2) The Wiener process and Modelling Stock Price

Question 1

What notation is used for changes in time and related quantities

Answer 1

• ∆t - Small but finite change in time
• ∆S, ∆V - Changes in quantities over a small finite change in time
• dt - Infinitesimally small change in time
• dS, dV - C hanges in quantities over an infinitesimally small change in time

Question 2

How do normal distributions combine through addition and linear transformation

Answer 2

For Y1 ∼ N(µY1, σ^2Y2), Y2 ∼ N(µY2, σ^2Y2)
* For Z = Y1 + Y2 ,we have
FZ ∼ N(µY1 + µY2, σ^2Y1 + σ^2Y2)

• For Z = a + bX where X ∼ N(0, 1), Z ∼ N(a, b^2)

Question 3

What is Random Walk

Answer 3

To create a random walk we take successive draws from a random distribution and add them together

Question 4

What is the Wiener Process

Answer 4

The continuous limit of a random walk process, e.g. Given the random walk we take the limit ∆t → 0

Question 5

What are the Properties of the Wiener Process

Answer 5

Question 6

What are the key expectations and relationships in a Wiener process

Answer 6

Question 7

Describe the proof of the key expectations and relationships in a Wiener process

Answer 7

Question 8

What is the probability density function of the Wiener Process

Answer 8

Question 9

How is the share price model approximated for a small time interval Δt

Answer 9