# An Analysis of Pricing Methods for Baskets Options

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2 TECHNICAL ARTICLE 5 Brownan motons and are therefore log-normally dstrbuted. As the sum of log-normally dstrbuted random varables s not log-normal, t s not possble to derve an (exact closed-form representaton of the basket call and put prces. Due to the fact that we are dealng wth a multdmensonal process, only Carlo or Quas- Carlo (and over mult-dmensonal ntergraton methods are sutable numercal methods to determne the value of these optons. As these methods can be very tme consumng we wll present alternatve valuaton methods whch are based on analytcal approxmatons n dfferent senses. 2...and here are the canddates! a Besser s condtonal expectaton technques Besser (1999 adapts an dea of Rogers and Sh (1995 ntroduced for prcng Asan optons. By condtonng on the random varable Z and usng Jensen s nequalty the prce of the basket call s estmated by the weghted sum of (artfcal European call prces, more precsely E([B(T K] + = E(E([B(T K] + Z E(E([B(T K Z] + ([ ] + = E w E[S (T Z] K Z := σ z T W(T = w S (σ W (T wth σ z approprately chosen. Note that n contradcton to S (T, all condtonal expectatons E[S (T Z] are log-normally dstrbuted wth respect to one Brownan moton W(T. Hence, there exsts an x, such that By defnng: w E[S (T W(T = x ] = K K := E[S (T W(T = x ] the event n w E[S (T Z] K s equvalent to E[S (T Z] K for all = 1,...,n. Usng ths argument we conclude that ([ ] + E w E[S (T Z] K = = w E([E[S (T Z] K ] + w [ F T N(d 1 K N(d 2 ] F T, K adjusted forwards and strkes and d 1, d 2 are the usual terms wth modfed parameters. b Gentle s approxmaton by geometrc average Gentle (1993 approxmates the arthmetc average n the basket payoff by a geometrc average. The fact that a geometrc average of lognormal random varables s agan log-normally dstrbuted allows for a Black- Scholes type valuaton formula for prcng the approxmatng payoff. More precsely after rewrtng the payoff of the basket opton as [ ( ] + P Basket (B(T, K,θ= θ w S (T K a = w F T, w F T S (T = S (T F T [ (( ] + = θ w F T a S (T K, ( = exp 1 2 T σ 2 ds + T σ dw (s we approxmate n a S (T by the geometrc average, thus ( n ( B(T = w F T S (T a. To correct for the mean, K = K (E(B(T E( B(T s ntroduced. As approxmaton for (B(T K +, ( B(T K + s used, whch as B(T s log-normally dstrbuted can be valued by the Black- Scholes formula resultng n ( V Basket (T = Df(Tθ e m+ 1 2 ṽ2 N(θd 1 K N(θd 2, (2.1 Df (T s the dscount factor, N( the dstrbuton functon of a standard normal random varable and d 1 = m log K + ṽ 2, ṽ d 2 = d 1 ṽ, ( m = E(log B(T = log w F T 1 2 ṽ 2 = Var(log B(T = j=1 a a j σ σ j ρ j T. c Levy s log-normal moment matchng a σ 2 T and The basc dea of Levy (1992 s to approxmate the dstrbuton of the basket by a log-normal dstrbuton exp(x wth mean M and varance ^ Wlmott magazne 83

3 V 2 M 2, such that the frst two moments of ths and of the orgnal dstrbuton of the weghted sum of the stock prces concde,.e. result n m = 2 log(m.5 log(v 2 v 2 = log(v 2 2 log(m and M E(B(T = w F (T V 2 E(B 2 (T = w w j F T FT j exp(σ σ j ρ j T E(B(T = E ( e X = e m+.5v2 and E(B 2 (T = E ( e 2X = e 2m+2v2 X s a normally dstrbuted random varable wth mean m and varance v 2. The basket opton prce s now approxmated by wth V Basket (T Df(T (MN(d 1 KN(d 2 m ln(k + v2 d 1 =, v d 2 = d 1 v. Note the subtle dfference to Gentle s method. Here, the dstrbuton of B(T s approxmated drectly by a log-normal dstrbuton that matches the frst two moments, whle n Gentle s approxmaton only the frst moment s matched. d Ju s Taylor expanson Ju (22 consders a Taylor expanson of the rato of the characterstc functon of the arthmetc average to that of the approxmatng lognormal random varable around zero volatlty. He ncludes terms up to σ 6 n hs closed-form soluton. Let ( A(z = F T exp 1 2 (zσ 2 T + zσ W (T be the arthmetc mean were the volatltes are scaled by a parameter z. Note that for A(1 we recover the orgnal mean. Let Y(z be a normally dstrbuted random varable wth mean m(z and varance v(z such that the frst two moments of exp(y(z match those of A(z. The approprate parameters are derved n secton c, only σ has to be replaced by zσ. Let X(z = log(a(z,then the characterstc functon s gven as: E [ e φx(z] = E [ e φy(z] E [ e φx(z] E [ e φy(z] = e φm(z φ 2 v(z/2 E [ e φy(z] = E [ e φy(z] f (z, f (z = E [ e φx(z] e φm(z+φ 2 v(z/2 Ju performs a Taylor expanson of the two factors of f (z up to z 6, leadng to f (z 1 φd 1 (z φ 2 d 2 (z + φ 3 d 3 (z + φ 4 d 4 (z, d (z are polynomals of z and terms of hgher order then z 6 are gnored. Fnally E [ e φx(z] s approxmated by E [ e φx(z] e φm(z φ 2 v(z/2 (1 φd 1 (z φ 2 d 2 (z + φ 3 d 3 (z + φ 4 d 4 (z. For ths approxmaton, an approxmaton of the densty h(x of X(1 s derved as ( d h(x = p(x + dx d 1(1 + d2 dx d 2(1 + d3 2 dx d 3(1 + d4 3 dx d 4(1 p(x 4 p(x s the normal densty wth mean m(1 and varance v(1. The approxmate prce of a basket call s then gven by, {[( ] V Basket (T = Df(T w F T N(d 1 KN(d 2 [ ]} dp(y d 2 p(y +K z 1 p(y + z 2 + z 3, dy dy 2 y = log(k, d 1 = m(1 y v(1 + v(1, d 2 = d 1 v(1 and z 1 = d 2 (1 d 3 (1 + d 4 (1, z 2 = d 3 (1 d 4 (1, z 3 = d 4 (1. Note that the frst summand s equal to Levy s approxmaton and the second summand gves the hgher order correctons. e The recprocal gamma approxmaton by Mlevsky and Posner Mlevsky and Posner (1998 use the recprocal gamma dstrbuton as an approxmaton for the dstrbuton of the basket. The motvaton s the fact that the dstrbuton of correlated log-normally dstrbuted random varables converges to the recprocal gamma dstrbuton as n. Consequently, the frst two moments of both dstrbutons are matched to obtan a closed-form soluton. Let G R be the recprocal gamma dstrbuton and G the gamma dstrbuton wth parameters α, β, then per defnton: G R (y,α,β= 1 G(1/y,α,β If the random varable Y s recprocally gamma dstrbuted, then E[Y ] = 1 β (α 1(α 2...(α and wth M and V 2 denotng the frst two moments as defned n the prevous secton, we get: α = 2V2 M 2 V 2 M 2 β = V2 M 2 V 2 M 84 Wlmott magazne

5 TABLE 1: VARYING THE CORRELATIONS SIMULTANEOUSLY ρ Besser Gentle Ju Levy MP-RG MP-4M Carlo CV StdDev,1 2,12 15,36 21,77 22,6 2,25 21,36 21,62 (,319,3 24,21 19,62 25,5 25,17 22,54 24,91 24,97 (,249,5 27,63 23,78 28,128,5 24,5 27,98 27,97 (,187,7 3,62 27,98 3,74 3,75 26,18 3,74 3,72 (,123,8 31,99 3,13 32,4 32,4 26,93 32,4 32,3 (,87,95 33,92 33,4133,92 33,92 27,97 33,92 33,92 (,24 Dev. 1,7 4,13,71,23 4,119,18 1 Dev. = n n (Prce MC Prce2. bad performance of MP-RG for hgh correlatons can be explaned by the fact, that wth effectvely one stock we are far away from nfntely many stocks, whch s the motvaton for ths method. A test wth fxed correlaton ρ 12 =.95 and varyng the remanng correlatons symmetrcally shows exactly the same result. In total the prces calculated by Ju s approach (whose method slghtly overprces and MP-4M are overall the closest to the Carlo prces. These approaches are followed by Levy s and Besser s approxmaton (whose approach slghtly underprces. The other two methods are not recommendable. b Varyng the strkes Wth all other parameters set to the default values, the strke K s vared from 5 to 15. Table 2 contans the results. The dfferences between the prces calculated by Carlo and the approaches of Ju and MP-4M are relatvely small. The prce curves of TABLE 2: VARYING THE STRIKE K Besser Gentle Ju Levy MP-RG MP-4M Carlo CV StdDev 5, 54,16 51,99 54,31 54,34 51,93 54,35 54,28 (,383 6, 47,27 44,43 47,48 47,52 44,4147,5 47,45 (,375 7, 41,26 37,93 41,52 41,57 38,1 41,53 41,5 (,369 8, 36,4 32,4 36,36 36,4 32,68 36,34 36,52 (,363 9, 31,53 27,73 31,88 31,92 28,22 31,86 31,85 (,356 1, 27,63 23,78 28,1 28,5 24,5 27,98 27,98 (,35 11, 24,27 2,46 24,67 24,7 21,39 24,63 24,63 (,344 12, 21,36 17,65 21,77 21,8 18,77 21,73 21,74 (,338 13, 18,84 15,27 19,26 19,28 16,57 19,22 19,22 (,332 14, 16,65 13,25 17,7 17,1 14,7 17,4 17,5 (,326 15, 14,75 11,53 15,17 15,19 13,1 15,14 15,15 (,32 Dev.,323 3,746,31,65 3,38,3 the method of Gentle and Mlevsky s and Posner s recprocal gamma approach (MP-RG run almost parallel to the Carlo curve and represent an under-evaluaton. The relatve and absolute dfferences of all methods are generally ncreasng when K s growng, snce the approxmaton of the real dstrbutons n the tals s gettng worse and the absolute prces are decreasng. Agan, overall Ju s approxmaton and MP-4M perform best, whle Ju s slghtly overprces. Levy s the thrd and Besser the fourth best. c Varyng the Forwards and Strkes The forwards on all stocks are now set to the same value F whch s vared between 5 and 15 n ths set of tests. Table 3 shows that MP-4M and Ju s method perform excellently, whle the second one agan typcally slghtly overprces. Levy and Besser s method also perform well and Besser agan slghtly underprces. The other methods perform worse. These effects can also be seen f some forwards are fxed and the remanng ones are vared. d Varyng the Volatltes The next set of tests nvolves varyng the volatltes σ. We start wth the symmetrcal stuaton at each step, σ s set to the same value σ, whch s vared between 5% and 1%. Table 4 shows the results of the test. The prces calculated by the dfferent methods are more or less equal for small values of the volatlty. They start to dverge at σ 2%. The Carlo, Besser, Ju and Levy prces reman close, as the prces calculated by the other methods are too low. The pcture obtaned so far completely changes f we have asymmetry n the volatltes, precsely f there are groups of stocks wth hgh and wth low volatltes enterng the basket. Ths s clearly demonstrated by Fgure 1 we fx σ 1 = 5% and vary the remanng volatltes symmetrcally. Ths tme the prces dverge much more. The method Levy s massvely overprcng wth all other methods underprcng. We note that Ju s and Besser s method performs best. Partcularly remarkable s the excellent performance of Ju for hgh volatltes. Snce t s a Taylor expanson around zero volatltes, one would not expect the valdty of ths expanson far away from zero. The same test but now wth σ 1 = 1% results n Table 5 and Fgure 2. Note the extremely bad performance for Levy s method for small values of σ whch t s even outperformed by Gentle s method! Besser the s only one who can deal wth ths parameters, whle both Mlevsky and Posner methods are also bad. e Implct dstrbutons In addton we plot the mplct dstrbuton of the partcular approxmatons and compare them to the real ones calculated by Carlo smulaton. Wth mplct dstrbuton we mean, that we derve the underlyng dstrbuton of the partcular method by an approprate portfolo of calls. Consder 86 Wlmott magazne

6 TECHNICAL ARTICLE 5 TABLE 3: VARYING THE FORWARDS SYM. WITH K = 1 F Besser Gentle Ju Levy MP-RG MP-4M Carlo CV StdDev 5, 4,16 3, 4,34 4,34 3,93 4,33 4,34 (,141 6, 7,27 5,53 7,517,52 6,56 7,5 7,5 (,185 7, 11,26 8,91 11,55 11,57 9,95 11,53 11,53 (,227 8, 16,4 13,13 16,37 16,4 14,1 16,34 16,35 (,268 9, 21,53 18,11 21,89 21,92 18,97 21,86 21,86 (,39 1, 27,63 23,78 28,1 28,5 24,5 27,98 27,98 (,35 11, 34,27 3,8 34,66 34,7 3,63 34,63 34,63 (,391 12, 41,36 36,91 41,75 41,8 37,32 41,73 41,71 (,433 13, 48,84 44,21 49,23 49,28 44,49 49,21 49,19 (,474 14, 56,65 51,92 57,4 57,1 52,8 57,3 57, (,516 15, 64,75 59,98 65,13 65,19 6, ,8 (,556 Dev.,316 3,989,31,72 3,516,22 TABLE 4: VARYING THE VOLATILITIES SYM. WITH K = 1 σ Besser Gentle Ju Levy MP-RG MP-4M Carlo CV StdDev 5% 3,53 3,52 3,53 3,53 3,52 3,53 3,53 (,14 1% 7,4 6,98 7,5 7,5 6,99 7,5 7,5 (,42 15% 1,55 1,33 1,57 1,57 1,36 1,57 1,57 (,73 2% 14,3 13,52 14,8 14,8 13,59 14,8 14,8 (,115 3% 2,91 19,22 21,8 21,9 19,49 21,7 21,7 (,237 4% 27,63 23,78 28,128,5 24,5 27,98 27,98 (,35 5% 34,15 27,1 34,84 34,96 28,51 34,73 34,8 (,448 6% 4,41 28,84 41,52 41, ,19 41,44 (,327 7% 46,39 29,3 47,97 48,5 33,72 46,23 47,86 (,49 8% 52,5 28,57 54,9 55,5 35,15 48,39 54,1 (,685 1% 62,32 24,41 64,93 67,24 36,45 47,9 65,31 (,996 Dev. 1,22 16,25,12,69 11,83 5,53 the payoff of the followng portfolo consstng only of calls: (B(T = α [P Basket (B(T, L 1α, 1 P Basket (B(T, L, 1 (P Basket (B(T, L + L, 1 P Basket (B(T, L + L + 1α ], 1 We notce that the payoff (B(T s explctly gven by : B(T <L 1 α α [ B(T ( ] L 1 : L 1 B(T L α α (B(T = 1 : L B(T L + L 1 α [B(T (L + L] : L + L B(T L + L + 1 α : B(T >L + L + 1 α (3.2 For α t s equal to: :B(T <L (B(T = 1:L B(T L + L :B(T >L + L So for a suffcently hgh α the value of our portfolo s approxmately the probablty that the prce of the basket s at maturty n [L, L + L]. To calculate the whole mplct dstrbuton, we shft the boundares stepwse by L. Instead of applyng the underlyng dstrbutons, we use ths procedure, because we can not drectly determne the dstrbuton for Besser s approxmaton. Besdes, ths procedure seems to be more objectve and consstent to compare the approxmatons. We examned the dstrbutons for the test cases a-d. The results confrmed our fndngs from the comparson of the prces. For the cases a-c the mplct dstrbutons of Ju, Levy and Besser were consstent wth Carlo, and the other ones not. But only Besser was able to deal wth nhomogeneous volatltes n case d, Levy showed massve devatons. We plot an example wth σ 1 = 9%, σ 2 = σ 3 = 5% and σ 4 = 1% n Fgure 3 to test f there s some balancng effect,.e. observe that (σ 1 + σ 4 /2 = σ 2. We see there s one except for Levy s approach. We dd not plot the graph for the state-prce densty method of Mlevsky and Posner, because t was runnng nto serous problems for small K. The parameter Q s defned as a + b log((k c/d, hence for all K < c the formula of Mlevsky and Posner s not well-defned (a smlar problem occurs for Type II. But for ths parameter set c s around 65, so we smply couldn t calculate all necessary prces. So whch method to choose? The tests confrm that the approxmaton of Ju s overall the best performng method. In addton t has the nce property, that t always overprces slghtly. Ju s method shows only a lttle weakness n the case of nhomogeneous volatltes, Besser s better. Even though t s based on a Taylor expanson around zero volatltes, t has absolutely no problems wth hgh volatltes, whch s qute contrary to both methods of Mlevsky and Posner. Besser s approxmaton underprces slghtly n all cases. The underprcng of Besser s approach s not surprsng snce ths method s essentally a lower bound on the true opton prce. Besser s approach s the only method whch s relable n the case of nhomogeneous volatltes. The performances of Mlevsky and Posner s recprocal gamma and Gentle s approach are mostly poor. A reason for the bad performance of MP-RG may be, that the sum of log-normally dstrbuted random varables s only dstrbuted lke the recprocal gamma dstrbuton as ^ Wlmott magazne 87

7 Prce 7, Carlo-CV 6, Besser 5, Gentle 4, Ju 3, Levy 2, MP-RG 1, MP-4M,,% 2,% 4,% 6,% 8,% 1,% 12,% vol Prce 7, Carlo-CV 6, Besser 5, Gentle 4, Ju 3, Levy 2, MP-RG 1, MP-4M,,% 2,% 4,% 6,% 8,% 1,% 12,% vol Fgure 1: Varyng the volatltes sym. wth σ 1 =5%,K = 1 Prce 8, 7, 6, 5, 4, 3, 2, 1,,,% 2,% 4,% 6,% 8,% 1,% 12,% vol Carlo-CV Besser Gentle Ju Levy MP-RG MP-4M Fgure 2: Varyng the volatltes sym. wth σ 1 = 1%, K = 1 (Table 5 TABLE 5: VARYING THE VOLATILITIES SYM. WITH σ 1 = 1%, K = 1 (FIGURE 2 σ Besser Gentle Ju Levy MP-RG MP-4M Carlo CV Std Dev 5,% 19,45 15,15 35,59 55,56 35,22 18,51 22,65 (,5594 1,% 2,84 16,6 36,19 55,52 35,23 18,64 21,3 (, ,% 22,6 18,8 36,93 55,61 35,24 18,81 22,94 (,266 2,% 24,69 19,56 37,8 55,71 35,26 19,1 25,24 (,2124 3,% 29,52 22,35 39,97 55,98 35,3 19,42 3,95 (,163 4,% 34,72 24,73 42,66 56,35 35,36 2,37 36,89 (,1156 5,% 39,96 26,52 45,84 56,89 35,44 2,6 41,72 (,894 6,% 45,5 27,59 49,39 57,68 35,56 21,72 46,68 (,472 7,% 49,88 27,87 53,2158,87 35,72 23,66 51,78 (,587 8,% 54,39 27,38 57,17 6,7 35,93 27,38 56,61 (,742 1,% 62,32 24,41 64,93 67,24 36,45 47,9 65,31 (,996 Dev. 1,92 19,18 8,96 22,7 14,48 17,84 Fgure 3: Denstes for the standard scenaro wth σ 1 = 9%, σ 2 = σ 3 = 5%, σ 4 = 1% n. But as n our case n = 4 or even n practce wth n = 3 we are far away from nfnty. The geometrc mean used n Gentle s approach also seems to be an napproprate approxmaton for the arthmetc mean. For nstance, the geometrc mean of the forwards equal to 1, 2, 3 and 4 would be wthout mean correcton 2.21 nstead of 2.5. Ths s corrected, but the varance s stll wrong. The MP-4M four moment method s recommendable only for low vols. The Ju method s the best approxmaton except for the case of nhomogeneous volatltes. The reason for ths drawback may be that all stocks are thrown together on one dstrbuton. Ths s qute contrary to Besser s approxmaton, every sngle stock keeps a transformed log-normal dstrbuton and the expected value of every stock s ndvdually evaluated. Ths s probably the reason why ths method s able to handle the case of nhomogeneous volatltes. A rule of thumb for a practtoner would be to use Ju s method for homogeneous volatltes and Besser s for nhomogeneous ones. But then the queston occurs, how to defne the swtch exactly. So we suggest the followng: Prce the basket wth Ju and Besser: If the relatve dfference between the two computed values s less than 5% use Ju s prce for an upper and Besser s prce for a lower bound. If t s bgger than 5% run a -Carlo smulaton or f ths s not sutable, keep the Besser result (keep n mnd that t s only a lower bound for the prces!. 88 Wlmott magazne

8 TECHNICAL ARTICLE 5 REFERENCES BEISSER, J. (1999: Another Way to Value Basket Optons, Workng paper, Johannes Gutenberg-Unverstät Manz. BEISSER, J. (21: Topcs n Fnance A condtonal expectaton approach to value Asan, Basket and Spread Optons, Ph.D. Thess, Johannes Gutenberg Unversty Manz. CURRAN, M. (1994: Valung Asan and Portfolo Optons by Condtonng on the Geometrc Mean Prce, Management Scence, 4, DEELSTRAY, LIINEV, VANMAELE (23: Prcng of arthmetc basket and Asan basket by condtonng,workng Paper, Ghent Unversty, Belgum. GENTLE, D. (1993: Basket Weavng, RISK, HYUNH (1993: Back to Baskets, RISK, JOHNSON, N.L. (1949: Systems of Frequency Curves Generated by Methods of Translaton, Bometrka, 36, JU, E. (1992: Prcng Asan and Basket Optons Va Taylor Expanson, Journal of Computatonal Fnance, 5(3, LEVY, E. (1992: Prcng European average rate currency optons, Journal of Internatonal Money and Fnance, 11, LEVY, EDMOND UND STUART TURNBULL (1992: Average Intellgence, RISK, 6, No. 2, February, 5 9. MILEVKSY, M.A. AND POSNER, S.E. (1998: A Closed-Form Approxmaton for Valung Basket Optons, Journal of Dervatves, MILEVKSY, M.A. AND POSNER, S.E. (1998b: Valung Exotc Optons by Approxmatng the SPD wth Hgher Moments, Journal of Fnancal Engneerng, 7(2, ROGERS L.C.G. UND Z. SHI (1995: The Value of an Asan Opton, Journal of Appled Probablty, 32, STAUNTON, M. (22: From Nuclear Power to Basket Optons, Wlmott Magazne, September 22, TURNBULL, ST. M. AND L.M. WAKEMAN (1991: A Quck Algorthm for Prcng European Average Optons, Journal of Fnancal and Quanttatve Analyss, 26, No. 3, September, W Wlmott magazne 89

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