Asan basket optons and mpled correlatons n ol markets Svetlana Borovkova Vre Unverstet Amsterdam, he etherlands Jont work wth Ferry Permana (Bandung)
Basket opton: opton whose underlyng s a basket (e a portfolo) of assets Partcular case of a basket opton: spread opton Payoff of a European basket call opton: B () X s the basket value at the tme of maturty, s the strke prce B ( ) X Payoff of an Asan basket call opton: B() s replaced by A(): the average basket value between tmes 0 and
Commodty baskets and spreads Crack spread: ku * Unleaded gasolne + kh * Heatng ol - Crude Energy company portfolos: k * E + k * E + + kn En where k s can be postve as well as negatve (e a portfolo can contan both long and short postons) 3
Motvaton: Commodty portfolos contan two or more assets, and often contan both long and short postons he valuaton and hedgng of optons on such portfolos (e basket optons) s challengng because the sum of lognormal rv s s not lognormal Such portfolos can have negatve values, so lognormal dstrbuton cannot be used, even n approxmaton Most exstng approaches can only deal wth regular basket optons or optons on a spread between two assets (Kemna and Vorst, Krk, urnbull and Wakeman, ) umercal and Monte Carlo methods are slow, do not provde closed formulae eed to extend prcng and hedgng to Asan basket optons 4
GL (Generalzed Lognormal) approach: Essentally a moment-matchng method Portfolo (e basket) dstrbuton s approxmated usng a generalzed famly of lognormal dstrbutons : shfted or negatve shfted lognormal l dstrbuton b t he man attractons: applcable to optons on portfolo wth several long and short postons naturally extendable to Asan-stylestyle optons allows to apply Black-Scholes formula provdes closed formulae for the opton prce and the greeks 5
Regular lognormal, shfted lognormal and negatve regular lognormal 6
Assumptons: Portfolo conssts of futures on dfferent (but related) commodtes he portfolo s value at tme of opton maturty B( ) where : the weght of asset (futures contract) t), : the number of assets n the portfolo, : the futures prce at the tme of maturty F a a F he futures n the portfolo and the opton on t mature on the same date 7
Indvdual assets dynamcs: Under the rsk adusted probablty measure Q, the futures prces are martngales he stochastc dfferental equatons for t s where W t F, W t t df F t t dw ( ) t :the futures prce at tme t :the volatlty of asset,,,3,, :the Brownan motons drvng assets and wth correlaton, F 8
he frst three moments and the skewness of the basket on t t d t b l l t d maturty date can be calculated: a F M B E 0 F F a a M B E, 0 exp 0 M B E 3 3 k k k k k k k F F F a a a 0 exp 0 0 3 ) ( 3 ) ( B B B E B E where : standard devaton of basket at the tme B 9
If we assume the dstrbuton of a basket s shfted lognormal wth parameters m, s,, the parameters should satsfy non-lnear equaton system : M M M exp ( m s expm s expm s 3 3 3 exp m s 3 expm s exp 3m s 9 If we assume the dstrbuton of a basket s negatve shfted lognormal, the parameters should satsfy non-lnear equaton system above by changng M ( ) to and to M M 3 M 3 0
Approxmatng dstrbuton: b t Skewness 0 0 Approxmatng dstrbuton shfted negatve shfted
Examples of termnal basket value dstrbuton: Fo Shfted lognormal [ 00;90]; [0;03]; a [ ;]; X 0; r 3%; year ; 09 egatve shfted lognormal Fo [ 05;00]; [03;0]; a [ ;]; X 5; r 3%; year; 09
Valuaton of a European call opton (shfted lognormal): Suppose that the dstrbuton of basket s lognormal hen an opton on such a basket can be valued by applyng the Black-Scholes (actually Black s (975)) formula Suppose that the relatonshp between basket and basket s B () ( t) B () t he payoff of a call opton on basket wth the strke prce s: () () () B X B X B X X It s the payoff of a call opton on basket wth the strke prce X 3
Valuaton of call opton (negatve lognormal): Suppose agan that the dstrbuton of basket s lognormal (an opton on such a basket can be valued by applyng the Black-Scholes formula) Suppose that the relatonshp between basket and basket s B () t B he payoff of a call opton on basket wth the strke prce s: () () B X B () t () X X B X It s the payoff of a put opton on basket wth the strke prce X 4
Closed form formulae for a (European) basket call opton: For eg shfted lognormal : c exp r M d X d where log d V d log M logx V M logx V V V M log M M It s the call opton prce wth strke prce X Dfferentate t wrt parameters analytc expressons for the greeks 5
Asan baskets Underlyng: average basket value over a certan nterval ote: t t A a t F a t B A n n ) ( ) ( ) ( ) ( So the average basket value s smply the basket of ndvdual assets averages wth the same weghts t k k t k k B A a t F a n t B n A ) ( ) ( ) ( ) ( averages, wth the same weghts assets averages are approxmated by lognormal dstrbutons, by matchng frst two moments (as n Wakeman method) the GL approach then apples drectly, only wth dfferent moments (calculated from the moments of the average asset prces) closed-form expressons for opton prces and greeks 6
Basket Basket Basket 3 Basket 4 Basket 5 Basket 6 Futures prce (Fo) Volatlty ( ) Weghts (a) [00;0] [50;00] [0;90] [00;60] [95;90;05] [00;90;95] [0;03] [03;0] [03;0] [03;0] [0;03;05] [05;03;0] [-;] [-;] [07;03] [-;] [; -08; -05] [06;08; -] Correlaton () 09 03 09 09,, 3,3 09 08,, 3,3 09 08 Strke prce (X) 0-50 04-40 -30 35 skewness () 0 Locaton parameter 0 ( ) = year; r = 3 % 0 0 0 0 0 0 0 0 0 0 7
Smulaton results (Asan call opton prces) Method Basket Basket Basket3 Basket 4 Basket 5 Basket 6 GL 59 8 85 45 595 7 neg shfted neg shfted shfted neg shfted neg shfted neg shfted Monte Carlo 59 75 8 45 595 706 (0) (00) (00) (00) (00) (003) 8
Greeks: correlaton vega Spread [00,0], vols=[0,03] 0 vega wth respect to corre elaton 4 6 8 0 40 30 0 strke prce 0 0 0 05 0 correlaton 05 9
Volatlty vegas vs volatltes same spread, X=0, correlaton=09 vega wth respect to sgma 50 40 30 0 0 0 0 0 30 40 04 vega wth respect to sgma 40 30 0 0 0 0 0 30 40 04 03 04 03 035 0 03 0 05 0 0 05 0 0 0 005 0 sgma sgma 005 sgma 0 05 0 sgma 05 03 035 04 0
Impled correlatons from spreads Basket opton prce formula can be nverted to obtan mpled correlaton Volatltes can be mpled from ndvdual assets optons, eg AM Correlatons mpled from YMEX Brent crude ol/heatng ol Asan spread optons n 009 (AM mpled vols for Brent and HO were used): Strke Oct Oct 3 Oct 6 Oct 7 Oct 8 Oct 9 5 05 0 048 06 03 00 45 037 037 044 044 067 067 05 05 054 054 035 035 4 057 064 08 069 073 059 35 073 073 079 079 093 093 083 083 088 088 076 076 096 097 098 096
Impled correlatons vs strkes 4 mpled correlaton 08 06 04 0 0 5 3 35 4 45 5 strke prce ($/bbl)
Conclusons Our proposed method: Has advantages of lognormal approxmaton Applcable to several assets, negatve weghts and Asan basket optons Provdes good approxmaton of opton prces Gves closed-form expressons for the greeks Performs well on the bass of delta-hedgng Allows to mply correlatons from lqud spread optons 3