The Omega Function -Not for Circulation-

Transcription

1 The Omega Function -Not fo Ciculation- A. Cascon, C. Keating and W. F. Shadwick The Finance Development Cente London, England July , Final Revision 11 Mach Intoduction In this pape we intoduce a bijection between a class of univaiate cumulative distibution functions and a class of monotone deceasing functions which we call Omega functions. Moe pecisely, if F is a univaiate cumulative distibution with domain D =(a, b)wheea may be and/o b may be, andf satisfies asimplegowthcondition,thee is a unique monotone function Ω F fom (a, b) to (0, ). With vey mild assumptions, each such function may be used to econstuct a unique cumulative distibution. (In fact, thee is a closed fom expession fo the distibution in tems of Ω F and its fist two deivatives [3].) The coespondence between cumulative distibutions and Omega functions is theefoe a natual duality. The global popeties of the distibution ae eflected in local popeties of the Omega function. Fo example, the mean is the unique point at which the Omega function takes the value 1. Fo a nomal distibution with mean µ and vaiance σ 2,theslopeoftheOmegafunction at µ is 2π σ.highemomentsae encoded in the shape of the Omega function. The latte popety of the Omega function makes it paticulaly well suited to the study of statistical data, such as financial time seies, in which the deviation fom nomality is often citical but is difficult to estimate eliably though the use of highe moments due to noisy o spase data, o is difficult to incopoate into models of makets and behaviou. This application was in fact what led Coesponding autho William.Shadwick@FinanceDevelopmentCente.com

2 to the discovey and development of the Omegafunction [1, 2]. We pesent the deivation in its oiginal fomhee, but note that it may be motivated equally well by any application in which benefits and costs, gains and losses ae to be compaed. In section 2 we pesent the definition of Omega and its most impotant popeties, with poofs of the latte esults. In section 3 we povide some examples of Omega functions fo standad univaiate distibutions. Section 4 contains an indication of futhe esults which we will epot on sepaately and a bief discussion of some open questions. 2 The Omega function of a cumulative distibution We motivate the definition of the Omega function using an example fom finance. We conside the etuns distibution F fo some financial instument and we let D=(a, b) denotethedomain of F.Nextweset a etun level =L in (a, b) which is to be egaded as a loss theshold. In othe wods, we conside a etun above the level L as a gain and a etun below L as a loss. With this stating point we now conside the quality of a bet on a gain. To evaluate this, we must conside both the odds, namely what we will win if we win and what we will lose if we lose, as well as the fom namely the pobabilities of winning and losing. As an example, we might egad the amount by which the conditional expectation E( >L)exceeds L as an indicato ofwhatwestand to win if we win. The amount by which the conditional expectation E( < L)falls below L is an indicato of what we stand to lose if we lose. If we weight the expected gain g = E( L >L)andlossl = E(L <L)bythe espective pobabilities of a gain and a loss, namely 1 F (L) andf (L),thei E( L >L)(1 F (L)) atio, E(L <L)F (L) is a ough measue of the quality of ou bet. Of couse the conditional expected gain and loss ae only one pai of infinitely many possible gains and losses. Fo example we may conside sepaately the possibility of the gains fom etuns of L + x and L + 2 x with pobability weights of 1 F (L) and1 F (L+ x) espectively. We also have the possibility of losses fom etuns of L x and L 2 x with pobability weights of F (L) and F (L x) espectively. Taking these possible etuns into account leads 2

3 (1 F (L)) x+(1 F (L+ x)) x to F (L) x+f (L x) x as an estimate of the quality of a bet on aetun above the level of L. If we continue in this vein, letting ou unit of gain and loss, x, tendtozeo, we ae led, in the limit, to the the integal I 2 (L) = b as the total pobability weighted gains and to the integal L 1 F (x)dx. (1) I 1 (L) = L a F (x)dx (2) as the total pobability weighted losses. The atio of these is ou measue of pobability weighted gains to losses fo a bet on a etun above = L (assuming, as we do hencefowad, that these integals ae finite.) If we now let the isk theshold L un ove the domain of the cumulative distibution of etuns, we obtain the function Ω F,whichwecalltheOmega function of the distibution F,defined on (a, b) by whee Ω F () = I 2() I 1 (). (3) I 1 () = and I 2 () = b a F (x)dx (4) 1 F (x)dx. (5) It is obvious that each cumulative distibution leads unambiguously to an Omega function, povided that the integals in question exist fo finite values of. Among the popeties of this function which we extablish in the theoem, is the fact that the map fom distibutions to Omega functions is one to one. (The invese poblem of econstucting a distibution fom an Omega function is the subject of a sepaate aticle [3].) 3

4 Theoem 1 If F and G ae continuous distibutions on (a, b) such that the integals (4) and (5) exist fo all in (a, b) then EF (C()) 1) Ω F () = E F (P ()) whee C()=max(x, 0), P () =max( x, 0) and E F denotes expectation with espect to the distibution F. 2)Ω F is diffeentiable and dωf d = F ()(I1() I2()) I1 2() 1 I 1() 3)Ω F is a monotone deceasing function fom (a, b) to (0, ). 4) If φ is an affine diffeomophism of (a, b), leth denote the induced distibution,h=f φ 1 dφ. If d > 0 then Ω dφ H(φ())=Ω F () fo all in (a, b). If d < 0 then Ω H (φ())=ω F () 1 fo all in (a, b). 5) If µ is the mean of the distibution F then Ω F (µ)=1. 6) F is symmetic about µ if and only if Ω F ( µ)=ω F ( + µ) 1 fo all in (a, b). 7) If F λ is a smooth 1-paamete family of defomations of F with F 0 := F and F (1) = df λ dλ λ=0, letω λ denote Ω Fλ.Then dω λ dλ λ=0= 1 I 1() ( 2 a F (1) (x)dxi 2 () b F (1) (x)dxi 1 ()). 8) Ω F =Ω G if and only if F = G. Poof. 1) This follows immediately fom the definition of Ω F and the fact that E F (φ(x)) = b 0 1 F φ(x)(y)dy 0 a F φ(x)(y)dy whee F φ(x) is the distibution induced by φ. 2) This follows diectly fom the definition of Ω F. 3) This follows fom 2) and the definition of Ω F. 4) This follows fom the definition of Ω F by making a change of vaiable in the integals I 1 () andi 2 (). 5) If µ =0itis immediate fom the definitions of I 1 () andi 2 () that I 2 (0) I 1 (0) = 0 which establishes 5) in this special case. The geneal case, fo adistibution F with µ 0nowfollows by using the tanslation φ(x) =x µ. The distibution H = F φ 1 induced by φ has its mean at 0 so Ω H (0) = 1. But Ω H (0) = Ω F (µ) by4). 6) This follows fom the definition of Ω F by changing vaiables in I 1 () and I 2 (). 7) This follows fom the definition of Ω Fλ by exchanging the odes of diffeentiation and integation. 4

5 8) It is immediate that when F = G, Ω F =Ω G. Next we assume that Ω F =Ω G.LetΩ F () = I2() I as usual and let Ω 1() G() = J2() J.Wenotefistthat 1() it follows fom Ω F =Ω G that µ F = µ G as the mean is the invese image of 1. We use µ to denote the common mean and will show that fo all >0, we have I 1 (µ ) =J 1 (µ ). Let I := I 1 (µ) =I 2 (µ) andj := J 1 (µ) =J 2 (µ). If we let A F = µ µ 1 F (x)dx and B F = µ µ F (x)dx, theni 2(µ ) =I + A F and I 1 (µ ) =I B F. With the obvious notation fo A G and B G we also have J 2 (µ ) =J + A G and J 1 (µ ) =J B G. Since A F + B F = and A G + B G =, theequality of Ω F and Ω G at µ gives J+ BG J B G = I+ BF I B F. It follows that (I B F )=(J B G )and, as 0,I B F = J B G. But this is the same as I 1 (µ ) =J 1 (µ ) andthiselation theefoe holds fo all >0. The same agument shows that this holds fo all <0aswelland, by continuity, it holds at =0andhence we have I 1 () =J 1 () foall. By diffeentiating this elation we obtain F = G. 3 The Omega functions fo some standad univaiate distibutions The Omega function is a non-local function of a distibution. This means that it encodes some popeties of the distibution in a way which is not immediately obvious fom the appeaance of the gaph of eithe the distibution o its density. This infomation lends itself natually to gaphical intepetation. Fo example, if we conside two symmetic distibutions with the same mean, the one with fatte tails will have a flatte Omega function asymptotically to the ight and left of the mean. Because they must agee at the mean, it follows that this will lead to an odd numbe of cossings of the Omega functions and thus to some significant changes in shape, even though the Omega function is monotone deceasing. In this section, we use the Omega functions fo some standad distibutions to illustate some of these featues. Because of the apid gowth and decay of these functions, and in ode to make some of thei symmeties moe appaent, it is convenient to pesent gaphs of the logaithms of the Omega functions. Likewise, we use the logaithmic deivative to illustate the changes in shapes. We begin withtheomegafunction fo the nomal distibution. It is easy to 5

6 veify fom the definition of Ω that fo a nomal distibution with mean µ and standad deviation σ, theslopeofωatµ is 2π σ. Figue 1 illustates the natual logaithm of the Omega function fo thee nomal distibutions with a common mean of 0 and σ =1,σ =0.75 and σ =0.5 espectively. The deivatives of log(ω) fo the same values of σ ae shown in the bottom panel. Next we illustate the Omegas fo the logistic distibution whose density exp( is L(, a, b) = ( a) b ). Figue 2shows the natual logaithms of the b(1+exp( ( a) b )) 2 Omegas fo L(, 0, 0.5), L(, 0, 0.6) and L(, 0, 1) togethe with the deivatives of log Ω. The logistic distibution has fatte tails than a nomal with the same mean and vaiance. Figue 3 contasts the logaithms of the Omegas fo a logistic and a nomal distibution with the same mean and vaiance. Because of its fatte tails, the Omega function fo the logistic distibution is dominated by that of the nomal as tends to and vice vesa as tends to. This poduces thee cossings which ae just discenable in Figue 3. In spite of the close ageement of the two Omega functions nea the mean, thei shapes ae quite diffeent. The bottom panel shows theextentofthe diffeences in shape of the two Omega functions. Now we tun to two distibutions defined ove the half line [0, ), the lognomal distibution 1 x 2πσ log(x µ) 2 e 2σ 2 σ2 µ+ whose mean is e 2 and vaiance is is x a 1 exp( b ) σ 2 and the Gamma distibution G(x, a, b) =x Γ(a)b whose mean is ab and a whose vaiance is ab 2. The natual logaithm of the Omega function fo the Gamma distibution G(x, a, b) isillustated in Figue 4 fo a = 1,b=4, a = 2, b =2anda =4,b =1. Thesedistibutions have a common mean of 4 and standad deviations of 16, 8 and 4 espectively. The deivatives of log(ω) ae shown inthebottom panel. Although the Omega functions fo lognomal and Gamma distibutions with a>bhave simila qualitative featues, as do the distibutions themselves,they ae quite diffeent in detail. We illustate this with a compaison of the two Omega functions fo the caseofthegamma distibution G(, 2, 1) and a lognomal with the same mean and vaiance in Figue 5. 6

7 Figue 1: Top panel LogΩ() fonomal distibutions with common mean of µ =1andσ =1.5 (solid), σ =2(dashed) and σ =2.5 (heavydashed line). 1 dω Bottom panel Ω() d fo the same values of σ. 7

8 Figue 2: Top panel LogΩ() fologistic distibutions with common mean of a =0fob =0.5 (dashed),b =0.6 (solid) and b =1(heavydashed line). 1 dω Bottom panel Ω() d fo the same values of b. 8

9 Figue 3: Top panel LogΩ() foalogistic distibution (solid) and a nomal distibution (dashed line) with the same mean and vaiance. Bottom panel fo the same distibutions. 1 Ω() dω d 9

10 Figue 4: LogΩ()fo Gamma(, 1, 4) (dashed line),gamma(, 2, 2) (solid line) 1 dω and Gamma(, 4, 1)(heavy dashed line). Bottom panel Ω() d fo the same distibutions. 10

11 Figue 5: LogΩ() fogamma(, 2, 1) (dashed), and a lognomal with mean 1 dω and vaiance of 2 (solid line). Bottom panel Ω() d fo the same distibutions. 11

12 4 An Example with Financial Retuns Data We illustate the use of the Omega function of a distibution fo the compaison of etuns fom two equity indices, the FTSE 100 index and the S&P500 index. The compaison is made on the basis of two yeas of daily index data fom 3 Januay 1995 to 31 Decembe The data sets consist of 505 etuns. (Hee the ith etun is i = Index(i+1) Index(i) Index(i),wheeIndex(i) istheclosing level on day i.) Compaison of the Omega functions fo the etuns distibutions ove this peiod povides an indication of the elative quality of investments in the two indices. As the top panel of Figue 6 shows, the Omega function fo the S&P500 index has heavie tails than the FTSE 100 index aswellasahighe mean etun. The obseved diffeence in thetwoomegafunctions is consistent with the standad statistics fo the two data sets. The skewness of the S&P 500 etuns set is 0.397, compaed with fo the FTSE 100. The kutosis of the S&P 500 etuns set is 5.31 compaed with 3.37 fo the FTSE 100. Ove most of the obseved ange of etuns, the S&P 500 Omega dominates the Omega function of the FTSE 100. This indicates that one might expect highe teminal values fom an investment in the S&P 500 index than fo an investment in the FTSE 100 index ove this peiod. This was indeed the case, as the fome poduced a two yea etun of 61% while the latte poduced a etun of 34% ove the same peiod. The subsequent yea poduced etuns of 31.6% in the S&P 500 compaed with 26.6% in the FTSE 100. The standad models in finance assume that etuns on financial instuments ae nomally distibuted. The Jaque Bea test statistic fo the FTSE 100 data is 4.93 while that fo the S&P 500 is 135. As the sample contains 505 data points, we may eject the hypothesis that these etuns ae nomally distibuted at the 95% confidence level. We may also compae the Omega functions fothesedata sets with those fo nomal distibutions with the same mean and vaiance. As the bottom panel of Figue 6 and the top panel offigue7show,thedepatue fom the Omega function of a nomal distibution is moe ponounced fo the S&P 500 etuns than fo those of the FTSE index. Moeove, both ae substantially geate than the depatue fom nomal of that of a data set consisting of 505 andom daws 12

13 Figue 6: Top Panel LogΩ() fotheftse100 index (dashed), and the S&P500 index fom 3 Januay 1995 to 31 Decembe The ange of etuns is the FTSE mean etun +/ 2 FTSE standad deviations. Bottom Panel LogΩ() fo the FTSE 100 index (dashed) and a nomal distibution with the same mean and vaiance (heavy dotted line). 13

14 Figue 7: Top Panel LogΩ() fothes&p500 index (solid), and a nomal distibution with the same mean and vaiance (heavy dotted line). Bottom Panel LogΩ() foasample of 505 data points fom a nomal distibution with the mean and vaiance of the FTSE 100 index (solid) andthetueomegafoa nomal with the same meanandvaiance. 14

15 fom a nomal distibution of the same mean and vaiance. This is illustated in the bottom panel of Figue 7. 5 Futhe wok Of the many questions which ae aised by this new way of looking at distibutions, pehaps the most obvious is the question of what distibutions aise fom natual conditions on Omega functions.the invese poblem of constucting the distibution which poduces a given Omega function is solved in [3] whee aclosedfomexpession fo the distibution in tems of Omega and its fist two deivatives is obtained. This, in tun, leads to some insights into the natue of the Ω-chaacteisation of natual distibutions. The question of the Ω-chaacteisation of the nomal distibution is a natual one, to which we do not as yet have a satisfactoy answe. We do, howeve, know the sampling distibution of the slope of the Omega cuve fo a nomal distibution at the mean, as this is 2π σ,asweobsevedinsection 3. Additional statistics fo Omega cuves ae the subject of continuing investigation. The Omega function extends natually to multivaiate distibutions and this is discussed in a sepaate pape [4]. 6 Acknowledgements We ae indebted to Michael Evans and Badley Shadwick fo helpful discussions. Refeences [1] C. Keating and W.F. Shadwick AUnivesal Pefomance Measue, J. Pefomance Measuement vol 6 no 3, pp 59-84, [2] A. Cascon, C. Keating and W.F. Shadwick, An Intoduction to Omega, Finance Development Cente Tech. Repot, [3] A. Cascon, B.A. Shadwick and W.F. Shadwick The Invese Poblem fo Omega, Finance Development Cente Tech. Repot, [4] A. Cascon and W.F. Shadwick The Omega Function fo a Multivaiate Distibution, Finance Development Cente Tech. Repot,

16 FTSE 100 Retuns Figue 8: Retuns fo the FTSE 100 index 3 Januay 1995 to 31 Decembe

17 S&P 500 Retuns Figue 9: Retuns fo the S&P 500 index 3 Januay 1995 to 31 Decembe