Chapter 20 Brownian Motion + Ito's lemma

Question 1

What are the four properties of brownian motion

Answer 1

1. Z(0) = 0
2. Z(t+s) - Z(t) Follows N(0, s)
3. non-overlapping intervals are independent
4. it’s continuous

Question 2

Martingale definition

Answer 2

E(Z(t+s)|Z(t)) = Z(t)

Question 3

The change in the value of brownian motion dZ(t) is

Answer 3

Y(t) * sqrt(dt) where Y(t) is +- 1 with prob .5

Question 4

Brownian motion will cross its starting point x times during a finite interval. X is

Answer 4

infinity

Question 5

The quadratic version of Brownian Motion or any higher order version of Brownian Motion is

Answer 5

Constant, T for the quadratic version

Question 6

Arithmetic Brownian Motion is

Answer 6

is brownian motion with a drift coeffiecient (mean) of At and variance of sigma

Written At + sigma * z(t)

Question 7

written with integrals Brownian motion at time T, Z(T) is

Answer 7

Z(0) + integral from 0 to T of dZ(t)

Question 8

Ornstein- Uhlenbeck Process

Answer 8

Arithmetic Brownian Motion with mean reversion. The mean term is lamda(Alpha - X(t)) lamda measures speed of mean reversion.

Question 9

When determining brownian motion over short time periods, the mean is BLANK while the SD is BLANK. Over long periods this is BLANK

Answer 9

less important, more important, reversed.

Due to the fact that sd relies on root h, while mean relies on h`

Question 10

two assets with the same dz have the BLANK sharpe ration

Answer 10

same

Question 11

prepaid forward of claim that pays S(t)^a

Answer 11

e^(-rt) * e^(a(r - delta - 0.5sigma^2)t + 0.5 * sigma^2 * a^2 * t)

Question 12

Ito’s Lemma

Answer 12

dC(s, t) = [(a - delta)S(dC/dS) + (0.5)sigma^2s^2(d2C/dS2) + C]dt + sigmaS(dc/ds)dZ