What exactly is Brownian Motion

Today we will talk about Brownian Motion,
More than 60% of CT8 syllabus moves around Brownian Motion.

What is stochastic process: Stochastic process is a sequence of some quantity where the future values cannot be predicted with certainty.


Brownian Motion:

Definition: It is a stochastic process with a continuous state space and in continuous time. This process has stationary, independent and normally distributed increments.

Understanding: Here in CT8 we are going to model the share prices that is we are looking at how can we make share prices models so that we can predict our future returns on the basis of risk level decided by investor and then invest accordingly. Brownian motion is one of the tool with the help of which we can do this.
Standard Brownian Motion has normal distribution which means it follows normal with mean “0” and Variance “t” at time t.
Under the General Brownian, it also follows the normal distribution with mean “u” and variance “σ2” but these mean and variance are linked with the length of time interval., and then there becomes an ultimate equation like this:
Wt = W0 + ut + σBt , which means the general Brownian motion at time t is the combination of stochasticity with standard Brownian motion (Bt) and changes its value with respect to time through mean (u).


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Example : Yes bank share price today is Rs.200 and I am saying this price can grow with respect to 10% p.a. then this thing is being captured by “u” in the formula of  “ Wt “ but there will be an uncertainty factor associated with it that it may be fall by 20% or rise by more than 10%, so this variation is being captured by “σ” in “ Wt
Thus, “u” represents “drift” and “σ” represents “ volatility”
Now we have seen that Brownian motion follows normal, so it means “u” can be negative and positive both, but share price can never be negative, then to use Brownian motion for modelling share price is a nonsense as share price can never be negative.
Then there comes the Geometric Brownian Motion: Under this we take the exponential of General Brownian motion and prevents share to go negative.


Understading : St = exp(Wt), now we know that ex value can never be negative thus here we are going to use this formula for modelling share prices as it prevents share price to go negative. Here W is the Brownian motion. Thus St becomes Geometric Brownian motion with Lognormal distribution.
St – LN( Wo+ut , σ2t)

So E(St) = exp(W0 + ut + 1/2σ2t ) and V(St) = {E(St)}2 {exp(σ2t)-1}




But the concept does not end here. There is something more known as Martingales

Martingale:

 It is a stochastic process whose current value is the optimal estimator of its future value. In other words, Expected change in the process is zero or the process has no drift 
( i.e. "u" = 0 )
Martingale approach is used for pricing and hedging of financial derivatives.





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