A STOCHASTIC GROWTH PRICE MODEL USING A BIRTH AND DEATH DIFFUSION GROWTH RATE PROCESS WITH EXTERNAL JUMP PROCESS *

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Page345 ISBN: 978 0 9943656 75; ISSN: 05-6033 Year: 017, Volume: 3, Issue: 1 A STOCHASTIC GROWTH PRICE MODEL USING A BIRTH AND DEATH DIFFUSION GROWTH RATE PROCESS WITH EXTERNAL JUMP PROCESS * Basel M.

1 Page345 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 A STOCHASTIC GROWTH PRICE MODEL USING A BIRTH AND DEATH DIFFUSION GROWTH RATE PROCESS WITH EXTERNAL JUMP PROCESS * Basel M. Al-Eideh Kuwait Uiversity, College of Busiess Admiistratio, Kuwait basel@cba.edu.kw Abstract Oe of the most importat fuctios of ecoomists is to provide iformatio o the future of prices, which is importat to pla for huma activities. There is a cosiderable iterest i stochastic aalogs of classical differeces ad differetial equatios describig pheomea i theoretical models ivolvig ecoomic structure. I this paper, a descriptio of growth price model usig a solutio of stochastic differetial equatio is cosidered. More specifically, the growth price process follows a Geometric progressio with growth rate follow a birth ad death diffusio process with radom exteral jump process is studied. The mea ad the variace approximatio, as well as the predicted ad the simulated sample path of such a growth price process are also obtaied. Numerical examples for the case of o jumps as well as the case of the occurrece of jump process that follow a uiform ad expoetial distributios are cosidered. Keywords : Growth Price Model, Birth-Death Diffusio Process with Jumps, Growth Rate, Stochastic Differetial Equatio. 1. Itroductio This paper shows how decisio makers cocers about model specificatio ca affect prices ad quatities i a dyamic ecoomy. We use this ew approach of stochastic growth process i price models for two reasos. The most importat oe is this model which is from the type of cotiuous time models which is differet from what we have already kow from the discrete-type models. The secod reaso is that this kid of model is ot widely used i various ecoomic models. David (1997) studies a model i which productio is liear i the capital stocks with techology stocks that have hidde growth rates. Veroesi (1999) studies a permaet icome model with a riskless liear techology. Divideds are modeled as a additioal cosumptio edow cet. Hidde iformatio was itroduced ito asset pricig models by Detemple (1986), who cosiders a productio ecoomy with Gaussia uobserved variables. Had (001) hasdeveloped methods usig statistical tools such as logistig regressio ad aïve Bayes, as well as eural etworks for assessig performace of the models to the cosumer credit risk. Al-Eideh ad Hasa (00) have cosidered growth price models uder radom eviromet usig a solutio of stochastic differetial equatio of the logistic price model, as well as the logistic price models with radom exteral jump process. They derived the steady state

2 Page346 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 probability ad the time depedet probability fuctios. Also, the mea ad the variace as well as the sample path of such a process are cosidered. Durig this past decade, there has bee icreasig effort to describe various facts of dyamic ecoomic iteractios with the help of stochastic differetial processes. Thus, stochastic differetial processes provide a mechaism to icorporate the iflueces associated with radomess, ucertaities, ad risk factors operatig with respect to various ecoomic uits (stock prices, labor force, techology variables, etc.) Therefore, the techiques of stochastic processes become relevat i pursuig quatitative studies i ecoomics. Stochastic modelig techiques ot oly eable us to obtai reliable estimates of certai useful ecoomical parameters, but also provides idispesable tools for estimatig parameters which are ofte associated with a high degree of o-samplig errors. Thus, it is proposed to itroduce a study of ecoomic structure usig the techiques of stochastic processes. Stochastic differetial equatio processes have bee itroduced i the study of three pricipal categories of ecoomic pheomea: (a) descriptio of growth of certai factors uder ucertaity (b) the ature of optio price variatios presetig certai market coditios ad (c) stochastic dyamic programmig ad cotrol objectives. Numerous researchers have worked o studyig various ecoomic uits from differet poits of view. For example, Aase ad Guttrop (1987) studied the role of security prices allocative i capital market, they preset stochastic models for the relative security prices ad show how to estimate these radom processes based o historical price data. The models they suggest may have cotiuous compoets, as well as discrete jumps at radom time poits. Also, two classical applicatios are Metro (1971) ad Black ad Scholes (1973). New refereces iclude Harriso ad Pliska (1981) ad Aase (1984). Whereas the first two works oly study processes with cotiuous sample paths, the other two allow for jumps i the paths as well. I other words, the processes have sample paths that are cotiuous from the right ad have left had limits (i fact, these processes are semi-martigales; for geeral theory of semi-martigales, see e.g. Kabaov et al., 1979 sec. ). May other authors have studied this problem from differet poits of view, such as Stei ad Stei (1991), Tauche ad Pitts (1983), Schwert (1990), Duffie ad Sigleto (1993), McGratta (1996), Calle ad Chag (1999), Karmeshu ad Goswami (001), etc. I this paper, we preset ew growth price models usig a solutio of stochastic differetial equatio. More specifically, the growth price process follows a Geometric progressio with growth rate follow a birth ad death diffusio process with radom exteral jump process is studied. The mea ad the variace approximatio, as well as the sample path of such a process are also obtaied. The mea ad the variace, as well as the predicted ad the simulated sample path of such a growth price process are also obtaied. Numerical examples for the case of o jumps, as well as the case of the occurrece of jump process that follow a uiform ad expoetial distributios are cosidered.. The Growth Price Model Usig a Birth ad Death Diffusio Growth Rate Process With Exteral Jump Process Now as a matter of fact, modelig's of Natural Prices have three basic characterizatios as follows: (1) The Prices over time may show the average desity of price beig maitaied at a

3 Page347 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 costat level over a log period of time, uless there is a major evirometal chage. () The growth of a Price eed ot ecessarily remai at a costat level; but the same may fluctuate aroud a costat mea value radomly. (3) The third type is agai a geeralizatio of the secod type, ad is give by superim-positio of radom cycle of oscillatio o the type of radom variatio. With this set up, we propose to develop a simple determiistic model. Here, we igore such factors as evirometal coditios etc. while developig a determiistic model. Further, as i the determiistic model, the probabilistic cosideratio relatig to the variatio of the price S( is igored, oe ca reasoably assume S( to be cotiuous variable. I other words, we structure the determiistic model as the followig geometric progressio: ds( S( B( D( dt (1) B( ad D( are the istataeous icreasig (birth) ad decreasig (Death) rates at time t. Note that they are idepedet of S(.. Now, by lettig G( B( D(, the we defie the Growth price process { S( ; t 0} which is modeled by the geometric progressio such that dx( G( X ( dt () G( represets the birth ad death diffusio growth rate with exteral jump process. Cosider the Growth rate process G ( ; t 0 i which the diffusio coefficiet a ad the drift coefficiet b are both proportioal to G ( at time t. The diffusio process is assumed to be iterrupted by exteral effects occurrig at a costat rate c ad havig magitudes with distributio H (). The G ( ; t 0 is a Markov process with State Space S 0, ad ca be regarded as a solutio of the stochastic differetial equatio dg( bg( dt ag( dw( G( t ) dz( (3) Here, W is a Wieer process with mea zero ad variace t Poisso process. N ( Y i i1. Also, ( Z is a compoud Z( (4) Here N ( is a Poisso process with mea rate c, c is the exteral jump rate, ad Y 1, Y,..., are idepedet ad idetically distributed radom variables with distributio fuctio

4 Page348 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 H (), with mea E( Y 1 ) ad variace v Var( Y 1 ). Note that the momets of determied from the radom sums formulas, ad are EZ ( ct (5) Z( ca be Ad t Var Z t cv ( ) (cf. Taylor ad Karli (1983), pp. 55, 01) (6) Now from equatio (3) we get, dg( bdt adw( dz( G( (7) Thus, the solutio of the stochastic differetial equatio i (7) is give by G( G(0)exp bt aw( Z( (8) G(0) is the iitial growth rate at time zero. Rewrittig equatio () we get ds( G( dt S( (9) Takig itegral o 0,t for both sides of equatio (9) we get t ds( t G( dt 0 S( 0 Note that t ds( S( l S( l S(0) l 0 S( S(0) (10) ad usig Karli ad Taylor (1981) ad Al-Eideh ad Al-Hussaia (00) ad after some algebraic maipulatios, it is easily show that t G( s) ds G(0) expbs aw( s) Z( ds 0 (1 b) G(0) expbt aw( Z( a a b (11) Therefore, the solutio of the stochastic differetial equatio i (9) is give by (1 b) S( S(0) exp G(0) expbt aw( Z( a a b (1) X (0) is the iitial populatio data at time zero.

5 Page349 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 3. Mea Ad Variace Approximatio of the Growth Price Process S( Usig the Birth ad Death Diffusio Growth Rate Process G( With Exteral Jump Process H(.) I this sectio, the mea ad the variace approximatio for the growth price process S( usig the birth ad death diffusio process G( with exteral jump process H() defied i equatio (1) are derived. Let M1( ES( ad V( VS( be the mea ad the variace of S( respectively. Now, usig the Machuria expasio, the S( i equatio (1) ca be approximated by (1 b) S( S(0) S(0) G(0) expbt aw( Z( a a b (13) or equivaletly; (1 b) S( S(0) S(0) G( a a b (14) G( is defied i equatio (8). Usig the results of fidig the momet approximatio of a birth ad death diffusio process with costat rate jump process (cf. Al-Eideh (001)), it is easily show that 1 1 E G( G(0) exp b a t 1 ct c( v ) t (15) ad E G ( G(0) exp b a t 1 ct c( v ) t (16) Therefore, the variace of G( is the give by V G( G(0) exp b a t 1 ct c( v ) t 1 ct v a t 1 e c t ct c t 4 (17) Now, usig the approximated growth price model S( i equatio (14), we get (1 b) M1( S(0) S(0) EG( a a b (18) ad V ( (1 b) a a b S(0) VG( G( E ad G( (19) V are defied i equatios (15) ad (17) respectively.

6 Page350 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 4. Predicted Ad Simulated Growth Price Model S( I this sectio, we will obtai the predicted ad the simulated sample path of the growth price process S( usig the birth ad death diffusio process G( with exteral jump process H() defied i equatio (1). Assumig writte as M ( t 1 t M 1( t t 1) be the oe-step predicted model of 1 (1 b) G(0) ) S(0) S(0) exp b a a b 1 1 ct c( v ) t t 1 1 a t For simulatio of the growth price process approximatio. S( t S(, the 1 M 1( t t 1) ca be (0) we used the followig discrete For iteger values k 1,,3,, ad 1,,3,, the birth ad death growth rate diffusio process G( with exteral jump process H() ca be simulated by * k 1 * k b * k a * k * k k k G G G G Z k1 G J C (1) Z (k) is a idepedet sequece of stadard ormal radom variables ad k C ; k 1,,... are idepedet ad idetically distributed with k PC 1 c k c PC 0 1 J 1 J ad,, are idepedet ad idetically distributed with distributio ( G H (). k, t,...,, For each set of positive itegers 1 t k the sequece of radom vectors * ),..., G ( tk )) ( G ( ),..., ( )) coverges i distributio to t1 G tk. * ( t1 Thus, the simulated Populatio growth model S( is give by

7 Page351 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 * k S S k S k (1 b) a a b 1 * * * * k 1 G i defied i equatio (1). k 1 G () X * Note that for each set of positive itegers k, the sequece of radom vectors (1),, ( ) * X k coverges i distributio to X ( 1),, X ( k). 5. Numerical Example Cosider, as a example, the followig sample paths of the above model S( i sectio (4) that represets the aual price of a item i US dollars whe S( 0, b 0. 0, a, 0, ad c 1 for the followig cases: Case 1 I this case, we cosider the sample path to the stochastic growth price model S( usig the stochastic birth ad death diffusio growth rate process G( with o exteral jump process, ote i this case the jump rate c 0. Figure 1 ad Figure represet this case for G( ad S( respectively. Figure 1: The Stochastic Growth Rate Process with o Exteral Jump Process Figure : The Associated Stochastic Growth Price Model

8 Page35 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: Case I this case, we cosider the sample path to the stochastic growth price model S( usig the Stochastic birth ad death diffusio growth rate process G( with Uiform exteral jump process, ote i this case the jump rate c 1. For simplicity, we take H() to be uiform o 0,1. Thus dh( y) 1, 0 y 1 (3) 1 1 Note that H (y) is idepedet of y v with mea, ad variace 1. Figure 3 ad Figure 4 represet this case for G( ad S( respectively. Figure 3: The Stochastic Growth Rate Process with Uiform Jump Process

9 Page353 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: Figure 4: The Associated Stochastic Growth Price Model Case 3 I this case, we cosider the sample path to the stochastic growth price model S( usig the Stochastic birth ad death diffusio growth rate process G( with Expoetial exteral jump process, ote i this case the jump rate c 1. For simplicity, we take H() to be expoetial with mea 1. Thus, ( ) y dh y e, y 0 (3) Note that H (y) depeds o y with mea 1, ad variace v 1. Figure 5 ad Figure 6 represet this case for G( ad S( respectively.

10 Page354 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 Figure 5: The Stochastic Growth Rate Process With Expoetial Jump Process Figure 6: The Associated Stochastic Growth Price Model Lookig to the above figures, we ca see the differece betwee these figures, also the differece betwee the uiform jump ad the expoetial jump is oted i the stochastic growth rate processes ad this shows the differece betwee the jump processes if they are depedet or idepedet of the growth rates ad fially this differece affects the stochastic growth price models. Ay way, the figures are reasoable ad suggested to be used i the modelig purposes for some prices.

11 Page355 ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 Coclusios I coclusio, this study provides a methodology for studyig the behavior of the prices. More specifically, the study departs from the traditioal before ad after regressio techiques ad the time series aalysis ad developed a stochastic model that explicitly accouts for the variatios ad volatilities i prices follow a geometric progressio usig a birth ad death diffusio growth rate process subject to radomly occurrig exteral jump processes, especially the uiform o [0, 1] ad expoetial with mea 1 processes. Ideally, a large class of exteral jump processes with geeral jump rate could be tackled i future researches. Also, some iferece problems could be doe for this model. I terms of future research, this methodology could be applied ot oly i prices, but o all aspects of ecoomics ad operatios research problems. * This work was supported by Kuwait Uiversity, Research Grat No. [IQ 0/15]. Refereces i. Al-Eideh, B. M. (001). Momet approximatios of life table survival diffusio process with geeral exteral effect. Iter. J. of Appl. Math., 5 (1), ii. iii. iv. Al-Eideh, B. M. ad Al-Hussaia, A. A. (00). A quasi-stochastic diffusio process of the Lorez curve. Iter. Math. J., 1 (4), Al-Eideh, B. M. ad Hasa, M. H. (00). Growth price model uder radom eviromet: A stochastic aalysis. Theory of Stoch. Proc., 8 (4), o. 1-, 14-. Aese, K. K. (1984). Optimum Portfolio diversificatio i a geeral cotiuous-time model. Stoch. Proc. Applic., 18, v. Black, F. ad Scholes, M. (1973). The pricig of optios ad corporate liabilities. Joural of Political Ecoomy, 81, vi. vii. Aase, K. K. ad Guttorp, P. (1987). Estimatio i Models for security prices. Scad. Actuarial J., 3-4, Calle, T. ad Chag, D. (1999). Modelig ad Forcastig Iflatio i Idia. IMF Workig Paper, WP/99/119. viii. David, A. (1997). Fluctuatig Cofidece i Stock Markets: Implicatios for returs ad volatility. Joural of Fiace ad Quatitative Aalysis, 3 (4), ix. Detemple, J. (1986). Asset pricig i a productio ecoomy with icomplete iformatio. Joural of Fiace, 41, x. Duffie, D. ad Sigleto, K.J. (1993). Simulated momets estimatio of Markov models of asset prices. Ecoometrica, 61, xi. xii. xiii. Had, D. J. (001). Modelig Cosumer Credit Risk. IMA Joural of Maagemet Mathematics, 1, Harriso, J.M. ad Pliska, S.R. (1981). Martigales ad stochastic itegrals i the theory of cotiuous tradig. Stoch, Proc. Appl., 11, Kabaov, Ju. M., Lipster, R.S. ad Shiryayev, A.N. (1979). Absolute cotiuity ad sigularity of locally absolutely cotiuous probability distributios. I. Math. USSR Sborik, 36,

12 Page356 xiv. ISBN: ; ISSN: Year: 017, Volume: 3, Issue: 1 S. Karli ad H. M. Taylor, A Secod Course i StochasticProcesses. Academic press. New York (1981). xv. Karmeshu ad Goswami, D. (001). Stochastic Evaluatio of Iovatio Diffusio i Heterogeeous groups: Study of Life Cycle Patters. IMA Joural of Maagemet Mathematics, 1, xvi. McGratta, E. R. (1996). Solvig the Stochastic Growth Model with a Fiite Elemet Method. Joural of Ecoomic Dyamics ad Cotrol, 0, xvii. Merto, R.C. (1971). Optimum Cosumptios ad Portfolio rules i a cotiuous-time model. J. Eco. Theory, 3, xviii. Schwert, G.W. (1990). Stock volatility ad the cash of 87. Review of Fiacial Studies, 3, xix. xx. xxi. Stei, E.M. ad Stei, J.C. (1991). Stock Price distributios with stochastic volatility: a aalytic approach. Review of Fiacial Studies, 4, Tauche, G.E. ad Pitts, M. (1983). The Price Variability volume relatioship o speculative markets Ecoometrica, 51, Taylor, H. M. ad Karli, S. (1984). A Itroductio to Stochastic Modelig. Academic Press, USA. xxii. Veroesi, P. (1999). Stock Market Overreactio to Bad News i Good Times: A ratioal expectatios equilibrium model. Review of Fiacial Studies, 1,

Subject CT1 Financial Mathematics Core Technical Syllabus

Subject CT1 Financial Mathematics Core Technical Syllabus Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig