MTH6121 Introduction to Mathematical Finance Lesson 5


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1 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif Geomeric Brownian moion Convergence of random walks o Brownian moions Convergence o Geomeric Brownian Moions Example Le W be he Wiener process. Find a E[W 2 + W 3]; b Cov W 2, W 3 ; c Var W 2 + W 3. Soluion. a Noe ha W 2 N 0, 2 and W 3 N 0, 3. Thus E[W 2 + W 3] = E[W 2] + E[W 3] = = 0. b By Lemma 2.8. CovW 2, W 3 = min2, 3 = 2. c Noe ha W 2 and W 3 are dependen, so we need o use a Proposiion from he previous chaper regarding he variance of he sum of random variables, Var W 2+W 3 = VarW 2+VarW 3+2Cov W 2, W 3 = = 9. When performing concree calculaions wih he Wiener process i is useful o remember ha, since W N 0,, we have 1 W N 0, 1. Example Le W be he Wiener process. Wha is PW 4 > 5? Soluion. Since W 4 N 0, 4, we have W 4 PW 4 > 5 = P > 5 = P 4 4 Z > 5 4 = 1 Φ 5 = =
2 Brownian moion wih drif In his shor secion we will discuss a generalizaion of he Wiener process ha will urn ou o be useful for our second model of sock marke prices. Definiion Brownian moion wih drif having drif parameer R and volailiy parameer > 0 is he sochasic process Y given by where W is he Wiener process. Y = + W, Noe ha if Y is a Brownian moion wih drif parameer and volailiy parameer, hen for every > 0, we have while E[Y ] = E[ + W ] = + E[W ] =, VarY = Var + W = VarW = 2 VarW = 2. Bu since for every > 0 he random variable Y is Normal, we have Y N, 2 > 0. Noe ha Y j Y j 1 = j j 1 + W j W j 1 for all j = 1,..., n. Using he independen incremens of he Brownian Moion see propery d in Theorem for he characerisaion of he Wiener process, we can conclude ha he random variables Y 1, Y 2 Y 1,..., Y n Y n 1 are independen. Calculaions for Brownian moion wih drif can be performed in he same way as for he Wiener process iself. Example Le Y be Brownian moion wih drif parameer = 0.2 and volailiy parameer = 0.1. Find E[Y 9], Var[Y 9] and PY 9 > 1. Soluion. Using he above resuls, we obain and Finally, we have E[Y 9] = 9 = 1.8, VarY 9 = 9 2 = PY 9 > 1 = P9 + W 9 > 1 = P W 9 > 1 9 = PW 9 > 8 W 9 = P > = P Z > 8 9 = 1 Φ 8 8 = Φ 3 3 =
3 Geomeric Brownian moion Brownian moion wih drif has a drawback, namely is propery ha i can become negaive. This feaure rules i ou as a suiable model for he movemens of he sock marke. The following varian does no suffer from his shorcoming, and provides a decen model of sock marke prices. Definiion Geomeric Brownian moion having drif parameer R, volailiy parameer > 0, and saring value = S is he sochasic process S given by where W is he Wiener process. S = S exp + W, I is easily verified from he above equaion ha = S exp0 + 0 = S which also explains he erminology saring value S of S. Moreover, so where Thus S S S S S log S = exp + W = + W + W N, 2. S LogNormal, 2. Observe also ha for any 0 1 < 2 < < n he successive raios S 2 S 1 = exp W 2 W 1 S 3 S 2 = exp W 3 W 2 S n S n 1 = exp n n 1 + W n W n 1 are all independen since Brownian moion has independen incremens. raios are all lognormally disribued. Thus, Furhermore, he Geomeric Brownian moion is an exension of he IID lognormal model o a coninuous ime parameer. Example Suppose ha he weekly price of chewing gum evolves according o Geomeric Brownian moion wih drif parameer = and volailiy parameer = 0.1. Wha is he probabiliy ha a he price of chewing gum is higher in 2 weeks and 5 days han i is now?
4 29 b he price of chewing gum rises from one day o he nex for he nex 3 days? Soluion. Le S denoe he price of chewing gum a ime, where is measured in weeks. We are old ha S = S exp + W, where W denoes he Wiener process, S is he saring parameer, = and = 0.1. a Since a week has 7 days, we need o find he probabiliy ha S2 + 5/7 >. Now S/7 P S/7 > = P > 1 = P log S/7 > 0. Bu so log S/7 P S/7 > = P log S/7 = P Z > = 1 Φ 7 N, 7 2, > 0 = P = Φ S/7 log 7 > = Φ0.07 = Thus, he probabiliy ha he price of chewing gum is higher in 2 weeks and 5 days han oday is 52.8%. b We shall firs calculae he probabiliy p ha he price of chewing gum is higher a he end of a day compared o he beginning. Now p = PS1/7 > = P log S1/7 > 0. Bu so p = P log S1/7 Bu since he random variables log S1/7 N 1 7, 1 2, S1/7 log 1 7 > 0 = P 1 > 7 1 = P Z > = 1 Φ = Φ = Φ0.02 = S1/7, S2/7 S1/7, S3/7 S2/7, are all independen and have he same disribuion, he probabiliy ha he price of chewing gum rises from one day o he nex for he nex 3 days is p 3 = 0.13 = 13%.
5 Convergence of random walks o Brownian moions. I urns ou ha he Wiener process is, in he sense o be made precise shorly, a rescaled version of a random walk. Recall ha Y n is a random walk if Y n = n X i, where he X i s are independen and idenically disribued random variables wih PX i = 1 = PX i = 1 = 1 2. Definiion Sochasic processes Y h converge in disribuion o a sochasic process Y as h 0, if for any 1 < 2 < < k, and x 1, x 2,... x k, for some k 1, lim PY h 1 x 1,..., Y h k x k = PY 1 x 1,..., Y k x k. 1 h 0 Recall ha, if 1 holds hen we wrie Y d = lim h 0 Y h. In order o sae he main resul of his subsecion, we also need he following definiion. Definiion Given a real number x, he floor of x, denoed by x, is he larges ineger less han or equal o x. Thus, for example, 1 = 1, 3.2 = 3, 5.9 = 5. We are now able o formulae he following ineresing resul. Theorem 2. Convergence of he random walk o he Wiener process. Le X 1, X 2,... be a sequence of independen and idenically disribued random variables wih common disribuion given by PX i = 1 = PX i = 1 = 1/2 for all i. For > 0, define he processes Y h = /h h X i. Then, here exiss a Wiener process Y, such ha Y d = lim Y h. h 0, h>0 Proof. We sar by showing ha Y N 0,. The proof follows from he Cenral Limi Theorem CLT. To see his, se n = /h = /h δ, for some 0 δ < 1 by he definiion of he floor funcion. Then h = /n + δ and hus Y h = /h h X i = n + δ n X i = n n + δ Since EX i = = 0 and VarX i = 2 = 1, we see ha X i = X i n n n X i n.
6 31 has he same form as he relevan expressions in he CLT. Noe ha, when h 0, we have n and hence CLT implies ha X i n d N 0, 1 as h 0. Also observe ha n n + δ 1 as h 0 which finally implies ha Y h = n n + δ X i n d N 0, 1 = N 0, as h 0. Therefore, we can se Y N 0,. Similarly, i is possible o show ha for any 0 1 < 2 Y 2 Y 1 d = lim h h 0,h>0 2 /h i= 1 /h +1 X i N 0, 2 1. Furhermore, i is possible o show ha for any 0 1 < 2 < < n he random variables Y 1, Y 2 Y 1, Y 3 Y 2,..., Y n Y n 1 are independen. By he Theorem 2.7 for he characerizaion of he Wiener process W, we mus have Y = W for every Convergence o Geomeric Brownian Moions We have seen ha he Wiener process can be obained as he limi of a random walk Theorem 2.. I urns ou ha a similar resul holds for he Geomeric Brownian moion, obained as he limi of a random walklike process. The laer is a muliplicaive equivalen o he random walk. Denoe by S h he price of a financial asse a imes = 0, h, 2h,... h > 0, which sars from a value S h 0 = S and evolves according o S h i + 1h = Sh ih Y i, for all i = 0, 1,... where Y i is a sequence of independen idenically disribued random variables aking one of he wo possible values: { u = e h, wih probabiliy p = h Y i = d = e h, wih probabiliy 1 p = h The above means ha afer each ime inerval of lengh h he price S h ih of he financial produc eiher goes up by a facor u or down by a facor d. Theorem As h 0 he muliplicaive random walk model described above converges o a Geomeric Brownian moion wih drif parameer, volailiy parameer, and sars from he value S. Proof. The proof relies on he applicaion of he CLT, wih similar argumens as in he proof of Theorem 2. above.
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