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#1
12-26-2009, 08:34 AM
 Noumenon84 Philosopher King Join Date: Jan 2009 Studying for FAP.... eventually Posts: 954
Arithmetic Brownian Motion

Basic question for ya.

I have read a couple of different source materials on this topic and each presented the material like this

Arithmetic Brownian motion is used to model stock prices by adding drift times t ($\mu t$) to a random normal element ($\sigma Z(t)$)

Then the next step is to move to Geometric Brownian motion by exponentiating this, because the previous form can be negative due to the random element and this just won't do for modeling of stock prices.

Here is what confuses me. In the first case $\mu$ is the "drift" which is multiplied by time to estimate the average growth of the stock and has little to do with the usual concept of [tex]\mu]/tex] from chapter 7, but by the use of the Geometric Brownian motion it appears then to be the mean of the expected growth rate.

Does Arithmetic Brownian motion model Stock prices or growth rate?
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Last edited by Noumenon84; 12-26-2009 at 08:39 AM..
#2
12-26-2009, 09:31 AM
 sohpmalvin Member CAS SOA Join Date: Aug 2007 Location: Singapore Studying for ACIM Favorite beer: It has been a while that I don't drink... Posts: 951

Arithmetic BM: St-S0 ~ normal(mt, st^0.5)
Geometric BM: St/S0 ~ lognormal(mt, st^0.5)
#3
12-26-2009, 10:41 AM
 Noumenon84 Philosopher King Join Date: Jan 2009 Studying for FAP.... eventually Posts: 954

Is the mu in abm the same mu that denotes the expected continuos growth rate?
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#4
12-26-2009, 10:47 AM
 Noumenon84 Philosopher King Join Date: Jan 2009 Studying for FAP.... eventually Posts: 954

When you take the differnce of abms it just seems to me we are modeling continuous growth rate which is normal and not stock prices
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#5
12-26-2009, 10:50 AM
 Noumenon84 Philosopher King Join Date: Jan 2009 Studying for FAP.... eventually Posts: 954

Furthermore if the mu which is drift is the sane as the expected growth rate does abm say that e^mu•t = mu•t
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#6
12-26-2009, 11:01 AM
 sohpmalvin Member CAS SOA Join Date: Aug 2007 Location: Singapore Studying for ACIM Favorite beer: It has been a while that I don't drink... Posts: 951

Maybe I should put it in this way:

Method 1 to model stock price movement, ABM:
St-S0 ~ normal(m_1t, s_1t^0.5)

Method 2 to model stock price movement, GBM:
St/S0 ~ lognormal(m_2t, s_2t^0.5)

Method 2 can also be expressed in ABM:
growth rate ~ normal(m_2 - 0.5s_2^2 , s_2t^0.5)
#7
12-26-2009, 11:05 AM
 sohpmalvin Member CAS SOA Join Date: Aug 2007 Location: Singapore Studying for ACIM Favorite beer: It has been a while that I don't drink... Posts: 951

dSt/S0 = a dt + s dZ is equivalent to dlnSt = (a- 0.5s^2) dt + s dZ.
#8
12-26-2009, 12:04 PM
 winvicta Member Join Date: Jun 2009 Posts: 77

Quote:
 Originally Posted by Noumenon84 Basic question for ya. Does Arithmetic Brownian motion model Stock prices or growth rate?
A basic answer is: the arithmetic Brownian motion models the growth rate, not the stock price.
#9
12-26-2009, 12:16 PM
 winvicta Member Join Date: Jun 2009 Posts: 77

Quote:
 Originally Posted by Noumenon84 Furthermore if the mu which is drift is the sane as the expected growth rate does abm say that e^mu•t = mu•t
That obviously is not true. In modeling asset price in continuous time, ie S(t)=S(0)exp(X(t)), where X(t) is the Ito process, the Ito process X(t) is to become the geometric Brownian motion if it is linearized. Once linearized, the positive coefficient of the linear term is your mu, which is alternately known as the Einstein coefficient, the instantaneous mean rate of return, or the growth rate. In the special case where the asset is of the risk-free type, mu is the force of interest. There are also a volatility term and a stochastic term. I would have tex them all up, but you can find them on p. 655 and 656 of MacDonald's DM.

Last edited by winvicta; 12-26-2009 at 05:27 PM.. Reason: for clarity (?)
#10
12-26-2009, 12:18 PM
 Force of Interest Member SOA Join Date: Sep 2009 Posts: 4,160

Quote:
 Originally Posted by winvicta A basic answer is: the arithmetic Brownian motion models the growth rate, not the stock price.
I agree with this.
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