We're ready to model ASSET prices - double click on right

Our goal is to carefully model "derivatives", i.e. puts and calls.  But first, let's model an asset price, say one with an annual mean drift (i.e., mean annual return) of 12%, and a volatility of σ = 20%:  dS = S(μ dt  +  σ  dX), our stochastic differential equation, is now  
    
                       dS = S (.12 dt + .2  dX)
To model this, we discretize S, and replace our random walk variable by a normal random variable whose variance is proportional to the time.  SO:
                S-S=S(.12 Δt+.2 Z),
where Z denotes a standard normal random variable (mean 0, variance 1).  Solving for S yields the first approximation below.  

Below we will present both the above ('incorrect') discretization and the 'more legitimate' one.  But the difference is, practically speaking, quite negligible.
      

It is remarkable how close the two plots above are; you can barely tell that they are different.  Now let's do it again (sometimes in demos, the stock price actually drops, even though "mu" was positive!):