Brownian Motion

Question 1

What 4 things do you need to know about Brownian Motion?

Answer 1

1. Z(0) = 0
2. Z(t+s) - Z(t) ~ N(0,s)
3. Changes in Z(t) are independant over time
4. E[Z(t+h)|Z(t)] = Z(t)

Question 2

What 3 multiplication rules do you need to remember when working with Brownian Motion?

Answer 2

1. dt x dt = 0
2. dt x dz = 0
3. dz x dz = dt

Question 3

ABM

dX(t) =

Answer 3

dX(t) = αdt + σdZ(t)

Question 4

ABM

X(t+s) - X(t) ~

Answer 4

X(t+s) - X(t) ~ N(αs, σ²s)

Question 5

GBM

Answer 5

Question 6

GBM

Answer 6

Question 7

GBM

ln[S(t)] ~ N(m, v²)

m =

v² =

Answer 7

ln[S(t)] ~ N(m, v²)

m = ln[S(0)] + (α - δ - 0.5σ²)t

v² = σ²t

Question 8

GBM

E[S(t)] =

Answer 8

E[S(t)] = S(0)e^(α-δ)t

Question 9

GBM

S(t) =

Answer 9

Question 10

Ornstein-Uhlenbeck Process

dX(t) =

Answer 10

dX(t) = λ[α - X(t)]dt + σdZ(t)

Question 11

What is Ito’s Lemma?

Answer 11

dV = V_sds + 0.5V_ss(ds)² +V_tdt

Question 12

Sharpe Ratio

Φ =

Answer 12

Φ = (α - r)/σ

Question 13

What do you know about two Ito Processes that depend on the same dZ(t) ?

Answer 13

Their Sharpe Ratios are equal.

Question 14

If Φ₁ > Φ₂, how do you exploit arbitrage?

Answer 14

1. Buy 1/(σ₁S₁) shares of S₁
2. Short 1/(σ₂S₂) shares of S₂
3. Invest (1/σ₂ - 1/σ₁) at the risk free rate

Question 15

Risk Free Portfolio

Answer 15

Question 16

Risk Free Portfolio

Answer 16

Question 17

Risk Free Portfolio

Answer 17

Question 18

Risk Free Portfolio

Three steps for creating a risk free portfolio?

Answer 18

Question 19

Risk Neutral Vs True

Relate to dZ(t) & Z(t)

Answer 19

Question 20

Answer 20

Question 21

Black Scholes Equation

Answer 21

Question 22

What is the Black Scholes PDE?

Answer 22