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

where Wt is a Wiener process with W0 = 0, μ is a "trend" factor, and
(Wt ˆ’ Ws) ˆ¼ N (0, (t ˆ’ s))
which says that the increments in Wt have zero mean and a variance equal to t ˆ’ s. Thus, at t the variance is equal to the time that elapsed since Ws is observed. We also know that these
Wiener increments are 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 = 0.01, σ = 0.15, and t = 1. Subdivide the interval [0, 1] into four subintervals and select four numbers randomly from:
x ˆ¼ N (0, 0.25)
(a) Construct the Wt and St over the [0, 1] 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, 1] into eight intervals.
(c) What is the distribution of

for "small" 0 (d) Let ˆ† = 0.25. What does the term

represent? In what units is it measured? How does this random variable change as time passes?
(e) Now let ˆ† = 0.000001. How does the random variable change as time passes?

f) If ˆ† †’ 0, what happens to the trajectories of the "random variable"

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

Transcribed Image Text:

St log (5-Δ log S, log S.-025 0.25 log St-log St-A log St-log St- 스