Optimal Hedging for Multivariate Derivatives Based on Additive Models

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1 0 Aercan Control Conference on O'Farrell Street San Francsco CA USA June 9 - July 0 0 Optal Hedn for Multvarate Dervatves Based on Addtve Models Yu Yaada Abstract In ths paper we consder optal hedes for a class of dervatve securtes whose underlyns are untraded usn the addtve su of sooth functons of traded assets that nzes the ean square error Based on the necessary and suffcent condton we derve a ethodoloy to copute optal sooth functons effcently by solvn a syste of lnear equatons Moreover we extend the dea to basket optons consstn of a portfolo of stocks where ndvdual payoff functons of traded assets are optally coputed We also provde nuercal experents to llustrate our ethodoloy I ITODUCTIO To explan the otvaton of ths work let us consder the standard nu varance hedn proble ven as nvar Y α S α where Y and S are prce chanes of two assets Y and S n a certan te perod and α s a hede rato In a typcal stuaton the asset Y can not be traded frequently n the arket whereas S ay be a lqudly traded asset It s known that the optal hede rato denoted by α provdes the nzer of n E { Y α S+c} αc whch ay be solved as an ordnary lnear reresson ven eprcal observaton data In ths sense the standard nu varance hede of s equvalent to the sple ordnary lnear reresson proble The standard nu varance hede ay be eneralzed for nonlnear case n whch a nonlnear sooth functon f s searched to nze the follown ean square error: E { Y f S} 3 In the case where ultple assets are avalable the proble ay be forulated as follows { } n E f S Y f S 4 where S are prce chanes of asset S and S s a set of sooth functons Obvously the proble addresses the standard nu varance hede or ore enerally the ultvarate nu varance hede usn ultvarate lnear reresson as a specal case when f s lnear and therefore we can expect to et the better hede Y Yaada s wth Graduate School of Busness Scences Unversty of Tsukuba Tokyo Japan -00 E-al: yu@ssotsukatsukubaacp UL: effect Ths s our basc dea n our prevous work 0 that we appled the eneralzed addtve odel GAM; see 5 9 for constructn optal payoff functons of weather dervatves ven eprcal observaton data The obectve of ths paper s to provde a theoretcal fraework for nonlnear nu varance hedn n contnuous te settn For ths obectve we frst forulate the nonlnear nu varance hedn proble and provde a necessary and suffcent condton for the optal sooth functons Then we derve an alorth to copute the optal sooth functons based on the sutable dscretzaton and deonstrate the optal hedes Moreover we extend the dea to the basket optons case whose underlyn s defned as the wehted averae of any stocks II POBLEM FOMULATIO Let Y t t 0 T be the value of an asset ben nontraded or llqud and S t the values of lqudly traded assets under a probablty space Ω F P and fltraton {F t } t 0T We consder the proble of hedn the payoff of dervatve securty on Y t usn the addtve su of sooth functons of S t To ths end we defne the nonlnear nu varance hedn proble as follows: { } n E f S Y T f S T 5 where Y T stands for the ternal payoff of a dervatve securty wth a ven payoff functon ote that Y t ay be the value of portfolo as n Secton V Also note that the proble forulaton of 5 s slhtly dfferent fro that of 4 as t nzes the ean square error between the ternal payoff Y T and the su of f S T Althouh we do not know how to fnd the optal sooth functons yet there exsts a necessary and suffcent condton for optal sooth functons f f as follows: Lea : Sooth functons f f provde nzers of the proble 5 f and only f the follown condtons are satsfed: EY T S T E f S T S T 0 6 Proof: For the proof see pp 08 n 5 In ths paper we deonstrate how to copute the sooth functons f f satsfyn Lea Before shown the soluton ethod for sooth functons we dscuss how to replcate f S T as cash values Snce each f s a sooth functon //$600 0 AACC 3856

2 there are two approaches to attan f S T The frst approach s to use European type calls and puts wth aturty T and any strkes where any twce contnuously dfferentable functon fx of the ternal stock prce S T x can be replcated by a unque ntal poston of f S 0 f S 0 S 0 unt dscount bounds f S 0 shares and f KdK out-of-the oney optons of all strkes K 3: f x f S 0 f S 0 S 0 + f S 0 x + S0 0 f KK x + dk + S 0 f Kx K + dk 7 The advantae of ths approach s that we do not have to estate any paraeters such as volatltes or ean rates of returns of the underlyn assets once the taret payoff functon f s specfed The second approach s to dynacally trade S t to replcate the ternal payoff f S T For ths approach to be applcable we need to ntroduce prce dynacs for Y t and S t naely the dynac hedn odel ote that n ths fraework althouh the total arket s ncoplete snce Y t s not tradable each payoff f S T ay be replcated by tradn S t dynacally We further dscuss ths approach n Secton IV III SOLUTIO METHOD FO OPTIMAL SMOOTH FUCTIOS ecall that fro Lea the proble reduces to fndn a set of real-valued functons f f satsfyn where E f S T ST x EYT S T x 8 E f S T S T x f x 9 Assue that there exst ont PDFs of pars S T S T and Y T S T denoted by and φ S S x x 0 φ YS y x respectvely Also let φ S S x x and φ Y S y x be condtonal PDFs defned as φ S S x x : φ S S x x φ S x φ Y S y x : φ YS y x φ S x where φ S x are arnal PDFs Then condton 8 ay be rewrtten as follows: f x + f x φ S S x x dx y φ Y S y x dy We would lke to fnd f such that holds for sutable doan of nput varables Here we provde a soluton ethod whch conssts of the follown three steps: Dscretze condton for y x and x densons to obtan a set of lnear equatons Solve the set of lnear equatons to fnd dscretzed ponts of sooth functons 3 Construct sooth functons usn cubc splnes ote that the above ethod ay be appled f the ont PDFs of pars n Y T and S T are specfed Frst we dscretze condton to approxate the nterals as f x + l l f x l φ S S y l φ Y S y l x δ y x l x δ x 3 for ven x where δ x and δ y are assued to satsfy l φ S S x l x δ x l φ Y S y l x δ y ote that δ x and δ y ay depend on x as well but we wll ot to specfy that dependence for brevty We then dscretze condton 3 for x densons e x k k as f l x k + l y l φ Y S f x l y l x k φ S S δ y x l x k δ x Let f and be vectors whose k-th entres are respectvely ven as f k : f x k k : y k k Also let Φ y and Φ be atrces whose k l-entres are ven as Φ k l : φ S S x l x k δ x Φ y k l : φ Y S y l k x δ y kl Wth these defntons and notatons we have the follown proposton: Proposton : For each condton ay be dscretzed as f + Φ f Φ y 4 Consequently we obtan the follown syste of lnear equatons wth respect to f : f f : Φf ĝ

3 where I Φ Φ 3 Φ Φ I Φ 3 Φ Φ : Φ 3 Φ 3 I Φ3 Φ Φ Φ 3 I Φ y 0 0 ĝ : 0 Φ y Φ y Althouh the soluton to 5 ay not be unque t can be expressed usn the eneralzed nverse atrx as f Φ ΦΦ ĝ 6 Then the optal sooth functons f ay be constructed usn cubc splnes f x c 0 + c x+ x k θ k x 3 7 k where c 0 c and θ k k are found to satsfy f x k f k and k θ k 0 k θ k x k 0 eark : Althouh we derved the set of lnear equatons based on the ont PDFs for Y t and S t t s often the case n dervatve prcn probles that the underlyn stochastc processes Y t and S t are expressed as the follown type of eoetrc processes: Y t Y 0 e Z t S t S 0 e X t 8 where Z t and X t are adopted to F t In ths case we can work on ont PDFs or correspondn condtonal PDFs for Z T and X T nstead of the ones for Y T and S T Let ont PDFs of pars X T X T and Z T X T be ven as and φ X X x x φ ZX z x respectvely Then condton s odfed to f S 0 e x + f S0 e x φ X X x x dx where Y 0 e z φ Z X z x dz 9 φ X X x x : φ X X x x φ φ X x Z X z x : φ ZX z x φ X x 0 and φ X x are arnal PDFs We see that the sae approach can be appled by dscretzn 9 for each denson to derve the slar set of lnear equatons IV DYAMIC HEDGIG MODEL In ths secton we ntroduce prce dynacs for S t that enable us to replcate the ternal payoff f S T usn dynac tradn stratey Assue that under the probablty space Ω F P the values of lqudly traded assets S t S t and nontraded asset Y t are overned by the follown stochastc dfferental equatons ds t µ S t dt + σ S t dw t dy t µ + Y t dt + σ + Y t dw +t where W t W +t are correlated Brownan otons wth dw t dw t ρ dt + For splcty let µ σ and ρ + be constant paraeters althouh the result can readly be eneralzed for the case of deternstc functons of t ote that the advantae of consdern the above odel s that there exsts a dynac tradn stratey see and 6 to replcate the ternal payoff f S T once the optal sooth functons are specfed A Case We frst derve the optal sooth functon for the case The follown proposton shows that f s expressed n a closed for for European call/put optons wth a strke prce K: Proposton : The optal sooth functon f s represented as { f x Y 0 exp µ ρ σ } T + ρ σ b x d x Kd x 3 when y y K + for European call optons or { f x Y 0 exp µ ρ σ } T + ρ σ b x d x+k d x 4 when y K y + for European put optons where and s the standard noral dstrbuton functon and b x : x {ln µ σ } T 5 σ S 0 Y0 d x : ln σ ρ T K +ρ σ b x+ µ + σ ρ σ T Y0 d x : ln σ ρ T K +ρ σ b x+ µ σ T Proof: For the proof see The proble settn n ths paper addresses the one n 8 when Also the proble s closely related to the poneern work of 4 for hedn the spot prce usn the self-fnancn portfolo of future prce ote that n our forulaton we ntend to hede the payoff of llqud asset dervatves usn lqudly traded asset dervatves 3858

4 B General case In the case wth traded assets S T and Y T are ven as S T S 0 e ν T+σ W T Y T Y 0 e ν +T+σ + W +T where ν : µ σ / + Snce the condtonal expectaton ven S t x corresponds to the one ven W t w wth sutable paraeter chanes condton 8 ay be rewrtten as E f S t W T w EYT W T w 6 for ote that the rht hand sde of 6 can be coputed based on Proposton as EY T W T w E Y T S T S 0 e ν T+σ w ĝ w 7 usn a sooth functon ĝ Also snce each S t s a functon of W t we wrte f S t ˆf W t and reforulate equaton 6 as follows: E ˆf W t W T w ĝ w 8 Let p w w be the condtonal probablty densty functon of W t ven W t e p w w : exp π ρ T w ρ w ρ T Then condton 8 ay be wrtten as follows: ˆf w + ˆf w p w w dw ĝ w Wth the slar aruent to the dervaton of condton 5 we can construct a set of lnear equatons by sutable dscretzaton for w w p w w ˆf w and ĝ w as I Φ Φ ˆf Φ I Φ ˆf Φ Φ I ˆf ĝ ĝ ĝ 9 Optal sooth functons f are then obtaned usn cubc splnes V BASKET OPTIOS In the prevous sectons we have assued that Y t stands for the value of nontraded asset and consdered to hede an opton on Y t usn lqudly traded assets S t Here we extend ths dea to the proble of hedn basket optons usn payoffs of optons on ndvdual assets Let us replace Y T n 5 by the wehted su of traded assets e { } n E f S Y T f S T Y T : α S T 30 where α are ven weht paraeters Then the nu varance hedn proble 30 s to fnd sooth payoff functons for ternal values of ndvdual assets S T that approxate the ternal payoff of basket opton as close as possble n the nu ean square sense otce that the left hand sde of equaton 8 s ndfferent even for basket optons and hence the left hand sde of 5 ay be constructed slar to Proposton f there are ont PDFs for S T usn the condtonal expectatons of Y T ven S T We wll deonstrate how to copute these condtonal expectatons when the value processes of S t are defned by Assue that S t follow the SDEs n and let ĝ be a functon satsfyn ĝ W T E Y T W T We would lke to express ĝ n a tractable for The follown proposton shows that ĝ ay be represented usn uncondtonal expectaton and thus be coputed effcently: Proposton 3: For each and a nonrando duy varable w there exst a functon h and ndependent Brownan otons B t B t t 0 T satsfyn ĝ w Eh w B T B T 3 Proof: Here we consder the case althouh the sae technque ay be appled for Let the covarance atrx of dst ds t S t S t be decoposed as LL dt where L s a lower tranular atrx defned by σ 0 0 L : σ σ 0 σ σ σ σ σ based on the Cholesky decoposton Then we obtan the follown equvalent representaton to : ds t /S t µ db t ds t /S t µ dt + L db t 3 where B t B t are ndependent Brownan otons and B t W t Snce S T s expressed as S T S 0 exp ν T + σ B T 3859

5 there exsts a functon h such that Y T Hence we have α S T E Y T S T E Y T W T h W T B T B T Eh W T B T B T W T 33 We dscuss soe propertes of the condtonal expectaton n 33 Frst we note that S T s a functon of W T and s ndependent of the other factors B T B T Ths ndcates that there exsts a sa alebra G F such that both W T and S T are G -easurable and B T B T are ndependent of G Then we can apply the Independence Lea that a functon ĥ of a duy varable w ĥ w : Eh w B T B T 34 satsfes the follown condton: ĥ W T Eh W T B T B T W T E Y T W T 35 Clearly condtons 34 and 35 ndcate that the stateent n the proposton holds wth and ĝ ĥ Slarly we can obtan h by reordern S t S t so that S t s the frst entry when applyn the Cholesky decoposton Ths copletes the proof We see that for any ven real nuber w ĝ w s coputed by the uncondtonal expectaton n 34 In eneral ths coputaton nvolves ultple nteraton but usually executed effcently based on the Monte Carlo ethod by eneratn ndependent Gaussan rando nubers for ndependent Brownan otons Once a set of rando nubers s enerated we can copute ĝ w for dfferent values of w w k k usn the sae set of rando nubers to construct a real-valued vector ĝ n the rht hand sde of equaton 9 Then we solve the set of lnear equatons for ˆf to fnd the optal sooth functons usn cubc splnes ote that other propertes of basket opton s hede s dscussed n 3 VI UMEICAL EXPEIMET In ths nuercal experent we frst consder a proble of hedn an opton whose underlyn s a arket ndex ben nontraded usn several stocks where each asset dynacs s odeled as and We wll forulate the proble as nonlnear nu varance hedn and solve t by applyn the proposed ethodoloy We use the eprcal data obtaned fro the Tokyo Stock Exchane TSE n the perod of for estatn the volatlty and correlaton paraeters of stock returns where the arket ndex s assued to be TOPIX and fve stocks S S 5 are chosen fro those lsted n the TSE The correlaton and volatlty paraeters of stock returns are estated as n Table I whereas we assue that each expected stock return correspondn to the drft paraeter TABLE I VOLATILITY AD COELATIO OF THE STOCK ETUS IDEX S S S 3 S 4 S 5 IDEX S 055 S S S S Volatlty TABLE II DIFT HAVIG THE SAME SHAP ATIO 05 IDEX S S S 3 S 4 S 5 Drft has the sae sharp rato 05 wth rsk free nterest rate r 005 Then drft paraeters are provded as n Table II We solve the proble 5 to fnd the nzers f f 5 for hedn an at-the-oney European call opton wth aturty T /4 where the ntal prces or ntal values are set to be Y 0 00 and S The correlaton coeffcent between Y T and 5 f S T ay provde a hede effect and for ths nuercal experent t s obtaned as Corr Y T 5 f S T ext we dscuss the ntal cost of the hede To copute the ntal cost we need to evaluate the ntal prce of the optons under a rsk neutral probablty easure Here we copute the nal arket prce of rsk see e to specfy a rsk neutral probablty easure P ote that the arket prces of rsk for the traded assets S S 5 are the sae and are ven by ther sharp ratos 05 whereas the nal arket prce of rsk for the nontraded asset denoted by ˆθ y s found to be ˆθ y 0306 We see that the arket prce of rsk for the ndex s hher than those of traded assets whch ay be nterpreted as a rsk preu for the nontradablty of the ndex Under the rsk neutral probablty easure we coputed the ntal value of call opton wrtten on Y t whch s ven as V 0 e rt ẼY T On the other hand ntal values of optons whose payoffs are deterned by f S T are ven as e rt Ẽ f S T 5 38 Snce each payoff ay be heded by the correspondn selffnancn portfolo wth the ntal cost ben equal to 38 the total cost of the replcatn portfolo s obtaned as X 0 5 e rt Ẽ f S T

6 In our nuercal experent the ntal value of portfolo X 0 s obtaned as X whch s alost the sae as V 0 n ths case When one takes a poston to sell the opton wth V 0 at te t 0 and construct a portfolo wth X 0 to hede the opton the ntal cost of the heded poston s ven by X 0 V 0 4 If X 0 V 0 s postve then she has to pay an extra cost to construct the portfolo and hence X 0 V 0 > 0 ay be nterpreted as a preu for the hede Here we evaluate the hede cost usn ts rato and defne the hede cost rato HC as HC : X 0 4 V 0 ote that HC > corresponds to X 0 V 0 > 0 F Correlaton coeffcents sold & Hede cost ratos dashed To exane the relaton of the hede effect and the HC wth respect to the nuber of traded underlyns we vared the nuber of traded underlyn fro to 5 and obtaned F where the sold lne refers to the values of correlaton coeffcents wth respect to 5 and the dashed to those of the HCs We see that both the hede effect and the HC are proved as the nuber of traded assets ncreases ote that the hede effect and the HC are both one for the coplete arket case as shown by the dotted lne n F ext we consder the basket opton s hede n whch the payoff depends on the wehted averae of fve stocks S S 5 wth the sae paraeter values n Tables I and II We solve the nonlnear nu varance hedn proble 30 to approxate the payoff of basket opton Y T Y T 5 S T + +S 5T by the su of ndvdual optons f S T 5 F shows the scatter plot of Y T vs 5 f S T Slar to the frst nuercal experent we can evaluate the hede effect by correlaton coeffcent between Y T and 5 f S T whch s obtaned as 5 Corr Y T f S T We see that the payoff of basket opton ay be approxated well usn ndvdual optons n ths exaple F Scatter plot for basket opton s hede n the ATM case VII COCLUSIO In ths paper we deonstrated optal hedes for a class of dervatve securtes whose underlyns are untraded usn the addtve su of sooth functons of traded assets that nzes the ean square error At frst we derved a ethodoloy to copute optal sooth functons effcently by solvn a syste of lnear equatons based on the necessary and suffcent condton Then we extended the dea to basket optons consstn of ultple stocks where ndvdual payoff functons of traded assets are optally coputed n the nu varance hedn proble We also provded nuercal experents to llustrate our proposed ethodoloy EFEECES T Bork 004 Arbtrae Theory n Contnuous Te nd edton Oxford Unversty Press F Black and M Scholes 973 The Prcn of Optons and Corporate Labltes Journal of Poltcal Econoy vol 8 637/654 3 P Carr and D Madan 00 Optal postonn n dervatve securtes Quanttatve Fnance vol 9/37 4 D Duffe and H chardson 99 Mean-varance hedn n contnuous te Annals Appl Probablty -5 5 T Haste and Tbshran 990 Generalzed Addtve Models Chapan & Hall 6 Merton 973 Theory of ratonal opton prcn Bell Journal of Econocs and Manaeent Scence 4 pp SE Shreve 004 Stochastc Calculus for Fnance II: Contnuous- Te Models Sprner 8 ES Schwartz and C Tebald 006 Illqud Assets and Optal Portfolo Choce BE Workn Paper o S Wood 006 Generalzed Addtve Models: An Introducton wth Chapan & Hall 0 Y Yaada 007 Valuaton and Hedn of Weather Dervatves on Monthly Averae Teperature Journal of sk vol 0 no pp 0 5 Y Yaada 008 Optal hedn of predcton errors usn predcton errors Asa-Pacfc Fnancal Markets vol 5 no pp Y Yaada 00 Optal Hedn wth Addtve Models to appear n ecent advances n fnancal enneern Proc of the KIE-TMU Internatonal Workshop on Fnancal Enneern World Scentfc 3 Y Yaada 0 Hedn of Multvarate Optons wth Addtve Models to appear n Proc of the 0 ppon Fnance Assocaton Annual Conference 386