Basket options and implied correlations: a closed form approach

Transcription

1 Basket optons and mpled correlatons: a closed form approach Svetlana Borovkova Free Unversty of Amsterdam CFC conference, London, January 7-8, 007

2 Basket opton: opton whose underlyng s a basket (.e. a portfolo) of assets. ( ) + Payoff of a European call basket opton: B ( T) X B(T) s the basket value at the tme of maturty T, X s the strke prce.

3 Commodty baskets Crack spreads: Qu * Unleaded gasolne + Qh * Heatng ol - Crude Soybean crush spread: Qm * Soybean meal + Qo * Soybean ol - Soybean Energy company portfolos: Q * E + Q * E + + Qn En, where Q s can be postve as well as negatve. 3

4 Motvaton: Commodty baskets consst of two or more assets wth negatve portfolo weghts (crack or crush spreads), Asan-style.\ The valuaton and hedgng of basket (and Asan) optons s challengng because the sum of lognormal r.v. s s not lognormal. Such baskets can have negatve values, so lognormal dstrbuton cannot be used, even n approxmaton. Most exstng approaches can only deal wth baskets wth postve weghts or spreads between two assets. Numercal and Monte Carlo methods are slow, do not provde closed formulae. 4

5 Our approach: Essentally a moment-matchng method. Basket dstrbuton s approxmated usng a generalzed famly of lognormal dstrbutons : regular, shfted, negatve regular or negatve shfted. The man attractons: applcable to baskets wth several assets and negatve weghts, easly extended to Asan-style optons allows to apply Black-Scholes formula provdes closed form formulae for the opton prce and greeks 5

6 Regular lognormal, shfted lognormal and negatve regular lognormal 6

7 Assumptons: Basket of futures on dfferent (but related) commodtes. The basket value at tme of maturty T B( T) N a. F ( T) a where : the weght of asset (futures contract), N : the number of assets n the portfolo, F ( T ): :the futures prce at the tme of maturty. The futures n the basket and the basket opton mature on the same date. 7

8 Indvdual assets dynamcs: Under the rsk adjusted probablty measure Q, the futures prces are martngales. The stochastc dfferental equatons for ( t) s df F ( t) () t σ. dw () t,,,3,..., N where F () t :the futures prce at tme t σ :the volatlty of asset () () ( j W t, W ) ( t ) :the Brownan motons drvng assets and j wth correlaton ρ, j ( ) F 8

9 Examples of basket dstrbuton: Fo Shfted lognormal [ 00;90]; σ [0.;0.3]; a [ ;]; X 0; r 3%; T year; ρ 0.9 Negatve shfted lognormal Fo [ 05;00]; σ [0.3;0.]; a [ ;]; X 5; r 3%; T year; ρ 0.9 9

10 The frst three moments and the skewness of basket on maturty date T : E E η σ B ( ) M ( T ). ( ) B T B( T ) N N N k j E N a F 0 N N ( B( T )) M ( T ) a. a. F ( 0 ). F ( 0 ).exp( ρ. σ. σ. T ) 3 ( B( T )) M ( T ) E 3 j k ( 0 ). F ( 0 ). F ( 0 ).exp( ρ. σ. σ. T + ρ. σ. σ. T + ρ. σ.. T ) a. a. a. F σ ( B( T ) E( B( T ))) σ 3 B( T ) j where ( T ): standard devaton of basket at the tme T j 3 j k j. j j, j. k j k j. k j k 0

11 If we assume the dstrbuton of a basket s shfted lognormal wth parameters m, s,τ, the parameters should satsfy non-lnear equaton system : M ( T ) exp ( m + s ( T ) τ +. τ.exp m + s + exp( m s ) M + M 3 3 ( T ) τ + 3. τ.exp m + s 3. τ.exp( m + 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 ( T ) and ( T ) to T. T M M 3 M 3 ( )

12 Shfted lognormal Fo [50;75]; σ [0.;0.3]; a [ ;]; X 50 ; r 3%; T year; ρ 0.8 Regular lognormal

13 Approxmatng dstrbuton: Skewness η > 0 η > 0 η < 0 η < 0 Locaton parameter τ 0 < 0 τ 0 τ τ < 0 Approxmatng dstrbuton regular shfted negatve negatve shfted 3

14 Valuaton of a call opton (shfted lognormal): Suppose that the dstrbuton of basket s lognormal. Then the opton on such basket can be valued by applyng the Black-Scholes formula. Suppose that the relatonshp between basket and basket s B () () ( t) B t ( ) + τ The payoff of a call opton on basket wth the strke prce s: () ( ( ) ) + () () B T X ( B ( T ) + τ ) X ) ( B ( T ) ( X τ )) + It s the payoff of a call opton on basket wth the strke prce X ( X τ ) 4

15 Valuaton of call opton (negatve lognormal): Suppose agan that the dstrbuton of basket s lognormal. Then the opton on such basket can be valued by applyng the Black-Scholes formula. Suppose that the relatonshp between basket and basket s B () () ( t) B ( t) The payoff of a call opton on basket wth the strke prce s: () () ( B ( T ) X ) B ( T ) () ( X ) ( X ) B ( T )) + + It s the payoff of a put opton on basket wth the strke prce X X 5

16 Closed form formulae of a basket call opton: For e.g. shfted lognormal : c exp ( rt )[( M ( T ) τ ). N ( d ) ( X τ ) N ( d )]. where d d log ( M ( T ) τ ) log( X τ ) + V V log ( M ( T ) τ ) log( X τ ) V V V M log ( T ). τ M ( ) + ( ( ) ) T τ M T τ. ( ) It s the call opton prce wth strke prce X τ. 6

17 Algorthm for prcng general basket opton: Compute the frst three moments of the termnal basket value and the skewness of basket. η If the basket skewness s postve, the approxmatng dstrbuton s regular or shfted lognormal. If the basket skewness η s negatve, the approxmatng dstrbuton s negatve or negatve shfted lognormal. By moments matchng of the approprate dstrbuton, estmate parameters m, s,τ. Choose the approxmatng dstrbuton on the bass of skewness and the shft parameter τ. 7

18 Basket 5: Fo [95;90;05]; σ [0.;0.3;0.5]; a [; 0.8; 0.5]; X 30; ρ, ρ,3 0.9; ρ T year; r 3%,3 0.8; Smulaton results: η < 0 ; τ < 0 (neg. shfted) Call prce: (7.799) 8

19 Basket 6 : Fo [00;90;95]; σ [0.5;0.3;0.]; a [0.6;0.8; ]; X 35; ρ, ρ,3 0.9; ρ T year; r 3%,3 0.8; η > 0 ; τ < 0 (shfted) Call prce : (9.0) 9

20 Basket Basket Basket 3 Basket 4 Basket 5 Basket 6 Futures prce (Fo) [00;0] [50;00] [50;75] [00;50] [95;90;05] [00;90;95] Volatlty (σ ) [0.;0.3] [0.3;0.] [0.;0.3] [0.;0.5] [0.;0.3;0.5] [0.5;0.3;0.] Weghts (a) [-;] [-;] [0.7;0.3] [-;] [; -0.8; -0.5] [0.6;0.8; -] Correlaton (ρ) ρ ρ,, 3 ρ, ρ ρ,, 3 ρ, Strke prce (X) skewness (η) Locaton parameter (τ ) η > 0 τ < 0 η < 0 η > 0 η < 0 η < 0 η > 0 τ < 0 τ > 0 τ > 0 τ < 0 τ < 0 T year; r 3 % 0

21 Monte carlo Krk Bacheler Our approach Method (0.0095) neg. shfted Basket (0.05).9663 (0.0044) 0.8 (0.083) (0.04) (0.043) shfted.9576 neg. regular regular neg. shfted shfted Basket 6 Basket 4 Basket3 Basket Basket

22 Performance of Delta-hedgng: Hedge error: the dfference between the opton prce and the dscounted hedge cost (the cost of mantanng the deta-hedged portfolo); computed on the bass of smulatons. Plot the rato between the hedge error standard devaton to call prce vs. hedge nterval. Basket : Fo [00;0]; σ [0.;0.5]; a [ ;]; ρ 0.9; X 0; T year ; r 3% Mean of hedge error s 4 % for basket and 7 % for basket. Basket : Fo [95;90;05]; σ [0.;0.3;0.5] a [; 0.8; 0.5]; ρ X 30; T, ρ,3 year; r 3% 0.9; ρ,3 0.8;

23 Greeks: correlaton vega Spread [0,0], vols[0.5,0.] 0 vega wth respect to correlaton strke prce correlaton 0.5 3

24 Correlaton vega vs. correlaton and vs. strke 0 0 vega wth respect to correlaton 4 6 vega wth respect to correlaton correlaton strke prce 4

25 Volatlty vegas vs. volatltes same spread, X0, correlaton0.8 vega wth respect to sgma vega wth respect to sgma sgma sgma sgma sgma

26 Volatlty vegas and call prce sgma0.3 sgma vega wth respect to sgma 0 0 call prce 9 8 vega wth respect to sgma call prce sgma sgma sgma sgma 6

27 7 Asan baskets Underlyng value: (arthmetc) dscrete average basket value over a certan nterval The same approach as above apples, because So the average basket value s smply the basket of ndvdual assets averages, wth the same weghts, so the above approach apples drectly, only wth dfferent moments! opton prces and greeks agan calculated analytcally. N t t k N k t t k k B T A a t F a n t B n T A n n ) ( ) ( ) ( ) (

28 Impled correlatons from spreads For spreads, when lqud opton prces are observed, the opton prce formula can be now nverted to obtan mpled correlaton (volatltes mpled from ndvdual assets optons). Correlatons mpled from NYMEX Brent crude ol/heatng ol Asan spread optons: Strke Oct. Oct. 3 Oct. 6 Oct. 7 Oct. 8 Oct *** 0.98 ***

29 Impled correlatons vs strkes.4. mpled correlaton strke prce ($/bbl) 9

30 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 30