Pricing and hedging coningen claims using variance and higher-order momen fuures by Leonidas S. Rompolis * and Elias Tzavalis * Absrac This paper suggess perfec hedging sraegies of coningen claims under sochasic volailiy and/or random jumps of he underlying asse price. This is done by enlarging he marke wih appropriae fuures conracs whose payo s depend on higher-order sample momens of he underlying asse price process. I also derives a model-free relaion beween hese higher-order momen fuures conracs and he value of a composie porfolio of European opions, which can be employed o perfecly hedge variance fuures conracs. Based on he heoreical resuls of he paper and on opions and variance fuures conracs price daa wrien on he S&P 500 index, i is shown ha, rs, random jumps are priced in he marke and, second, hedging sraegies for European opions employing variance and higher-order momen fuures considerably improves upon he performance of randiional dela hedging sraegies. This happens because hese sraegies accoun for volailiy and jump risks. Keywords: Variance fuures conracs, higher-order momens, volailiy and jump risks, hedging sraegies. JEL: C4, G, G3 The auhors would like o hank Jens Carsen Jackwerh for useful commens on an earlier version of he paper. * Deparmen of Accouning and Finance, Ahens Universiy of Economics and Business. 76 Paission sree, 0434 Ahens, Greece. Email: rompolis@aueb.gr. * Deparmen of Economics, Ahens Universiy of Economics and Business. 76 Paission sree, 0434 Ahens, Greece. Email: e.zavalis@aueb.gr.
Inroducion Since he seminal papers of Black and Scholes (973), and Meron (973) on pricing and hedging coningen claims, here has been a vas amoun of sudies rying o develop parameric opion pricing models based on more general assumpions of he underlying asse price sochasic process. Prominen examples of hese sudies include he sochasic volailiy (SV) opion pricing model of Heson (993) and is exension allowing for random jumps in he underlying asse price (denoed as SVJ), suggesed by Baes (996), as well as is various exensions (see, e.g., Eraker, Johannes and Polson (2003)). Despie he plehora of sudies exending he Black-Scholes (BS) model for opion pricing, here are a few sudies focused on hedging derivaives under more general assumpions abou he sochasic process of he underlying asse price (see, e.g., Pham (2000) for a heoreical survey and Bakshi, Cao and Chen (997) for an empirical applicaion). This happens because hedging derivaives under more general model speci caion assumpions is no so sraighforward, compared o he BS framework. Indeed, as he marke is incomplee perfec hedging agains all sources of risks and, in paricular, volailiy and jump risks associaed wih a posiion in derivaives is no feasible. In his paper, we exend he BS framework o provide perfec hedging sraegies of coningen claims under more general assumpions of he underlying asse price allowing for sochasic volailiy and/or random jumps. This is done by enlarging he marke wih appropriae fuures conracs, whose payo s depend on higher-order sample momens of he asse price process. These conracs can e cienly hedge agains volailiy and jump risks. The paper conribues ino he lieraure on many frons. I derives new hedging raio formulas of he above fuures conracs, he underlying asse and he zero-coupon bond of he self- nancing hedging porfolio for boh SV and SVJ models. For he SV model, we demonsrae ha enlarging he marke wih a variance fuures (or swap) conrac, which is nowadays raded in he marke, can perfecly hedge posiions in coningen claims wrien on he underlying asse and/or is volailiy. This makes he marke complee. For his model, i is also shown ha he price of a variance fuures conrac can be perfecly hedged by he value of a composie porfolio of ou-of-he money (OTM) European call and pu These sudies develop he so-called quadraic hedging sraegies which are based on risk minimizaion and mean-variance crieria. 2
opions (see, e.g., Gaheral (2006a)). The paper shows how o calculae he value of his porfolio, numerically, based on a discree se of daa and i derives approximaion error bounds of is value. The laer can be proved very useful in pracice o assess he magniude of he numerical errors encounered when calculaing composie porfolios of OTM opions. For he SVJ model, he paper shows ha, in addiion o variance fuures, higher han second-order momen fuures conracs should be included in he self- nancing porfolio in order o hedge he exposure of a coningen claim price agains random jumps. If his claim is a variance fuures conrac, his can be done ogeher wih he composie porfolio of OTM opions. Since he size of jumps is random, a su cienly large number of higherorder momen fuures conracs is required for perfec hedging. As his number goes o in niy, he paper shows ha he value of he self- nancing hedging porfolio converges o he price of he coningen claim, hus making he marke approximaely (or quasi) complee in he sense of Björk, Kabanov and Runggaldier (997) and Jarrow and Madan (999). In pracice, someone can approximae he value of his hedging porfolio wih a nie number of higher-order momen fuures conracs. The above resuls of he paper can nd applicaions o di eren markes. These include he sock marke (see Bakshi and Kapadia (2003), and Bollen and Whaley (2004)), he xed-income securiy marke (see Li and Zhao (2006) and Jarrow, Li and Zhao (2007)), he morgage-backed securiy marke (see Boudoukh, Whielaw, Richardson and Sanon (997)) and he credi defaul swap marke (see Brigo and El-Bachir (200)). The resuls of he paper are used o empirically address wo quesions, which have imporan porfolio managemen implicaions. The rs is if random jumps are priced in he marke. According o heory, he price of jump risk mus be re eced in he di erence beween he price of a variance fuures conrac and he value of he composie opions porfolio. This will be examined wihou relying on any parameric model of he marke. The second quesion is if a posiion in a European call, or pu, can be e cienly hedged based on a wo-insrumens hedging sraegy, which considers a variance fuures conrac as a hedging vehicle, or i requires hedging sraegies which rely on higher-order momen fuures. This can shed ligh on he abiliy of variance fuures conracs o hedge agains volailiy risk and he number of higher-order momen fuures conracs needed for approximaely compleing he marke in he presence of random jumps. To answer he above quesions, we rely on opions and variance fuures price daa wrien on he S&P 500 index. 3
The empirical resuls of he paper lead o he following main conclusions. Firs, random jumps are indeed priced in he variance fuures marke. The prices of hese conracs are found o be signi canly bigger in magniude han hose of he composie opions porfolio, which spans variance fuures conracs under he assumpion of no jumps. Second, a woinsumens hedging sraegy, which also includes a variance fuures conracs in he self- nancing porfolio, is found o considerably improve upon he performance of he radiional dela hedging sraegy ofen used in pracice, which only includes he underlying asse and he zero-coupon bond in his porfolio. The improving performance of his sraegy comes from he fac ha variance fuures conracs can e cienly hedge he exposure of a call, or a pu, opion o volailiy risk. The inclusion of a nie (up o he 4h-order) number of higherorder momen fuures conracs ino he self- nancing porfolio is found o furher improve he performance of he above wo-insrumens hedging sraegy, especially for shor-erm a-he-money (ATM) calls and OTM pus which are more sensiive o jump risk. The paper is organized as follows. Secion 2 inroduces he variance and higher-order momen fuures conracs and gives he relaion beween hem, and he value of he composie porfolio of OTM opions. Secion 3 shows how hese derivaives can be used o perfecly replicae he price of a coningen claim under he SV and SVJ models. Secion 4 addresses he empirical quesions of he paper. Secion 5 concludes he paper. All derivaions are given in a echnical Appendix. 2 Variance and higher-order momen fuures conracs 2. Variance fuures conracs Variance fuures (or swaps) are derivaive conracs in which one counerpary agrees o pay he oher a noional amoun imes he di erence beween a xed level and a realized level of variance. The xed level is he variance fuures price. Realized variance is deermined by non-cenral second-order sample momen of he underlying asse over he life of he conrac. More precisely, le T e = 0 < < ::: < n = T be a pariion of ime inerval [T e; T ] ino n equal segmens of lengh, where e is accrual period of he conrac, given as n=252. Then, he payo of his variance fuures conrac is given as V (n) T e;t = 252 n nx i= 2 Si ln, S i 4
where S i is he price of he underlying asse a ime i. Since his conrac worhs zero on he incepion dae, no arbirage dicaes ha he ime- price of he above variance fuures conrac is given by he risk-neural expeced value of he conrac payo, i.e., F V (n) = E Q h V (n) T e;t i, where Q denoes he risk-neural measure. If ime is coninuous, i.e. n!, payo V (n) T e;t converges in probabiliy o he annualized quadraic variaion of he log-price process of he underlying asse, denoed as e hx; Xi T e;t (see Proer (990)), where X = ln S. Then, he variance fuures price F V (n) converges o is coninuous-ime limi, de ned as F V, given by he risk-neural expeced value of e hx; Xi T see Jarrow e al (20). 2 lim F V (n) n! = E Q e hx; Xi T e;t, i.e., e;t F V, () Under he assumpion ha log-price X has no disconinuous componen, variance fuures conracs can be priced and perfecly hedged based on a porfolio of European call and pu conracs (see, e.g. Carr and Lee (2009), Friz and Gaheral (2005) and Gaheral (2006)). However, his is no rue if X conains a disconinuous componen, which is found o characerize he price dynamics of many nancial asses (see, e.g., Barndor -Nielsen and Shepard (2006b)). To consruc such a porfolio, nex proposiion gives he price of variance fuures conracs F V under more general assumpions of he underlying asse price allowing for a disconinuous componen in log-price X. The proposiion gives exac formulas of price F V for he following wo cases: (i) T e 6 6 T, and (ii) < T e. The second case is no considered in he lieraure. I assumes ha ime- is prior o he saring dae of he accrual period. Proposiion Le he sochasic process of he underlying asse log-price (X u ) u2[0;t ] be a 2 As aply shown by Broadie and Jain (2008), he e ecs of assuming coninuous ime on pricing variance fuures conracs is negligible. This allows us o rely on coninuous-ime models in pricing and hedging coningen claims. 5
semimaringale. Then, he price of variance fuures conrac F V is given as (i) F V = e hx; Xi T e; + 2er e + 2 Z T ds u S T S e EQ S u S Z + S 2 e Z S K 2 C (; K)dK + 0 2 X j=3 j! EQ 4 X <u6t K 2 P (; K)dK 3 + (X u ) j 5 (2) for T (ii) e 6 6 T and Z + Z F V = 2er S e S K 2 C (; K)dK + 0 K 2 P (; K)dK 2e r Z + Z S e S K 2 C ( ; K)dK + 0 K 2 P ( ; K)dK + 2 Z T ds u S T S T e e EQ T e S u S 2 3 X 4 X (X u ) j 5 (3) 2 e j=3 j! EQ T e<u6t for < T e, where = T and = T e, and C (; K) and P (; K) respecively denoe he prices of European call and pu prices a ime, wih srike price K and mauriy inerval. Boh formulas (2) and (3), given by Proposiion, indicae ha variance fuures price F V is deermined by similar erms. Their inerpreaion is as follows. Consider, for insance, formula (2). The inegral erms, de ned as V o; = 2er e Z + Z S S K 2 C (; K)dK + 0 K 2 P (; K)dK ; (4) give he value of a composie porfolio of opions, V o;, which depends on he values of wo subporfolios of OTM European call and pu opions wih expiraion dae T. Term h R i E Q T ds u S T S + S u S of formula (2) depends on he underlying asse of he variance fuures conrac. If his asse is a sock (or a sock marke index) ha pays a consan dividend yield, hen his erm becomes E Q Z T + ds u S T S = + (r ) e (r ). S u S If he underlying asse is a fuures or swap conrac, his erm becomes zero. Finally, erm P h P j=3 j! EQ <u6t (X u) ji of formula (2) is due o he disconinuous componen of process X. 6
The decomposiion of price F V ino he hree erms discussed above provides he correc framework o replicae and, hus, hedge variance fuures conracs. For insance, under he assumpion ha he log-price process X does no conain a disconinuous componen, formula (2) implies ha a long posiion in a variance fuures conrac aken a ime, where T e 6 6 T, can be perfecly replicaed by a porfolio consruced by holding: (a) a saic long posiion in 2 calls a srikes K > S ek 2 and 2 pus a srikes K < S ek 2 wih ime o mauriy, (b) a dynamic posiion in 2 e S u S shares a any ime u 2 [; T ], and (c) e r e hx; Xi T e; in cash (see, e.g., Carr and Lee (2009)). In case (ii) of Proposiion, where < T e, he replicaing porfolio should include: (a) a saic long posiion in 2 ek 2 calls a srikes K > S and 2 ek 2 pus a srikes K < S wih ime o mauriy, (b) a saic shor posiion in 2 calls a srikes K > S ek 2 and 2 pus a srikes K < S ek 2 wih ime o mauriy, and (c) a dynamic posiion in 2 e S u S shares a any ime u 2 [T e; T ]. Proposiion clearly indicaes ha he above porfolios can no perfecly replicae a long posiion in variance fuures if process X conains a disconinuous componen, due o he sum of higher-order momen erms P h P j=3 j! EQ <u6t (X u) ji for case (i), or P h P j=3 j! EQ T e<u6t (X u) ji for case (ii). Ignoring hese erms will lead o an imperfec hedge of variance fuures conrac by means of raded opion conracs. The magniude of he hedging errors encounered will be invesigaed in Secion 4, based on marke daa. To replicae hese erms, in he nex secion we will inroduce fuures conracs whose payo s are de ned by higher han second-order sample momens of X. 2.2 Higher-order momen fuures conracs Le de ne he jh-order momen fuures conrac, for j 3, as a conrac in which one counerpary agrees o pay he oher a noional amoun imes he di erence beween a xed level (conrac s price) and a realized level of he jh-order non-cenral sample momen of process X. The payo of his conrac is given as V (n) (j);(t e;t ) = 252 n where n are segmens of ime inerval [T nx i= j Si ln, for j 3, S i e; T ] of lengh. The price of his conrac under he no arbirage principle is given by he risk-neural expeced value of V (n) (j);(t e;t ), 7
i.e. F (n) (j); = EQ h i V (n) (j);(t e;t ). In coninuous ime (i.e. probabiliy o e P T n! ), i can be shown ha payo V (n) (j);(t e;t ) converges in e<u6t (X u) j (see Lepingle (976), for a proof). Thus, he coninuousime value of he above conrac, denoed as F (j);, can be de ned as lim F (n) n! (j); = EQ 2 4 e T 3 X (X u ) j 5 F (j); : (5) e<u6t This de niion of F (j); implies ha he prices of odd-order momen fuures will be negaive, if large drops in X occurring beween T e and T dominae is movemens. In his case, posiions in hese fuures conracs should be reversed. Tha is, a long posiion in a higherorder momen fuures will be aken as a shor posiion, and vice versa. If we assume ha a nie number of j = 3; 4; :::; N higher-order momen fuures are raded in he marke, hen formulas (2) and (3) of Proposiion imply ha long posiions in variance fuures conracs can be hedged agains disconinuous movemens in X, capured by erm P h P j=3 j! EQ T e<u6t (X u) ji. This will happen, if he replicaing porfolios, de ned in he previous secion for cases (i) and (ii) of Proposiion, include also a shor posiion in 2 e j! jh-order momen fuures conracs, for j = 3; 4; :::; N. For su cienly large N, i can be easily proved ha hese replicaing porfolios can approximaely hedge long posiions in variance fuures conracs, even when he underlying asse price conains disconinuous componens. These resuls mean ha exisence of higher-order momen fuures can make he marke approximaely (or quasi) complee (see Björk, Kabanov and Runggaldier (997), or Jarrow and Madan (999)). 3 Noe, a his poin, ha he above higher-order momen fuures conracs can no capure he variaion of he disconinuous componen of X, de ned as P e T e<u6t (X u) 2. This variaion is par of he annualized quadraic variaion of X, e hx; Xi T have e hx; Xi T e;t = e hx; Xic T e;t + e T X e<u6t e;t, since we (X u ) 2 ; (6) 3 Noe also ha Proposiion implies ha he jh-order momen fuures conrac can be hedged by means of European calls and pus, he variance fuures conrac and a nie number of sh-order momen fuures conracs, for s 6= j. This can be done by solving (2) or (3) for F (j);, using he de niion of F (j); given by (5). 8
where hx; Xi c denoes he coninuous par of hx; Xi. Since n e p X 2= ln Si i= S i ln Si+ S i p! e hx; Xic T e;t, where " p! " signi es convergence in probabiliy (see Barndor -Nielsen and Shephard (2003, 2004)), and, hence, 252 n nx 2 n Si X ln p S i 2= ln Si i= i= S i ln Si+ S i! p! e T X e<u6t (X u ) 2, one can inroduce a new fuures conrac o price he disconinuous componen of hx; Xi. This conrac will be henceforh referred o as bipower variaion fuures conrac and is discree-ime payo a ime T is given as 252 n nx 2 n Si X ln p S i 2= ln Si i= i= S i ln Si+ S i! : As can be seen in Secion 3, his conrac can help us o separae he volailiy from he jump risk boh of hem encounered when invesing in variance fuures conracs. Furhermore, i is necessary o perfecly hedge coningen claims, like European opions and opions wrien on volailiy swaps, agains he above wo sources of risk. Under he risk-neural measure Q, he ime- price of a bipower variaions fuures conrac is given as 4 F (2); = E Q 2 4 e T 3 X (X u ) 2 5 : (7) e<u6t 2.3 Numerical errors in calculaing he value of he composie opions porfolio Proposiion indicaes ha, in pricing and hedging variance fuures conracs, we need o calculae value V o; of he composie porfolio of OTM European calls and pus. This is 4 Noe ha subracing F (2); from F V yields F V F (2); = E Q e hx; Xic T e;t, which shows ha a long posiion in a variance fuures conrac and a shor posiion in a bipower variaion fuures conrac forms a porfolio whose payo depends only on he coninuous par of quadraic variaion hx; Xi c. The price of his porfolio can be hus used o deermine he marke price of volailiy risk. 9
given by formula (4), for he case (i) of Proposiion. 5 To his end, we need o rely on an e cien numerical mehod which s he inegral funcions of (4) ino nie, discree crosssecional ses of OTM European call and pu opion prices. Following recen lieraure, a mehod which can be used o his end employs an inerpolaion-exrapolaion scheme of he implied by he BS volailiies of he above opion prices. 6 This scheme inerpolaes hese volailiies over he observed closed inerval of srike prices [K min ; K max ], where K min and K max denoe he minimum and maximum srike prices available in he opions marke, respecively, and i exrapolaes hem over inervals (0; K min ] and [K max ; +), where opion prices are no available. The inerpolaion of he above implied volailiies can be based on cubic splines. These consiue smooh funcions which can provide accurae esimaes of implied volailiies over inerval [K min ; K max ]. On he oher hand, he exrapolaion of implied volailiies over inervals (0; K min ] and [K max ; +) can be done eiher based on a linear, or consan, funcion. Boh of hese funcions are runcaed a srike prices K 0, which is considered as an approximaion of a zero srike price, and K, which is considered as an approximaion of a srike price which ends o in niy. The srike prices K 0 and K are calculaed o correspond o OTM pu and call opion prices, respecively, which are very close o zero, e.g., smaller han 0 3. An alernaive o he above exrapolaion schemes is o se K 0 =K min and K =K max, which means o rely only on available opion daa. Tha is, o choose no o exrapolae. This can be jusi ed economically by he fac ha invesing in he composie porfolio of OTM opions in order o hedge variance fuures does no imply rading on ari cial opions, bu only marke available opions de ned over srike price inerval [K min ; K max ]. According o Proposiion, he value V o; of such a porfolio should be also re eced in variance fuures prices. The numerical inerpolaion-exrapolaion procedures suggesed above can lead o an approximaion error relaed o he curve- ing scheme of implied volailiies over inerval [K 0 ; K ]. This error depends on he number of opion prices available, he lengh of inerval [K min ; K max ] and he exrapolaion scheme chosen. Given ha his error can no 5 Noe ha he analysis of his secion applies also o case (ii) of Proposiion. 6 See Dennis and Mayhew (2002) and Jiang and Tian (2005), iner alia. Noe ha inerpolaion and exrapolaion of implied volailiies, insead of opion prices, has been suggesed in he lieraure in order o avoid numerical di culies in ing smooh funcions ino opion prices. To convey he observed opion prices ino implied volailiies and vise versa, we use he BS formula. This mehodology does no require he BS model o be he rue opion pricing model. 0
be measured because he rue heoreical implied volailiy funcion is no known, he nex proposiion enables us o appraise is size e ecs on V o;, by deriving upper and lower bounds for i. This can shed some ligh on he imporance of his numerical error on possible biases of V o; in replicaing marke prices of variance fuures conracs, especially relaive o ha due o he disconinuous componen of he log-price process X. Proposiion 2 Le b C (; y) and b P (; y) denoe he approximaed values of he rue call and pu prices C (; y) and P (; y), wih respec o variable y = ln(k=s ), respecively, based on an inerpolaion-exrapolaion scheme. If d Vo; denoes he approximaed value of he composie porfolio of opions V o;, given by formula (4), based on b C (; y) and b P (; y), hen we can derive he following approximaion error bound of d V o; : where C error = d V o; max y2(0;y ) V o; 6 2e r es Cerror e y + P error e y 0 + ", (8) C (; y) C b (; y) and P error = max y2(y 0 ;0) P (; y) P b (; y) consiue upper bounds of he approximaion errors of he European call and pu opion pricing funcion, respecively, y 0 = ln(k 0 =S ), srike prices K a endpoins K and K 0, respecively, de ned as y = ln(k =S ), and " is a runcaion error of " = Z + K K 2 C (; K)dK + Z K0 0 K 2 P (; K)dK. The approximaion error bound formula (8), given by Proposiion 2, can be used as a meric of choosing beween alernaive inerpolaion-exrapolaion schemes o esimae V o; (see our empirical Secion 4). As shown by (8), he upper and lower bounds of approximaion error d V o; V o; depends criically on he call and pu opion pricing funcion approximaion errors C error and P error. These are muliplied by funcions ( e y ) and (e y ), respecively. To invesigae more rigorously he e ecs of hese wo funcions on error bound V d, o; V o; in he nex corollary we analyze some of heir properies, while in Figure we plo heir values over y. Noe ha his gure gives he values of funcion 8 < e y if y < 0 f(y) = : e y if y > 0; (9) which ness he wo funcions ( e y ) and (e y ).
Corollary Taking he limi y! + or y!, we have lim e y = and lim e y = +, (0) y!+ y! respecively. We can also prove ha e y 6 8 y 2 R + : () The resuls of he above corollary indicae ha funcion e y, which muliplies C error, is bounded by. This means he size e ecs of his erm on aproximaion error V d o; V o; is bounded. This can be con rmed by he graph of Figure. On he oher hand, Corollary indicaes ha funcion e y, which muliplies P error, is no bounded. This funcion ends o in niy as y!, which means ha he size e ecs of approximaion error P error on V d o; V o; can be unbounded. This resul implies ha, in implemening formula (4) o a discree se of opion prices, we should avoid exrapolaing implied volailiies up o exreme values of K 0, or we should no exrapolae implied volailiy funcions a all. Oherwise, approximaion error P error may be considerably magni ed. 3 Pricing and hedging coningen claims using variance and higher-order momen fuures conracs In he previous secion, we show how o price and perfecly hedge a variance fuure conracs in he presence of a disconinuous componen in he underlying log-price process X. In his secion, we generalize hese resuls o he case of any coningen claim wrien on he price of a sock and/or is volailiy. These coningen claims include plain vanilla European opions, volailiy swaps or opions wrien on hese swaps. The laer consiue more complex derivaives, which have non-linear payo s wih respec o volailiy (see Carr and Lee (2009) and Friz and Gaheral (2005)). Mos of hese coningen claims are raded in he marke. To price and hedge hem, we will iniially assume ha sock prices follow he sochasic volailiy (SV) model (see, e.g., Heson (993)). Then, our analysis will be exended o he SV model allowing for jumps (SVJ) (see, e.g., Baes (996)). This model has been found o beer describe he dynamics of sock prices or heir volailiy and, hus, is considered as a common speci caion in he lieraure. Our analysis sars wih he SV 2
model, as his model enables us o more clearly see he role of variance fuure conracs o perfecly hedge posiions in coningen claims. Nex, we demonsrae ha in order o complee he marke for he SVJ model we need o enlarge i wih he bipower variaion and higher-order momen fuures conracs. 3. The SV model Assume ha he underlying asse is a sock which pays dividends a rae. Then, he SV model assumes ha is price S and volailiy V obey he following processes: ds S = S d + p V dw () (2) dv = ( V ) d + p V dw (2), (3) where S = (r + S V ) is he expeced reurn of he underlying asse, where r is he rae of reurn of a zero-coupon bond and S V is he marke price of risk, and processes W () and W (2) are wo correlaed Brownian moions, wih correlaion coe cien. 7 The las assumpion implies ha here exiss a Browian moion process W (3) which is independen of W () and sais es he following relaionship: dw (2) = dw () + p 2 dw (3). Under he SV model, he sochasic discoun facor (SDF) process, used for asse pricing, is given as d p = r d S V dw () p V V p (3) dw 2 ; (4) where parameer V accouns for he marke price of volailiy risk (see Chernov and Ghysels (2000)). This marke is incomplee in he sense ha a posiion in a coningen claim can no be perfecly replicaed by a self- nancing porfolio, referred o as hedging porfolio, consising of a posiion in he underlying sock and he zero-coupon bond. In his secion, we will show ha his marke can be compleed if he hedging porfolio conains, in addiion o he above wo asses, a posiion in a variance fuures conrac. To his end, our analysis sars wih pricing variance fuures conracs. 7 Noe ha our resuls do no depend on he speci caion of he volailiy process V. They hold under he following more general speci caion of V : where and are appropriaely de ned funcions. dv = (; V )d + (; V )dw (2), 3
3.. Pricing variance fuures conracs Consider ha, apar from he sock and zero-coupon bond, a variance fuures conrac is raded in he marke. This is wrien on sock price S and is de ned over ime inerval [0; T ]. The payo of his conrac a ime T is given as R T T 0 V udu. The following proposiion gives analyic formulas of he price of his conrac and is expeced reurn, E h df V F V i, under he SV model. Proposiion 3 Assume ha no-arbirage opporuniies exis in he marke. Then, under he SV model, he price of a variance fuures conrac a ime 2 [0; T ] is given as F V = T Z 0 V u du + F (; V ), (5) where F (; V ) = V + ( ) Q wih = ( e Q )= Q and = T, while is expeced reurn, denoed F V, is given as df F V V d E = @ ln F V F V @ ln V e V d, (6) where Q and Q consiue risk-neural counerpars of and, respecively, and e V = S + V. Proposiion 3 shows ha he expeced reurn of a variance fuures conrac F V depends on he marke price of risk S, due o he correlaion beween S and V, and he price of volailiy risk V. If V (200)), 8 hen F V and < 0, as shown in many empirical sudies (see, e.g., Carr and Wu will be negaive, given ha S > 0, < 0 (due o he leverage e ec) @ ln F V=@ ln V > 0. The negaive expeced reurn of his conrac is somehing o expec, since i pays when volailiy unexpecedly increases. Due o aversion o his even, invesors wish o pay more o acquire variance fuures conracs. Anoher ineresing resul of Proposiion 3 is ha, under he SV model, he price of volailiy risk V can be uniquely deermined by expeced reurns F V and S, since V = e V S. This resul implies 8 The negaive sign of V can be also explained heoreically based on Lucas consumpion asse pricing model in which he represenaive invesor has a coe cien of risk aversion. Under hese assumpions, discoun facor becomes = C, where C denoes consumpion a ime. Based on his model, i can be shown ha E h dc C dv i = E h d dv i = V V d. The las relaionship implies ha, if E h d dv i > 0, meaning ha he represenaive invesror subsiues fuure wih curren consumpion a a greaer rae when volailiy end o increase, hen V < 0. 4
ha a posiion in any coningen claim can be perfecly replicaed by a porfolio consising of a number of socks, bonds and a posiion in variance fuures conrac. This will be shown more rigorously in he nex secion. 3..2 Pricing and hedging coningen claims Consider now a coningen claim C. Is price a ime, denoed C, depends on boh he sock price and he spo variance, i.e. C = C (; S ; V ), where funcion C(:) has coninuous second order parial derivaives. As noed before, his claim can be a plain vanilla European opion, a volailiy swap or an opion wrien on his swap. The self- nancing porfolio needed o hedge (replicae) a posiion in his claim consiss of a posiion in he underlying sock, he zero-coupon bond and a variance fuures conrac. The numbers of hese hedging insrumens a ime 2 [0; T ], known as "delas", will be de ned by he following vecor of real-value adaped processes = ( S ; B ; F V ) 0. A ime, he value of his porfolio will be equal o he price of he coningen claim C, i.e. C = S S + B B, (7) where B denoes he price of zero-coupon bond. Noe ha he number of variance fuures conacs F V hem. is omied from equaion (7), since i coss nohing o ake a posiion in Nex proposiion derives he values of he vecor of delas. These can be employed o perfecly hedge changes in coningen claim price C driven by sochasic movemens in sock price S and volailiy V. The proposiion also derives he insananeous expeced reurn of coningen claim C, de ned as E h dc C i. Proposiion 4 Consider a coningen claim C wrien on he underlying sock and/or is volailiy, wih price funcion C = C (; S ; V ), where C(:) is a coninuous funcion which has second order parial derivaives. Then, under he SV model and he no-arbirage principle, he delas of he self- nancing porfolio replicaing coningen claim price C, a 2 [0; T ], are given as follows: S = @C=@S and B = B C S S 5
for he underlying sock and zero-coupon bond, respecively, and F V = @C=@F V for he variance fuures conrac. The insananeous expeced reurn of C is given as where S and F V dc E = r d + @ ln C @ ln C C @ ln S S + r d + @ ln F V F V d; (8) are he expeced reurns of he underlying sock and variance fuures conrac, respecively, de ned in Proposiion 3. Proposiion 4 proves ha, under he SV model, a coningen claim C wrien on a sock and/or is volailiy can be perfecly hedged by a self- nancing porfolio consising of a zero-coupon bond, he underlying sock and a variance fuures conrac. The price of his coningen claim, C, is uniquely deermined by he prices (or expeced reurns) of hese hree nancial insrumens. As shown in he appendix, his price can be obained by solving Heson s (993) parial di erenial equaion (PDE). In conras o Heson s approach, which obains his PDE relying on equilibrium approach, we have derived i explicily by eliminaing all sochasic erms deermining he coningen claim price. These resuls mean ha, under he SV model, he marke enlarged wih a variance fuures conrac becomes complee. As menioned before, coningen claim C can also include volailiy swaps and opions wrien on hem. The resuls of Proposiion 4 imply ha hese asses can be perfecly hedged by variance fuures conracs, whose price equals he value of he composie porfolio of OTM opions V o;, under he SV model (see Proposiion ). Thus, hese more complex derivaives, wih non-linear payo s, can be perfecly hedged wih respec o volailiy by a porfolio of OTM opions. Formula (8) of Proposiion 4 indicaes ha he expeced reurn of coningen claim C, E h dc C i, in addiion o he risk-free rae r and expeced reurn of he sock (implied by he BS model), also depends on he volailiy risk premium. The laer is re eced in he expeced reurn of he variance fuures conrac F V (see equaion (6)). This resul means ha, if he hedging porfolio of coningen claim C does no include he variance fuures conrac, hen his will lead o non-zero dela-hedged gains, as i will no perfecly replicae claim C. As shown by Bakshi and Kapadia (2003), hese dela-hedged gains are 6
expeced o be negaive under he physical measure, when F V < 0 and @ ln C @ ln F V > 0. These resuls are consisen wih he empirical ndings of Coval and Shumway (200), which show ha BS model overesimae a-he-money (ATM) and long-erm European opions reurns. This happens because @ ln C @ ln F V opions. is posiive and larger in magniude for hese wo caegories of The resuls of Proposiion 4 have also an ineresing empirical implicaion. They can faciliae esimaion of he parameers of he SV model under risk-neural measure Q. This can be done by exploiing informaion from opions, variance fuures and sock price daa, joinly. In paricular, he values of laen variable V, for all, can be subsiued by a linear funcion of variance fuures prices F V, which are observable in he marke, using formula (5). This will reduce he number of parameers required in he esimaion of he model and will increase he e ciency of heir esimaes, given ha i explois addiional marke informaion. 3.2 The SVJ model Under he SVJ model, he processes driving sock price and volailiy changes are given as follows: ds S = S d + p V dw () + J dn d (9) dv = ( V ) d + p V dw (2), (20) where S now is de ned as S = r + S V + Q, J is he random percenage jump condiional on a jump occurring disribued as ln(+j ) N J ; 2 J, N is a Poisson process wih inensiy parameer, i.e., prob(dn = ) = d, = exp J + 2 2 J and Q is he risk neural counerpar of (see, e.g., Baes (996)). As wih he SV model, W () and W (2) are wo Brownian moion processes which are correlaed wih each oher. For he SVJ model, he SDF process is given as where ln(+j ; ) N d p = (r + ) d S V dw () p V V p (3) dw 2 + J ; dn ; J ; 2 and J denoes he mean jump size = exp J + 2 2 J. Following Pan (2002) and Broadie, Chernov and Johannes (2007), we se = 0. 7
Under he SVJ model, a coningen claim C wrien on a sock and/or he volailiy can no be perfecly replicaed by a porfolio consising of he underlying sock and he zerocoupon bond. To perfecly hedge C, in he nex subsecions we show ha he self- nancing porfolio mus conain, in addiion o he above wo insrumens, a variance fuures conrac, as happens wih he SV model, and he higher-order momen fuures conracs de ned in Secion 2. Before proving his resul, we need o derive he price and expeced reurn of hese fuures conracs, under he SVJ model. 3.2. Pricing variance and higher-order momen fuures conracs Under he SVJ model, he annualized quadraic variaion of log-price process X as T hx; Xi 0;T = T Z T 0 Z T V u du + 0 ej 2 udn u is given where e J = ln ( + J). The following proposiion gives he price and insananeous expeced reurn of a variance fuures conrac. Proposiion 5 Under he SVJ model and he no-arbirage principle, he price of a variance fuures conrac a ime 2 [0; T ] is given as F V = T Z 0 Z V u du + 0 ej udn 2 u + F (; V ) + G (), (2) where F (; V ) is de ned in Proposiion 3, G() = Q (2), where Q (2) is he second-order non-cenral momen of e J under he risk-neural measure Q. Is expeced reurn is given as df F V V d E = @ ln F V F V @ ln V e V d + F V (2) Q (2) d, (22) where e V = S + V and (2) is he second-order non-cenral momen of e J under he physical probabiliy measure P. Proposiion 5 indicaes ha he expeced reurn of a variance fuures conrac F V depends, in addiion o he marke and volailiy risk premium, on a risk premium relaed o he jump componen of he underlying sock price S. Given recen evidence ha Q J < 2 J < 0 and Q J > 2 J, where Q J and Q J are respecively he risk-neural counerpars of J and J (see Broadie, Chernov and Johannes (2007)), his premium mus be negaive, 8
implying (2) Q (2) < 0. This means ha is presence will reduce he expeced reurn of a variance fuures furher han he volailiy premium due o invesors aversion owards jumps. The prices and heir expeced changes of he bipower variaion and higher-order momen fuures conracs are given in he nex proposiion. Insead of he expeced reurns, he proposiion gives he expeced price changes of he above conracs, since heir expeced reurns depend on he sign of he curren prices F (j); which can be negaive for odd j. The sign of he expeced price changes of an invesmen in a higher-order momen fuures conrac is only deermined by ha of jump risk premium. Noe ha, under he SVJ model, he payo s of hese nancial insrumens are respecively de ned as T and T R T 0 ej j udn u, for j 3. R T 0 ej 2 udn u Proposiion 6 Under he SVJ model and he no-arbirage principle, he price of he bipower variaion and higher-order momen fuures conracs a ime 2 [0; T ] are given as F (j); = T Z 0 ej j udn u + Q (j) (23) for j = 2 and j > 3, respecively. The expeced changes of he prices of hese derivaives are given as F (j) d E df(j); = (j) Q (j) d, (24) where (j) and Q (j) are he jh-order non-cenral momen of e J under he physical and risk-neural measure, respecively. These momens can be calculaed respecively as (j) = @ (j) M(x) x=0 and Q @x (j) (j) = @(j) M Q (x) x=0, where M(x) = exp @x (j) J x + (=2) ( J ) 2 x 2 and 2 M Q (x) = exp QJ x + (=2) Q J x 2 are he momen-generaing funcions of random variable e J under measures P and Q, respecively. Proposiion 6 implies ha he expeced change of he price of he bipower variaion fuures F (2) is negaive and lower in magniude han ha of he variance fuures. This can be obviously aribued o he fac ha, by de niion, he bipower variaion fuures pays only when jumps occur in he marke during ime period [0; T ], whereas he variance fuures conrac accouns also for an increase in volailiy. Using formulas (22) and (24), for j = 2, i can be easily seen ha e V = S + V, which depends on he price of marke 9
and volailiy risk, can be calculaed as e V = F V @F V=@ ln V F V @F V=@ ln V F (2) : (25) This relaionship implies ha he expeced gains of a porfolio consruced by holding = (@F V=@ ln V ) of he noional in a long posiion of a variance fuures and = (@F V=@ ln V ) of he noional in a shor posiion of bipower variaion fuures, respecively, will be relaed o he volailiy risk premium. This comes from he fac ha he reurn of his porfolio does no depend on he jump risk premium, as noed in Secion 2 (see fn 4). The resuls of Proposiion 6 indicae ha he sign of he expeced changes of he prices of higher-order momen fuures F (j), for j 3, depends on he sign of momens di erence (j) Q (j). Noe a his poin ha momens (j) and Q (j), for all j, exis due o he normaliy assumpion of he log-jump size ~ J. For j = 3, he sign of (3) Q (3) is expeced o be posiive, which implies ha he expeced price change of he 3rd-order momen fuures conrac F (3) will be posiive. This compensaes invesors aking a long posiion in his fuures conrac for bearing he risk of possible negaive jumps occurring during ime period [0; T ]. 9 Following analogous argumens, we can generalize he above resuls o he case of higher-order odd or even momens as follows: 8 < F (j) = : > 0, if j is odd < 0, if j is even (26) 3.2.2 Pricing and hedging coningen claims To price and hedge coningen claim C under he SVJ model, we will assume ha he self- nancing porfolio, which replicaes he price of C, C = C (; S ; V ) for 2 [0; T ], consiss, in addiion o he nancial insrumens of he corresponding porfolio for he SV model, of higher-order momen fuures conracs. Tha is, he vecor of delas now is de ned as = ( S ; B ; F V ; F (2) ; :::; F (N) ) 0, where is elemens denoe he number of he 9 The posiive sign of di erence (3) Q (3) can be more clearly seen by wriing i as 3 2 (3) Q (3) = 3 J Q J + 3 J 2 J Q J Q J : Given evidence ha Q J < J < 0 and 2 Q J > 2 J, he above equaion implies (3) Q (3) > 0. 20
underlying sock, zero-coupon bond, variance fuures, bipower variaion fuures (for j = 2) and higher-order momen fuures conracs (for j 3), a ime, respecively. Noe ha his replicaing porfolio is assumed ha consiss of a nie, bu su cienly large number N of higher-order momen fuures conracs, which is adequae o approximaely replicae C in he sense ha lim dv ( ) = dc (27) N! for all 2 [0; T ], where V (:) denoes he value of he self- nancing porfolio. In he nex proposiion, we derive analyic formulas of he elemens of vecor of delas and insananeous expeced reurn of coningen claim C. Proposiion 7 Consider a coningen claim C wrien on he underlying sock and/or is volailiy, wih price funcion C = C (; S ; V ), where C(:) is a coninuous funcion which has parial derivaives of any order. Then, under he SVJ model and he no-arbirage principle, he delas of he self- nancing porfolio replicaing coningen claim price C, a 2 [0; T ], are given as follows: S = @C @S, B = B C S S, F V = @C @F V F (2) = 2! @ (2) C @ ln S (2) @C @ ln S! @C @F V and F (j) = j! @ (j) C @ ln S (j) (28)! @C, for j 3, @ ln S while he insananeous expeced reurn of C is given as dc E C = r d + @ ln C @ ln C @ ln S S + r d + @ ln F V F V d +!! + 2 + lim N! j=3 @ (2) C=C @ ln S (2) NX j! @ ln C @ ln S @ (j) C=C @ ln S (j) @ ln C @ ln S @ ln C @F V! F (2) d F (j) d. (29) The resuls of Proposiion 7 imply ha enlarging he marke wih variance, bipower variaion and higher-order momen fuures conracs enables us o approximaely replicae coningen claim C. For su cienly large N, his marke becomes approximaely (or quasi) complee, hus implying ha he dela-hedged gains of he self- nancing porfolio given by he proposiion converge o zero. In such a marke, here is a unique risk-neural measure Q (see Björk, Kabanov and Runggaldier (997), or Jarrow and Madan (999)). Under 2
his, we can derive he price of coningen claim C by solving Baes s (996) PDE (see Appendix). As wih he SV model, we have derived his PDE by eliminaing all sochasic erms deermining sochasic movemens in C, and no based on equilibrium approach. The formula of he expeced reurn of he coningen claim E h dc C i, given by Proposiion 7, indicaes ha, in addiion o he expeced reurns of he underlying asse and he variance fuures conrac, i also depends on hose of he bipower variaion and higherorder momen fuures conracs. The bipower variaion fuures conrac has wo e ecs on h i h i E dc C which are complemenary o each oher. The rs adjuss E dc C for is bias due o esimaing he volailiy premium based on he expeced reurn of a variance fuures conrac, which also depends on he jump risk (see equaion (22)). The second accouns for he exposure of coningen claim price C o he jump risk, especially, is componen relaed o J e h i 2 (see (24), for j = 2). The oal of he above wo e ecs on E dc C depend on he sign and magniude of dela coe ciens F (j), for j > 2. For European opion conracs, which is he focus of our empirical analysis in nex secion (Secion 4), hese coe ens depend on he mauriy and moneyness of he opion conrac. Since he above de niions of F (j) are new in he lieraure, below we give a more deailed discussion abou heir sign and magniude. This is done across di eren caegories of moneyness of European calls and pus, and over shor and long mauriy inervals. To help his discussion, in Figure 2 we graphically presen esimaes of F (j), for j = 2; 3; 4, across di eren srike prices K and mauriy inervals of one and six monhs. These rely of parameer esimaes of he SVJ model, based on our empirical resuls of Secion 4. Noe ha he pu-call pariy relaion implies ha F (j) will be he same for calls and pus of he same mauriy inerval and srike price, for all j. This discussion is also very helpful in undersanding sources of dela-hedged gains which can be found in pracice (see Secion 4). For deep-otm pus and calls, dela coe ciens F (j) will be almos zero, for all j = 2; 3; 4. In paricular, F (2) will be close o zero, because boh vega and gamma are very close o zero given ha srike price K akes very small or large values. 0 F (3) and F (4) will be also close o zero, because deep-otm pus and calls are very lile in uenced by he expecaion 0 This can be easily seen by wriing F (2) as F (2) = 2 S2 @ (2) C @S (2) @C @F V : 22
of a jump occurrence, which is priced by he 3rd and 4h-order momen fuures. The pracical implicaion of hese resuls is ha a wo-insrumens hedging sraegy, involving a shor posiion in he underlying sock and a variance fuures conrac, will be su cien o hedge he exposure of deep-otm opions o all sources of risk, as jump risk is negligible for his caegory of opions. This is also rue for long-erm opions of any moneyness. As long as volailiy risk is hedged hrough variance fuures, he exposure of hese opions o jump risk becomes also negligible. In conras o long-erm, he sign of F (j) for shor-erm opions changes signi canly across moneyness. In paricular, for shor-erm OTM pus F (2) will be negaive. This can be aribued o he fac ha a posiion held in variance fuures conracs is in excess of ha required o hedge he exposure of he opion o a jump, independenly of he sign of he jump. OTM pus bene more by an increase in volailiy, which can cause he price of he underlying sock o decrease due o leverage e ec, compared o an equally possible posiive or negaive jump e ec on he sock price, measured by J e2. This means ha he opion s vega should be larger han he opion s gamma, implying F (2) < 0. For shor-erm OTM calls, he sign of F (2) is expeced o be posiive. In his case, gamma should be larger han vega, given ha a posiive or negaive jump is preferable han an increase in volailiy. Regarding he sign of higher-order delas F (3) and F (4), his will be respecively negaive and posiive for shor-erm OTM pus. For F (3), his happens because his caegory of opions is considered by invesors as a securiy insrumen agains severe adverse movemens of he sock marke. Thus, a long posiion on hem embody a shor posiion in a 3rd-order momen fuures conrac. The posiive sign of F (4) can be explained by he fac ha OTM pus also embody a long posiion in he 4h-order momen fuures conrac, which pays when exreme negaive or posiive jumps occur. For shor-erm OTM calls, F (3) and F (4) are boh posiive. In his case, an invesor who akes a long posiion in his caegory of opions is averse o jumps, and hus embodies a long posiion in he 3rd and 4h-order momen fuures conracs. The long posiion in he 3rd-order momen fuures conrac sells proecion for a possible negaive jump in he marke. The long posiion in he 4h-order momen fuures conrac compensaes he invesor for an exreme posiive or negaive jump. Finally, for shor-erm ATM opions, he sign of F (j) is expeced o be negaive, for all j = 2; 3; 4. More speci cally, F (2) will be negaive, since a posiive or negaive jump is 23
preferable han an increase in volailiy causing leverage e ecs, as i happens wih OTM pus. The negaive value of F (3) can be aribued o he fac ha aking a long posiion in ATM opions eiher leads o immediae gains in case of a pu or o limied loses in case of a call. On he oher hand, he negaive sign of F (4) can be explained by he fac ha an exreme jump can render ATM opions unpro able. 4 Can volailiy and jump risks be e cienly hedged by variance and higherorder momen fuures? Using daa on variance fuures conracs and European call and pu prices, in his secion we answer he following wo quesions based on he heoreical relaionships derived in he previous secion. The rs is if random jumps are priced in he variance fuures marke. As shown by Proposiion 2, his mus be re eced in he di erence beween he price of a variance fuures conrac and he value of he composie opions porfolio, i.e. F V V o;. This quesion will be examined wihou relying on any parameric model of he marke. The second quesion is if a long posiion in a European call or pu can be e cienly hedged based on he wo-insrumens hedging sraegy which considers a variance fuures conrac as a hedging vehicle, or i requires hedging sraegies which rely on higher-order momen fuures. To address his quesion, we will rely on esimaes of delas of he above hedging sraegies based on he formulas of Proposiion 7, for he SVJ model. Answering he above quesions has imporan porfolio managemen implicaions. They can indicae if i is bene cial, rs, o include variance fuures conracs in he radiional dela hedging sraegy and, second, o consider higher-order momen fuures for hedging opions and variance fuures posiions agains random jumps. 4. The daa The opions price daa used in our empirical analysis are aken from he OpionMerics Ivy daa base. These are daily marke closing prices of European ATM and OTM calls, wih K > S, and OTM pus, wih K < S, wrien on he S&P 500 index. We use ATM and OTM calls and pus because, as is well known in he lieraure, hese opions are more acively raded compared o in-he-money (ITM). We have excluded opion prices violaing he boundary condiions and opions wih mauriy inervals less han 2 weeks. Finally, 24
following Driessen, Maenhou and Vilkov (2009), we have removed opions wih zero open ineres for liquidiy reasons. Apar from opion prices, we have also used he repored implied volailiy surfaces (IVS) of he above daa base, wih mauriy inervals, 2, 3, 6 and 9 monhs. This is done in order o esimae he parameers of he SVJ model needed in our analysis and calculae he values of he composie opions porfolio. This daa se conains implied volailiies on boh call and pus on a grid of 3 srike prices. The variance fuures price daa used in our analysis are wrien on he S&P 500 index. These conracs are raded over-he-couner. This daa se consiss of closing prices of variance fuures ne of he accrued realized variance wih mauriy inervals of, 2, 3, 6 and 9 monhs. Thus, if we have a raded variance fuures conrac during period [T e; T ], h i hen he recorded quoe of F V is adjused as follows: F g V E Q e hx; Xi ;T = F V e hx; Xi T e;. Finally, he series of he risk-free ineres rae and dividend yield are also aken from he OpionMerics daa base. The ineres rae is derived by Briish Banker s Associaion LIBOR raes and selemen prices of Chicago Mercanile Exchange Eurodollar fuures. conracs. The dividend yield is esimaed by he pu-call pariy relaion of ATM opion Our daa se covers he period from March 30, 2007 o Ocober 29, 200. This sample inerval consiss of 900 rading days and i includes he period of he recen nancial crisis, he mos serious a leas since he 930 s. Our analysis presens resuls for subsamples before and afer he dae ha his crisis seems o be inensi ed. This dae is found o be closely relaed o ha of he collapse of Lehman Brohers. 4.2 Fiing he SVJ model ino he daa To esimae he parameers of he SVJ model, we rely on daily values of IVS of pu opions wrien on he S&P 500 index, wih consan mauriy inervals, 2, 3 and 6 monhs. Fiing he SVJ model ino consan-mauriy IVS provides esimaes of is parameers which are less sensiive o available mauriy inervals every day. Since he following parameers of he SVJ model:,, and are he same under he physical and risk neural measures, in he esimaion procedure we will se hem o cerain values provided in he lieraure, following Broadie, Chernov and Johannes (2007). In paricular, hese are se Noe ha he choice beween European calls and pus in esimaing he SVJ model does no a ec he resuls, due o pu-call pariy relaion. 25