Arithmetic Brownian Motion

[Noumenon84]

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 () to a random normal element ()

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 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?

[sohpmalvin]

Arithmetic BM: St-S0 ~ normal(mt, st^0.5)
Geometric BM: St/S0 ~ lognormal(mt, st^0.5)

[Noumenon84]

Is the mu in abm the same mu that denotes the expected continuos growth rate?

[Noumenon84]

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

[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

[sohpmalvin]

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)

[sohpmalvin]

dSt/S0 = a dt + s dZ is equivalent to dlnSt = (a- 0.5s^2) dt + s dZ.

[winvicta]

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.

[winvicta]

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.

[Force of Interest]

Originally Posted by winvicta

A basic answer is: the arithmetic Brownian motion models the growth rate, not the stock price.

I agree with this.