DELTA-GAMMA-THETA HEDGING OF CRUDE OIL ASIAN OPTIONS

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1 ACA UNIVERSIAIS AGRICULURAE E SILVICULURAE MENDELIANAE BRUNENSIS Volume Number 6, 05 hp://dx.doi.org/0.8/acaun DELA-GAMMA-HEA HEDGING OF CRUDE OIL ASIAN OPIONS Juraj Hruška Deparmen of Finance, Faculy of Economics and Adminisraion, Masaryk Universiy, Žeroínovo nám. 67/9, Brno, Czech Republic Absrac HRUŠKA JURAJ. 05. Dela-gamma-hea Hedging of Crude Oil Asian Opions. Aca Universiais Agriculurae e Silviculurae Mendelianae Brunensis, 63(6): Since Black-Scholes formula was derived, many mehods have been suggesed for vanilla as well as exoic opions pricing. More of invesing and hedging sraegies have been developed based on hese pricing models. Goal of his paper is o derive dela-gamma-hea hedging sraegy for Asian opions and compere is efficiency wih gamma-dela-hea hedging combined wih predicive model. Fixed srike Asian opions are ype of exoic opions, whose special feaure is ha payoff is calculaed from he difference of average marke price and srike price for call opions and vice versa for he pu opions. Mehods of sochasic analysis are used o deermine delas, gammas and heas of Asian opions. Asian opions are cheaper han vanilla opions and herefore hey are more suiable for precise porfolio creaion. On he oher hand heir delas are also smaller as well as profis. ha means ha hey are also less risky and more suiable for hedging. Resuls, conduced on chosen commodiy, confirm beer feasibiliy of Asian opions compering wih vanilla opions in sense of gamma hedging. Keywords: Asian opion, dela, gamma, hea, hedging, invesmen decision making INRODUCION Exoic opions are mosly considered as plainly mahemaical issue, since hey are no widely raded on world derivaive markes. In case hey are raded i is usually on OC markes. Asian opions are one of hose which are a leas occasionally lised on he greaes derivaive markes such as CBO or EUREX. heir underlying asses are for example crude oil, ehanol, iron ore and places on he cargo ships beween Europe and eas coas of USA and beween Persian Gulf and Japan. Papers published abou he opic of Asian opions are focused eiher on heir pricing, wih usage of various mahemaical mehods or heir applicaion in invesmen process. Black-Scholes mehodology was used for pricing Asian opions wih discree averaging (Zhang, 998). his mehod is someimes considered as ou of dae and is being replaced wih more sophisicaed mehods. Dubois is using parial differenial equaions for heir pricing (Dubois, 004). hese equaions are solved wih various approaches; perurbaion mehod (Zhang, 003) for example. In modern research jump processes are preferred o simple Wiener process in derivaive pricing, because hey can beer reflec he naure of sock price movemens (Bayrakar, 00). Invesmen sraegies in paricular asses could be hedged wih Asian opions or hey can be used for direc invesmen. Lévy s models wih jump processes were considered as effecive possibiliy for saic hedging (Albrecher, 005). Gamma hedging iself was used in several researches evaluaing invesmen opporuniies of his mehod (Dengler, 996 and Jarrow, 994). I is also used as a ool for hedging moraliy and ineres raes (Luciano, 0). In his paper Asian opions are being applied in invesmen sraegy based on creaing dela and gamma neural porfolio. his mehod was developed for hedging plain-vanilla opions (Šurc, 00). Invesmen sraegy is enhanced wih simple decision model based on pas movemen of underlying asse. Gamma hedging had been proven o be successful wih plain vanilla opions. he main disadvanage of gamma hedging, as well as in all porfolio opimizing sraegies, are he opimal weighs in porfolio. hey are difficul o achieve in pracice, because of rading in los and limied available capial. his can be solved by using Asian opions insead of European. he fac ha he spo 897

2 898 Juraj Hruška price is averaged in pay-off funcion makes hem much cheaper han plain-vanilla opions (Haug, 007). Alhough his sraegy is called hedging i carries cerain amoun of speculaion, because we are always choosing price direcion, which is supposed o be hedged. he opposie price direcion should bring us profi all he ime, if all he requiremens hold. MEHODOLOGY AND DAA We can disinguish beween Asian opions wih floaing srike and wih he fixed srike. Posiion of averaged price in pay-off funcion is he main difference. Average price replaces srike price in floaing srike opions and spo price in fixed srike opions. he average can be calculaed in arihmeic or geomeric form. In he nex analyses only hose wih fixed srike are considered, because, hey are more common in pracice. Hence more daa are available for adequae model selecion. Mehodology All he calculaions in his paper are based on Zhang s model for pricing Asian opions wih fixed srike and coninuous geomeric averaging (Zhang, 998). his model was chosen afer comparison wih oher known models for pricing opions wih coninuous as well as discree averaging. ess were conduced on real opions daily prices wih large variey of srike prices and maximal available observaions. Pay-off funcion of hese opions is using arihmeic discree averaging. Models using geomeric averaging are commonly used for pricing opions wih arihmeic averaging as an approximaion (Zhang, 995), because pricing models using arihmeic averaging have no closed form (Hull, 0). Bu i is sill surprising, ha models for geomeric averaging are more suiable. Menioned Zhang s model is defined as: rg 6 r V ke S N( kd) ke KNkd where S ln r g K d. 3, () 3 Spo price of underlying asse is noed as S, K represens srike price, r is used for risk-free ineres rae. Volailiy is marked as, cos of carry g and ime o mauriy as. In case call opion is calculaed k =. On he oher hand, if pu Asian opion price is calculaed k =. N is he disribuion funcion of normal sandardized disribuion. Using firs and second derivaion of opion price value wih respec o spo price and firs derivaion wih respec o ime o mauriy he formulas for dela, gamma and hea of his opion were idenified. hea has been added o he model, because one day observaions are used for esing. his ime sep is long enough o cause decline of he ime value of Asian opions. rg 6 ke N ( kd ), () rg 6 e N'( d), (3) S 3 rg 6 r g ( ) '( ) Se kn kd N kd d 3 r kke rnkdk N' kdk d, (4) 3 3 where d S K ln r g he main idea was o creae porfolio of call and pu opions ha would be hedged agains he price movemen in chosen direcion. herefore he weighs of opions should reflec he change in price of he opposie opion. o esimae anicipaed change in he opion price we can use aylor polynomial wih delas, gammas and heas as funcion derivaions. Expeced changes in call and pu opion prices in up and down direcion can be expressed as: call call callup dsup ( dsup) calld, (5)!! call call calldown dsdown dsdown calld!! ( ), (6) pu pu puup dsup ( dsup) pud, (7)!! pu pu pu down dsdown ( dsdown) pud. (8)!! hese changes are largely depended on anicipaed movemen of underlying asse (ds up and ds down ). In his case hey were calculaed as average decrease

3 Dela-gamma-hea Hedging of Crude Oil Asian Opions 899 in prices and average increase in prices of pas 0 adequae observaions (days) before he porfolio creaion. If he change of he asse price is consan in boh direcions gamma hedged porfolio can keep is value in one seleced direcion and increase is value in opposie direcion. All parameers of call and pu opions mus be he same. Dela-gamma hedging is no aking ime ino accoun. Hence porfolio needs o be rebalanced wih high frequency or he sraegy mus be closed in shor ime, oherwise hedging would no be effecive. High frequency of rebalancing is ineffecive, because he gained profi is only small fracion of commissions paid for opening and closing posiions. I used daily price changes for my analysis and included heas o reflec ime value decay. Fridays have been excluded, because weekend is already oo long ime difference, beween opening and closing posiions. All porfolios creaed before non-rading days were omied. If we wan o hedge our porfolio agains decline in price i should be consruced as: pu down V call call down V pu, (9) or as pu up V call + call up V pu, (0) if we are anicipae prices o decrease and wan o hedge agains he increasing prices. his mehod was originally used for plain-vanilla opions (Šurc, 00), bu here are no boundaries for is applicaion on Asian opions. Cerainly, his is only heoreical concep and real price changes could srongly affec he sraegy performance. herefore I had o opimize he choice of he opions so he profi from movemen opposie o he hedged one, would be maximal. his can be accomplished by maximizing funcion: pu down call up call down pu up max. () Afer applying firs order condiion wih respec o he spo prices of underlying asse we obain he formula for he opimal spo price a which should be he profi from he sraegy maximal. 5 rg 6 S Ke. () I is impossible o choose he spo price, because i is given by marke. I is also possible o wai for he spo price o fulfill his rule. On he oher hand, we can find opion pair wih appropriae srike price, which holds he relaionship () in accordance o observed spo price. Opions were chosen if he difference beween he real and opimal srike price was less han 0 cens. Expeced changes of he underlying asse price have no effec on he value of opimal spo price. ha means ha hey affec only he weighs in porfolio. Even when he posiions are heoreically hedged, i is sill possible o gain loss afer all, because he price changes ac as was expeced raher occasionally. Simple predicive model was creaed o decide which price movemen is anicipaed in nex observaion (day in his case) k i i j ij k l l i j k l, (3) up S up where up is a dummy variable represening upward movemen of underlying asse, S is he spo price of underlying asse, depics ime (in number of observaions) elapsed since he las occurrence of upward movemen and κ is dummy variable idenifying levels of variable. Maximal ime beween wo upward momens was 7 days, hence i was se as maximal level for variables up and. Variables and are applied wih accordance o he mehodology of dealing wih ime dependency in dummy ime series (Carer, 00). Logi ransformaion was applied and he coefficiens where esimaed wih MLE (Heij, 004). his model was rebalanced every 00 rading days. hese models were se o sugges growh porfolio in cases when prediced probabiliy of price increase was greaer han 0.55, decline porfolio if he prediced probabiliy was lower han 0.45 and in he oher cases invesor have o wai ill nex day. Models were esimaed in 5 seps. Firs esimaion was based on 560 observaions prior he firs rade realizaion. hese esimaions were applied nex 00 rading days. Afer his ime, new observaions were added o daa se and model was revaluaed. his procedure was repeaed afer every 00 days, unil he end of simulaion. Daa All he calculaion are based on he prices of call and pu WI Asian opions wih fixed srike price and wih expiry dae on (G9J4C Comdy). he underlying asse of all analyzed derivaives was Ligh Swee Crude Oil fuures wih mauriy on (CSSJ4). Fig. suggess ha here were no huge changes in rend movemen. Even volailiy seems o be sable for enire ime, wihou any peaks, ha would cause radical changes in sraegy performance. Same variables for prices of Asian opions wih mauriy on have been used for comparison. Averaged price is calculaed from he prices from he end of las calendar rading day. Including new price ino average is called fixaion. Opions wih srike prices from $64 o $5 were chosen ino selecion sample. Bu only hose where boh call and pu opions were available. Fig. shows how many days before mauriy were he opions wih cerain srike price issued on he marke. As we can see mos of opions were issued for approximaely one year (from 00 o 50 rading days). However opion pair wih srike price $00 was available more han 800

4 900 Juraj Hruška 05 CrudeOilFuurePrices May Aug Nov Feb3 Jun3 Sep3 Dec3 Apr4 : Developmen of Crude Oil fuures prices wih mauriy dae on (in American dollars) from o Numberofdaysomauriy : Availabiliy of Crude oil Asian opions (in days) days before mauriy. Also opions wih srike prices $95 and $99 were available nearly 600 days before mauriy. hese hree opion pairs were used in majoriy of cases, during he firs year of simulaion. Risk-free ineres rae is represened by 0-year US governmen bonds. Volailiy of underlying asse was esimaed from implied volailiy calculaed from plain vanilla opion wih same underlying asse. Coss of carry are considered o be 0. All daa were obained from he Bloomberg daabase. Simulaed rading was evaluaed from he firs observaion, when here were acually exising opions wih desired parameers. ha occurred on (476 rading days before mauriy). Some of he opions were available even before ha dae, bu he opimal srikes defined by relaionship () were always much higher. Simulaion was sopped one monh before he expiry dae, because even pricing models performed srong inaccuracies during he las days before he opions mauriy. ha could have cause imbalance in porfolio weighs, which are derived from hese pricing models. RESULS he hedged porfolios are cerainly unable o perform he same resuls in pracice. During he whole observed period he sraegy based on gamma hedging wihou predicive model was raher inconsisen and oucomes were volaile, which was caused by he errors in price changes predicions. Hedging was no working perfecly, bu i was able o reduce significan par of he losses. Hence, he growh porfolios managed o creae profi 4.5% and he decline porfolios were slighly worse wih profi 6.48%. his posiive oucome was mosly creaed by leverage effec, parial hedging and minimal ime beween opening and closing of posiions. o enhance he profi and lower he volailiy of reurns I have combined his sraegy wih predicive model. Chosen model () did no show asonishing predicive power, bu is resuls were significan. he success rae and coefficien esimaions of predicion models for seleced periods when hey were applied are shown in ab. I. Firs and second lag of dependen variable were omied due o srong correlaion wih ime variable and ohers due o insignificance. he value of coefficiens is changing over ime, bu heir characer seems o say he same. hird lag of dependen variable has always posiive effec on he probabiliy of he nex upward movemen of he crude oil fuures prices. On he oher hand longer lags have compleely opposie effec. his suggess ha he dependen variable is moving in approximaely 3 day rends, during he observed period. Nex hree ime variables confirms exponenially rising

5 Dela-gamma-hea Hedging of Crude Oil Asian Opions 90 I: Predicive models Cons. up( 3) up( 5) up( 7) ime ime^ ime^3 Period (0.7858) (0.0038) (0.049) (0.0648) (0.44) (0.047) (0.0859) (0.0534) (0.0057) (0.003) (0.009) (0.4970) (0.0573) (0.085) (0.5098) 0.5 (0.088) (0.047) (0.0074) (0.0076) (0.0080) (0.055) (0.0) day afer growh (0.090) Accuracy 58.3% 56.% 54.5% 56.3% 55.8% Sommer s D II: Summary saisics of gamma hedging sraegies Porfolios Num. of rades Avg. Reurn Sd. deviaion Maximal Drawdown growh decline predicive probabiliy of nex up movemen wih he rising ime since he las upward movemen. he las variable idenifying day afer upward movemen has also posiive effec on he explained variable. his can be explained by he herd invesing, when mos of he invesors are expecing he rising rend in prices o coninue. Reconfiguraion of he model is definiely necessary; probably wih even higher frequency han every 00 days. Afer he implemenaion of he predicive model ino gamma hedging sraegy, significan improvemen in is performance is observed. During he observed period of 3 monhs 6 porfolios were creaed. Posiions were closed nex day o minimize ime impac on he esimaed changes in opion prices. Iniial capial was increased by 33.4%, wih average profi.44% per realized porfolio. I means ha he average profi was eigh imes higher han he average profi of growh porfolio and 5 imes higher han decline porfolios. All hree sraegies are relaively similarly risky (measured by sandard deviaion of reurns). If we compare sraegies by he maximal drawdown, sraegy wih predicive model is definiely less risky han oher wo sraegies. Deail comparison can be seen in ab. II. Prices of he underlying asse and he prices of Asian call opion wih srike price $00 (his opion was used in porfolios in mos of he cases) were used as benchmark for gamma hedging sraegy. Invesmen o crude oil fuures would bring profi 6.9% during observed period (average reurn was 0.07% wih riskiness.0377%). In case of Asian call opion he oucome will be loss 78.67% of iniial value (average reurn was 0.30% wih riskiness.55%). In boh cases invesmens were also combined wih he same predicive model used wih gamma hedging. Under hese circumsances average reurn of underlying asse is 0.045% wih riskiness % and average reurn of Asian call opion is 0.904% wih riskiness 7.648%. Hence, he riskreward raio of hese wo alernaives was 50.6 and If we compare i o he growh porfolio wih risk-reward raio 4., decline porfolio wih 0.58 and predicive porfolio 3.33, we can see ha, even hough gamma hedging carries a cerain amoun of risk, i is sill much beer opion han invesing o he crude oil fuures and crude oil Asian opions direcly wihou hedging. Gamma hedging was working nearly ideally during he firs year of esing. he original value of he porfolio was sable. On he oher hand, oucomes of sraegies became much more volaile in ime closer o he opions mauriy as can be seen in Fig. 3. his could no be creaed by he increasing volailiy of he underlying asse. here are no evidences of such influences in Fig.. he cause of he sraegy errors mus be inrinsic. One of he explanaions could be ha he specificaion of he pricing model is inappropriae. I was using

6 90 Juraj Hruška Porfoliosvaluedevelopmen May Aug Nov Feb3 Jun3 Sep3 Dec3 Apr4 Predicive Growh Decline 3: Developmen of he performance of he sraegies from o model for pricing Asian opions wih coninuous averaging, because his model bes fi he real daa. Bu hese opions are using discree monhly averaging, which may no play role when only a few prices were included ino he average and many of he prices remain uncerain. Mos of all, ime value is much higher and covers he differences in averaging ype. However, when he ime o mauriy was shrinking, his model could no bring adequae resuls, because he ime value of he opions were largely reduced and difference beween coninuous and discree average became significan; especially afer one before he las fixaion. During his period daily profis of he sraegies were in range from 30% o 40% (wih sandard deviaion.%, which means four imes higher volailiy). If I would coninue wih rading even afer he las price fixaion, he predicive porfolio would creae 38.58% of oal profi, he growh porfolios 4.4% and he decline porfolios 70.40%. he profis are much higher bu he risk conneced wih hem is no worh of i. Sraegy has been also esed on he oher se of opions wih he expiry dae on Resuls were significanly worse han in firs case. Growh porfolios creaed loss 4.85% and decline porfolio loss 6.99%. he porfolios wih predicive model have again performed beer, bu he overall profi was only.0%. Effecs of he commissions on he sraegy performance have also been esed. he commissions and rading coss varies depending on he broker providing he marke access. he average commission for rading plain vanilla opions is around 50 cens per conrac (according o Bloomberg saisics). hose for Asian opions are lower, because even he prices are lower. Gammadela-hea hedging sraegy is srongly sensiive o he effecs of spreads and commissions. Even commission such as cen eliminaed possibiliy of any profi. heirs effec is even sronger during he las monh before mauriy, when ime value of boh call and pu opions is declining faser. DISCUSSION AND CONCLUSION Resuls presened in his paper confirm ha Asian opion wih fixed srike price can be hedged successfully using he firs and second derivaion of he opion price. he profi of 33.4% in 6 realized rades is really saisfying oucome. his mehod could be used in long posiion in call or pu opions o assure posiion for shor ime or o profi from he shor erm speculaions. Hence I have explained ha Asian opions are as good for gamma based sraegies as plain-vanilla opions. Also i is more suiable for hese purposes because hey are cheaper and herefore much suiable for fiing he porfolios wih specific weighs. Gamma hedging could be upgraded o Speed hedging, bu adding hird derivaion wih respec o price of he underlying asse would have only slide effec on he porfolios. Amending he model wih hea would heoreically allow us o incorporae he ime effec ino he weighs. ha would mean ha porfolios could be hold for he longer ime and hence reduce he commissions originally needed for porfolio rebalancing. Spreads and commissions are one major phenomenon ha has been considered in analyses and has he srong poenial o reduce or compleely eliminae he profi and efficiency of he sraegy.

7 Dela-gamma-hea Hedging of Crude Oil Asian Opions 903 As he ime changes are considered o be zero, gamma hedging is mos suiable as high-frequency sraegy for invesors wih he possibiliy o inves wih minimal commissions on he marke wih low spreads. More sophisicaed mehods for esimaing price changes (han simple average) should be used on daa wih daily frequency. For example Box-Jenkins mehodology or oher predicive economerical models or models working wih sochasic volailiy. Acknowledgemen Suppor of Masaryk Universiy wihin he projec MUNI/A/7/04 Analýza, vorba a esování modelů oceňování finančních, zajišťovacích a invesičních akiv a jejich využií k predikci vzniku finančních krizí (Suden Projec Gran a MU, Faculy of Economics and Adminisraion, Deparmen of Finance) is graefully acknowledged. REFERENCES ALBRECHER, H., DHAENE, J., GOOVAERS, M., SCHOUENS, W. and PANG, L Saic Hedging of Asian Opions under Lévy Models. he Journal of Derivaives, (3): BAYRAKAR, E. and XING, H. 00. Pricing Asian opions for jump diffusion. Mahemaical Finance, (): CARER, D. B. and SIGNORINO, C. S. 00. Back o he fuure: Modeling ime dependence in binary daa. Poliical Analysis, 8(3): 7 9. DUBOIS, F. and LELIÈVRE, Efficien pricing of Asian opions by he PDE approach. Journal of Compuaional Finance, 8(): DENGLER, H. R., JARROW, A., CAPINSKI, M. and KOPP, E Opion pricing using a binomial model wih random ime seps (A formal model of gamma hedging). Review of Derivaives Research, (): HULL, J. C. 0. Opions, fuures, and oher derivaives. 8 h ediion. Harlow: Pearson Edusaion. HAUG, E. G he complee guide o opion pricing formulas. nd ediion. New York: McGraw-Hill. HEIJ, C. e al Economeric mehods wih applicaions in business and economics. s ediion. New York: Oxford Universiy Press Inc. JARROW, R. A. and URNBULL, S. M Dela, gamma and bucke hedging of ineres rae derivaives. Applied Mahemaical Finance, (): 48. LUCIANO, E., REGIS, L. and VIGNA, E. 0. Dela and Gamma Hedging of Moraliy and Ineres Rae Risk. SSRN Elecronic Journal, 50(3): [Online]. Available a: hp:// com. [Accessed: 05, February 5]. ROGERS, L. C. G., SHI, Z., DAHL, L. O., and BENH, F. E he Value of an Asian Opion. Journal of Applied Probabiliy, 3(4): 0 6. ŠURC, B. and ČURLEJOVÁ, L. 00. eoreické asapeky využiia gama hedgingu opcií. In: Finance a managemen v eorii a praxi: proceedings of he Inernaional Conference. Univerzia Jana Evangelisy Purkyně, 30 April. Úsí nad Labem: Univerzia Jana Evangelisy Purkyně, ZHANG, J. E Pricing coninuously sampled Asian opions wih perurbaion mehod. Journal of Fuures Markes, 3(6): ZHANG, P. G Flexible Arihmeic Asian Opions. he Journal of Derivaives, (3): ZHANG, P. G Exoic opions. nd ediion. Singapore: World Scienific Publishing Co.Pe.Ld. Juraj Hruška: [email protected] Conac informaion