brownian motion

Question 1

List the desirable characteristics of a model that can be used for modelling a share price
process.

Answer 1

• The model should return non-negative share price values.
• There should be an increasing trend in the share prices produced by the model.
• The volatility of the share prices should increase in magnitude with increasing price.
• The model should be relatively simple and have a good empirical fit to past data.

Question 2

State Ito’s lemma

Answer 2

d(log St) = σdBt +(µ −σ^2/2)dt

Question 3

When dSt= St[σdBt +µdt]
Using Ito’s lemma on the function f(t, St) = log St
show that:
St = S0 exp [σBt +(µ −1/2σ^2)t]
.

Answer 3

St = S0 exp [σBt +(µ −1/2σ^2)t]

Question 4

List the defining properties of a standard Brownian motion.

Answer 4

Standard Brownian motion is a stochastic process {Bt
: t ≥ 0}, on state space S = R
(set of real numbers) with the following defining properties:
(i) B0 = 0.
(ii) Independent increments: Bt − Bs is independent of {Br : r ≤ s}, where s < t.
(iii) Stationary increments: Distribution of Bt − Bs depends only on (t − s), where s < t.
(iv) Gaussian increments: Bt − Bs ∼ N(0, t − s).
(v) Continuity: Bt has continuous sample paths.

Question 5

Let Bt be a standard Brownian motion and Fu denote the filtration (history) up to time u.
Show that: Bt = Bt+1 − B1, is also a standard Brownian motion.

Answer 5

Bt+1 and B1 are Gaussian, so B*t is Gaussian.
• E[B∗t] = E[Bt+1] − E[B1] = 0.
•Cov[B∗t, B∗s] = Cov[Bt+1 − B1, Bs+1 − B1]
= Cov[Bt+1, Bs+1] − Cov[Bt+1, B1] − Cov[Bs+1, B1] + V [B1]
= min(s + 1, t + 1) − 1 − 1 + 1
= min(s, t).
• As Bt has continuous sample paths, B∗t will also have continuous sample paths, as B∗t is a linear combination of Bt+1 and B1