Histogram: Daily Closings Log Returns, f (sp) Frequency, f (sp) S&P500 Log Returns, DLog(S)

Transcription

1 Jump-Duson Stock Return Models n Fnance: Stochastc Process Densty wth Unform-Jump Ampltude Floyd B. Hanson Laboratory for Advanced Computng Unversty ofillnos at Chcago and 85 Morgan St. M/C 49 Chcago, IL , USA hanson@math.uc.edu 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 ttng a theoretcal jump-duson 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-duson model combnes a the usual geometrc Brownan moton for the duson 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-duson densty sderved, leadng to a jump-duson smulator approxmaton for the case the the log-return tme s a small fracton of a year. There are ve jump-duson parameters that need to be determned, the means and varances for both duson and jumps, as well as the jump rate, gven the average log-return tme. Aweghted least squares s used to t the theoretcal jump-duson 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 tted 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. Introducton A classcal model of nancal market return process, such as the Black-Scholes [, 8], s the lognormal duson 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: () the model must permt large random uctuatons such as crashes or sudden upsurges, () 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-duson process wth log-unform jump-ampltude Posson process s used to t 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-duson model results of Merton [8] and the authors [, 4, 5]. The computatonal ssues are prncpally dscussed n another paper of the authors at ths conference [6].

2 Densty for Jump-Dusons Let S(t) be the prce of a stock or stock fund satses a Markov, contnuous-tme, geometrc, jump-duson stochastc derental equaton (SDE), ds(t) =S(t)[ d dt + d dz(t)+j(q)dp (t)] S(0) = S 0 S(t) > 0 (.) where d s the mean return rate, d s the dusve volatlty, Z(t) s a one-dmensonal stochastc duson process, J(Q) s a log-return mean j and varance j random jump-ampltude and P (t) s a smple Posson jump process wth jump rate. It s assumed that the stock prce parameters d,, d j, and j are constants. The derental duson process wth drft ddt + d dz(t) s has mean d dt and d dt varance. The space-tme jump process J(Q)dP (t) hasmeane[j(q)]dt, varance E[J (Q)]dt and dp (t) has the dscrete dstrbuton p k (dt) =Prob[dP (t) =k] = exp(;dt)(dt) k =k! k =0: : (.) 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 (.) has a geometrc or lnear form t can can be transformed to the smpled log-return form usng the stochastc process chan rule, d[ln(s(t))] = ld dt + d dz(t)+ln( + J(Q))dP (t) (.3) where ld dt = d ; d = s the log-duson drft and ln(+j(q)) s the log-return jump-ampltude. For nte log-return jump-ampltude and to avod complete nvestment loss, J(Q) > ;, so the underlyng random jump mark ampltude Q =ln(+j(q)) on (; +)schosen 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 nancal market dstrbutons, as can seen n the hstogram of the log-returns for S & P 500 Index [0] daly closngs n the decade from n Fgure. Snce large jumps n the log-returns seem to be rare events relatve to the background ups and downs modeled by the duson 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 Q (q) (u) (q Q a Q b ) U(q Q a Q b ) Q b ; Q a (.4) where U(x a b) denotes a unt step functon on [a b], such that j =(Q a + Q b )= and j =(Q b ; Q a ) =: (.5) Thus, the combned log-normal duson, log-unform jump densty derves from a trad form of random processes +, wth duson = ld dt+ d dz(t), jump-ampltude = Q and jump-tme = dp (t) processes. Ths densty s proven n our tme-dependent nance paper [5] and s gven here n the moded form, Theorem. The probablty densty for the log{normal duson log{unform jump{ampltude log{return derental d[ln(s(t))] speced n the SDE (.3) s gven by d ln(s(t)) (x) = p 0 (dt) (n) (x ld dt ddt) (.6) + X k= p k (dt) (n) (x ; kq b x; kq a ld dt d dt) k(q b ; Q a )

3 300 Hstogram: Daly Closngs Log Returns, f (sp) 50 Frequency, f (sp) S&P500 Log Returns, DLog(S) Fgure : Hstogram of log-return of daly closngs n the S & P 500 Index for the decade 99{00, usng 00 bns. ; < x < +, where the Posson dstrbuton p k (dt) s speced n (.) and the normal dstrbuton on [x y] s (n) (x y ld dt ddt) Z y x (n) (z ld dt ddt)dz Z y x exp(;(z ; ld dt) =( dt)) d q dz (.7) d dt where the ntegrand s the normal densty of the duson process = ld dt + d dz(t) n (.3). In the theorem there s no menton that dt s the nntesmal of tme, snce t can be used for small but non-nntesmal tme ncrements t as needed n the nancal markets. In the S& P 500 Index the average tme between closngs s t =0: years, so (t) =0: s neglgble n comparson to t, f that would be sucently accurate. Hence, the two-term asymptotc form of (.6) wll be used: Corollary. As t! 0 +, (.6) can be asymptotcally approxmated as ln(s(t)) (x) (jd) (x) (.8) neglectng O((t) ). ( ; t) (n) (x ld t dt)+t (n) (x ; Q b x; Q a ld t d t) Q b ; Q a Eq. (.8) s consstent wth the usual zero-or-one jump denton of the nntesmal Posson dstrbuton gven n full form by (.), such that there are zero jumps wth probablty (; t) and one jump wth probablty t. Note that n (.8) the zero-jump densty s just the duson densty, whle the one-jump densty can be called the secant-normal densty snce t s the rato of the derence n normal dstrbutons dvded by the derence n arguments. Eq. (.8) s also consstent wth the small tme form of the log-return n (.3), such that ln(s(t)) = Z t+ t d ln(s()) ld t + d Z(t)+QP (t) (.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. (.9) can also for the jump-duson smulatons usng p t tmes a normal random number generator for Z(t), a standard unform generator on [0 ] parttoned nto [0 t] for one-jump and (t ] for no-jump n P (t), and a unform generator on [Q a Q b ]forsmulatng Q provded a one jump s selected by the smulaton of P (t). 3 Jump{Duson Parameter Estmaton For nancal market modelng purposes, t s necessary to have an estmate of the parameters of the market dstrbuton. For the log-normal duson, log-unform jump-ampltude jump-duson theoretcal model, there s a set of ve parameters, f d d d d g, assumng the tme-step t s known. The object of ths paper s to estmate these parameters by ttng the theoretcal model to the decade worth of log-returns of the S & P 500 Index from 99 to 00 portrayed n N (bn) = 00 hstogram of Fgure, subject to some constrants to keep the parameter estmaton computatonally reasonable. There are a total of 5 daly closngs S (sp), so that there are N (sp) = 5 log-returns, (ln(s (sp) )) ln(s (sp) + );ln(s(sp) ). The constrants used are matchng the decade mean M (sp) ' 4:050 ;4 and varance M (sp) ' 9:8740 ;5. Relatve to the normal dstrbuton, the hgher order moment coecents are (sp) 3 M (sp) 3 =(M (sp) ) :5 ' ;0:93 for skewness and b (sp) 4 M (sp) 4 =(M (sp) ) ; 3 ' 4:804 for kurtoss, subtractng three for the unshfted normal kurtoss coecent. 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 coecents. Hence, t s mportant that the ttng of the dstrbutons be sucently weghted so that the tals are sucently 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 t (see for nstance the summares n [9]), = N X (bn) =! f (jd) ; f (sp) (3.0) where! 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-duson bn frequency correspondng to the same sample sze N (sp) = 5. 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. If f (jdsm) = P N j= U(S (jdsm) j th bn [x x + ) and S (jdsm) j for (.9), then the bn frequency expectaton and varance are =E hf (jdsm) = f (jd) =Var f (jdsm) x x ; + ) for =:N (bn) are the frequences of the s the jth jump-duson smulaton, usng N samples, as prescrbed and f (jdsm) hf (jdsm) = N respectvely, where the th expected bn frequency after N smulatons s f (jd) = N Z x+ x (jd) (x)dx:. ; f (jd) (jd) N f (3.) 4

5 The bn weghts are chosen as the theoretcal values,! = = f (jd), N (bn) X j= = f (jd) j (3.) for =: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 fq a Q b tg subject to constrants, M (jd) = ld t + j t = M (sp) and M (jd) = dt +( j + j)t = M (sp) (3.3) servng as elmnants of ld t and t, wth the jump-moments denton (.5) of d j and j relatng them to Q a and Q b (n rare case, non-negatvty must be enforced on the varances). The global mnmzer Golden Super Fnder (GSF) [7], developed for nancal problems n [4, 5], was used to estmate the t (3.0). Ths method s an extensve modcaton of the method of golden secton search (see [9]) and s descrbed more n [6]. The nal parameter results are d ' 0:06386 d ' 0:00553 j ' 0: j ' 0: ' 55:46 (3.4) wth mnmum mn ' :6 0;5 wth a relatve value-locaton hybrd stoppng crteron of 5 0 ;3 n a total of 6 GSF-teratons. The nal successful mnmum weghted least squares teraton results are llustrated n Fgure, 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 : Hstogram of log-returns from the log-normal duson, log-unform jump-duson model tted to the S & P 500 Index log-returns for the decade 99{00 shown n Fg., usng 00 bns. The gure on the left s the tted theoretcal jump-duson hstogram, whle the gure on the rght s the correspondng smulated jump-duson hstogram usng the same nal parameter results and the same number of samples as the S & P

6 Conclusons In ths paper, sgncant progress has been made toward ttng the theoretcal log-normal duson, log-unform jump-duson model to realstc nancal 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-duson smulaton. The resultng estmated jump-duson parameter set can add more realsm to nancal 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{99{733. References [] F. Black and M. Scholes, \The Prcng of Optons and Corporate Labltes," J. Poltcal Economy, vol. 8, , 973. [] F. B. Hanson and J. J. Westman, \Optmal Consumpton and Portfolo Polces for Important Jump Events: Modelng and Computatonal Consderatons," Proceedngs of 00 Amercan Control Conference, pp , 5 June 00. [3] F. B. Hanson and J. J. Westman, \Stochastc Analyss of Jump{Dusons for Fnancal Log{ Return Processes," Proceedngs of Stochastc Theory and Control Workshop, Sprnger{Verlag, New York, pp. -5, accepted, March 00. [4] F. B. Hanson and J. J. Westman, \Optmal Consumpton and Portfolo Control for Jump- Duson Stock Process wth Log-Normal Jumps," Proceedngs of 00 Amercan Control Conference, pp. -6, 08 May 00, to appear. [5] F. B. Hanson and J. J. Westman, \Portfolo Optmzaton wth Jump{Dusons: Estmaton and Applcaton," Proceedngs of 00 Conference on Decson and Control, pp.-5, 07 March 00, submtted for an nvted sesson. [6] F. B. Hanson and J. J. Westman, \Computatonal Methods for Portfolo and Consumpton Polcy Optmzaton n Log-Normal Duson, Log-Unform Jump Envronments," Proceedngs of the 5th Internatonal Symposum on Mathematcal Theory of Networks and Systems, pp. -6, August 00, to appear. [7] F. B. Hanson and J. J. Westman, \Golden Super Fnder: Multdmensonal Modcaton of Golden Secton Search Unrestrcted by Intal Doman," under testng and n preparaton, Aprl 00. [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 Scentc Computng, Cambrdge Unversty Press, Cambrdge, UK, 99. [0] Yahoo! Fnance, \Hstorcal Quotes, S&P 500, Symbol bspc," URL: February 00. 7