The Brownian Motion of visible particles suspended in a fluid   led to one of the first accurate determination   of the mass   of the visible molecules.  Mathematical model of Brownian motion   has numerous real world applications. For instance stock market fluctuations.  The Black- Scholes  model  for calculating  the premium of  an option  was introduced  in 1973 in a paper   published in  Journal  of Political Economy  developed by three Economists –Fisher Black, Myron Scholes  and Robert Merton and  is world’s  most  well known  Option Pricing Model . In 1997 all was awarded    Nobel Prize in Economics. Keywords: Brownian  Motion, Market fluctuations, Arbitrage Theorem, Random Walk, Hitting Time, Betting

Singh, Parvinder

English

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Research Journal of Finance and Accounting                                                                                                                                    www.iiste.org ISSN 2222-1697 (Paper) ISSN 2222-2847 (Online) Vol.6, No.3, 2015  22 Brownian Motion and the Black-Scholes Option Pricing Formula  Parvinder Singh P.G. Department of Mathematics, S.G.G. S. Khalsa College ,Mahilpur. (Hoshiarpur).Punjab. Email: parvinder070@gmail.com  Abstract The Brownian Motion of visible particles suspended in a fluid   led to one of the first accurate determination   of the mass   of the visible molecules.  Mathematical model of Brownian motion   has numerous real world applications. For instance stock market fluctuations.  The Black- Scholes  model  for calculating  the premium of  an option  was introduced  in 1973 in a paper   published in  Journal  of Political Economy  developed by three Economists –Fisher Black, Myron Scholes  and Robert Merton and  is world’s  most  well known  Option Pricing Model . In 1997 all was awarded    Nobel Prize in Economics. Keywords: Brownian  Motion, Market fluctuations, Arbitrage Theorem, Random Walk, Hitting Time, Betting.  1. Introduction:  In 1827 The English Botanist Robert Brown observed that the microscopic pollen grains suspended in water   perform a continual swarming motion. The phenomenon was first explained by Einstein in 1905 who said the motion comes from the pollen being hit by the molecules in the surrounding water.  The mathematical derivation of the Brownian motion   was first done by Wiener in 1918 and in his honor   it is often called Wiener Process.   2. Definition: A stochastic process [X (t), t ] is   said to be Brownian motion   process if 1. X(0) = 0 2. [ X(t),  t ] has  stationery  and  independent increments. 3.  For any t  X (t) is normally distributed with mean 0 and variance . When =1 the process is called Standard Brownian Process.  3. Random Walk:  Considering a walk in which each time unit is equally likely   to take a unit step either to the left or to the right i.e.  If  be the Markov chain with =  = , i= 0      More precisely    if each   time   we take a step of size  either to the left or to the right   with equal probabilities. If we let X (t) denote the position at time t then X (t) = (  +.........+ )                                                                       ..................(1.1)  Where   =          [t/    is the largest integer less than or equal to t/  and  are assumed independent with P [ = P[ =    Further as X (t) is normal with mean 0 and variance t the probability density function of X is .  Then joint density function of   X ( , X ( ......, X (   for   is   f ( = . .............  = ............................... (1.2) Where   X( ) =  ,   X( ) =  , .................., X( ) = are equivalent to X( ) = , X( )-  X( ) = ....... ,   X( ) - X( ) =   From the equation (1.2) we can compute desired   probabilities. Such as conditional distribution of X(s) given that X (t) = B where s   is  Research Journal of Finance and Accounting                                                                                                                                    www.iiste.org ISSN 2222-1697 (Paper) ISSN 2222-2847 (Online) Vol.6, No.3, 2015  23 (x/B)= = exp = exp =  exp =     where   K1, K2 and K3   do not depend on x. Hence conditional distribution of X(s) given that X (t) = B is for s  is normal with mean given by E  = B and Variance given by Var   3.1. Hitting Time: Let  denote the first hitting time the Brownian Process hits . When  we compute P { } by considering P[X (t)   and considering whether or not . This gives P[X(t) = P P{ }+P{ P{ }                                                         .............(1.3)                  Now if , then the process hits  at some point  in [0,t] and by symmetry  it is just above X  or below . Then P { =    As second right hand side  term of  (1.3) is clearly  equal to zero we see that P { }= P[X(t)   = dx = dy  , ................................(1.4)  If  the distribution of  is same as of  due to symmetry. Thus from (1.4) we get  P { = dy                                                                          .............................. (1.5)  The maximum value of the process attains at [0,t] as follows:  For , P[ ] = P[  = 2P{ } =  dy . Probability that  Brownian Motion  hit A  before B   where   A  0 and  B 0 { As  Brownian motion    is the  limit  of    symmetric Random walk }By  Gambler’s ruin problem Probability that  the symmetric random walk   goes up A  before going  down B when each step is  equally   likely of distances     with N =    )   equal to  =  .   Letting    .  3.2. Definition: Brownian Motion with Drift: [ ] is a Brownian Motion with drift co-efficient  and variance   if  1. X(0)=0 2. [ ] has stationery and independent increments. 3. X(t) is  normally distributed  with  mean  and  variance  4.  3.3. Definition: Geometric Brownian Motion: If [ ] is a Brownian Motion Process with drift co-efficient  and variance parameter    , then the process [ ] defined by  is called Geometric Brownian Motion. Art. (1): To compute Expected value of the process at time t given the history of the process upto time s .i.e.  For s < t consider E [ , 0 u  s] E [ , 0 u  s] = E [ , 0  u  s] = E [ /Y(u) ,0 u  s] = [ E { /Y(u) ,0 u  s}] = X(s) E[  Art. (2):  Moment Generating Function of a random variable  is given by  Research Journal of Finance and Accounting                                                                                                                                    www.iiste.org ISSN 2222-1697 (Paper) ISSN 2222-2847 (Online) Vol.6, No.3, 2015  24 E [ ] = .   Since Y (t)- Y(s) is normal with mean  and  variance  (t-s) ,   for   a = 1   E [ ] =   We get     E[X (t)/X(u) ,0 u s] = X(s) ......................(1.6) It is useful in  the  modeling of stock  prices  over time  when  percentage of  charges    are independently  and identically distributed . If  be the price of some stock at time n , then it is  reasonable to suppose   , n are  independently and identically distributed. Let    and so  Iterating we get   =  =  ...............=   Then +  , since  , t  are   independently and identically distributed, [  will be suitably normalized. So  will be approximately Geometric Brownian Motion.  4. Arbitrage Theorem:  Exactly one of the following statements is true: a). There exist a probability vector P = ( p1,p2,.....,.pn ) for which = 0 for all i= 1,2,....,m b). There exist a betting vector X = ( X1, X2,......,Xm) for   which  for all  J=1,2,....,n  In other words  if X be the outcome of the experiment, then the arbitrage theorem states that  either   is a probability vector  P for X such that  Ep[ri(X)] = 0 for I = 1,2,....,n.  Or else there is betting   scheme that leads to a sure win. Proof: The arbitrage theorem can be proved in several ways. Here we prove it by means of the duality of linear programming. Consider the standard primal and dual linear programming Problems: Pr        Primal Du        Dual  Az         Az= b              0           →min! z                                     yA c     max!  According to the duality theorem of the linear programming if the primal and the dual problems have feasible solutions then both problems have optimal solutions and the minimal value of the primal objective function is equal to the maximal value of the dual objective function.  Consider the following linear programming problem.                                                                             ..........................( 2.1) X m+1  max According to the condition of the problem we would like to reach at least an amount X m+1 for all outcomes and beside we want that this amount should be maximal. This problem can be transformed to the standard dual linear programming problem and we can write the primal problem as follows:                                              ................(2.2)                                               pj 0,  for  j= 1, 2, ...,       0  min  Note that the condition of problem (2.2) is the same as in the arbitrage theorem. It can be easily observed that the problem (2.1) has feasible solution (e.g. x = 0 and X m+1 = 0). We distinguish two cases according that the problem (2.2) has or hasn’t got any solution.  If the problem (2.2) has feasible solution (there exists a probability vector) then according to the duality theorem both problems have optimal solutions, the optimal value is zero. So X m+1 = 0 means that there is no sure win opportunity.  If the problem (2.2) has no feasible solution (there doesn’t exist a probability vector) then according to the Research Journal of Finance and Accounting                                                                                                                                    www.iiste.org ISSN 2222-1697 (Paper) ISSN 2222-2847 (Online) Vol.6, No.3, 2015  25 duality theorem there isn’t any optimal solution and the objective function of problem (2.1) is not bounded from above. It means that X m+1 can be positive. In this case there is sure win for all outcomes, so there is arbitrage. The arbitrage theorem has been proved.  The arbitrage theorem has a weak form, which gives a connection for the sure not-lose opportunity instead of the sure win.  4.1. Weak arbitrage theorem: Exactly one of the following statements is true: a).  There exists a probability vector p = (p1, p2, ..., pn), all of whose components are positive for          which     0       for all i = 1, 2, ..., m, b). there exists a betting vector x = (x1, x2, ..., xm) for which     for all j= 1, 2, ..., n, but for at least one index the strict inequality holds.  5. Main Result: The Black-Scholes Option Pricing Formula: Consider first wager of observing the stock for a time s and then purchasing (or selling), one share with the intention that of selling (or purchasing) it, at time t, 0 s  t   T.  The present value of the amount paid for the stock is  whereas the present value of the amount received is . Hence in order for  the expected return of this wager  to be 0  when P is the probability  measure on X(t) , 0  t T ,we  must  have  / X (u),   0  u   s] =   Now consider the wager   of the purchasing an option.  Suppose the option gives us the right to buy one share of   the stock at time t for a price K, at time t, the worth of this option will be as follows. Worth of option at time t =   At time t worth of the option is . Therefore present value of the worth of the option is .  If c  is the (time 0)  cost of the option ,  Then in order  for purchase   the option  to have expected ( present value) return 0  we mut have   ]=c                                                     ........... (3.2)   By Arbitrage theorem we can find a probability measure P on the set of outcomes of the equation (2.1). Then if c be the cost of an option   to purchase one share at a time t at the fixed price K given in the equation (2.2) .Then no arbitrage is possible.  On the other hand if for a given prices ,  i  = 1,2,...., N. There is no probability measure P that satisfies (3.1). And the equality [  , for I = 1, 2 ,....., N.  then a sure  win is possible.  We will now present a Probability measure P on the outcome X (t), 0  t  T that satisfies (3.1).   Suppose that X (t) = , whrere [   is a Brownian Motion Process with drift co- efficient  and variance parameter   . That is [   is a Geometric Brownian Motion Process. From the equation (1.7) we have for s < t   E[X (t) / X (u), 0 u s]  =  X (s) .......  (3.3)   If we chose  and   such that  =  . Then equation (2.1) will be satisfied. That is by letting P be the probability measure  governing  the stochastic process [ , 0  t  T],  where  is the  Brownian Motion  with drift parameter  and variance parameter   and where   =  then the  equation (2.1) is satisfied .  From the preceding if we price an option  to  purchase  a share of stock at  time  t  for a fixed price K by C = Research Journal of Finance and Accounting                                                                                                                                    www.iiste.org ISSN 2222-1697 (Paper) ISSN 2222-2847 (Online) Vol.6, No.3, 2015  26 ] , then no arbitrage  is possible . Since X (t) = , where  is normal with  mean  and variance parameter  ,  Then = =   Put   = (y-    we have                           ......................... (3.4) Where a =  Now = d =    P[N(  ,1) P[  N(  ,1)  P[  N(  ,1) =  where N(m,  is a  normal  random variable with mean m and  variance  .  Thus from (2.4) we get   =   Using   c= -k . ................. (3.5)    where b =   The option price formula  given by (2.5) depends  on the  initial price  of the stock  , the option exercise  time t , the option  exercise  price k ,  the discount ( or interest rate ) factor   , and the value .  Note that  for  any value of , if the  options  are  priced according  to the formula  of equation (2.5)  Then no arbitrage is  possible.  Therefore price of a stock actually follows a Geometric Brownian Motion - That is X (t) =  where   is Brownian motion with parameter   and . The formula (2.5) is known as Black Scholes Option Cost Valuation.  6. Refrences:  B.B. Mandelbrot and H.W. Van Ness, "Fractional Brownian motions, fractional noises and applications,"SIAM Rev., Vol. 10, pp. 422-436, oa. 1968. N. Tamas. Arbitrage Theorem and its Applications. “Club of Economics in Miskolc” TMP Vol. 1., pp. 27–32. 2002. J. Audounet, G. Montseny, and B. Mbodje, Optimal models of fractional integrators and application to systems with fading memory, 1993, IEEE SMC's conference, Le Touquet(France). J. Beran and N. Terrin, Testing for a change of the long-memory parameter, Biometrika 83 (1996), no.   3, 627{638). S.M. Ross.Introduction to Probability Models.10thedition.Academic  Press,San Diego,California.USA. V.S. Borkar, Probability theory: an advanced course, Universitext, Springer, 1995. Yu.A. Rozanov, Stationary random processes, Holden{Day, San Francisco, 1966.  The IISTE is a pioneer in the Open-Access hosting service and academic event management.  The aim of the firm is Accelerating Global Knowledge Sharing.  More information about the firm can be found on the homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS There are more than 30 peer-reviewed academic journals hosted under the hosting platform.   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Brownian Motion and the Black-Scholes Option Pricing Formula

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Brownian Motion and the Black-Scholes Option Pricing Formula

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