]. This holds for an arbirary corridor, so he opimal corridor includes exacly hose K which make a posiive conribuion o he sum. 4 Variance call: Upper bound In his secion le X := log(y /Y 0 ). Consider a variance call wih srike Q 0 and expiry T > 0. Our sraegy o superreplicae ((X) T Q) + comes from he following inuiion. Le τ b be he exi ime of Y from some fixed inerval (b d, b u ). Alhough we canno perfecly replicae ((X) T Q) +, we can perfecly replicae ((X) τb Q) + by rading a porfolio having iniial value BP (Y 0, 0; Q), as shown in Proposiion If τ b T hen he shorfall is covered by creaing he remaining variance (X) T (X) τb. To do so, we follow Example 2.18 and include in our holdings a claim on L(Y T ) L(Y τb ), where L(y) := 2 log(y) + A 1 y + A 0. By choosing (A 0, A 1 ) such ha L(b d ) = L(b u ) = 0, we make he L(Y τb ) erm vanish, so he claim s payoff simplifies o L(Y T ). If τ b > T hen a ime T we are long a claim on ((X) τb Q) + bu we also hold L(Y T ) < 0, a liabiliy which we canno always afford. We can always afford o accep he smaller liabiliy BP (Y T, 0; Q) L(Y T ) and sill superreplicae, because ((X) τb Q) + ((X) τb (X) T Q) + ((X) T Q) +. So in he inerval b d < Y T < b u, le us replace he L(Y T ) payoff by a BP (Y T, 0; Q) payoff. This increase in he payoff preserves superreplicaion in he case τ b T. The following proof makes his argumen precise, and exends i o forward-saring variance. 4.1 Forward-saring variance call: Superreplicaion Le he forward-sar dae be a consan θ [0, T ). Proposiion 4.1 (Forward-saring variance call superreplicaion). Consider a forward-saring variance call which pays ((X) T (X) θ Q) +. Choose any b d (0, Y 0 ] and any b u [Y 0, ). Le BP (y, q; Q) := BP (y, q; b d, b u, Q), 17

18 which has he Fourier represenaion given in Proposiion Define 2 log(y/b u ) + 2 log(bu/b d) b u b d (y b u ) if b d = b u L(y) := L(y; b d, b u ) := 2 log(y/y0 ) + 2y/Y 0 2 if b d = b u = Y 0 and L(y) if y / (b d, b u) L ( y ) := L ( y; b d, b u, Q) := (4.1) BP (y, 0; Q) if y (b d, b u ). Le τ b := inf{ θ : Y / (b d, b u )}. Le BPy (Y, (X) (X) θ ; Q) if θ τ N := Ly (Y ) if > τ b. Assume ha claims on L (Y θ ) and L (Y T ) are radeable. Then he following sraegy superreplicaes he forward-saring variance call: a each ime [0, θ] hold b (4.2) 1 claim on L (Y T ) 1 claim on L (Y θ ) (4.3a) and a each ime (θ, T ) hold 1 claim on L (Y T ) L (Y θ ) + N s dy s N Y bonds. θ The superreplicaing porfolio has ime-0 value V 0 [L (Y T ) L (Y θ )]. Proof. The sraegy clearly self-finances and has he claimed ime-0 value. If τ b T, hen he porfolio has ime-t value N shares (4.3b) L (Y T ) + BP (Y T, (X) T (X) θ ; Q) = BP (Y T, 0; Q) + BP (Y T, (X) T (X) θ ; Q) (4.4) by (2.19). If τ b < T hen he porfolio has ime-t value θ τ b ((X) T (X) θ Q) +. (4.5) BP y (Y s, (X) s (X) θ ; Q)dY s L (Y θ ) L y (Y s )dy s + L (Y T ) (4.6) T = ((X) τb (X) θ Q) + L (Y θ ) + L y (Y s )dy s + L (Y T ) (4.7) τ b T ((X) τb (X) θ Q) + L(Y τb ) L y (Y s )dy s + L(Y T ) (4.8) τ b = ((X) τb (X) θ Q) + + (X) T (X) τb ((X) T (X) θ Q) + (4.9) τ b T 18

19 as desired. In case τ b = θ, lines (4.7) and (4.8) use L (Y θ ) = L(Y τb ) 0. In case τ b > θ, line (4.7) uses L (Y θ ) = BP (Y θ, 0, Q) and Proposiion 2.10 (applied o he semimaringale Y +θ relaive o filraion {F +θ }); and line (4.8) uses L (Y θ ) + = L(Y τb ) = 0. In boh cases, line (4.8) also uses L (Y T ) = BP (Y T, 0, Q) BP (Y T, 0, 0) = L(Y T ) (4.10) by (2.20). The equaliy in line (4.9) holds by Example Superreplicaion under a margin consrain The value V super of he superreplicaing porfolio is an upper bound on he price of he variance call, in he sense ha if he variance call is bid a a price above V super, hen shoring he variance call and going long he porfolio produces an arbirage ha is well-behaved in he following way. We prove ha he superreplicaion sraegy (4.3) saisfies a naural margin consrain on [0, T ]. Specifically, we show ha V super a all imes T exceeds he inrinsic value of he variance call, as defined by he RHS of (4.11). There are a leas wo plausible ways o define inrinsic value, so le us clarify: we prove ha a all imes our porfolio value exceeds (q Q) + evaluaed no merely a q = (X) (X) θ, bu indeed ha i exceeds (q Q) + evaluaed a he sum of (X) (X) θ and he marke s expecaion of he remaining variance (X) T (X) θ. This consrain prevens he inrinsic value of he call (which we are shor) from becoming oo large, relaive o he collaeral ha we own, having gone long he superreplicaing porfolio. Definiion 4.2 (Call seller s margin consrain). Assume ha claims on log(y T ) and log(y θ ) are radeable. We say ha a self-financing rading sraegy wih ime- value V saisfies he call seller s margin consrain if for all [0, T ], V ((X) (X) θ 2V log(y T /Y θ ) Q) +. (4.11) Remark 4.3. Our seller s margin/collaeral consrain (4.11) is he naural analogue (for realized variance conracs) of a ameness noion (for European conracs on price) se forh in Cox-Hobson [11]. Their Definiion 5.1 defined he fair seller s price of an opion wih payoff H(S T ) and wih collaeral requiremen funcion G o be he smalles iniial forune needed o consruc a self-financing wealh process W saisfying he superreplicaion condiion W T H(S T ) and he collaeral condiion W G(S ) for all < T. In paricular, for he case of a European call payoff H(s) = (s K) +, i is naural o impose a collaeral consrain of he call payoff funcion iself, hus G(s) = H(s). In oher words, he requiremen is simply ha seller of he opion mus, a each ime, have wealh sufficien o cover he inrinsic value of he opion, (S K) +. Cox-Hobson cied pracical preceden o jusify his crierion, saing ha (modulo noaional differences): 19

20 European call opions on socks canno be exercised before mauriy, bu he erms and condiions of opions on Inerne socks ofen included he proviso ha if he firm was subjec o a akeover a ime < T, hen he opion paid (S K) +. In order o super-replicae he call opion i is necessary o have a wealh process which saisfies boh a condiion a mauriy and his condiion a inermediae imes. In our seing, wih a call on variance insead of price, i is appropriae o replace he European call s inrinsic value (S K) + wih insead he inrinsic value of he variance call (for noaional convenience in his remark, le us say a spo-saring variance call wih θ = 0). The variance call s inrinsic value could be defined as ((X) Q) +, bu we will show ha indeed our sraegy saisfies he sronger consrain (4.11), which defines he margin/collaeral requiremen o be he forward-looking inrinsic value which replaces (X) by (X) 2V log(y T /Y ). Similar reasoning explains our definiion of buyer s margin/collaeral consrain (3.10). Proposiion 4.4 (Superreplicaing sraegy saisfies he margin consrain). Assume ha claims on log(y T ) and log(y θ ) are radeable. Then, under he hypoheses of Proposiion 4.1, he superreplicaing sraegy (4.3) saisfies he call seller s margin consrain. Therefore, he (4.3) sraegy s value V super is an upper bound on he seller s price of he variance call, where seller s price is defined as he infimum of he prices of all superreplicaing sraegies saisfying he call seller s margin consrain. Proof. If θ hen V super = V [L (Y T ) L (Y θ )] 0 because L is convex; moreover, V super = V [L (Y T ) L (Y θ )] (4.12) V [L(Y T ) L(Y θ ) + L(Y θ ) L (Y θ )] (4.13) = V [L(Y T ) L(Y θ ) + I (b d,b u) (Y θ )( BP (Y θ, 0; 0) + BP (Y θ, 0; Q))] (4.14) V [L(Y T ) L(Y θ ) Q]. (4.15) The remaining calculaions use he resuls referenced in he proof of Proposiion 4.1. If θ < τ b, hen V super = V L (Y T ) L (Y θ ) + BP y (Y s, (X) s (X) θ ; Q)dY s (4.16) θ = V L (Y T ) + BP (Y, (X) (X) θ ; Q) (4.17) V L(Y T ) L(Y ) + L(Y ) + BP (Y, (X) (X) θ ; Q) (4.18) = V L(Y T ) L(Y ) BP (Y, 0; 0) + BP (Y, 0; Q ((X) (X) θ )) (4.19) V L(Y T ) L(Y ) + (X) (X) θ Q. (4.20) 20

21 If > τ b hen V super is θ τ b BP y (Y, (X) (X) θ ; Q)dY L y (Y s )dy s + V L (Y T ) L (Y θ ) (4.21) τ b = ((X) τb (X) θ Q) + L y (Y s )dy s + V L (Y T ) L(Y θ ) + (4.22) τ b ((X) τb (X) θ Q) + L y (Y s )dy s + V L(Y T ) L(Y τb ) + L(Y ) L(Y ) (4.23) τ b ((X) τb (X) θ Q) + + (X) (X) τb + V L(Y T ) L(Y ) (4.24) as claimed. ((X) (X) θ Q + V L(Y T ) L(Y )) + (4.25) 4.3 Forward-saring variance call: Upper bounds Each choice of (b d, b u ) gives an upper bound V 0 [L (Y T ; b d, b u, Q) L (Y θ ; b d, b u, Q)] on he ime-0 price of he variance call. Hence V SUPER := inf V 0 [L (Y T ; b d, b u, Q) L (Y θ ; b d, b u, Q)] (4.26) (b d,b u) gives an opimized upper bound. Remark 4.5. Because ((X) T Q) + (X) T, he spo-saring variance call has a naive upper bound, namely he value of he (X) T -replicaing porfolio: V 0 [ 2 log(y T /Y 0 )]. Taking b d = b u = Y 0 in our upper bound recovers he naive upper bound, because V 0 L (Y T ; Y 0, Y 0, Q) = V 0 [ 2 log(y T /Y 0 )]. Since our bound opimizes over all pairs (b d, b u ), i never does worse han he naive upper bound. Likewise, for forward-saring variance calls, our upper bound never does worse han he naive upper bound V 0 [ 2 log(y T /Y θ )]. Remark 4.6. Figure 1 shows four examples of model-independen superreplicaing porfolios for a variance call. Remark 4.7. If barrier opions are available, hen we can improve he upper bound. In (4.3), replace he claim on L (Y T ; b d, b u, Q) by a double knock-in claim on L(Y T ; b d, b u ) plus a double knock-ou claim on BP (Y T, 0, Q), where each claim has barriers a b d and b u, moniored on he ime inerval [θ, T ]. Remark 4.8. The difference beween a variance pu wih payoff (Q (X) T ) + and he variance call wih payoff ((X) T Q) + is a claim on (X) T Q, which is perfecly replicable by Example Hence subreplicaion and superreplicaion sraegies for he variance pu follow direcly from he corresponding sraegies for he variance call. 21

22 Figure 1: Superreplicaing porfolios for a variance call Payoff Y T Le Y 0 = 100. A claim on any one of hese ime-t payoffs, ogeher wih dynamic rading of shares, model-independenly superreplicaes a spo-saring T -expiry variance call wih srike Q = Referring o (4.1), he plos show L (Y T ) L (Y 0 ) for hree paricular choices of (b d, b u ). Each superreplicaing payoff is universally valid for all coninuous semimaringales wih Y 0 = 100. The marke prices of Europeans expiring a T deermine which of he infinie family of superreplicaing porfolios is cheapes; of course he cheapes need no be among hese hree examples. 22

23 Remark 4.9. We have used only he European opions informaion available a incepion (ime 0), bu an analysis from he sandpoin of he European opions informaion available a ime > 0 follows immediaely. In paricular, suppose ha we have a call on [θ, T ] variance, sruck a Q 0. If, a ime T, he running variance (X) (X) θ exceeds Q, hen we are guaraneed o finish in-he-money, so he call reduces o a variance swap paying (X) T (X) θ Q, which can be priced and replicaed exacly, by Example If, on he oher hand, (X) (X) θ Q, hen he seasoned Q-srike call given running variance (X) (X) θ is equivalen o a newly-issued call (given zero running variance) wih an effecive srike Q ((X) (X) θ ), o which our analysis applies direcly. Remark As quadraic variaion accumulaes during he life of a variance call, he call s effecive srike decreases. Eiher he call finishes ou-of-he-money and pays nohing, or i finishes in-he money and he effecive srike approaches zero a some ime. In he laer case, our upper and lower bounds converge (o he price of a variance swap) as he effecive srike approaches zero (equivalenly, as running variance approaches srike). Thus, even if he observed Europeans daa may generae a incepion a significan gap beween our upper and lower bounds for a paricular variance conrac, our resuls can noneheless offer furher insigh for hedging and risk managemen a laer imes, because he gap approaches zero as running variance approaches he srike. Moreover, even if a wide inerval exiss a incepion (or any oher ime), our bounds addiionally offer immediaely usable informaion: he size of he inerval gives an upper bound on he model risk presen if one aemps o price he variance call by specifying a model and calibraing o Europeans. 5 Numerical examples In order o specify and o compue some examples of variance call values and bounds, his secion assumes he exisence of a maringale measure ha prices all European conracs and variance conracs. We ake he European prices as given, bu his will no uniquely deermine he maringale measure. Each model meaning each choice of maringale measure consisen wih he Europeans generaes an arbirage-free variance call valuaion. We compare he valuaions generaed by various models agains he bounds arising from our sub/superreplicaion sraegies. Suppose ha he ime-0 prices of T -expiry European conracs paying (Y T K) + are given by E P Heson (Y T K) + for all K, where T = 1 and he expecaion is wih respec o a measure P Heson, under which he pahs of Y have disribuion given by he Heson dynamics dy = V Y dw 1, dv = 1.15(0.04 V )d V dw 2, V 0 = (5.1) where W 1 and W 2 are independen Brownian moions. The prices of variance conracs may or may no be given by P Heson -expeced payoffs. Some maringale measure P does price, via expeced 23

24 payoffs, he Europeans and he variance conracs, bu P need no be P Heson ; i may agree wih P Heson on expecaions of European payoffs bu no variance payoffs. In oher words, he Heson dynamics (5.1) are one way o generae hose paricular observed prices of Europeans, bu no he only way for example, here exis local volailiy models which imply, for all T -expiry Europeans, he same prices as (5.1). Therefore, pah-dependen conracs, such as variance calls, admi a range of values consisen wih he given European prices. Regard he process Y as a random variable aking values in he space consising of all posiive coninuous price pahs on [0, T ], and define on his space he family P of probabiliy measures P such ha he Y is a P-maringale saisfying he consisency condiion for all K E P Heson (Y T K) + = E P (Y T K) +. (5.2) Each of he measures P can be described as a model, in he sense of Con [10]. By (5.2) he models agree on he value of he observable Europeans, bu hey produce a range of differen values for E P ((X) T Q) +. One value in ha range is he Heson-model (5.1) expecaion E P Heson ((X) T Q) +. The middle curve in Figure 2 plos his Heson variance call price for srikes 0.0 Q 0.1. Aside from he Heson model, we shall exhibi wo oher models a Roo model and a Ros model consisen wih he European values E P Heson (Y T K) +. Equivalenly, leing ν denoe he P Heson -disribuion of Y T, we shall exhibi wo oher models under which Y T has disribuion ν. Boh consrucions are ideas of Dupire [17], adaped by us o he case of logarihmic quadraic variaion. In boh cases, suppose G is a drifless uni-volailiy geomeric Brownian moion wih respec o some measure P G, and le G 0 = Y 0. Firs consider he Roo consrucion. Ros [25] Theorem 1 and Corollary 3 imply ha here exiss a space-ime barrier B Roo (0, ) [0, ) such ha (i) τ roo := inf{u 0 : (u, G u ) B Roo } is a finie sopping ime ha saisfies G τroo ν. (ii) For each y > 0, here exiss u roo (y) [0, ] such ha (u, G u ) B Roo for all u > u roo (y) and (u, G u ) / B Roo for all u < u roo (y). Choosing any such barrier, define he Roo model P Roo by specifying ha he pahs of Y have P Roo -disribuion idenical o he P G -disribuion of he pahs of he process G τroo (/(T )) for T, wih he convenion /(T ) := for = T. For numerical compuaion purposes, we obain B Roo and he ransiion densiy p roo (u, y) of he process u G u τroo (a ransiion densiy in he sense ha p roo (u, y) = P G (G u τroo dy)) by seing up he forward Kolmogorov equaion, and solving numerically he following free boundary 24

25 problem o find he business ime u roo (y) a which he barrier begins for each y: p roo (0, y) = δ(y Y 0 ) 1 p y 2 2p roo roo 2 u < u y y 2 roo ( ) = (5.3) u 0 u > u roo (y) p roo (u roo (y), y) = p ν (y) where p ν denoes he densiy funcion of ν. Then + P G + + E P Roo ((X) T Q) + = E P G (( log G) τ roo Q) = E (τ roo Q) = (u roo (y) Q) p ν (y)dy. (5.4) The lower dashed curve in Figure 2 plos he Roo-model variance call price as a funcion of Q. Similarly, we compue a reversed barrier B Ros (0, ) [0, ) such ha (iii) τ ros := inf{ 0 : (, G ) B Ros } is a finie sopping ime ha saisfies G τros ν. (iv) For each y > 0, here exiss u ros (y) [0, ] such ha (u, G u ) B Ros for all u < u ros (y) and (u, G u ) / B Roo for all u > u ros (y). by solving numerically he free boundary problem p ros (0, y) = δ(y Y 0 ) p ros 0 u < u ros (y) = u p y ros u > uros (y) p ros (u ros (y), y) = p ν (y) 2 y 2 (5.5) o find he business ime u ros (y) a which he barrier ends for each y. Define he Ros model P Ros by specifying ha he pahs of Y have P Ros -disribuion idenical o he P G -disribuion of he pahs of he process G τros (/(T )) for T, and compue E P Ros ((X) T Q) + = (u ros (y) Q) + p ν (y)dy. (5.6) The upper dashed-doed curve in Figure 2 plos he Ros-model variance call price as a funcion of Q. Inuiively, he Ros model embeds he given disribuion ν in he geomeric Brownian moion G by sopping some G pahs very early and sopping oher G pahs very lae, leading o high variance of business ime (equivalenly, high variance of realized variance), hence high prices for calls on realized variance. In conras, he Roo model does he embedding by sopping he G pahs neiher early nor lae, leading o low variance of business ime, hence low prices for calls on realized variance. See Dupire [16] which esablished he link beween volailiy derivaive pricing and Skorokhod embedding, and Obloj [24] which surveyed he Skorokhod embedding problem. Finally, he op and boom solid curves in Figure 2 show, respecively, he model-free upper bound V SUPER and lower bound V SUB, given in (4.26) and (3.12), and enforceable by saic posiions 25