IJRSS Volume 2, Issue 2 ISSN:

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1 A LOGITIC BROWNIAN MOTION WITH A PRICE OF DIVIDEND YIELDING AET D. B. ODUOR ilas N. Onyango _ Absrac: In his paper, we have used he idea of Onyango (2003) he used o develop a logisic equaion used in naural science o develop a logisic geomeric Brownian moion wih a price of dividend yielding asse since in realiy, asses do pay dividends o shareholders a regular inervals afer a mauriy period which normally afer one year. We have firs derived a deerminisic Walrasian price-adjusmen model wih price of dividend yielding asse afer which we have inroduced he sochasic effec of noise o derive a logisic geomeric Brownian moion wih a unique volailiy funcion (, such ha he observed opion price is consisen wih he price of he dividend yielding asse model. Keywords: Volailiy, Modeling, Geomeric Brownian moion, upply and Demand funcions, Equilibrium price, Walrasian excess demand funcion, deerminisic Walrasian price-adjusmen model, Logisic Brownian moion. Deparmen of Mahemaics and Applied aisics, Maseno Universiy, P.O Box 333 Maseno Kenya. Faculy of Commerce and Disance Learning, KCA Universiy, P.O Box 56808, Nairobi Kenya. 44

2 1. Inroducion: The supply and demand curves deermine he price and quaniy a which asses are bough and sold. The demand curve shows he quaniies buyers wan o buy a various prices and he supply curve shows he quaniies he sellers wan o sell a various prices. A siuaion in which here is no endency for change in securiy price and securiy quaniy a a given poin in ime is known as marke equilibrium. Tha is, here is no reason for he marke price of producs o rise or fall. In sock markes, he price of an asse is assumed o respond excess demand and is expressed as; ED ( Q ( Q ( 1.1 D Where ED ( is he excess demand, Q D ( and Q ( are quaniies demanded and supplied respecively a a given ime,, and price (. The logisic equaion was firs used by Verlhus (1838) and Reed (1920) in populaion growh. In Verlhus's model for sudying dynamics of human populaion growh in Unied aes, he ook Prepresen he environmenal carrying capaciy in which a populaion lives, which favorably compares o he Walrasian equilibrium marked price, a price a which he marke supply and demand are equal. Onyango (2003) using Verhluss logisic equaion developed a logisic equaion for asse securiy prices, considering he fac ha naurally asse securiy prices would no usually shoo indefiniely (exponenially) due o a regulaing facor ha may limi he asse prices. He inroduced he excess demand funcions and applied hem in he Walrasian (Walrasian amuelson) price adjusmen mechanism o obain a non-linear Brownian moion of he form d( ( ( ( ) ( ( ( ) dz( 1.2 Where is he equilibrium marked price, ( is he marke price of he asse or securiy, µ is he rae of increase of he asse and is he volailiy while dz( is he sandard wiener process. 45

3 2. Preliminaries: The law of demand The law of demand gives he relaionship beween demand price of an asse and quaniy demanded (ceeris paribus). The law saes ha as prices of produc decreases, he quaniy of he produc ha buyers are able and willing o purchase in a given period of ime increases if oher facors remain unchanged (ceeris paribus.) The law of supply The law of supply indicaes he relaionship beween supply price and quaniy supplied (Ceeris paribus). Tha is suppliers will supply less of goods or services a low price and as price rises he quaniy supplied will increase if oher facors remain unchanged (ceeris paribus.) Equilibrium price (Walrasian Price Equilibrium) This is a sae of sabiliy or balance where he quaniy of a good or service supplied is equal o he quaniy of he same good or service demanded. This sae leads o equilibrium price and equilibrium quaniy Q. Tha is Q D () =Q () and Q D (Q) =Q (Q) 2.1 where Q D (Q) he quaniy is demand funcions and Q (Q) is quaniy supply funcions. Walrasian price equilibrium saes ha he oal demand does no exceed oal supply and vice versa. 3. Wiener Process (Brownian moion): Wiener process is a paricular ype of Markov ochasic process wih mean change of zero and variance 1.0. If X ( follows a sochasic process where he mean of he probabiliy disribuion is and is he sandard deviaion. Tha is, X ( ~ N (, ) hen for Wiener process 46

4 X ( ~ N (0, 1) which means X ( is a normal disribuion wih =0 and =1. Expressed formally, a variable Z follows a Wiener process if i has he following properies: PROPERTY 1: The change Z during a small period of ime i Z = 3.1 where has a sandardized normal disribuion; (0,1) PROPERTY 2: The values of Z for any wo differen shor ime inervals of ime,, are independen. Tha is, Var ( Z i, Z j )=0, i j. I follows from he firs propery ha iself has a normal disribuion wih Mean of Z=0, andard deviaion of Z= and Variance of Z=. The second propery implies ha Z follows a Markov process. Consider he change in he value of Z during a relaively long period of ime T. This can be denoed by Z (T)-Z (0). I can be regarded as he sum of he changes in Z in N small ime inervals of lengh (, where T N Thus Z (T)-Z (0) = i, 3.2 i1 where he i (i=1,2,3...n) are disribued (0,1). From he second propery of Weiner process, i are independen of each oher. I follows ha Z (T)-Z (0) is normally disribued wih Mean =E (Z (T)-Z (0)) = 0, Variance of (Z (T)-Z (0)) = n = T hus, andard deviaion of (Z (T)-Z (0)) is T Hence Z (T)-Z (0) ~ N (0, T ). I^o process is a generalized Wiener process in which he parameers a and b are funcions of he value of he underlying variable X and ime. An I^o process can be wrien algebraically as, dx =a(x, +b(x, dz. 3.3 I^o lemma is he formula used for solving sochasic differenial equaions. I is a reamen of wide range of Wiener-like differenial process ino a sric mahemaical framework. 47

5 4. Geomeric Brownian moion: A specific ype of Io s process is he geomeric Brownian moion of he form dx=ax+bxdz 4.1 where a(x,=ax and b(x, =bx (samuelson, 1965 Black and choles,1973) The geomeric Brownian moion has been applied in sock pricing and is given as d dz 4.2 Where is he sock price, µ is he expeced rae of reurn per uni ime and is he volailiy of sock price. 5. Deerminisic Walrasian price-adjusmen model: Onyango (2003) borrowed from he Walrasian amuelson model core principle ha sellers call ou prices of commodiies and here is no immediae rade unil excess demand is zero and brough in he idea ha securiy prices are relaed o excess demand If we consider fracional rae of increase of asse price 1 d( (, i can be shown o be proporional o excess demand funcion, hence we have 1 d( QD( Qs( 5.1 ( The general expression of a linear demand funcion can be given as Q D ( a( b 5.2 Where ( is he price of asse a ime and is he equilibrium price for some appropriae parameers a and b. imilarly a linear supply funcion is expressed as 48

6 Q ( c ( d 5.3 For some appropriae parameers c and d (Jacques 1992) A linearise demand and supply curves abou equilibrium, gives equaions (5.2) and (5.3) as Q D ( ( ( ( ) and ( ( ( ( ) 5.4 Q Where is he demand elasiciy and is he supply elasiciy. Equaion (5.4) requires ha Q D ( is a decreasing linear funcion of (, > 0 and Q ( is an increasing linear funcion of (, 0, The excess demand funcion is given as ED ( Q D ( Q ( ( )( ( ) 5.5 From proporional rae of increase in equaion (5.1) and excess demand funcion in equaion (5.5) we have Or Or d( h( )( ( 1 d ( ( ( d( ( ( ( ) ( ) ( ) Where h ( ) and is a growh consan. The asse price growh rae is also equal o zero when (=o. This is a deerminisic logisic (firs order) ordinary differenial equaion in sock price ( and he limiing consan ( is also known as Verhlus logisic equaion. In realiy, asses do pay dividends o hareholders. In his case we consider paymens of dividends o be coninuous. The Verhlus logisic equaion in (5.7) becomes d ( q) ( ( ( ( ) : q h( )

7 Where q is he dividend paying rae. d( Or ( q) d( 5.10 ( ( ( The L.H. of equaion (5.10) can be done using Heaveside cover up mehod, leing ( 0 and ( afer inegraing one has 1 ( ln ( o ( q) ( ( 0 ) o In In ( q) ( 0 ) 5.12 ( ( ) 0 Rearranging and simplifying equaion (5.12) and solving for ( we ge ( (0) ( ( (0)) (0) e ( q) ( 0 )), if P( 0 ) P(0) 5.13 This is a deerminisic logisic equaion in asse price (, wih iniial price (0), equilibrium price, asse price growh rae and q dividend yield rae. 6. Logisic geomeric Brownian moion model wih dividend yielding asse: From equaion (5.6) we exend Walrasian Price adjusmen model by inroducing acual asse prices wih noise effec which is a Weiner process. We consider rading ground where bargains gives small random effec in sensiiviies of and in supply and demand funcions. We le and be accumulaive changes in price sensiiviies of and respecively during rading period. Then equaion (5.6) becomes. d( h( )( ( 1 ( ) h( )( ( ) h( )( ( )

8 Wiener process on supply and demand from cumulaive effecs on random shock on and is pu as h( ) dz 6.2 Where he sandard deviaion (volailiy) of underlying asse, Z is is he sandard Wiener process ( Z, N(0,1 ) ubsiuing (6.2) in (6.1) we have 1 d( ( q) dz ( ( ( ) P( Using he same process for solving (5.13). Equaion (6.3) becomes P 6.3 ( ( 0 ) In In ( q) ( 0 ) dz( ( ( ) Rearranging and simplifying equaion (6.4) we have ( (0)) (, 6.5 ( q) ( 0 ) dz ( ) (0) ( (0) e This is logisic Brownian moion model in sock price wih dividend yielding asse. 7. Conclusion: In his paper we have developed logisic geomeric Brownian moion wih a price of dividend yielding ass since in realiy asses do pay dividend o shareholders. This can be used o show a unique volailiy funcion (, from observed opion price which is consisen wih dividend yielding asse model in (6.5) 51

9 References: Akira, Y; N (1996). Mahemaics Economics 2nd Ediion-Cambridge Universiy press. Black, F and choles, N (1973). The pricing opions and corporae liabiliies, Journal of poliical Vol 18, Feller, W (1991). An inroducion o probabiliy heory and is applicaion, Vol 2 2nd Ediion, New York John Willey and sons. Hull J, (2000). Opion, Fuure and oher price derivaives, Fourh ediion, Hull Inernaional Hull, J and Whie a, (1987). The pricing of opion all asses wih sochasic volailiy of finance, Volume 2; Jacques, I N (1992). Mahemaics Economics and Business, Wokingham England; Addision-Wesley publisher company. Marin B and Andrew R. (2000). An inroducion o derivaive pricing, Cambridge Universiy press. Meron R (1973). Theory of raional opion pricing-the Bell of Economics and managemen science, Vol 4: Onyango (2003). Exracing sochasic processes from marke price daa; a paern recogniion approach, PhD Thesis, Universiy of Hadderfield UK. Onyango. (2005). On he Linear sochasic price adjusmen of securiy., The Eas Africa journal of saisics. Jomo Kenyaa Universiy press. pp