Jump-Diffusion Stock Return Models in Finance: Stochastic Process Density with Uniform-Jump Amplitude

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1 Jump-Dffuson Stock Return Models n Fnance: Stochastc Process Densty wth Unform-Jump Ampltude Floyd B. Hanson Laboratory for Advanced Computng Unversty of Illnos at Chcago 851 Morgan St.; M/C 249 Chcago, IL , USA hanson@math.uc.edu and J. J. Westman Department of Mathematcs Unversty of Calforna Box Los Angeles, CA , USA jwestman@math.ucla.edu Abstract The stochastc analyss s presented for the parameter estmaton problem for fttng a theoretcal jump-dffuson model to the log-returns from closng data of the Standard and Poor s 500 (S&P500)stock ndex durng the pror decade The jump-dffuson model combnes a the usual geometrc Brownan moton for the dffuson and a space-tme Posson process for the jumps such that the jump ampltudes are unformly dstrbuted. The unform jump dstrbuton accounts for the rare large outlyng log-returns, both negatve and postve n magntude. The log-normal, log-unform jump-dffuson densty s derved, leadng to a jump-dffuson smulator approxmaton for the case the the log-return tme s a small fracton of a year. There are fve jump-dffuson parameters that need to be determned, the means and varances for both dffuson and jumps, as well as the jump rate, gven the average log-return tme. A weghted least squares s used to ft the theoretcal jump-dffuson model to the S&P500 data optmzng wth respect to three free parameters, wth the two other parameters constraned by the mean and varance of the S&P500 data. The weght dstrbuton derves from stochastc methods. The deal ftted model determnes the three free parameters, but the correspondng smulated results resemble the orgnal S&P500 data better. Ths stochastc analyss paper s a companon to a computatonal methods and portfolo optmzaton paper at ths conference. 1Introducton A classcal model of fnancal market return process, such as the Black-Scholes [1, 8], s the lognormal dffuson process, such that the log-return process has a normal dstrbuton. However, real markets exhbt several devatons from ths deal, although useful, model. The market dstrbuton, say for stocks, should have several realstc propertes not found n the deal log-normal model: (1) the model must permt large random fluctuatons such as crashes or sudden upsurges, (2)the logreturn dstrbuton should be skew snce large downward outlers are larger than upward outlers, and (3)the dstrbuton should be leptokurtc snce the mode s usually hgher and the tals thcker than for a normal dstrbuton. For modelng these extra propertes, a jump-dffuson process wth log-unform jump-ampltude Posson process s used to ft the S & P 500 Index log-returns. A reasonable estmaton of the parameters of the log-return process can be made usng a weghted least squares approxmaton that s an mprovement over earler jump-dffuson model results of Merton [8] and the authors [2, 4, 5]. The computatonal ssues are prncpally dscussed n another paper of the authors at ths conference [6]. 1

2 2 Densty for Jump-Dffusons Let S(t)be the prce of a stock or stock fund satsfes a Markov, contnuous-tme, geometrc, jump-dffuson stochastc dfferental equaton (SDE), ds(t) =S(t)[µ d dt + σ d dz(t)+j(q)dp (t)], S(0)= S 0, S(t) > 0, (2.1) where µ d s the mean return rate, σ d s the dffusve volatlty, Z(t)s a one-dmensonal stochastc dffuson process, J(Q)s a log-return mean µ j and varance σj 2 random jump-ampltude and P (t) s a smple Posson jump process wth jump rate λ. It s assumed that the stock prce parameters µ d, σd 2, µ j, σj 2 and λ are constants. The dfferental dffuson process wth drft µ ddt + σ d dz(t)s has mean µ d dt and σ d dt varance. The space-tme jump process J(Q)dP (t)has mean E[J(Q)]λdt, varance E[J 2 (Q)]λdt and dp (t)has the dscrete dstrbuton p k (λdt) =Prob[dP (t) =k] =exp( λdt)(λdt) k /k!, k =0:. (2.2) The processes Z(t)and P (t)are parwse ndependent, whle J(Q)s also ndependent except that t s condtoned on the exstence of a jump n dp (t). Snce the SDE (2.1)has a geometrc or lnear form t can can be transformed to the smplfed log-return form usng the stochastc process chan rule, d[ln(s(t))] = µ ld dt + σ d dz(t)+ln(1+j(q))dp (t), (2.3) where µ ld dt = µ d σd 2 /2 s the log-dffuson drft and ln(1+j(q)) s the log-return jump-ampltude. For fnte log-return jump-ampltude and to avod complete nvestment loss, J(Q) > 1, so the underlyng random jump mark ampltude Q =ln(1+j(q)) on (, + )s chosen for convenence. For ths paper, we are nterested n a unformly dstrbuted mark varable Q to account for the exceptonally long negatve and postve tals n fnancal market dstrbutons, as can seen n the hstogram of the log-returns for S & P 500 Index [10] daly closngs n the decade from n Fgure 1. Snce large jumps n the log-returns seem to be rare events relatve to the background ups and downs modeled by the dffuson process, the jump-ampltude dstrbuton wll be assumed to be unformly dstrbuted on [Q a,q b ], Q a < 0 <Q b, wth tme-ndependent densty φ Q (q) φ (u) (q; Q a,q b ) U(q; Q a,q b ), (2.4) Q b Q a where U(x; a, b)denotes a unt step functon on [a, b], such that µ j =(Q a + Q b )/2 and σ 2 j =(Q b Q a ) 2 /12. (2.5) Thus, the combned log-normal dffuson, log-unform jump densty derves from a trad form of random processes ξ +η ζ, wth dffuson ξ = µ ld dt+σ d dz(t), jump-ampltude η = Q and jump-tme ζ = dp (t)processes. Ths densty s proven n our tme-dependent fnance paper [5] and s gven here n the modfed form, Theorem 2.1. The probablty densty for the log normal dffuson log unform jump ampltude log return dfferental d[ln(s(t))] specfed n the SDE (2.3)s gven by φ d ln(s(t)) (x) = p 0 (λdt)φ (n) (x; µ ld dt, σd 2 dt)(2.6) + p k (λdt) Φ(n) (x kq b,x kq a ; µ ld dt, σd 2dt), k(q b Q a ) k=1 2

3 300 Hstogram: Daly Closngs Log Returns, f (sp) 250 Frequency, f (sp) S&P500 Log Returns, DLog(S) Fgure 1: Hstogram of log-return of daly closngs n the S & P 500 Index for the decade , usng 100 bns. < x < +, where the Posson dstrbuton p k (λdt) s specfed n (2.2)and the normal dstrbuton on [x, y] s y y Φ (n) (x, y; µ ld dt, σd 2 dt) φ (n) (z; µ ld dt, σd 2 dt)dz x x exp( (z µ ld dt) 2 /(2σd 2 2πσ dt)) dz, (2.7) 2d dt where the ntegrand s the normal densty of the dffuson process ξ = µ ld dt + σ d dz(t) n (2.3). In the theorem there s no menton that dt s the nfntesmal of tme, snce t can be used for small but non-nfntesmal tme ncrements t as needed n the fnancal markets. In the S & P 500 Index the average tme between closngs s t = years, so ( t) 2 = s neglgble n comparson to t, f that would be suffcently accurate. Hence, the two-term asymptotc form of (2.6)wll be used: Corollary 2.1. As t 0 +, (2.6)can be asymptotcally approxmated as φ ln(s(t)) (x) φ (jd) (x)(2.8) neglectng O(( t) 2 ). (1 λ t)φ (n) (x; µ ld t, σ 2 d t)+λ tφ(n) (x Q b,x Q a ; µ ld t, σ 2 d t) Q b Q a, Eq. (2.8)s consstent wth the usual zero-or-one jump defnton of the nfntesmal Posson dstrbuton gven n full form by (2.2), such that there are zero jumps wth probablty (1 λ t) and one jump wth probablty λ t. Note that n (2.8)the zero-jump densty s just the dffuson densty, whle the one-jump densty can be called the secant-normal densty snce t s the rato of the dfference n normal dstrbutons dvded by the dfference n arguments. Eq. (2.8)s also consstent wth the small tme form of the log-return n (2.3), such that ln(s(t)) = t+ τ t d ln(s(τ)) µ ld t + σ d Z(t)+Q P (t), (2.9) 3

4 provded the parameters are constant and hgher order jumps are neglected, wth P (t)playng the role of an ndcator functon for ether zero or one jump. Eq. (2.9)can also for the jump-dffuson smulatons usng t tmes a normal random number generator for Z(t), a standard unform generator on [0, 1] parttoned nto [0,λ t] for one-jump and (λ t, 1] for no-jump n P (t),anda unform generator on [Q a,q b ]forsmulatngq provded a one jump s selected by the smulaton of P (t). 3 Jump Dffuson Parameter Estmaton For fnancal market modelng purposes, t s necessary to have an estmate of the parameters of the market dstrbuton. For the log-normal dffuson, log-unform jump-ampltude jump-dffuson theoretcal model, there s a set of fve parameters, {µ d,σd 2,µ d,σd 2,λ}, assumng the tme-step t s known. The object of ths paper s to estmate these parameters by fttng the theoretcal model to the decade worth of log-returns of the S & P 500 Index from 1992 to 2001 portrayed n N (bn) = 100 hstogram of Fgure 1, subject to some constrants to keep the parameter estmaton computatonally reasonable. There are a total of 2522 daly closngs S (sp), so that there are N (sp) = 2521 log-returns, (ln(s (sp) )) ln(s (sp) +1 ) ln(s(sp) ). The constrants used are matchng the decade mean M (sp) and varance M (sp) Relatve to the normal dstrbuton, the hgher order moment coeffcents are η (sp) 3 M (sp) 3 /(M (sp) 2 ) for skewness and η (sp) 4 M (sp) 4 /(M (sp) 2 ) for kurtoss, subtractng three for the unshfted normal kurtoss coeffcent. The dstngushng feature of real markets are the thcker tals that are longer on the negatve sde compared to normal dstrbutons, leadng to negatve skew and larger kurtoss coeffcents. Hence, t s mportant that the fttng of the dstrbutons be suffcently weghted so that the tals are suffcently detectable. In our papers [4, 5], an unweghted least squares was used whch resulted n the negatve tals over-domnatng the postve tals. Here, we use a weghted least squares or χ 2 ft (see for nstance the summares n [9]), χ 2 = N (bn) =1 ( ω f (jd) ) f (sp) 2, (3.10) where ω s the weght of the th bn, f (sp) s the th emprcal S & P 500 bn frequency data and f (jd) s the th theoretcal jump-dffuson bn frequency correspondng to the same sample sze N (sp) = An estmate of the weghts correspondng to a errors n measurements s not easy to get, but we wll use the followng theoretcal result to be proved elsewhere: Theorem 3.1. If f (jdsm) = N j=1 U( S(jdsm) j ; x,x +1 ) for =1:N (bn) are the frequences of the th bn [x,x +1 ) and S (jdsm) j s the jth jump-dffuson smulaton, usng N samples, as prescrbed for (2.9), then the bn frequency expectaton and varance are [ ] [ ] µ (jdsm) f =E f (jdsm) = f (jd) and σ 2 =Var f (jdsm) f (jdsm) ( = N respectvely, where the th expected bn frequency after N smulatons s f (jd) = N x+1 x φ (jd) (x)dx. 1 f (jd) / ) 2 (jd) N f, (3.11) 4

5 The bn weghts are chosen as the theoretcal values, ω = ( 1/σ 2 f (jd) ) / N (bn) j=1 ( ) 1/σ 2 f (jd) j, (3.12) for =1:N (bn) bns, normalzed to a unt sum for convenence of small mnma. The problem s reduced to a 3-dmensonal global mnmzaton for the transformed parameter set {Q a,q b,λ t} subject to constrants, M (jd) 1 = µ ld t + µ j λ t = M (sp) 1 and M (jd) 2 = σ 2 d t +(σ2 j + µ 2 j)λ t = M (sp) 2, (3.13) servng as elmnants of µ ld t and σ 2 d t, wth the jump-moments defnton (2.5)of µ j and σ 2 j relatng them to Q a and Q b (n rare case, non-negatvty must be enforced on the varances). The global mnmzer GoldenSuperFnder(GSF)[7], developed for fnancal problems n [4, 5], was used to estmate the ft (3.10). Ths method s an extensve modfcaton of the method of golden secton search (see [9])and s descrbed more n [6]. The fnal parameter results are µ d , σ 2 d ,µ j , σ 2 j , λ 55.46, (3.14) wth mnmum χ 2 mn wth a relatve value-locaton hybrd stoppng crteron of n a total of 16 GSF-teratons. The fnal successful mnmum weghted least squares teraton results are llustrated n Fgure 2, wth both theoretcal and smulaton hstograms. The hstogram on the rght for the smulatons more closely resembles the S & P 500 data hstogram, the S & P 500 beng a large realstc smulaton. 350 Hstogram: Post GSF Jump Dffuson Theory Ft, f (jdth) 350 Hstogram: Post GSF Jump Dffuson Smulaton Ft, f (jdsm) Frequency, f (jdth) Frequency, f (jdsm) Log Returns, x +0.5 Log Returns, x +0.5 Fgure 2: Hstogram of log-returns from the log-normal dffuson, log-unform jump-dffuson model ftted to the S & P 500 Index log-returns for the decade shown n Fg. 1, usng 100 bns. The fgure on the left s the ftted theoretcal jump-dffuson hstogram, whle the fgure on the rght s the correspondng smulated jump-dffuson hstogram usng the same fnal parameter results and the same number of samples as the S & P

6 Conclusons In ths paper, sgnfcant progress has been made toward fttng the theoretcal log-normal dffuson, log-unform jump-dffuson model to realstc fnancal market data, here the log-returns of the S & P 500 Index. The log-unform jump dstrbuton s a bg mprovement over the lognormal jump dstrbuton used n [4]. The crucal advance was to use a least squares method wth weghts and to establshng a method for computng the least square weghts from the theoretcal bn frequences. In essence, the S & P 500 Index data s treated as a large scale jump-dffuson smulaton. The resultng estmated jump-dffuson parameter set can add more realsm to fnancal market applcatons, such as the optmal portfolo and consumpton polcy problem treated n a computatonal companon paper [6] of the authors at ths conference. Acknowledgement: Work supported n part by the Natonal Scence Foundaton Computatonal Program Mathematcs Grant DMS References [1] F. Black and M. Scholes, The Prcng of Optons and Corporate Labltes, J. Poltcal Economy, vol. 81, , [2] F. B. Hanson and J. J. Westman, Optmal Consumpton and Portfolo Polces for Important Jump Events: Modelng and Computatonal Consderatons, Proceedngs of 2001 Amercan Control Conference, pp , 25 June [3] F. B. Hanson and J. J. Westman, Stochastc Analyss of Jump Dffusons for Fnancal Log Return Processes, Proceedngs of Stochastc Theory and Control Workshop, Sprnger Verlag, New York, pp. 1-15, accepted, March [4] F. B. Hanson and J. J. Westman, Optmal Consumpton and Portfolo Control for Jump- Dffuson Stock Process wth Log-Normal Jumps, Proceedngs of 2002 Amercan Control Conference, pp. 1-6, 08 May 2002, to appear. [5] F. B. Hanson and J. J. Westman, Portfolo Optmzaton wth Jump Dffusons: Estmaton and Applcaton, Proceedngs of 2002 Conference on Decson and Control, pp. 1-15, 07 March 2002, submtted for an nvted sesson. [6] F. B. Hanson and J. J. Westman, Computatonal Methods for Portfolo and Consumpton Polcy Optmzaton n Log-Normal Dffuson, Log-Unform Jump Envronments, Proceedngs of the 15th Internatonal Symposum on Mathematcal Theory of Networks and Systems, pp. 1-6, August 2002, to appear. [7] F. B. Hanson and J. J. Westman, Golden Super Fnder: Multdmensonal Modfcaton of Golden Secton Search Unrestrcted by Intal Doman, under testng and n preparaton, Aprl [8] R. C. Merton, Contnuous Tme Fnance, Basl Blackwell, Cambrdge, MA,

7 [9] W. H. Press, S. A. Teukolsky, W. T. Vetterlng and B. P. Flannery, Numercal Recpes n C: The Art of Scentfc Computng, Cambrdge Unversty Press, Cambrdge, UK, [10] Yahoo! Fnance, Hstorcal Quotes, S&P 500, Symbol SPC, URL: February