An Analysis of Pricing Methods for Baskets Options


 Job McDowell
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1 An Analyss of Prcng Methods for Baskets Optons Martn Krekel, Johan de Kock, Ralf Korn, TnKwa Man Fraunhofer ITWM, Department of Fnancal Mathematcs, Kaserslautern, Germany, emal: Abstract: Ths paper deals wth the task of prcng basket optons. Here, the man problem s not pathdependency but the multdmensonalty whch makes t mpossble to gve exact analytcal representatons of the opton prce. We revew the lterature and compare sx dfferent methods n a systematc way. Thereby we also look at the nfluence of varous parameters such as strke, correlaton, forwards or volatltes on the performance of the dfferent approxmatons. Keywords and Phrases: Exotc optons, basket optons, numercal methods 1 Introducton Whle wth many exotc optons t s even hard to fully understand the way ther fnal payoff s bult up, the constructon of the payoff of a (European basket opton s very smple. We defne the prce of a basket of stocks by B(T = w S (T,.e. t s the weghted average of the prces of n stocks at maturty T. Here the weghts w are usually assumed to be postve and to sum up to 1, but also be qute arbtrary. Our task s to determne the prce of a call (θ = 1 or a put (θ = 1 wth strke K and maturty T on the basket,.e. to value the payoff P Basket (B(T, K,θ= [θ(b(t K] +. We prce these optons wth the BlackScholes Model. Note that by the form of the payoff t s not necessary to dstngush between the tradng date and the valuaton date to calculate the values of these optons, snce they are not pathdependent. Hence wthout loss of generalty we can set t = and denote the remanng tme to maturty wth T. In order to ease the calculatons we use the socalled forward notaton. The Tforward prce of stock s gven by F T ( T = S ( exp (r(s d (s ds r(. and d (. are determnstc nterest rates and dvdend yelds. Wth ts help the stock prces can be represented as ( T S (T = F T 1 exp 2 σ 2 ds + T σ dw (s the W (. are correlated onedmensonal Brownan motons wth correlaton of ρ j. Further, we defne the dscount factor as ( T Df(T = exp r(sds. The forwardorented notaton has two advantages: Frstly, n opposte to short rates and dvdend yelds, forward prces and dscount factors are marketquotes. Secondly, from a computatonal pont of vew, t s less costly to work wth sngle numbers,.e. the forward prces and the dscount factor, nstead of several termstructures, namely the short rates and the dvdend yelds. The problem of prcng the above basket optons n the BlackScholes Model s the followng: The stock prces are modelled by geometrc 82 Wlmott magazne
2 TECHNICAL ARTICLE 5 Brownan motons and are therefore lognormally dstrbuted. As the sum of lognormally dstrbuted random varables s not lognormal, t s not possble to derve an (exact closedform 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 multdmensonal 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 lognormally 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 lognormally 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 lognormally 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 lognormal moment matchng a σ 2 T and The basc dea of Levy (1992 s to approxmate the dstrbuton of the basket by a lognormal 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 lognormal 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 closedform 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 lognormally dstrbuted random varables converges to the recprocal gamma dstrbuton as n. Consequently, the frst two moments of both dstrbutons are matched to obtan a closedform 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
4 TECHNICAL ARTICLE 5 Basc calculatons yeld: V Basket (T Df(T (MG(1/K,α 1,β KG(1/K,α,β Note, that we use the parametrsaton of the gamma dstrbuton found n Staunton (22, snce ths produces more accurate results than that from the orgnal paper by Mlevsky and Posner (1998. f Mlevsky and Posner s approxmaton va hgher moments Mlevsky and Posner (1998b use dstrbutons from the Johnson (1994 famly as stateprce denstes to match hgher moments of dstrbuton of the arthmetc mean. More precsely, they wrte the prce of a call on a basket as: [ ] V Basket (T = Df(T (x K + h(xdx h(x s the state prce densty. Note that, we would end up n Levy s approxmaton, f we were usng the lognormal densty wth the frst two moments matchng those of the mean. Mlevsky and Posner however use two members of the Johnson famly, whch s a collecton of statstcal dstrbutons, that can be represented by a transformaton of the normal dstrbuton Z: ( Z a Type I: X = c + d exp or b ( Z a Type II: X = c + d snh b The parameters a, b, c and d are chosen, so that the four moments of the arthmetc mean are approxmated (snce there are no closedform solutons for them. If the kurtoss of the Type I s close enough to the kurtoss of the mean, they use Type I, otherwse Type II. The closedform soluton for Type I s gven by: ( 1 2ab V Basket (T Df(T M = a F T ( K c Q = a + b log d ω = [ M K + (K cn(q d exp 8 + 4η η 2 + η b 2 ( N Q 1 ] b η η 2 + η 4 a = 1/ log(ω, b = 1 2 log(ω(ω 1/ξ 2, d = sgn(η, c = dm e ( 1 2b a/b ξ s the varance, η the skewness and κ the kurtoss. 3 Test Results As the advantage of analytcal methods compared to Carlo or numercal ntegraton s of course speed of computatons, we only have to compare the accuracy of the analytcal methods presented n the foregong secton. We wll perform a systematc test by lookng at the effect of varyng correlatons, strkes, forward and strkes and volatltes. Our standard test example s a call opton on a basket wth four stocks and parameters gven by T = 5., Df (T = 1., ρ j =.5 (for = j, K = 1, F T = 1, σ = 4% and w = 1 4. As reference values we compute the prces of all the optons below by a Carlo smulaton usng antthetc method and geometrc mean as controle varate for varance reducton. The number of smulatons was always chosen large enough to keep the standard devaton below.5. We dd not test the method of Hyunh (1993, because t s an applcaton of the method of Turnbull & Wakeman (1991 for Asan Optons (Edgeworth expanson up to the 4th moment and t s a wellknown problem that ths approxmaton gves really bad results for long maturtes and hgh volatltes. See also Ju (22, who ponted out that the Edgeworth expanson dverges f the approxmatng random varable s lognormal. We also tested Curran s (1994 approxmaton whch computes the prce by condtonng on the geometrc mean. But we do not show the numercal results here, because f we transformed the forwards to one (smply by multplyng the weghts wth them the prces were exactly the same as those of Besser (1999. If dd not transform the forwards to one, Besser and Curran gave dfferent prces, but on the other hand Curran s results were mostly worse. For further readng we refer to Deelstray, Lnev, Vanmaele (23 and Besser (21 who developed a general framework for the prcng of baskets and asan optons va condtonng. a Varyng the correlatons Table 1 below shows the effect of smultaneously changng all correlatons from ρ = ρ j =.1 to ρ =.95. Note that, except for Mlevksy and Posner s recprocal gamma (MPRG and Gentle, all methods perform reasonably well. Especally for ρ.8, the methods of Besser, Ju, Levy, the four moments method of Mlevsky and Posner (MP4M and Carlo gve vrtually the same prce. The good performance of Besser, Ju, Gentle and Levy for hgh correlatons can be explaned as follows: All four methods provde exactly the BlackScholes prces for the specal case that the number of stocks s one. For hgh correlatons the dstrbuton of the basket s approxmately the sum of the same (for ρ = 1 exactly the same lognormal dstrbutons, whch s ndeed agan lognormal. As Levy uses a lognormal dstrbuton wth the correct moments, t has to be a good approxmaton for these cases. The same argumentaton apples for Gentle. If we have effectvely one stock the geometrc and the arthmetc average are the same. The ^ Wlmott magazne 85
5 TABLE 1: VARYING THE CORRELATIONS SIMULTANEOUSLY ρ Besser Gentle Ju Levy MPRG MP4M 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 MPRG 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 MP4M 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 MP4M are relatvely small. The prce curves of TABLE 2: VARYING THE STRIKE K Besser Gentle Ju Levy MPRG MP4M 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 (MPRG run almost parallel to the Carlo curve and represent an underevaluaton. 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 MP4M 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 MP4M 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 MPRG MP4M 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 MPRG MP4M 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 ad. The results confrmed our fndngs from the comparson of the prces. For the cases ac 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 stateprce 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 welldefned (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 MPRG may be, that the sum of lognormally dstrbuted random varables s only dstrbuted lke the recprocal gamma dstrbuton as ^ Wlmott magazne 87
7 Prce 7, CarloCV 6, Besser 5, Gentle 4, Ju 3, Levy 2, MPRG 1, MP4M,,% 2,% 4,% 6,% 8,% 1,% 12,% vol Prce 7, CarloCV 6, Besser 5, Gentle 4, Ju 3, Levy 2, MPRG 1, MP4M,,% 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 CarloCV Besser Gentle Ju Levy MPRG MP4M 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 MPRG MP4M 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 MP4M 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 lognormal 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 GutenbergUnverstät 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 ClosedForm 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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