Pricing exotic options. with an implied integrated variance

Transcription

1 Pricing exoic opions wih an implied inegraed variance Ruh Kaila Ocober 13, 212 Absrac In his paper, we consider he pricing of exoic opions using wo opion-implied densiies: he densiy of he implied inegraed variance and he risk-neural price densiy. We presen a Bayesian mehod o esimae boh densiies and an opionimplied correlaion coefficien beween he sock price and volailiy shocks. We discuss how he convexiy adjusmen can be avoided when pricing volailiy derivaives and discuss he ypes of addiional informaion ha can be provided by he disribuion of he implied inegraed variance when pricing digial and barrier opions. We hen presen a simple algorihm o esimae he implied inegraed variance beween wo fuure momens in ime and o price wo pahdependen opions, namely, compound and clique opions. Aalo Universiy, School of Science, Insiue of Indusrial Engineering and Managemen, 76 AALTO, Finland. ruh.kaila@aalo.fi. The auhor would like o hank Professor Erkki Somersalo, Case Wesern Reserve Universiy and Professor Rama Con, Columbia Universiy and Universié Paris VI-VII for discussions, ideas, commens and suppor. The auhor would also like o hank Alex Lipon, Peer Carr, Jim Gaheral, Nicole el Karoui and Dilip Madan. 1

2 1 Inroducion One of he cenral problems in financial mahemaics is he pricing and hedging of derivaives such as opions. The accuracy of correc pricing and hedging is subsanially dependen on how well he volailiy of he underlying asse has been esimaed. In his paper, we esimae wo opion-implied densiies, namely he densiy of an implied inegraed variance and a risk-neural price densiy, and use hese densiies o price hree ypes of exoic opions: volailiy opions, semi-pah-dependen opions and pah-dependen opions. We adop a Bayesian approach where, insead of an absolue ruh, we search for evidence of he consisency or inconsisency of a given hypohesis. The Black-Scholes implied volailiy or variance and he model-free volailiy or variance are common opion-based forecass of he volailiy or variance. The model-free volailiy, presened by Brien-Jones and Neuberger (2), is defined as he risk-neural expeced realized volailiy implied by opions and can be compued as a weighed average of a coninuum of European opions wih differen srike prices and one mauriy. The modelfree volailiy is independen of srike prices and can be compued beween wo fuure momens in ime. The performance of he implied volailiy and he model-free volailiy in forecasing he fuure volailiy is considered by Muzzioli (21), where addiional lieraure on he opic is also presened. The implied inegraed variance can be seen as a sochasic exension of he Black-Scholes implied variance. The disribuion of he implied inegraed variance can be esimaed from opion prices using eiher a general opion payoff formula or he Hull-Whie formula (1987). Similar o he model-free variance, he disribuion of he implied inegraed variance is also independen of srike prices and can be compued beween wo fuure momens in ime. The main difference beween he implied inegraed variance and he 2

3 model-free variance is ha we consider a disribuion of he variance, raher han a fixed value, in he former. We can exend he Hull-Whie approach of pricing European opions o pricing some exoic opions and compuing hedge raios using an opion implied inegraed variance. The inverse problem of implied inegraed variance is considered by Friz and Gaheral (25) and, laer, by Kaila (28), assuming an uncorrelaed volailiy. In his paper, we will simulaneously esimae an implied inegraed variance and an implied correlaion coefficien by applying a mehod presened by Kaila (212). An esimae of he riskneural price densiy is obained direcly from hese esimaes. The inverse problem of an implied inegraed variance is ill-posed, as explained by Friz and Gaheral (25). In heir paper, he auhors coped wih he ill-posedness by firs regularizing he ill-posed problem and hen using a Bayesian lognormal prior. Insead of regularizing, Kaila (212) recass he deerminisic ill-posed inverse problem of an implied inegraed variance in he form of a saisical inference of he disribuion of he unknowns and adops a Bayesian approach. In addiion o providing an esimae of he unknowns, his approach provides informaion on he reliabiliy of he esimaes. Following Friz and Gaheral (25), he disribuion of he inegraed variance is assumed o be lognormal, such ha esimaing he enire disribuion reduces o esimaing he mean and variance of he disribuion. A saisical approach o esimaing an opion-implied volailiy is also adoped by Con and Ben Hamida (25), who solved an unregularized, ill-posed problem of implied volailiy wih evoluionary opimizaion and Mone Carlo sampling. The main conribuion of his paper is ha we propose pricing formulas for several exoic opions assuming a correlaed sochasic volailiy, using boh he opion payoff formula and he correlaed Hull-Whie formula. A saisical approach akes ino accoun he effec of noisy daa. The use of disribuions insead of fixed values in pricing provides 3

4 addiional informaion on he possible pricing errors we are making. The problem of missing daa is ofen encounered when esimaing model-free volailiies or risk-neural price densiies, bu i can be avoided o some exen when he disribuion of he inegraed variance is already refleced in a few opion prices wih differen srike prices. Esimaes of he implied inegraed variance can be used when pricing volailiy derivaives, assuming eiher a correlaed or uncorrelaed volailiy. The convexiy adjusmen (relaed o volailiy swaps and calls, for example) is avoided by using a disribuion of he implied inegraed variance insead of he corresponding expecaion when pricing hese derivaives. Finally, wo differen pah-dependen opions are priced using he disribuions of he implied inegraed variance and he risk-neural price densiies beween wo fuure momens in ime. This paper is organized as follows: Secion 2 solves he inverse problem of an implied inegraed variance and an implied correlaion coefficien and provides he corresponding esimaes of he risk-neural price densiy. Secion 3 addresses volailiy derivaives, while digial opions and barrier opions are priced in Secion 4. In Secion 5, we consider compound opions and clique opions, and we offer a conclusion in Secion 6. 4

5 2 Implied inegraed variance We assume a coninuous-ime economy wih a finie rading inerval [, T ]. Le (Ω, F, P ) be a probabiliy space, equipped wih a filraion (F ) T. We denoe a risk-neural measure by P, equivalen o he naural measure P. The sock price process (X ) T and he volailiy process (σ ) T are modeled as he following diffusion processes: dx = rx d + σ X ( 1 ρ 2 dw + ρdz ), dσ = θσ d + νσ dz, (1) where r is he riskless rae of reurn, and θ and ν are he drif of he volailiy and he volailiy of he volailiy, respecively. The P -Brownian moions W and Z are independen and T σ2 sds <. We denoe by ρ = [ 1, 1] he consan correlaion coefficien beween he shocks of he sock price process and he volailiy process. When ρ, we say ha he volailiy is correlaed and when ρ =, we say ha he volailiy is uncorrelaed. We define he inegraed variance, denoed by σ 2 T, as σ T 2 = 1 T σ T sds, 2 < T, condiional on he filraion F. The inegraed variance is approximaed in he lieraure by he realized variance. We call he square roo of he inegraed variance, s T = σ 2 T, he inegraed volailiy. We begin wih a variaion of he approach presened by Romano and Touzi (1997) and Willard (1997), apply he wo-dimensional Iô formula o log(x ) and inegrae from o 5

6 T, < T, obaining he following: ( ) XT log X = ρϕ T + ( (r 1 2 σ2 ρ,t )(T ) + T ) 1 ρ 2 σ s dws, where (2) ϕ T = 1 2 ρ σ2 T (T ) + T σ s dz s and σ 2 ρ,t = (1 ρ 2 ) σ 2 T. (3) Raher han ϕ T, Romano and Touzi (1997) and Willard (1997) apply ξ T = exp(ρϕ T ) and show how X T /X is lognormally disribued condiional on X, σ 2 T and ξ T. We now have wo lognormal disribuions under he measure P : 1. Condiional on X, σ 2 T and ϕ T, he reurn X T /X is lognormally disribued: X T X ln (ρϕ T + ( r 1 ) 2 σ2 ρ,t (T ), σ 2 ρ,t (T )). 2. Condiional on X, σ and σ 2 T, ϕ T is shifed lognormally disribued: ϕ T ln ( log ( σ ) 1 + (θ ν 2 ν2 )(T ), ν 2 (T ) ) σ ν eθ(t ) 1 2 ρ σ2 T (T ), (4) where θ and ν can be esimaed from he disribuion of σ T 2. The proof is given in he Appendix and in he paper by Kaila (212). As a resul, given X, he disribuion π( σ T 2 ) and ρ, we can esimae he condiional disribuion π(x T X ). Now consider a European call on he sock, wih price U = U (x; K, T ; σ 2, r) given by he following equaion: U = e r(t ) E {(X T K) + x = X, σ 2, F } = e r(t ) E {h(x T ) x, σ 2, F }, (5) 6

7 where K is he srike price and T is he mauriy. We denoe by E he expecaion wih respec o he measure P. Hull and Whie (1987) showed ha when he volailiy is sochasic and uncorrelaed, he prices of European opions are given as expecaions of he corresponding Black- Scholes prices (1973) wih respec o he inegraed variance. Renaul and Touzi (1996) explained how he Hull-Whie formula produces a symmeric volailiy smile when he implied volailiies are ploed as a funcion of he log-moneyness log(k/x e r(t ) ). The asymmeric smiles ha are observed in he marke can herefore be explained by a correlaion beween he sock price process and he volailiy process; as a resul, he assumpion of an uncorrelaed volailiy is unjusified in pracice. Romano and Touzi (1997) and Willard (1997) exended he Hull-Whie formula o opions wih correlaed volailiy. The correlaed Hull-Whie call price is given by U HW,ρ (x; σ 2 ) = U HW,ρ (x; K, T ; σ 2, r) = E { E { U BS (ξ T ( σ T 2, ρ)x; σ2 ρ,t, r) σ2 T, F } } σ 2, F, where he Black-Scholes call price U BS (x; σ 2 ) = U BS (x; K, T ; σ 2, r) is (x; σ 2 ) = xφ(d + ) e r(t ) KΦ(d ), d ± = log(x/k) + (r ± σ2 /2)(T ) σ, (6) (T ) U BS wih Φ(z) = 1/ 2π z e y2 /2 dy. We subsiue ρϕ T = log(ξ T ) ino he correlaed Hull-Whie formula so ha U HW,ρ (x; σ 2 ) = E { } E {U BS (e (ρϕ T ( σ T 2,ρ)) x; σ ρ,t 2, r) σ T 2, F } σ 2, F. (7) Two common inverse problems are o esimae he risk-neural price densiy, i.e., he densiy of he sock price X T implied by opion marke prices u obs under a risk-neural measure P defined by he markes, and o esimae he Black-Scholes implied volailiy 7

8 1 call price K/x variance Dela Variance K/x IV phi X X cond T1 T2 Figure 1: Due o he convexiy of he Black-Scholes formula (6) wih respec o he volailiy and he underlying and he convexiy of he payoff pricing formula (5) wih respec o he underlying, differen disribuions of σ T 2 and ϕ T resul in differen opion prices for he same srike prices and mauriies. As a resul, assuming ha opion marke prices coincide wih he prices given by he Hull-Whie formula (7) or he payoff pricing formula, opion marke prices should provide informaion on he disribuions of σ T 2 and X T. In he firs row and in he lef panel, a surface indicaing he convexiy (yellow) and concaviy (red) of he Black-Scholes formula wih respec o σ 2 is ploed. In he righ panel, we plo he surface of he Black-Scholes dela. The convexiy of he Black- Scholes formula wih respec o he underlying is indicaed wih differen colors. In he second row, we plo in he lef panel wo randomly chosen disribuions of he inegraed variance wih he same expecaion (verical line), namely, σ T,1 2 ln (.15,.3) (blue) and σ T,2 2 ln (.125,.5) (red). In he righ panel, he condiional disribuions π(ϕ T σ T 2 ) compued wih differen values of ρ, specifically ρ 1 =.85 (black) and ρ 2 =.4 (yellow), are shown, assuming ha σ T 2 [,.6]. In he boom row, wo randomly chosen densiies of X wih he same expecaion are ploed in he lef panel, and in he righ panel, he corresponding condiional densiies π(x T X ) are shown, compued wih (2), σ T,2 2 and ρ =. We also plo in he righ panel he densiies X T compued wih σ T,2 2 and ρ 1 (blue) and ρ 2 (red), assuming ha X = 1 (T 1 =.5 years, r = ). 8

9 I. As a sochasic exension of he Black-Scholes implied variance, we define he disribuion of he implied inegraed variance and he implied correlaion coefficien o be, respecively, he disribuion of σ 2 T and he value of ρ for which he correlaed Hull-Whie opion prices equal he rue opion prices U rue : U HW,ρ (x; K, T ; σ 2, r) = U rue (K, T ), given opions wih srike prices K i, 1 i L, and a mauriy T. In he inverse problem of he implied inegraed variance, he parameer o esimae is a probabiliy densiy funcion π( σ T 2 ) implied by a series of opion prices wih differen srike prices and he same mauriy. Then, using he uncorrelaed Hull-Whie formula, for example, we esimae π( σ 2 T ) from U rue,i T = E {U,i BS (x; K i, T ; σ T 2, r) σ T 2, F } = U,i BS (x; K i, T ; σ T 2, r)π( σ T 2 )d σ T 2, where he subscrip i refers o he srike price K i, 1 i L, and he inegral is compued wih respec o all possible realizaions of he inegraed variance. In pracice, we will have o esimae he unknowns from he noisy opion prices observed in he markes. Figure 1 illusraes why Hull-Whie opion prices wih differen srike prices conain informaion on he disribuions of σ T 2 and ϕ T. I also illusraes how an iniial disribuion π(x ) affecs he reurn (X T /X ) similarly o he disribuions of ϕ T and ξ T. The inverse problem of implied inegraed variance was firs considered by Friz and Gaheral (25), who assumed an uncorrelaed volailiy. They showed how his problem is unique if a coninuum of opions wih differen srike prices is provided. However, he problem is ill-posed in he sense ha small errors in he daa may resul in large errors in he esimaes of he unknown. Friz and Gaheral (25) applied leas-squares opimizaion using radiional regularizaion echniques and general assumpions regarding he probabiliy densiy and hen use a lognormal prior disribuion in he Bayesian sense. 9

10 Raher han using regularizaion echniques, we will adop a Bayesian saisical approach. In his saisical approach, he enire deerminisic inverse problem is recased in he form of a saisical inference of he disribuion of he unknown. We hen model boh he unknown X R n and he observaion Y R m, as well as he noise ϵ R k, as random variables or vecors. Then, given he model funcion F, Y = F (X) + ϵ, he unknown X is esimaed from a realizaion y R m of Y. In addiion o an esimae of he unknown, his approach provides informaion on he reliabiliy of he esimae. A saisical approach for he inverse problem of opion-implied volailiies is presened by Con and Ben Hamida (25), where he unregularized ill-posed problem is solved using evoluionary opimizaion and Mone Carlo sampling. As an example, he local volailiy surface implied by opions wih differen srike prices and mauriies is esimaed. In he Bayesian framework prior informaion on he unknown x = X, encoded in a prior disribuion P prior (x), is combined wih informaion on he realizaion y of Y, modeled as a likelihood funcion P lik (y x). The soluion of he Bayesian inverse problem is he poserior densiy, given by he Bayes formula: P pos (x y) P prior (x)p lik (y x), where means up o a normalizing consan. The poserior densiy can be visualized by poin esimaes, such as Maximum-A-Poseriori (MAP) esimaes and Condiional Mean (CM) esimaes. A brief inroducion o saisical Bayesian inverse problems is given in he Appendix, and hey are discussed in more deph by Kaipio and Somersalo (25). Jacquier, Polson and Rossi (1994) and (25) apply a Bayesian approach when esimaing he sochasic uncorrelaed or correlaed volailiy process from opion prices. A lieraure review on he more recen use of Bayesian mehods in quaniaive finance is provided by Jacquier and Polson (21). When esimaing he implied inegraed variance, we will no search for one probabiliy densiy π( σ T 2 ). Insead, given he opion daa and he noise level, we will ry o esimae 1

11 he mos likely disribuion, as well as he probabiliy of oher disribuions. Based on prior knowledge, we know ha he oal probabiliy mus equal one. Following Friz and Gaheral (25), we make a prior assumpion ha he disribuion of σ 2 T is lognormal: σ 2 T ln (µ, ς), such ha he problem of esimaing he enire disribuion reduces o esimaing µ and ς, which are random variables in our saisical approach. The lognormaliy of he realized variance is repored by Andersen, Bollerslev, Diebold and Ebens (21), Andersen, Bollerslev, Diebold and Labys (21) and Zumbach, Dacorogna, Olsen and Olsen (1999). A shifed lognormal disribuion of realized variance is assumed by Carr and Lee (27) and a gamma disribuion is assumed by Carr, Madan, Geman and Yor (25). Because opions have differen marke prices for bids and offers and here is no unique price o be used when esimaing he inegraed variance, we will model opion prices as normally disribued random processes U rue N (Û, Var ). The mean Û is given as he average of he realized bid and offer prices, Û = (u bid is Var = V(u bid observed opion price u obs + u offer )/2 and he variance u offer ), where V is a posiive finie consan. We assume ha he is a realizaion of he arbirage-free rue opion price U rue and ha each observed opion price coincides wih he correlaed Hull-Whie price (7) or he corresponding opion payoff price (5) up o a small normally disribued error e,i N (, Var,i ), such ha u obs,i = U HW,ρ,i (x; K i, T ; σ 2, r) + e,i or u obs,i = E {h i (X T )} + e,i, (8) where 1 i L. Nex, we briefly presen an algorihm o compue MAP esimaes of σ 2 T ln (µ, ς), assuming an uncorrelaed volailiy. A similar algorihm ha can be used o solve he 11

12 enire inverse problem wih correlaed volailiy and unknown ρ using eiher he payoff formula or he correlaed Hull-Whie formula is presened in deail in he Appendix. The laer algorihm is from he sudy by Kaila (212). We fix ime and wrie u obs = u obs, U BS = U BS, U rue = U rue. For he compuaion, we mus firs discreize all of he disribuions and he pricing funcion. We assume ha he disribuion of he implied inegraed variance σ 2 = σ 2 T conains posiive values in he inerval [a, a + M σ ], divide his inerval ino n σ poins, σ 2 j and denoe by z R n σ he discreized probabiliy disribuion of σ 2, such ha z j is he probabiliy of σ 2 j, 1 j n σ. We assure ha n σ j=1 z j = 1 wih z j = z j /(M σ /n σ nσ j=1 z j). We denoe A = M σ n σ n σ j=1 U BS (x; K, T ; σ 2 j, r), u obs = Az + e. According o he prior assumpion, z ln (µ, ς). The prior densiy P prior (z) consiss of all possible lognormal disribuions z, denoed by z log (µ, ς). We model he hyperprior of he pair θ = (µ, ς) as a uniform disribuion: P hyper (µ, ς) = U((µ min, µ max ) (ς min, ς max )). Because we have assumed a normally disribued error e, he likelihood funcion is given by he Gaussian kernel: ( P lik (u obs z) exp 2( 1 (u obs Az) T Γ 1 (u obs Az) )), where he covariance marix Γ = diag(var 1, Var 2,..., Var L ). The poserior densiy P pos (z u obs ) P lik (u obs z)p prior (z)p hyper (µ, ς), which is he soluion of he inverse problem, reduces under he lognormaliy assumpion of z o P pos (z u obs ) P lik (u obs z log (µ, ς))p hyper (µ, ς). 12

13 In our case, he Maximum-A-Poseriori (MAP) esimaor z MAP (µ MAP, θ MAP ), which is characerized by z MAP = arg max z R nσ P pos (z u obs ), leads o he following opimizaion problem: z MAP (µ MAP, ς MAP ) = arg min zlog (µ,ς))( (u obs Az log (µ, ς)) T Γ 1 (u obs Az log (µ, ς)) ), (9) where (µ, ς) is given by he hyperprior. A simple mehod o esimae z MAP or θ MAP = (µ MAP, ς MAP ) is o draw a sample S = {θ 1, θ 2,..., θ N } of independen realizaions θ j R 2 from he hyperprior P hyper (µ, ς) and compue he θ j ha minimizes he righ-hand side of (9). However, his mehod can be compuaionally cosly because he sample size N mus be large enough such ha S represens he uniform disribuions of µ and ς. Alernaively, we can compue esimaes wih he grid search mehod, where we approximae he uniformly disribued coninuous hyperprior P hyper (µ, ς) by a discreized hyperprior. This discreized hyperprior is a grid consising of n µ evenly spaced values of µ wihin [µ min, µ max ] and n ς values of ς wihin [ς min, ς max ]. The norm on he righ-hand side of (9) is evaluaed a all of he grid poins, and θ MAP and z MAP are given by he grid poin (µ i, ς j ), 1 i n µ, 1 j n ς minimizing he norm. If we know ρ and have good hyperpriors, an esimae z MAP is obained in less han a minue (compued wih 2.2 GHz Inel Core i7). We explain in he Appendix how he hyperpriors can be chosen. We canno expec he approximae soluion o yield a smaller residual error han he measuremen error e and should accep as suiable approximaions of he unknowns all he esimaes saisfying u obs Az e = u obs U rue. We denoe such disribuions z log by z err, and hose wih he smalles and larges expecaions by z min and z max, respecively. This discrepancy principle, he effec of noisy daa and he bid-ask spread in he volailiy esimaion have also been considered by Con and Hamida (25). 13

14 Theoreically, he disribuion of he implied inegraed variance is he same for all srike prices and is already refleced in a couple of opion prices due o he convexiy of he Black-Scholes formula and he payoff formula wih respec o he underlying and he volailiy, as illusraed in Figure 1. If he opion daa are very noisy or he invesors in in-he-money opions have differen expecaions of he volailiy han hose invesing in a-he-money or ou-of-he-money opions, esimaes of he implied inegraed variance could vary according o he srike prices of he opions. The accuracy of he ou-ofhe-money calls and in-he-money pus is essenially lower han ha of he res of he opions. In Figure 2, we illusrae he performance of he grid search mehod wih he MAP esimaes of σ 2 T implied by S&P 1 call opions mauring in Augus 29. We plo z MAP and z err, as well as he pricing errors relaed o hese esimaes. As suppors for he discree probabiliy disribuions, we use σ T 2 [1e 5, 1], n σ = 5 and ϕ T [.5,.5], n ϕ = 1. The discree hyperprior of he pair (µ, ς) consiss of n µ = 1 evenly disribued poins µ on he inerval [log(.5iatm 2 ), log(2i2 ATM )], where I ATM is he implied a-he-money volailiy, and n ς = 6 prior values ς prior ha are unevenly chosen on he inerval [.5,.3]. We have chosen he prior values of ς unevenly in order o minimize he required compuaion; he opion prices are no sensiive o wheher he variance of σ 2 T is ς =.5 or ς =.7, for example. Addiionally, based on prior informaion on he ypical values of ρ implied by S&P 1 opions in 29, he discreized hyperprior of ρ is he se [.7,.75,.8,.85,.875,.9,.925]. Because we do no have acual informaion on he bid-ask spreads, we assume ha e N (, (.2u obs ) 2 ). To close he secion, le us consider he inegraed variance beween wo fuure momens 14

15 in ime T 1 < T 2 : σ 12 2 = σ T 2 1 T 2 = 1 T2 σ T sds, 2 T 12 = T 2 T T 1 Given wo esimaes of σ 2 T i ln (µ i, ς i ) implied by he opions u i wih differen srike prices and wo mauriies T i, i = 1, 2, we can compue an esimae of σ For σ 2 12 ln (µ 12, ς 12 ), he esimaes µ 12 and ς 12 are obained from µ i and ς i, i = [1, 2] by µ 12 = ( log( T 2 e µ 2+.5ς 2 T 1 e µ 1+.5ς 1 ) ) 1 T 12 T 12 2 ς 12, ( T 2 T ς 12 = log (e ς 2 1)e 2µ 2+ς 2 T 1 T 12 (e ς 1 1)e 2µ 1+ς 1 ( T 2 T 12 e µ 2+.5ς 2 T 1 T 12 e µ 1+.5ς 1) 2 The corresponding proof is provided in he Appendix. We will need σ 2 12 when pricing some pah-dependen opions in Secion 5. ). 15

16 Figure 2: MAP esimaes. In he firs row, we plo in he lef panel as disribuions and expecaions (verical lines) he esimaes z MAP,all compued from all L = 54 call opions (blue) and z MAP,nATM from L = 16 near-a-he-money calls wih moneyness.85 M 1.3 (green). In he righ panel, he corresponding L = 54 rue call prices u obs (red) and he call prices u MAP compued wih he correlaed Hull-Whie formula and he esimaes z MAP in he lef panel are ploed using he same colors. In he second row, we plo in he lef panel he relaive errors beween he esimaed prices and he rue prices, (u obs u MAP )/u obs, compued wih z MAP,all (blue) and z MAP,nATM (green). As he accuracy of he ou-of-he money prices ends o be less han he accuracy of in-he-money and a-he-money prices, we plo in he righ panel he same informaion on he relaive errors bu for he in-he-money and a-he-money opions only. The MAP esimaes of he correlaion coefficien implied by all L = 52 opions and L = 16 near-a-he-money opions are ρ MAP,all =.85 and ρ MAP,nATM =.8, respecively. In he boom row, we plo esimaes z min (green) and z max (red) ha should be acceped according o he discrepancy principle, as well as z MAP (blue). In he lef panel, we have used all L = 54 opions and ρ MAP,all and in he righ panel, we have used he L = 16 near-a-he-money opions and ρ MAP,nATM (he esimaes are compued from S&P 1 calls mauring in Augus 29 and he assumed noise is e N (, (.2u obs ) 2 ) ). 16

17 3 Volailiy derivaives Inegraed and realized variances and volailiies are closely relaed o some volailiy derivaives, such as variance and volailiy swaps and opions. These insrumens provide he possibiliy o speculae on or hedge risks associaed wih he size of he movemen of an underlying produc, such as a sock index, an exchange rae or an ineres rae. Variance and volailiy swaps are forward conracs ha pay he difference beween he realized variance or volailiy and an agreed fixed amoun, he srike, a mauriy. Variance and volailiy calls and pus are similar o European calls and pus, wih an underlying realized variance or volailiy. There are wo main approaches in pricing volailiy derivaives, based on eiher he underlying sock price process or he informaion provided by he marke prices of European opions. Opion-implied volailiy derivaives are discussed in deail by Carr and Lee (28), and volailiy derivaives more generally are discussed by Dupire (1992), Carr and Madan (1998), Derman e al. (1999a), (1999b) and, more recenly, by Carr e al. (25) and Carr and Sun (28). The volailiy derivaives markes are considered by Carr and Lee (29), who have also provided a lieraure review. The payoff funcions of a variance swap wih srike price Kvar, H denoed by H var ( σ T 2 ), and of a variance call wih price Kvar, h denoed by h var ( σ T 2 ), are given by H var ( σ 2 T ) = σ 2 T K H var and h var ( σ 2 T ) = ( σ 2 T K h var) +, and he corresponding payoff funcions of a volailiy swap wih a srike price K H vol and a 17

18 volailiy call wih a srike price K h vol, denoed by H vol( s T ) and h vol ( s T ), are given by H vol ( s T ) = s T K H vol and h vol ( s T ) = ( s T K h vol) +. In his paper, for simpliciy, we will assume ha he swap srikes K H var = and K H vol =. Based on he work of Neuberger (1994), Demeerfi, Derman, Kamal and Zou (1999) show how variance swaps can be replicaed by holding a saic posiion in European calls and pus wih all srikes and rading dynamically on he underlying asse. The logarihm of sock price reurn is obained by ( ) XT log = X T X X X X (K X T ) + dk K 2 X (X T K) + K 2 dk. (1) The inegraed variance is replicaed by a porfolio of calls and pus where each of he opions is weighed by he squared srikes. Carr and Ikin (21) noe hree ypes of errors relaed o he synheizing of log(x T /X ) wih (1): errors due o jumps in asse prices, exrapolaion and inerpolaion errors due o a finie number of available opion quoes insead of he coninuum of opions ha is needed in he replicaion, and errors in compuing he realized reurn variance. Javahari, Wilmo and Haug (24) value volailiy swaps using he GARCH process. Gaheral (26) proposes ha he value of he variance swap can be expressed as a weighed average of Black-Scholes implied variances I 2 = I 2 (log(x /K)), wih weighs given by he log-moneyness, such ha E {I 2 } = Φ (z)i 2 (z)dz, R where z = log(x /K) and Φ (y) = e y2 /2 / 2π. Carr and Lee (27) and (28) consruc 18

19 a synheic swap by rading a log conrac of he underlying and bonds. Their approach does no make prior assumpions on he dynamics of he volailiy and is robus wih respec o correlaed volailiy. Carr and Ikin (21); Carr, Geman, Madan and Yor (21); and Carr, Lee and Wu (212) propose an asympoic mehod wih he assumpion ha he underlying process is modeled by a Levy process wih sochasic ime change. Friz and Gaheral (25) price volailiy and variance swaps wih he expecaion of he implied inegraed variance, assuming an uncorrelaed volailiy. Furhermore, based on he Black-Scholes formula, hey derive a closed-form pricing formula for variance calls. Our approach is an exension of ha of Friz and Gaheral (25). We assume a correlaed volailiy and use he disribuion of he implied inegraed variance o price volailiy opions. We show how he disribuion provides informaion on he pricing error we may be making and explain how noisy opion daa should be aken ino accoun when pricing volailiy derivaives. We also esimae he disribuions of variance and volailiy swaps beween wo fuure momens in ime. The prices of various volailiy derivaives compued wih implied inegraed variances and volailiies are arbirage-free prices. Unless he markes for hese volailiy derivaives are liquid, he corresponding marke prices may differ subsanially from he arbirage-free prices. The error beween he arbirage-free prices and he marke prices, as well as possible sysemizaion in he error, is a subjec of furher research. Variance swaps and calls can be perfecly replicaed under he classical derivaives pricing heory and are o some exen sraighforward o price and hedge. The same is no rue for volailiy swaps and calls. Hedging a volailiy swap using variance swaps requires a dynamic posiion in he log conrac replicaing he variance swap. For example, according o (1), his posiion depends on a srip of vanilla opions. Because some of hese opions will be very far ou-of-he-money, he rading is performed a large bid-offer 19

20 spreads and he re-balancing of he hedge becomes expensive. Le us look a he problem of pricing volailiy swaps and calls. Denoe by ˆσ 2 T and ŝ T = E { s T } = E { σ T 2 } he expecaions of he inegraed variance and volailiy, respecively. As he expecaion of he square roo of a random variable is less han he square roo of he expecaion, we have ŝ T = E { σ 2 T } E { σ 2 T } = ˆσ 2 T, (11) The difference beween he expecaion of he inegraed volailiy and he square roo of ˆσ T 2 is called he convexiy adjusmen. Someimes, E { σ T 2 } is called he model-free implied volailiy. The VIX index approximaes his volailiy. Brockhaus and Long (2) sugges a volailiy convexiy correcion ha relaes variance and volailiy producs. Heson and Nandi (2) provide an analyical soluion o price volailiy swaps and opions and demonsrae how o hedge differen volailiy derivaives by rading only on he underlying asse and a risk-free asse. Broadie and Jain (28) derive parial differenial equaions o price volailiy derivaives and o compue Greeks in order o hedge hese derivaives, and Windcliff, Forsyh and Vezal (26) sugges improving he dela hedge of he volailiy derivaive by an addiional gamma hedge. As explained by Friz and Gaheral (25), under he prior assumpion σ 2 T ln (µ, ς), he inegraed volailiy s T ln (µ/2, ς/4) and he convexiy adjusmen is given by E { σ 2 T } E { σ 2 T } = (e.5ς 1)E { σ 2 T }. This formula provides a convexiy correcion for he VIX index. Carr and Lee (28) add o (11) a lower bound ( 2π/X )E {(X T X ) + } and an upper bound based on he 2

21 log-reurn log(x T /X ) and show how o superreplicae he payoff of σ 2 T. We sugges ha when pricing variance and volailiy swaps, no only he expecaions bu he enire disribuions of he inegraed variances and volailiies are imporan. In paricular, he volailiy call is easily compued from he disribuion of he corresponding swap. In Figure 3, we presen examples of he MAP esimaes of variance and volailiy swaps, H var ( σ 2 T ) and H vol( s T ), respecively, as well as he corresponding payoffs of variance and volailiy calls h var ( σ 2 T ) and h vol( s T ) as disribuions and expecaions, compued from S&P 1 ATM calls mauring in June 29 wih ρ MAP =.85, assuming ha he error e 1 N (, (.2u obs ) 2 ) and K var =.4, K vol =.4 =.2. We also indicae he discree poins σ T,j 2 1 j n σ ha are.67 sandard deviaions from ˆσ T 2. Wih he probabiliy P =.5, he inegraed variance is beween hese poins. The disribuions z MAP, z min and z max in Figure 2 provide an idea of he effec of noisy daa on he prices of variance and volailiy derivaives. According o he discprepancy principle, boh he esimaes z min and z max should be considered as suiable esimaes of σ 2 T. To close he secion, we plo he surfaces of he implied variance and he implied volailiy beween wo fuure momens in ime, esimaed from S&P 1 call opions wih differen srike prices and he mauriies T 1 in March and T 2 in April 21. The esimaes were compued four imes, every 15 days, saring 6 days before he mauriy T 1. Informaion provided by hese surfaces can be used when pricing variance and volailiy swaps and opions. 21

22 Inegraed variance Inegraed volailiy Days.15.1 Variance Days.15.1 Volailiy.5 Figure 3: Variance and volailiy derivaives. In he op row and in he lef panel, esimaes of a variance swap σ T 2 (blue) and he payoff h var( σ T 2 ) of a variance call (red) are ploed as expecaions (verical lines) and disribuions, based on S&P 1 calls mauring in June 29 wih T = 1/1 years. The righ panel presens he corresponding daa for volailiy swaps and calls, as well as E{ σ T 2 } (green). Wih he probabiliy P =.5, he implied inegraed variance and volailiy are wihin he poins indicaed wih magena lines. The srike prices are K var =.4 and K vol =.4 =.2. In he boom row, esimaes of he implied inegraed variance σ 12 2 (lef panel) and he corresponding implied inegraed volailiy s 12 = sqr σ 12 2 (righ panel) beween wo fuure momens in ime are ploed. Based on S&P 1 calls mauring in June and July 29, we have esimaed boh disribuions every 15 days, beginning 6 days before he firs mauriy in March. 22

23 4 Pricing digial opions and barrier opions In his secion, we use he risk-neural price densiy and he implied inegraed variance o price digial opions and one ype of semi-pah-dependen opions, namely, he barrier opions. The pricing of boh ypes of opions when he volailiy is sochasic has been considered by Fouque, Papanicolaou and Sircar (2), where an asympoic mehod exploiing volailiy clusering is applied, and by Tahani (25), where hese opions are priced wih a Taylor expansion around wo average volailiies and he Hull-Whie formula. Our conribuion o he lieraure is o presen how boh ypes of opions can be priced using he correlaed Hull-Whie paradigm. We also show how boh he disribuion of he implied inegraed variance and knowledge of he noise level in he opion prices provide addiional informaion on he size of he pricing error ha we are poenially making. The price D (x; σ 2 ) = D (x; K, T, Q; σ 2, r) of a digial call is given by D (x; σ 2 ) = e r(t ) QE {(X T > K) x, σ 2, F } = e r(t ) QE {h D (X T ) x, σ 2, F }, where Q is a posiive consan. When he volailiy is consan, he price of a digial call is given by D BS (x; σ 2 ) = e r(t ) QΦ(d (σ 2 )). We sugges ha in he case of sochasic correlaed volailiy, he corresponding price would be D HW,ρ (x; σ 2 ) = e r(t ) QE {Φ(d ( σ 2 ρ,t )) σ 2, F }, applying he Hull-Whie paradigm. Figure 4 presens esimaes of such digial opions as funcions of moneyness, compued from he S&P 1 calls ha are mauring in November 23

24 21. Now consider he pricing of digial opions using he risk-neural price densiy. The disribuion of σ 2 T provides addiional informaion on he reliabiliy of he risk-neural price densiy and he digial opion prices. According o (2), he risk-neural price densiy is he expecaion of lognormal disribuions π(x T σ T 2 ) condiional on σ2 T. In Figure 4, we plo such lognormal disribuions, condiional on he discree values σ 2 T,j, 1 j n σ wihin one and wo sandard deviaions from he expecaion ˆσ 2. Wih he probabiliies of P 1.7 and P 2.95, he risk-neural price densiy is wihin he range of hese disribuions. We hen plo he expeced disribuion of X T and he digial opion prices corresponding o each disribuion. Nex, consider he pricing of barrier opions. Barrier opions are semi-pah-dependen opions whose payoff depends on wheher he underlying asse price his a specified value during he lifeime of he opion. For example, a down-and-in opion becomes a European call or pu if he underlying sock price X reaches a predeermined level B a any ime before he mauriy T, and a down-and-ou opion is a European opion ha becomes worhless if X falls below a level B. The payoffs of a down-and-in call, h DI, and of a down-and-ou call, h DO, are given by h DI = (x; K, B, T ; σ 2, r) = E {(X T K) + 1 [X >B for any T ](X )}, h DO = (x; K, B, T ; σ 2, r) = E {(X T K) + 1 [X<B for all T ](X )}, where 1 E (x) = 1 if x E and 1 E (x) =, oherwise. A common model for pricing and hedging barrier opions, based on a reflecion heorem in he Black-Scholes model, was iniially considered by Carr, Ellis and Gupa (1998) 24

25 Digial call price Relaive price difference K/x K/x RNPD digial call, K= digial pu, K= digial call, K=.85 Figure 4: In he lef panel in he op row, we plo he digial opion prices D MAP compued using he esimae of z MAP (red), prices D.14 and D.86 compued using he discree values σ T,.14 2 and σ2 T,.86 ha are a 1.5 sandard deviaion from he expecaion ˆσ2 T (blue and green), and prices D I compued using he implied variance I 2 (magena). The prices compued wih z min and z max are ploed wih dashed lines. In he righ panel, we presen he corresponding relaive differences beween each price and D MAP, (D MAP D I )/D MAP, for example (S&P 1 mauring in November 28, T = 1.5 monhs, Q = 1). In he second row, we consider esimaes of he risk-neural price densiy compued wih σ T 2 ln (.15,.5) and ρ =.15. We plo in red he expeced disribuion of X T (lef panel) and he corresponding digial opion payoff h D (X T ) (righ panel), boh as disribuions and expecaions. We hen plo in darker and ligher blue lognormal densiies of X T,j condiional on each poin of σ T,j 2, 1 j n σ ha is wihin 1 or 2 sandard deviaions from ˆσ 2 (lef), as well as he corresponding digial opion payoffs as densiies (righ). Wih he probabiliy P.95, he payoff of he digial opion price is wihin he wo black lines. In he boom row, we plo he same informaion for a pu opion (lef panel) and a call opion when σ T 2 ln (.12,.1) and ρ =.45 (righ panel) (Q = 1, T =.5 years, r = ). 25

26 Barrier price Relaive price difference K/x K/x Figure 5: We plo in he lef panel he prices of a down-and-in call (green), down-and-ou call (red) and a vanilla call (blue), compued using z MAP implied by S&P 1 calls mauring in November 21 (T 2 monhs, B =.95K). The corresponding prices compued wih he implied variance I 2 are indicaed wih dashed lines. The relaive differences beween he down-and-ou prices compued wih z MAP and he prices compued wih σ 2 T,.14 (magena), σ2 T,.86 (red) and I2 (green), z min (black) and z max (blue) are ploed in he righ panel. and Carr and Chou (1997a), (1997b). Barrier opions wih deerminisic volailiy are considered by Andersen, Andreasen and Eliezer (22), Nalholm and Poulsen (26), Poulsen (26) and Carr and Nadochiy (211), who also provide a lieraure review of barrier opions. When he volailiy is consan, barrier opions can be priced based on he Black-Scholes formula. For example, he price of he down-and-in barrier opion, DI BS (x; σ 2 ) = DI BS (x; K, B, T ; σ 2, r) wih B K is given by DI BS (x; σ 2 ) = x ( B x ) 2Ψ ( ) 2Ψ 2 B Φ(y) Ke r(t ) Φ(y σ (T )) x where Ψ = r σ , y = ln(b2 /(xk)) σ (T ) + Ψσ (T ). and he price of he down-and-ou opion DO BS DO BS (x; σ 2 ) = U BS (x; σ 2 ) = DO BS (x; K, B, T ; σ 2, r) is (x; σ 2 ) DI BS (x; σ 2 ). When B > K, he price of a down-ou barrier 26

27 opion is given by DO BS (x; σ 2 ) = xφ(x 1 ) Ke r(t ) Φ(x 1 σ (T )) ( ) 2Ψ ( ) 2Ψ 2 B B x Φ(y 1 ) + Ke r(t ) Φ(y 1 σ (T )), x x where and DI BS x 1 = log(x/b) σ (T ) + Ψσ (T ), y 1 = log(b/x) σ (T ) + Ψσ (T ), (x; σ 2 ) = U BS (x; σ 2 ) DO BS (x; σ 2 ). We sugges ha when he volailiy is sochasic, he barrier opion prices could be given as expecaions of he Black-Sholes barrier opion prices wih respec o he inegraed variance and ϕ T. For example, he prices of a down-and-in and a down-and-ou opion would be as follows: DI HW,ρ (x; σ 2 ) = E {E {DI BS (e ρϕ T x; σ 2 ρ,t ) σ 2 T, F } σ 2, F } DO HW,ρ (x; σ 2 ) = E {E {DO BS (e ρϕ T x; σ 2 ρ,t ) σ 2 T, F } σ 2, F }. We presen a proof of hese formulas wih sochasic volailiy in hese Appendix, closely following he proof by Carr and Chou (1997b) associaed wih consan volailiy. In paricular, because he implied inegraed variance is independen of he srike prices, his variance is quie robus wih respec o he changes in he prices of he underlying during he lifeime of he opion, compared wih using he implied volailiy corresponding o a cerain srike price. Figure 5 illusraes barrier opion prices based on esimaes z MAP, z min and z max compued from S&P 1 calls mauring in November 21. We also compued wo prices wih he 27

28 discree values σ T,j 2 ha are wihin 1.5 sandard deviaions from ˆσ2 T = E{ σ2 T }, denoed by σ 2.14 and σ Boh he expecaion and he shape of he disribuion of σ 2 T affec he prices of he barrier opions. 28

29 5 Pah-dependen opions In his secion, we consider he pricing of wo ypes of pah-dependen opions: compound opions and clique opions. Boh ypes of opions are sensiive o changes in volailiy. When pricing hese opions, we will need esimaes of σ 12, 2 he implied inegraed variance beween wo fuure momens in ime, namely, T 1 and T 2, T 1 < T 2. Figure 6 presens such esimaes and he corresponding risk-neural price densiies π(x 12 ) = π(x T2 X T1 ). Firs, for a fixed ime, we esimae hese densiies four monhs ino he fuure based on he prices of opions mauring in five consecuive monhs. We hen ake wo opions wih consecuive mauriies T 1 and T 2 and presen how he esimaes of σ 12 2 and he corresponding densiy of X 12 change as a funcion of ime. We esimae boh densiies every 15 days, beginning 6 days before T 1. Le us hen price compound opions, i.e., opions-on-opions, wih wo srike prices and wo expiraion daes. On he dae of he firs mauriy T 1, he holder of a compound opion has he righ o buy or sell a price K 1 a new opion wih srike price K 2 and mauriy T 2. Analyical pricing formulas have been proposed by Geske (1979), Hodges and Selby (1987) and Rubinsein (1992), provided ha he volailiy is consan, and Fouque and Han (25) sugges a perurbaion approximaion when he volailiy is sochasic. We consider here he pricing of a pu-on-call opion; he pricing of a pu-on-pu, a callon-call or a call-on-pu is similar. To keep he noaion simple, we assume ha ρ =. Exension o correlaed volailiy is sraigh forward. Suppose ha we have esimaes of σ 12 2 and X T1. Then, condiional on X T1, he price of he pu on a call opion is given by C PoC,COND = e r(t 1 ) max(, K 1 E {U BS (X T1 ; K 2, T 2 T 1 ; σ 2 12, r) X T1, F }), 29

30 Inegraed variance Risk neural price densiy IV 2 4 Days X T Days IV Days X T Days Figure 6: Esimaes of σ T 2 and X T beween wo fuure momens in ime. In he lef panel in firs row, we plo σ T 2 1 (red), σ T 2 2 (green), and σ 12 2 (blue), as well as he corresponding expecaions, esimaed a. The corresponding risk-neural price densiies are ploed in he righ panel. In he second row, we esimae a ime he implied inegraed variance and he risk-neural price densiy during 9 fuure days. We separaely esimae each ime inerval [T i, T i+1 ], 1 i 4, using opions wih he corresponding mauriies. We plo in he lef panel he surface of he MAP esimaes of σ m, 2 m = [12, 23, 34, 45], and in he righ panel, we plo he corresponding esimaes of X m ( =Ocober 14, T 1 =November 2, T 2 =December 18, T 3 =January 15, T 4 =February 19, T 5 =March 19, 29). In he boom row, we plo esimaes of σ 12 2 (lef panel) and X T (righ panel) using opions mauring in June and July 29. The densiies are esimaed every 15 days, beginning on March 24, 29. 3

31 and he price C PoC is obained by inegraing C PoC,COND wih respec o X T1. The chooser is an opion where, a ime T 1, he holder of he claim can decide o choose eiher a call or a pu opion, wih mauriy T 2 and srike price K 2. If we denoe he payoffs of a call and a pu condiional on X T1 by h U = U BS (X T1 ; K 2, T 2 T 1 ; σ 2 12, r) and h P = P BS (X T1 ; K 2, T 2 T 1 ; σ 12, 2 r), respecively, he price of a chooser pu opion, condiional on X T1, is hen given by C Ch,COND = e r(t 1 ) (K 1 max(h U, h P )), and he price C Ch is obained by inegraing C Ch,COND wih respec o X T1. Clique opions are exoic opions consising of a series of forward saring opions whose erms are se on he rese daes. Haug and Haug (21) presen a mehod o price clique opions based on a binomial ree developed by Cox, Ross and Rubinsein (1979), and Gaheral (26) prices clique opions wih local volailiy. Numerical pricing mehods and various volailiies are invesigaed by Windcliff, Forsyh and Vezal (26) and den Iseger and Oldenkamp (25). We price a clique opion wih σ 2 12 and X 12. We consider one paricular clique, namely, a European-syle rese pu opion wih a single rese dae, wih payoff CL(x; T 1, T 2 ) given by CL(x; T 1, T 2 ) = X T1 X T2, if (X T1 > x, X T2 X T1 ), x X T2, if (X T1 x, X T2 x), (12), oherways. A closed-form pricing formula for his ype of clique opion wih consan volailiy is given by Gray and Whaley (1999), where he expeced erminal payoffs of he clique are 31

32 divided ino hree possible oucomes and he probabiliies for each oucome are compued. Consider now he pricing of clique opions using he implied inegraed variance. Assume ha we know he disribuions of X T1 and σ 2 12 and denoe he Black-Scholes pu opion price by P BS. Condiional on X T1 and σ 2 12, he discouned payoff corresponding o X T1 X T2 in he firs row in (7) is given as follows: h 1 = e r(t 1 ) E {P BS (X T1 ; X T1, T 2 T 1 ; σ 2 12, r)1 XT1 x(x T1 ) x, X T1, σ 2 12, F }, and he discouned payoff corresponding o x X T2 is given by h 2 = e r(t 1 ) E {P BS (X T1 ; x, T 2 T 1 ; σ 2 12, r)1 XT1 <x(x T1 ) x, X T1, σ 2 12, F } The price of he rese pu opion is obained by inegraing h 1 and h 2 firs wih respec o X T1 and hen wih respec o σ Figure 7 presens an example of rese pu opion prices, based on informaion provided by S&P 1 calls. We also illusrae he effec of noisy daa and, based on he discrepancy principle considered in Secion 2, we plo a minimum and maximum price for his clique opion, compued using z min and z max. Because we do no have informaion on he bid-ask spread of he opions considered, in order o illusrae he phenomenon, we assume a high noise level of e N (, (.45u obs ) 2 ). 32

33 Inegraed variance Risk neural price densiy K/x Clique opion Days Compound Clique K1.4.6 K X T 1 IV Figure 7: Compound and clique opions. In he lef panel of he firs row, we plo he MAP esimae z MAP of σ 12 2 (blue), as well as he esimaes z min (green) and z max (red) acceped by he discrepancy principle when e N (, (.45u obs ) 2 ). We esimaed he disribuions from S&P 1 call opions mauring in June and July 29 so ha T 1 = 6 and T 2 = 8 days. The corresponding esimaes of he risk-neural price densiy are ploed in he righ panel. In he second row, he prices of a pu-on-call and a pu-on-pu opion (doed line) are ploed in he lef panel and he prices of a clique opion are ploed in he righ panel. All of hese opions were compued using he densiies of he firs row and ploed wih he corresponding colors. The effec of noisy daa can paricularly be observed in he esimaes of clique prices. In he boom row, we plo surfaces of compound pu-on-call and pu-on-pu opions as funcions of srike prices (lef panel), and we plo clique opions as funcions of he underlying and he inegraed variance (righ panel). 33

34 6 Concluding remarks In his paper, we have considered he pricing of exoic opion, namely, some volailiy opions, semi-pah-dependen opions and pahdependen opions, wih wo opion-implied densiies: he densiy of he inegraed variance and he risk-neural price densiy. We have assumed a sochasic correlaed volailiy and proposed a Bayesian approach for he ill-posed inverse problem of an implied inegraed variance and an implied correlaion coefficien. We have presened an algorihm wih which o calculae Maximum-A-Poseriori (MAP) esimaes of he disribuion and expecaion of his variance from a series of European opions wih differen srike prices and, possibly, differen mauriies. Based on he discrepancy principle, we have explained how noisy daa affec hese esimaes. Due o he convexiy of he Black-Scholes formula wih respec o he volailiy and he underlying, he disribuion of he inegraed variance and he risk-neural price densiy can be esimaed from a couple of opion prices wih differen srike prices, hus avoiding he problem of missing opion price daa ofen encounered when esimaing he riskneural price densiy. Because he implied inegraed variance is independen of srike prices, i is robus wih respec o changes in he price of he underlying. We have considered he pricing of variance and volailiy swaps and opions and showed how he convexiy adjusmen can be avoided by compuing volailiy swaps and opions from he disribuion of he inegraed variance insead of he corresponding expecaion. We hen priced digial opions and barrier opions and showed wha addiional informaion can be obained from he disribuion of he implied inegraed variance. Finally, we presened a simple algorihm wih which o esimae he implied inegraed variance beween wo fuure momens in ime and used his variance o price compound and chooser opions as well as clique opions. 34

35 7 Appendix proof of (4), page 6 We wrie σ T = σ + and denoe I T = T T θσ s ds + T σ s dz s. Now, νσ s dz s = σ exp((θ 1 2 ν2 )(T ) + νz T ) I T = 1 ν (σ exp((θ 1 2 ν2 )(T ) + νz T ) σ and I T has a shifed lognormal disribuion, condiional on T (I T T T σ s ds: θσ s ds), (13) σ s ds) ln ( log ( σ ) 1 + (θ ν 2 ν2 )(T ), ν 2 (T ) ) σ ν θ T σ s ds. ν Inegraing wih respec o T gives σ s ds and noicing ha E {θ T σ s ds} = σ (e θ(t ) 1) ϕ T ln ( log ( σ ) 1 + (θ ν 2 ν2 )(T ), ν 2 (T ) ) σ ν eθ(t ) 1 2 ρ σ2 T (T ). Now, we wan o esimae ϕ T condiional on σ T 2. We can expec o have informaion on he iniial volailiy σ a ime. Le us explain how he facors θ and ν in (13) are relaed o σ 2 T. Assuming ha σ 2 T ln (µ, ς), he expecaion E { σ 2 T } = e(µ+.5ς) and he variance Var ( σ 2 T ) = (eς 1)e (2µ+ς). The ph momen of a lognormally disribued random variable R ln (A, B 2 ) is given by E{R p } = exp(pa +.5p 2 B 2 ). Denoe A = (ν.5θ 2 )(T ) 35