We consider the random process St, which plays a fundamental role in BIack-Scholes analyses:

St = S0e[μ1+σWt]

Where Wt is a Wiener process with W0 = 0, μ is 'a "trend" factor, and

(Wt - Ws) ≈ N(0, (I - s)),

Which says that the increments in Wt have zero mean and a variance equal to t - S Thus, at t the variance is equal to the time that elapsed since Ws is observed We also know that these Wiener increments arc independent over time. 

According to this, St, can be regarded as a random variable with log-normal distribution. We would like to work with the possible trajectories followed by this process.

Let μ = .01, σ = .15 and t = 1. Subdivide the interval [0, 1] into subintervals and select 4 numbers randomly from:

x ≈ N(0, .25)

(a) Construct the W1 and S, over the 10, ii using these random numbers

Plot the Wt and St. (You will obtain piecewise linear trajectories that will approximate the true trajectories.)

(b) Repeat the same exercise with a subdivision of [0, l] into 8 intervals.

(c) What is the distribution of log (St / St-Δ)  For "small"0 < Δ.

represent? In what units is it measured? How does this random variable change as time passes?

(e) Now let Δ = .000001, How does the random variable,

logSt - logSt-Δ / Δ

Change as time passes?

(f) If Δ → 0, what happens to the trajectories of the "random variable"

logSt - logSt-Δ / Δ

(g) Do you think the term in the previous question is a well-defined random variable?