Robust Hedging Performance and Volatility Risk in Option Markets. Chuan-Hsiang Han 1

Transcription

1 Robus Hedging Performance and Volailiy Risk in Opion Markes Chuan-Hsiang Han Deparmen of Quaniaive Finance, Naional sing-hua Universiy., Secion, Kuang Fu Road, aiwan, 3, ROC. FAX: EL: E- mail:

2 Absrac We invesigae daily robus hedging performance wih rading coss for markes of S&P 5 Index opion (SPX) and aiwan Index opion (XO). Robus hedging refers o minimal model dependence on he risky asse price. wo hedging caegories including model-free and volailiy-model-free, and nonparameric mehods for volailiy esimaion are considered in our empirical sudy. In paricular, he insananeous volailiy is esimaed by a proposed nonlinear correcion scheme of Fourier ransform mehod, jusified by a simulaion sudy for a local volailiy model. An asymmeric phenomenon of hedging performances is documened. Hedging porfolios consruced from he volailiy-model-free caegory induce much higher Sharpe raios han hose from he model-free caegory on SPX, while hey perform comparably on XO. Moivaed from he price limi regulaion in aiwan, we furher develop a imescale change mehod o explain his phenomenon. Asympoic momen esimaes of differences of some hedging porfolios are consisen wih our empirical findings. JEL classificaion: C4; C5; G5. Keywords: Opion Hedging Sraegies, Volailiy Esimaion, Fourier ransform mehod, Momen Esimaion.

3 . Inroducion I has been recognized from empirical sudies ha complicaed asse pricing models may no have beer hedging performance han he ad hoc Black-Schole model. Bakshi e al. (997), Lam e al. () and Yung e al. (3) documen ha sochasic volailiy models, variance gamma models, and EGARCH (GARCH) models, respecively, are superior in volailiy forecas and/or opion pricing, bu hese models perform jus comparably or even worse han he ad hoc Black-Scholes model (Dumas e al. (998)) in opion hedging. hese observaions indicae he imporance of robus hedging. ha is, model dependence of opion hedging sraegies should be minimized. We idenify and classify several hedging sraegies according o heir level of model dependence. Since implemenaion of many hedging sraegies essenially requires volailiy as he inpu variable, nonparameric mehods for volailiy esimaions are incorporaed for our empirical sudy o keep he spiri of reducing model errors. Fourier ransform mehod proposed by Malliavin e al. (9) provides a new and alernaive nonparameric esimaion o measure he insananeous volailiy risk wihou imposing any specific volailiy models. his line of invesigaion also differeniaes our research from he curren lieraure of merely using he implied volailiy, a Black-Scholes modeldependen volailiy. As a whole, opion hedging performances given robus hedging sraegies and nonparameric volailiy esimaions are comprehensively sudied in his paper. Opion markes including SPX in US and XO in aiwan are chosen for empirical sudies. A surprising difference on hedging performances beween hese wo markes is 3

4 documened. he price limi effec posulaed in aiwan is aribued o such difference according o our asympoic analysis. his paper sudies inensively on hedging performance wih rading coss for index opion markes of SPX and XO. We begin wih an empirical sudy on hedging performances of hese index opions by various rading sraegies wih ransacion coss and axes. wo caegories of hedging sraegies are considered. () Model-Free caegory includes he sop-loss sraegy and an adjused sop-loss sraegy. () Volailiy- Model-Free caegory includes he dela hedging, an adjused dela hedging sraegy, and he dela-gamma sraegy. Each hedging sraegy doesn depend on eiher any specific asse pricing model or any specific volailiy model. Combinaions of hese wo hedging caegories wih hree volailiy esimaions of he hisorical volailiy, he insananeous volailiy and he implied volailiy are seleced o compare hedging performances. A key parameer for implemening hese hedging sraegies excep he sop loss is volailiy. Besides convenional mehods of volailiy esimaion by he hisorical volailiy and he implied volailiy, a recen progress using a nonparameric Fourier ransform mehod (Malliavin and Mancino (9)) o esimae volailiy marix dynamics paves a way for esimaion of he insananeous volailiy. In comparison wih anoher popular nonparameric mehod for volailiy esimaion by quadraic variaion formulas (see Zhang e al. (5) and references herein), Malliavin and Mancino (9) claimed ha Fourier ransform mehod is more sable because i relies on he inegraion of Fourier coefficiens of he variance process as opposed o a numerical differeniaion of he hisorical volailiy and he insananeous volailiy are esimaed based on nonparameric mehods. he implied volailiy does depend on he Black-Scholes model. I is incorporaed because of is populariy in heory and pracice. 4

5 quadraic variaions. However, Reno (8) alers ha he Fourier algorihm performs badly near ime boundaries of esimaed volailiy ime series daa, i.e. esimaed volailiy of he firs and las % ime series are no accurae enough. o avoid his boundary effec pifall, Han e al. () provided an effecive price correcion scheme based on a linear regression derived from he disribuion of esimaed volailiy given observed price reurns. hey jusified ha sochasic volailiy models calibraed o he insananeous volailiy ouperform GARCH (,) model based on backesing resuls of Value-a-Risk (Joridon (7)). In his paper, we propose a new correcion scheme based on a nonlinear regression. A Mone Carlo simulaion sudy for a local volailiy model is used o demonsrae he accuracy of volailiy esimaion by hese wo correcion schemes. We documen an asymmeric phenomenon of hedging performances beween SPX and XO. For SPX, hedging porfolios associaed wih he volailiy-model-free caegory induce much higher Sharpe raios han hose associaed wih he model-free caegory. (In fac, he dela hedging wih he insananeous volailiy ouperforms oher combinaions including he well-perceived dela hedging wih he implied volailiy.) However for XO, Sharpe raios of hedging porfolios associaed wih hese wo caegories are comparable. ha is, using sop-loss like sraegies in daily hedge perform as good as dela hedging like sraegies in aiwan. We furher invesigae he sample mean and sandard deviaion of P/L differences beween he sop-loss and he dela hedging porfolios. I is clear o observe ha hese wo saisics from XO are very small compared wih hose from SPX. See Figures 3 and 4 in Secion 3 for graphically 5

6 demonsraions. hese resuls furher moivae our addiional sudy of momen esimaes for differences of hedging sraegies. Noice ha here is a sric price limi consrain in he marke of aiwan Weighed Sock Index (AIEX). Such price limi conrols he flucuaion of each sock price in daily basis. Enormous lieraure has been sudying price limi effecs, which include cooling-off, volailiy spillover, delay in price discovery, rading inerference, and magne effec. See discussions from Kim and Rhee (997), Chen (998), Cho e al. (3) and references herein. he relaionship beween hedging performance and he price limi has received surprisingly lile aenion despie is highly pracical relevance in emerging markes such as aiwan. We develop a heory ha qualiaively explains small values of mean and sandard deviaion menioned above on XO, whereas hedging performance of SPX is considered as a conrol group of no price limi. Moivaed from he cooling-off effec 3 from he price limi, we apply a ime-scale change mehod o he Black-Scholes model, and analyze differences of hedging P/L induced from he sop-loss sraegy and a rescaled dela hedging sraegy. We obain an asympoic resul o show ha he P/L difference beween hese wo hedging porfolios is small when he ime change variable is small. his heoreical resul is consisen o empirical findings in XO. We shall remark ha he ime change mehod has been exensively sudied in probabiliy and mahemaical finance. See an overview by Geman (5), Fouque e al. (3) and references herein. he organizaion of his paper is as follows. In Secion, we inroduce procedures of various rading sraegies o hedge index opions on SPX and XO. In Secion 3, 3 he cooling-off effec means ha he price limi helps dampen volailiy and sabilize rading volumes paricularly during urbulen rading days. 6

7 volailiy esimaions including he hisorical volailiy, he insananeous volailiy, and he implied volailiy are inroduced. Fourier ransform mehod wih price correcion schemes is used for he insananeous volailiy esimaion. A local volailiy model is examined as a simulaion sudy o validae he effeciveness of our proposed nonlinear regression correcion scheme. In Secion 4, daa ses, ransacion coss and axes of each rade, and empirical resuls of hedging performances are demonsraed. Comparisons of iner and inra opion markes are discussed. In Secion 5, a ime-scale change mehod is developed o he Black-Scholes model in order o mimic a cooling-off effec of price limi. We analyze momens of P/L differences of wo hedging sraegies associaed wih he dela and he sop loss, and confirm ha our heoreical resul in a qualiaive sense is consisen wih empirical findings on XO.. Hedging Sraegies wo caegories of dynamic hedging sraegies are invesigaed in his paper. hey are Model-Free caegory and Volailiy-Model-Free caegory. he firs caegory consiss of wo hedging sraegies including he sop loss and an adjused sop loss. he laer sraegy is designed o explore he persisency and he mean-revering propery of volailiy in order o improve he sop-loss sraegy. he second caegory consiss of hree dynamic hedging sraegies including he dela hedging, an adjused dela hedging, and he dela-gamma hedging. he adjused dela hedging sraegy is based on he correced dela hedging formula derived in Fouque, Papanicolaou, and Sircar (). heoreically, his sraegy is helpful o improve he dela hedge by aking he smile or smirk effec of implied volailiy ino accoun. his 7

8 paper provides an empirical examinaion for such sraegy. he dela-gamma hedging incorporaes an addiional opion ino he rading porfolio in order o eliminae he volailiy risk. A number of pracical ways o manage he volailiy risk can be found in Gaheral (6).. Model-Free Caegory wo sraegies are considered: he sop loss and an adjused sop loss. Boh sraegies are fully independen of any pricing model. he sop-loss sraegy is even independen of he volailiy.. Sop Loss: his sraegy akes a hedging posiion as fully covered when he underlying price S is in he money; oherwise fully naked. I can perfecly replicae he opion payou bu may suffer a huge ransacion cos when S is wandering around he srike price. See for example in Hull (9) for a discussion.. Adjused Sop Loss: Based on one sylized fac of volailiy (Engle (9)) propery of mean reversion, we spli he ad hoc sop-loss hreshold K (he srike price) o an upper hreshold such as.k and a lower hreshold such as.99k. When he curren volailiy level is low enough, he index price is likely o be in he money for a call opion due o he leverage effec. Hence, i migh be favorable o lower he sop-loss hreshold K o, say.99k, for an early assess ino a hedging posiion. Analogously when he volailiy is high enough, he sop-loss hreshold migh be changed o.k for an early exi posiion. he volailiy used o measure he deph of moneyness is chosen as he hisorical volailiy, he insananeous volailiy or VIX. 8

9 We remark ha VIX may no exis in some opion markes or is leverage doesn appear srongly. For example AIFEX didn' announce aiwan VIX unil December 6. Even aiwan VIX has been calculaed in AIFEX for some ime since hen, he correlaion beween reurns of AIEX and reurns of aiwan VIX during December 6 and May 9 is only Compared wih he hisorical correlaion -.73 beween S&P 5 Index prices and he CBOE VIX during our sample period, aiwan VIX provides a relaively weak leverage for is index price. Hence we use he hisorical volailiy or he insananeous volailiy as oher volailiy measures.. Volailiy-Model-Free Caegory hree dynamic sraegies wihin his caegory are he dela hedging, an adjused dela hedging, and he dela-gamma hedging. Derivaions of hese hedging sraegies are all rooed from he Black-Scholes pricing model bu no specific volailiy model is acually posulaed. See Fouque e al. () for deailed discussions. As a resul, hese sraegies permi sraighforward implemenaions wihou a full esimaion of any volailiy models such as he coninuous-ime Heson model (Heson (993)) or discreeime ARCH/GARCH models (say (5)). Given he following noaions: he curren ime, he curren index price S, he volailiy!, he ime o mauriy!, he srike price K, he risk-free ineres rae r, and he opion price P(,S ), hree dynamic hedging sraegies are inroduced below.. Dela Hedging: his sraegy is effecive o reduce he risk of marke price. According o he Black-Scholes heory, an opion price can be approximaed by he dynamic porfolio! S " e r, where! = " is defined by ( )! P S " =! S, and! denoes he ne 9

10 posiion invesed in he money marke accoun afer ransacion cos and ax. In he case of call opions,! = ( ) N d e dx! d " x = $, where d "# (!,S ) = "! [log S K (r " )!].. Adjused Dela Hedging: heoreically, his sraegy is able o reduce no only he marke price risk, bu also parially he volailiy risk. Fouque e al. () applied a singular perurbaion echnique o derive an opion price approximaion such ha an adjused dela hedge sraegy! can be deduced as follows: ~ "P(, x)! = # V 3$ "x x (4 x " P(, x) " 3 P(, x) " 4 P(, x) 5x 3 x 4 ), "x "x 3 "x 4 where he addiional parameer V 3 can be esimaed from a linear regression of implied volailiies over he logarihm of marke o money raio (LMMR). his adjused dela is capable of aking he volailiy smile or smirk ino accoun. 3. Dela-Gamma Hedging: his sraegy can reduce boh he marke price risk and he volailiy risk, bu such rading porfolio coss more han he previous wo sraegies because of exra posiions in longer-daed opions. In order o furher reduce he volailiy risk of an opion price P () (,S ) wih a shorer mauriy, anoher opion P () (,S ) wih a longer mauriy, >, can be raded in he hedging porfolio. Assuming ha boh opions have he same srike prices, we can consruc a dynamic porfolio of! S " e r c P () (,S ), where c ( ) # P ( ) ( ) "! % = # = = & ( ) ( ) ( ) # P! % #" $ $ ( ) ( ) a = ' $ c & '

11 and ( )! P, S!" # = =! S! S. his sraegy can approximae he opion payou of P () by eliminaing he marke price risk and he volailiy risk simulaneously. In summary, he dela hedging, an adjused dela hedging, and he dela-gamma hedging correspond o rading porfolios in he dela neural posiion, he dela and parially gamma neural posiion, and he dela-gamma neural posiion, respecively. hese posiions are useful o disinguish he effeciveness of eliminaing he marke price risk wih or wihou he volailiy risk. 3. Volailiy Esimaion Almos all hedging sraegies menioned above, excep he sop loss, require an inpu of volailiy for implemenaion of hedging porfolios. he sudy of volailiy esimaion eiher from he hisorical daa and/or from he derivaives daa has drawn remendous aenions in pas decades. See say (5), Gaheral (6), Malliavin e al. (9) and references herein. Is high-dimensional exension, i.e., volailiy marix esimaion or correlaion esimaion, has been recenly challenged by rapid developmens in credi derivaives, credi porfolio risk managemen, ec. See Engle (9) for deails. In he spiri of reducing model dependence, volailiy esimaion mehods considered in his paper are mosly nonparameric. For example, quadraic variaion and Fourier ransform mehod have no dependence on volailiy specificaion, and hey are used for esimaion of he hisorical volailiy and he insananeous volailiy, respecively in his paper. hough he implied volailiy, defined as an inversion of he Black-Scholes formula, does depend on he Black-Scholes model, i is addiionally employed ino our empirical sudy due o is populariy in heory and pracice.

12 Nex we review he Fourier ransform mehod and is price correcion scheme. A new correcion scheme by a nonlinear regression mehod is proposed. We use a local volailiy esimaion problem for a simulaion sudy o examine effeciveness of correced Fourier ransform mehods. 3. Insananeous Volailiy Esimaion by Fourier ransform Mehod Malliavin and Mancino (, 9) proposed a nonparameric Fourier ransform mehod for esimaion of he volailiy process. he volailiy ime series can be reconsruced in erms of sine and cosine basis under he following coninuous semimaringale assumpion. Le u be he log-price of a one-dimensional risky asse S, i.e., which follows a diffusion process u = ln( ) S a ime where du = µ d! dw, () µ and W denoe he insananeous growh rae and a one-dimensional sandard Brownian moion, respecively. When he ime inerval [,] of he daa period is rescaled o [,! ], i is known ha he underlying u can be reconsruced as he Fourier series expansion ( " ( ) = a! b du k ( ) $ u ( ) cos( k) a du % k ) sin( k) ', k= # $ k k &' in which Fourier coefficiens of a s and b s are defined as follows:! a ( du )! du = ", ()

13 ! ak ( du) = cos( k) du! ", (3)! bk ( du) = sin( k) du! ", (4) for any k!. Mallianvin and Mancino derived he Fourier coefficiens of he variance by for N # k! * * * * ak (" ) = lim * & as ( du) as k ( du) bs ( du) bs k ( du) N $% N ( ' ), (5) s=# N! b * ( a du b du b du a du ' ), (6) N # k * * * * k (" ) = lim & s ( ) s k ( ) # s ( ) s k ( ) N $% N s=# N * * a s du b s du are defined as k, in which ( ) and ( ) " as ( du), if s > " bs ( du), if s > * # * # as ( du) = $, if s = and bs ( du) = $, if s = # a! s ( du), if s # % < %! b! s ( du), if s <. A smoohing echnique is convenionally applied so ha he ime series of variance is approximaed by where! x ( ( ) = sin x) N!! "!(! k)[a k (! )cos(k) b k (! )sin(k)], (7) k= x is a smooh funcion wih he iniial condiion! ( ) = and is a smooh parameer ypically specified as! = 5 (Reno 8). Maiussi e al. () furher invesigae sensiiviies of he number of Fourier series and he smoohing parameer by simulaion sudies. Several advanages of his Fourier ransform mehod can be readily observed from Equaions ()-(6). Firs, he inegraion error of Fourier coefficiens is adversely proporional o daa frequency so his Fourier ransform mehod is suiable for highfrequency daa. Second, his mehod is easy o implemen because, as shown in (5) and 3

14 (6), Fourier coefficiens of he variance ime series can be approximaed by a finie sum of muliplicaions of * a and * b. hird, his inegraion mehod avoids he insabiliy inheried from hose radiional mehods based on he differeniaion of quadraic variaion. See Zhang e al. (5) for deails. 3. Price Correcion Schemes One key drawback of his Fourier ransform mehod is documened by he boundary effec, i.e., a Gibbs phenomenon caused by he Fourier mehod. Reno (8) noed ha Fourier algorihm provides inaccurae esimae for volailiy ime series near he ime boundary of simulaed daa. o remedy his boundary defici, Han e al. () ook advanage of he relaionship beween asse reurns and volailiy, and proposed a price correcion scheme based on a linear regression. In his paper, we propose anoher correcion scheme based on a nonlinear regression. In our numerical simulaion for esimaing a local volailiy ime series, we find ha he nonlinear regression scheme performs beer. o fix noaions, recall ha u defined in () is he naural logarihm of asse price. Based on he Euler discreizaion, he incremen of log-price u can be approximaed by! " #, where! denoes a small discreized ime inerval and! denoes a sequence of i.i.d. sandard normal random variables. his approximaion is derived from neglecing he drif erm of small order! and using he incremen disribuion of Brownian moion!w =! ". Le ˆ! denoe he volailiy ime series esimaed from he original Fourier ransform mehod. We review he linear regression 4

15 correcion scheme proposed by Han e al. () for bias reducion of volailiy esimaion. A new nonlinear regression correcion scheme is also devised below. () Linear Regression Correcion Scheme (Han e al. ()): his scheme consiss of a log-linear ransformaion on he esimaed variance process ˆ! by Fourier ransform mehod in order o guaranee posiiveness of volailiy. ha is, we ransform ˆ Y = ln ˆ! o a by ˆ so ha he correced volailiy! = exp a b ˆ (( Y ) ) saisfies!u " exp a b ˆ (( Y ) / )! ", where!u = u " u, and a and b denoe he correcion coefficiens. Hence, one can use he maximum likelihood mehod o regress ou hese wo coefficiens via he relaionship beween he logarihm of he squared sandardized reurn!u! and he driving volailiy process a b ˆ Y :! ln!u $ " # % &! = a b ˆ Y ln!. (8) () Nonlinear Regression Correcion Scheme: By aking a direc linear ransformaion on esimaed volailiy ˆ! from he original Fourier ransform mehod, we end up solving a nonlinear regression equaion for esimaion of correcion coefficiens a and b. ha is, he rue volailiy! = a b!! saisfies!u " (a b!! )!" so ha a nonlinear regression equaion is obained: ln (!u! ) = ln(a b!! ) ln!. (9) Noe hese wo price correcion schemes (8) and (9) mus be solved numerically by he maximum likelihood mehod due o he complex disribuion of a log-chi square 5

16 ln!. heir compuaional coss are he same. hough here is no guaranee ha he correced volailiy esimaion! = a b!! based on a nonlinear regression scheme remains posiive, no negaive volailiy has been found in eiher our simulaion sudy or empirical sudy. In fac, his nonlinear correcion scheme ouperforms he linear correcion scheme according o he following simulaion sudy for a local volailiy model. 3.3 A Simulaion Sudy: Local Volailiy Esimaion Since he rue insananeous volailiy is no known, we es wo proposed correcion schemes by simulaion. A local volailiy model of he following form ( ) ds = " m $ S d # S dw! is considered. In Jiang (998), hose model parameers were esimaed as! =.93, m =.79,! =.794 and! =.474. We employ his se of parameers, hen simulae he price process S wih is volailiy process! = "S #. he simulaion is done by he Euler discreizaion wih ime sep size! = /5 and he oal sample number is 5. Based on he original Fourier ransform mehod and hose wo proposed price correcion schemes, hree volailiy ime series can be esimaed and used o compare wih he acual volailiy series. We use wo crierions for error measures including Mean squared errors (MSE) and Maximum absolue errors (MAE). Comparison resuls are lised below: 6

17 . MSE: 7.5E-4 (Fourier mehod),.9e-5 (Linear Regression Correcion Scheme), 7.6E-6 (Nonlinear Regression Correcion Scheme).. MAE:.4 (Fourier mehod),. (Linear Regression Correcion Scheme),. (Nonlinear Regression Correcion Scheme). Noiceably, he price correcion schemes (8) and (9) are able o reduce effecively boh error crierions a leas by half in his simulaed example. Our newly proposed nonlinear regression correcion scheme performs beer han he linear correcion scheme. We will use his nonlinear scheme for esimaion of he insananeous volailiy in our empirical sudy of hedging performance. 4. Empirical Sudy of Hedging Performance: SPX and XO We consider he hedging performance for call opions of S&P 5 Index and aiwan Index. Profi and loss (P/L) and Sharpe raio are used as wo measures for hedging performance. Various sraegies wihin he wo hedging caegories discussed in Secion are possibly combined wih hree volailiy esimaions, discussed in Secion 3, including he hisorical volailiy, he insananeous volailiy and he implied volailiy. he hisorical volailiy is esimaed from hiry-day hisorical reurns, he insananeous volailiy is esimaed by he proposed nonlinear regression correcion scheme of Fourier ransform mehod, and he implied volailiy is esimaed from an inversion of he Black- Scholes formula. 4. Daa Descripion 7

18 he neares conrac monhs of opion prices wih mauriy imes ha are greaer han one day bu less han or equal o hiry days are seleced in his empirical sudy. We avoid one-day opion prices because some implied volailiies on XO canno be solved from he Black-Scholes formula. Opion prices wih hiry-one days o mauriy and beyond are also excluded because of low rading volumes on XO. Such selecion crierions are applied o SPX for daa consisency. he sample period of S&P 5 Index prices and prices of SPX, raded in he Chicago Board Opions Exchange (CBOE), is from January o June 6. Daily daa were rerieved from he Ivy Daabase of OpionMerics. he oal number of call opions wihin ha sample period is 5,5. One conrac of SPX is on US dollars (USD) imes he opion price. he ransacion cos of rading opions is se as.5 USD per conrac. he risk-free ineres rae is chosen as he hree-monh U. S. reasury Bill. aiwan Index opion (XO) has been raded in aiwan Fuures Exchange (AIFEX) since. Is underlying is aiwan Sock Exchange Capializaion Weighed Sock Index (AIEX). Our sample period is from July 3 o March 9 including he recen financial crisis. Daily prices of AIEX and XO are downloaded from aiwan Sock Exchange (WSE) and AIFEX, respecively. he ime o mauriy of XO lass from wo rading days o hiry rading days. he oal number of call opions wihin ha daa sample period is 43,993. One index opion conrac in XO is on 5 New aiwan Dollars (ND) imes he opion price. he ransacion cos in AIFEX is 9 ND for buying and selling each opion conrac wih an addiional.% ax rae. he risk-free 8

19 ineres rae is chosen as he average of one-monh CD raes from five large domesic banks 4 in aiwan. Noaions of hedging sraegies are shown in he following:!-h,!-f and!-imp denoe he dela hedging sraegy combined wih he hisorical volailiy, he insananeous volailiy and he implied volailiy, respecively. We denoe by ad! and!-! he adjused dela hedging and he dela-gamma hedging, respecively. Boh use he hisorical volailiy. SL denoes he sop-loss sraegy, and adsl-h, adsl-f and adsl-v denoe he adjused sop-loss sraegy using he hisorical volailiy, he insananeous volailiy and VIX, respecively. 4. Hedging Performance of SPX Resuls of wo measures (P/L and Sharpe raio) for hedging call opions wih ime o mauriy = days are repored in able. On each row of ha able, he year period, he oal number N of hedged call opions and heir hedging performances are repored. he bes measure of hedging performances wihin each row is highlighed in bold face wih an underline. he sample mean of P/L, i.e. averaged P/L, wih a parenhesis means a loss; oherwise i means a profi. For example, Panel A-() illusraes ha here are 44 call opions hedged in. he bes hedging performance is obained by he sop-loss sraegy, which makes a profi of USD 96 on average per conrac. he las row of his subpanel records he hedging performance for he whole sample period from year o 7. 4 Bank of aiwan, aiwan Cooperaive Bank, Firsbank Commercial Bank, Hua Nan Bank, and Chang Hwa Bank 9

20 Given such fixed hedging period =, he P/L average of each hedging sraegy is posiive. In general, hedging sraegies wihin he model-free caegory generae larger profis han hose in he volailiy model-free caegory (see Panel A-()), so do heir sandard deviaions (see Panel A-()). his implies hedging performance of he modelfree caegory is less sable han he volailiy-model-free caegory. In erms of Sharpe raio as a measure of hedging performance, he volailiy model-free caegory ouperforms he model-free caegory in general. Wihin hese wo caegories, adjused sraegies including he adjused sop loss and he adjused dela hedging perform roughly he same as heir ordinary sraegies including he dela hedging and sop loss, respecively. Nex, we demonsrae he ime evoluion of hedging performance of Sharpe raios from = o 3 in Figure. his figure shows dynamic behaviors of Sharpe raios associaed wih all hedging sraegies wihin he volailiy-model-free caegory and he model-free caegory. I is observed ha firs, he dela hedging using he insananeous volailiy (Del-F) performs bes wihin he volailiy-model-free caegory. his resul is a new finding in hedging lieraure o our knowledge and consisen wih he use of he insananeous volailiy in Value-a-Risk esimaion (Han e al. ()). However, i is no so clear o deermine he bes sraegy wihin he model-free caegory. Second, all hese Sharpe raios increase wih mauriy ime. ha is, longer he ime period a hedging posiion is formed, higher Sharpe raio is obained. hird, he volailiy-model-free caegory ouperforms in general han he model-free caegory excep when he ime o mauriy is shor. he las observaion, i.e., a separaion of hedging performances beween he wo hedging caegories on SPX will have a conradic resul on XO.

21 4.3 Hedging Performance of XO able records resuls of wo measures including P/L and Sharpe raio for he performance of hedging call opions on XO. he ime o mauriy is chosen as rading days. Several observaions can be made below. Firs, aggregae hedging performances, shown in he las lines of each panel, of he volailiy-model-free caegory and he model-free caegory are roughly of he same numeric order. his means ha hedging performances of all sraegies are comparable in XO. his phenomenon is significanly conradicory o wha observed a well separaion of hedging performances on SPX. Second, in he volailiy-model-free caegory he dela hedging using he implied volailiy performs worse han he oher hree hedging sraegies using he hisorical volailiy. his is also differen from wha observed in able on SPX. Figure demonsraes dynamic behaviors of several hedging performances. We observe ha firs all hedging sraegies perform raher comparable excep he dela hedging using he implied volailiy. Second, Sharpe raios decrease wih mauriy ime. his means ha shorer he ime period of he hedging posiion is formed, higher he Sharpe raio is gained. I is worh noing ha hese wo phenomena are significanly differen from wha observed on SPX. 4.4 Comparisons of Hedging Performances for SPX and XO A summary of hedging performances on SPX and XO is lised below.. Dynamics of Sharpe raios for hedging call opions are differen. Sharpe raions of SPX end o increase wih mauriy ime while hey end o decrease on XO.

22 . In boh measures of P/L and Sharpe raios, he volailiy-model-free caegory dominaes he model-free caegory on SPX, while hese wo caegories perform comparably on XO. We furher invesigae he P/L difference beween he dela hedging sraegy and he sop-loss sraegy for a comparison. Each sraegy is considered as he delegae of he volailiy-model-free caegory and he model-free caegory. he empirical hedging differences of P/L are demonsraed in Figures 3 and 4 for heir means and sandard deviaions, respecively. Mauriy imes span from wo o hiry rading days. In each figure, HE(,3) represens he averaged hedging difference beween he sop loss and he dela hedging in use of he hisorical volailiy (he insananeous volailiy, he implied volailiy, respecively). he dollar uni is USD. Hedging differences on XO are all close o zero raher uniformly for boh mean and sandard deviaion. In conras, hese wo saisics are relaively large on SPX. he nex secion is managed o provide a heoreical jusificaion abou small hedging differences beween he dela and he sop loss on XO. 5. A Momen Analysis for Hedging Differences We have seen from able ha hedging performances beween model-free caegory and volailiy-model-free caegory are comparable on XO. More specifically Figures 3 demonsraes ha he hedging differences beween he dela hedging and he sop loss are relaively small compared wih hose differences induced from SPX. his secion is devoed o jusify hese small hedging differences on XO by a mahemaical momen analysis.

23 Noice ha here is a sric price limi consrain on AIEX while S&P 5 Index doesn'. Moivaed from he volailiy cooling-off effec of he price limi (Kim and Rhee (997)), we develop a ime-scale change mehod for he classical Black-Scholes model, and analyze he hedging difference beween he sop-loss sraegy and a rescaled dela hedge sraegy. he Black-Scholes-Meron's opion pricing heory assumes ha he dynamic of he underlying risky asse price S follow a geomeric Brownian moion. ha is, under he physical probabiliy measure, he asse price saisfies ds S = µd!dw, where µ is he reurn rae,! is he volailiy, and W is he Brownian moion. he ## soluion of his sochasic differenial equaion is S = S exp µ! " & %% (! $ $ ' ( ) "W! given ha! and S!. he ime-scale change mehod posulaes a variable change in ime as he follows: & ( ' ds S = µd! "dw! = µ!d "!dw () where! > is a small ime scale, which conrols he speed of he new ime variable!. he ime scale! can be eiher deerminisic or random, assuming independen of he Brownian moion ( B )!. In his sudy we assume a deerminisic! for ease of ## explanaion. hus, he soluion of Eq. () is S = S exp µ! " & %% ()! $ $ ' ( ) " )W! & (. ' 3

24 Under he risk-neural probabiliy measure, he marke price risk or risk premium is chosen as r! µ" # ". Le ( p!,s ) denoes he European opion price under he scaled dynamic () wih he payoff h( S ). By applicaions of Io's lemma used in he opion pricing heory, i is sraighforward o obain he following resuls. he scaled pricing parial differenial equaion is L! P! (, x) = wih he erminal condiion P! (, x) = h x L! = " " #!x ( ), where he parial differenial operaor " "x rx " $ r. Hence a he curren ime, he vanilla call opion price "x wih he srike price K and he mauriy is p! ( ( )) " e! (, x) = xn d,x ( )! KN ( d (, x) ), "r "! where d (, x) = ( ) # ln( z /K ) r "! "! # ( )! and d (, x) = d! "#! ", and is dela is!p " (, x)!x " = N ( d (, x) ). We remark ha he ime-scale change model proposed can be possibly exended o random ime change. For example in he case of he variance gamma model (see Geman (5)), he scaled ime! has a gamma disribuion, so ha he opion pricing formula can be carried ou by Fourier ransformaion. We leave his heoreical issue for a fuure research opic. 4

25 Suppose one hedges he opion using he rading sraegy (!," ), where his sraegy invess! unis of he index a ime and! = ( P " (,S ) #$ S ) e r. We define he insananeous hedging error by! ds " re r d # dp $ (,S ). he accumulaive hedging errors from ime o ime, denoed by HE, is defined by % % HE =! ds " re r d # P $,S = P!,S ( ) " ( ( ) # P $ (,S )) # ds # $ re r d % P!,S ( ) ( Assume wo differen hedging sraegies! ) ( and! ) ( ) are used for hedge. Le HE and ( ) HE denoes heir accumulaive hedging errors respecively. heir difference equals o ( HE ) (! HE ) ( = " ) ( ) ( (!" )ds! " ) ( ) # # (!" )rs d ( ) ( =! ) "! ( ) ( # ( µ$ " r)s d % $! ) # "! ( ) S dw. ( In cases of he sop-loss sraegy! ) = I S > e ( ) "r ( " ( ) K ) and he dela hedging ( sraegy! ) " = N ( d (, x) ), we will prove ha any momen of he accumulaive hedging ( difference HE ) ( )! HE converges o zero when he ime scale! approaches o zero. o obain his resul, we firs prove he following lemma. Is proof is showed in he Appendix. ( Lemma. E! ) "! {( ( ) ) } # C $ e" {( ( ) ) } ( scale!. I implies ha E! ) "! $ for some consan C independen of ime and he converges o zero as! goes o zero. By applicaions of he Cauchy-Schwarz inequaliy on he hedging error equaion, any momen of he accumulaive hedging errors is bounded by # {( ( ) ) } d ( C E! ) "! 5

26 for some consan C independen of!. By Lemma, i is easy o obain he following heorem. heorem (Momens Bound) For any posiive ineger n, {( ( ) ) } n " C # e! # for some consan C independen of!. ( E HE )! HE We obain an asympoic resul o show ha he difference beween wo hedging sraegies is small when he ime change variable is small. his heoreical resul is consisen o observed hedging performance in aiwan. 6. Conclusion his paper exends previous empirical sudies on opion hedging performance. Robus hedging sraegies and nonparameric volailiy esimaions are comprehensively sudied. I shows ha he insananeous volailiy esimaed from a correced Fourier ransform mehod may play an imporan role in hedging on SPX. An asymmeric phenomenon arising from our empirical sudy is also observed as follows. he volailiy-model-free hedging caegory generally ouperforms he model-free hedging caegory on SPX; while hese wo caegories perform roughly he same on XO. he second par of his paper aims o explain his documened phenomenon by a deailed comparison beween he dela hedging and he sop-loss sraegy as delegaes of wo hedging caegories. We propose a ime-scale change mehod o accoun for he price limi, which is ypically regulaed in emerging markes such as aiwan. he SPX marke serves as a conrol group of no price limi. An asympoic analysis confirms esimaed momens of hedging porfolio differences wih our empirical finding. 6

27 Appendix A: Proof of Lemma Recall ha he soluion of () is = S exp ( r ) S ( Z), where Z denoes he sandard normal random variable. Subsiuing his ino α and α, we deduce Le z * α α ( ) ( ) S = Ι ln K S ln K = Ν r S ln r = K, hen we deduce E = ( ) ( ) {( α α ) } * ( ) z z ( ) ( α ) e dz α z * π r ( ) Z > Z. z ( ) e dz () S We firs consider he convergence resul in he firs erm. Assuming ha ln r >, K * z as. o analyze he firs erm in (), we divide he inegraion domain * * ( z, ) ino hree regions (, ε ), ( ε, ε ), ( ε, ) π z for some ε >, hen sudy he convergen resul for each corresponding sub-inegral. Because α ( ) S ln K = Ν r z, he hird sub-inegral equals o ( ) 7

28 8 / ln exp ln ln ln Ν Ν > r K S r K S dz e r K S dz e z r K S z z ε ε π ε π ε ε If one choose ε = e, hen his erm is bounded above by e C for some consan C, independen of and. Nex we consider he second sub-inegral, ln ε π ε ε Ν dz e z r K S z where we use he bound of he normal inegral funcion. Nex we proceed o he firs subinegral by a furher division as z *,!! ( ) = z *, z * /! ( )" z * /!,!" ( ). / ln / * * * * * ln Ν r K S z z z z z z z e e dz e dz e z r K S π

29 9 Noe ha = r K S r K S z * ln ln when is small. Nex we consider he oher sub-inegral and obain an upper bound ( ) z z e dz e z r K S Ν ε π * ln. Noe ha we have used he following resul: when ε < < z z *, ( ) s z r K S z r K S a = > ln ln which is approximaely when is small enough, so ha ( ). ln e z r K S Ν Ν he procedure o prove he oher case ln < r K S, is similar, so we skip he proof here. We conclude his lemma by ( ) ( ) ( ) { } α α e C E for some consan C independen of ime and he scale.

30 able : P/L and Sharpe raio of hedging performance on SPX (ime o mauriy = rading days) Panel A : P/L of Hedging Sraegies () Mean (Average) US Dollars Mean (=) Volailiy-Model-Free Model-Free Year N -H -F -Imp ad - SL adsl-h adsl-f adsl-v ~ () Sandard Deviaion S.D. (=) Volailiy-Model-Free Model-Free Year N -H -F -Imp ad - SL adsl-h adsl-f adsl-v ~

31 Panel B : Sharpe Raio of Hedging Sraegies S.R. (=) Volailiy-Model-Free Model-Free Year N -H -F -Imp ad - SL adsl-h adsl-f adsl-v ~

32 able : P/L and Sharpe raio of hedging performance in XO (ime o mauriy = rading days) Panel A : P/L of Hedging Sraegies () Mean (Average) New aiwan Dollars Mean (=) Volailiy-Model-Free Model-Free Year N -H -F -Imp ad - SL adsl-h adsl-f adsl-v (9) (69) (63) (4) () (35) 5 9 (35) (4) (437) (35) (675) (7) (43) (99) (5) 6 3 (78) (34) (366) (78) (536) (98) (73) (4) (47) 7 36 (8) (4) (95) (8) 57 (3) (5) (66) (365) (36) (385) (77) (36) (944) () (434) (46) (434) 3 ~ () Sandard Deviaion S.D. (=) Volailiy-Model-Free Model-Free Year N -H -F -Imp ad - SL adsl-h adsl-f adsl-v ~ he calculaion of VIX before is announcemen from AIFEX in December 6 is based on a formula given by SinoPac Fuures. 3

33 Panel B: Sharpe Raio of Hedging Sraegies S.R. (=) Volailiy-Model-Free Model-Free Year N -H -F -Imp ad - SL adsl-h adsl-f adsl-v (.9). (.3496) (.43) (.84).9.3 (.45) (.7) 5 9 (.3556) (.3538) (.445) (.3556) (.6966) (.) (.64) (.68) (.94) 6 3 (.36) (.394) (.658) (.37) (.498) (.367) (.7) (.759) (.4) 7 36 (.33) (.437) (.96) (.33).366 (.56) (.98) (.454) (.637) (.9) (.7) (.6) (.9) (.94) (.4) (.7) (.78) (.7) 3 ~

34 Figure : Evoluion of Sharpe raios of hedging sraegies on SPX given ime o mauriy from o 3. S&P Index Opion 34

35 Figure : Evoluion of Sharpe raios of hedging sraegies on XO given ime o mauriy from o 3. aiwan Index Opion 35

36 Figure 3: Comparisons of differences of hedging performances beween Dela and Sop- Loss sraegies. (a) Mean of Hedging Differences (b) Sandard Deviaion of Hedging Differences 36

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