Basket options and implied correlations: a closed …

1 1 Basket optionsand implied correlations : a closed form approach Svetlana BorovkovaFree University of AmsterdamCFC conference, London, January 17-18, 20072 Basket option: option whose underlying is a Basket ( a portfolio) of assets. Payoff of a European call Basket option: is the Basket value at the time of maturity ,is the strike price.()+ XTB)()(TBTX3 Commodity baskets Crack spreads: Qu * Unleaded gasoline + Qh * Heating oil - Crude Soybean crush spread:Qm * Soybean meal + Qo * Soybean oil - Soybean Energy company portfolios:Q1 * E1 + Q2 * E2 + .. + Qn En,where Qi s can be positive as well as : Commodity baskets consist of two or more assets with negative portfolio weights (crack or crush spreads), Asian-style.

2 \ The valuation and hedging of Basket (and Asian) options is challenging because the sum of lognormal s is not lognormal. Such baskets can have negative values, so lognormal distribution cannot be used, even in approximation. Most existing approaches can only deal with baskets with positive weights or spreads between two assets. Numerical and Monte Carlo methods are slow, do not provide closed approach : Essentially a moment-matching method. Basket distribution is approximated using a generalized family of lognormal distributions : regular, shifted, negative regular or negative main attractions: applicable to baskets with several assets and negative weights, easily extended to Asian-style options allows to apply Black-Scholes formula provides closed form formulae for the option price and greeks6 Regular lognormal, shifted lognormal and negative regular lognormal7 Assumptions: Basket of futures on different (but related) commodities.

3 The Basket value at time of maturity T where: the weight of asset (futures contract) i,: the number of assets in the portfolio,: :the futures price iat the time of maturity . The futures in the Basket and the Basket option mature on the same date.() ==NiiiTFaTB1.)(iaN()TFi8 Individual assets dynamics:Under the risk adjusted probability measure Q, the futures prices are martingales. The stochastic differential equations for iswhere:the futures price iat time t:the volatility of asset i:the Brownian motions driving assets iand jwith correlation()tFi()()()NitdWtFtdFiiii,.., 3,2,1,.)(== ()tFii ()()()()tWtWji,ji, 9 Examples of Basket ;1%;3;10];1;1[]; ; [];90;100[=== = === ;1%;3;5];1;1[]; ; [];100;105[=== = === yearTrXaFoShifted lognormalNegative shifted lognormal10 The first three moments and the skewness of Basket on maturity date T:where : standard deviation of Basket at the time T() () ()() ===++= ()()()()3)(3)(TBTBTBETBE =()TB () ()() ===NiiiFaTMTBE110.

4 ()( )()() ()() ====NjNijijijijiTFFaaTMTBE11, ()()()==TMTBE3311 If we assume the distribution of a Basket is shifted lognormal with parameters , the parameters should satisfy non-linear equation system : If we assume the distribution of a Basket is negative shifted lognormal, the parameters should satisfy non-linear equation system above by changing to and to . ,,sm() +=2121(expsmTM()() ++ ++= ()() ++++ ++= )(1TM()TM1 ()TM3()TM3 12 Shifted lognormalRegular ;1%;3;150];1;1[]; ; [];175;50[==== === yearTrXaFo13 Approximating distribution:negative shiftednegativeshiftedregularApproximati ng distributionLocation parameterSkewness0> 0> 0< 0 0 0< 0< 0< 14 Valuation of a call option(shifted lognormal): Suppose that the distribution of Basket 1 is lognormal.)

5 Then the option on such Basket can be valued by applying theBlack-Scholes formula. Suppose that the relationship between Basket 2 and Basket 1 is The payoff of a call option on Basket 2 with the strike price is:It is the payoff of a call option on Basket 1 with the strike price () +=tBtB)1()2()(X()()()()()() ( )()++ = += XTBXTBXTB)1()1()2(() X15 Valuation of call option (negative lognormal): Suppose again that the distribution of Basket 1 is lognormal. Then the option on such Basket can be valued by applying theBlack-Scholes formula. Suppose that the relationship between Basket 2 and Basket 1 is The payoff of a call option on Basket 2 with the strike price is:It is the payoff of a put option on Basket 1 with the strike price ()()tBtB)1()2( =X()()()()() ()()++ = = TBXXTBXTB)1()1()2(X 16 closed form formulae of a Basket call option: For shifted lognormal :whereIt is the call option price with strike price.

6 ()()()()( )()[] =()()()VVXTMd21121loglog+ = ()()()VVXTMd21221loglog = ()()()() + = TMTMTMV() X17 Algorithm for pricing general Basket option: Compute the first three moments of the terminal Basket value andthe skewness of Basket . If the Basket skewnessis positive, the approximating distribution is regular or shifted lognormal. If the Basket skewnessis negative, the approximating distribution is negative or negativeshifted lognormal. By moments matching of the appropriate distribution, estimate parameters . Choose the approximating distribution on the basis of skewness and the shift parameter.

7 ,,sm 18 Simulation results:%3;1; ; ;30]; ; ;1[]; ; ; [];105;90;95[:53,13,22,1===== = ===ryearTXaFoBasket 0;< 0< Call price: ( )(neg. shifted)19%3;1; ; ;35];1; ; []; ; ; [];95;90;100[:63,13,22,1====== ===ryearTXaFoBasket 0> 0;< (shifted)Call price : ( )20 Location parameterskewness35-30-140104-5020 Strike price(X) [ ; ; -1][1; ; ][-1;1][ ; ][-1;1][-1;1]Weights(a)[ ; ; ][ ; ; ][ ; ][ ; ][ ; ][ ; ]Volatility[100;90;95][95;90;105][200;50][50;175][150;100][100;120]Futures price(Fo) Basket 6 Basket 5 Basket 4 Basket 3 Basket 2 Basket 10> 0< )( )( )( )( ==3,22,1 =3,1 0< 0> ==3,22,1 =3,1 0> 0> 0< 0< 0> 0< 0< 0< T=1 year.

8 R = 3 %21 Monte carloKirkBachelierOur ( ) shiftedBasket ( ) ( ) ( ) ( ) ( ) 6 Basket 4 Basket3 Basket 2 Basket 122 Performance of Delta-hedging: Hedge error: the difference between the option price and the discounted hedge cost (the cost of maintaining the deta-hedged portfolio); computed on the basis of simulations. Plot the ratio between the hedge error standard deviation to call price vs. hedge interval. Mean of hedge error is 4 % for Basket 1 and 7 % for Basket ;1;10; ];1;1[]; ; [];110;100[:1==== ===ryearTXaFoBasket %3;1;30; ; ]; ; ;1[] ; ; [];105;90;95[:23,13,22,1== ==== ===ryearTXaFoBasket 23 Greeks: correlation vegaSpread [110,10], vols=[ , ] 1 10010203040 10 8 6 4 20correlationstrike pricevega with respect to correlation24 Correlation vegavs.

9 Correlation and vs. strike 1 10 8 6 4 202correlationvega with respect to correlation 10 50510152025303540 10 9 8 7 6 5 4 3 2 10strike pricevega with respect to correlation25 Volatility vegas vs. volatilitiessame spread, X=10, correlation= 40 30 20 1001020304050sigma 1sigma 2vega with respect to sigma 1 40 30 20 10010203040sigma 2sigma 1vega with respect to sigma 2 26 Volatility vegas and call pricesigma1= sigma2= 40 30 20 100102030sigma 2vega with respect to sigma 2call 40 30 20 10010203040sigma 1vega with respect to sigma 1call price27 Asian baskets Underlying value: (arithmetic) discrete average Basket value over a certain interval The same approach as above applies, becauseSo the average Basket value is simply the Basket of individual assets averages, with the same weights, so the above approach applies directly, only with different moments!

10 Option prices and greeks again calculated analytically. =======NiiittkNikiittkkBTAatFantBnTAnn11 )()(1)(1)(1128 implied correlations from spreadsFor spreads, when liquid option prices are observed, the option price formula can be now inverted to obtain implied correlation (volatilities implied from individual assets options ). correlations implied from NYMEX Brent crude oil/heating oil Asian spread ** ** 19 Oct. 18 Oct. 17 Oct. 16 Oct. 13 Oct. 12 Strike29 implied correlations vs price ($/bbl) implied correlation30 ConclusionsOur proposed method: Has advantages of lognormal approximation Applicable to several assets, negative weights and Asian Basket options Provides good approximation of option prices Gives closed - form expressions for the greeks Performs well on the basis of delta-hedging Allows to imply correlations from liquid spread options