Lecture 2: Fundamentals of stochastic calculus & deprivative pricing

Question 1

Price of the underlying asset (e.g. stock price)
ST St or S0
Note the time subscript

μ = mean return on the underlying asset
σ = historical standard deviation of the underlying asset
Δ = discrete time step
d = infinitesimal time step
ε = random drawing from a probability distribution
g = general term for a function of s
e.g. g  f(S, t)

Answer 1

Price of the underlying asset (e.g. stock price)
ST St or S0
Note the time subscript

μ = mean return on the underlying asset
σ = historical standard deviation of the underlying asset
Δ = discrete time step
d = infinitesimal time step
ε = random drawing from a probability distribution
g = general term for a function of s
e.g. g  f(S, t)

Question 2

A variable whose value changes over time in an uncertain way follows a ? process. Such a process may be ? or ?.

Answer 2

A variable whose value changes over time in an uncertain way follows a STOCHASTIC process.
Such a process may be DISCRETE or CONTINOUS.

Question 3

MARKOV PROCESS:
A particular type of ? process where only the ?? of a variable is relevant for predicting the future. Stock prices are said to be Markov, consistent with ????.
The change in the value of a variable following a continuous-time Markov process during a very short time period is:
? = …?

Answer 3

MARKOV PROCESS:
A particular type of STOCHASTIC process where only the PRESENT VALUE of a variable is relevant for predicting the future. Stock prices are said to be Markov, consistent with WEAK-FORM MARKET EFFICIENCY.
The change in the value of a variable following a continuous-time Markov process during a very short time period is:
Zt = Z0 + ∑_(i=1)^n(ε_i)

Question 4

WIENER PROCESS (aka BROWNIAN MOTION)
- Discrete & Continuous time
Zt - Zo = ?
n = no of random shocks = ?? = Total time period / length of each discrete timestep
As ∆t –> 0 (shrink timesteps to an instant of time):
1) dz = ?? , ε ~ N(0,1)
2) Values of dz for any 2 different short time intervals are ?.
=> Mean dz = ?
Variance dz = ?
Standard deviation dz = ??

Answer 4

WIENER PROCESS (aka BROWNIAN MOTION)
- Discrete & Continuous time
Zt - Zo = ∑_(i=1)^n (ε_i) * (∆t)^(1/2)
n = no of random shocks = T/∆t = Total time period / length of each discrete timestep
As ∆t –> 0 (shrink timesteps to an instant of time):
1) dz = ε * (dt)^(1/2) , ε ~ N(0,1)
2) Values of dz for any 2 different short time intervals are independent.
=> Mean dz = 0
Variance dz = dt
Standard deviation dz = (dt)^1/2

Question 5

Generalised Wiener process 
	(a.k.a. arithmetic Brownian motion)
Contains ?, μ and ?, σ
dx = ?????
x ~ N(μ, σ)
dz = ε * (dt)^(1/2)

Answer 5

Generalised Wiener process (gWp)
	(a.k.a. arithmetic Brownian motion)
Contains drift, μ and volatility, σ
dx = μdt + σdz
x ~ N(μ, σ)
dz = ε * (dt)^(1/2)

Question 6

Geometric Brownian motion
Includes the ?? in the ? and ?terms:

dx = ????

Answer 6

Geometric Brownian motion (gBm)
Includes the asset value in the drift and volatility terms:

dx = μxdt + σxdz

Question 7

Itô process:
A process where the drift and variance rate can be a function of both ? and ?:
dx = ?????
The change in x over a short period of time is normally distributed.
The ? and ? are examples of Itô processes.

Answer 7

Itô process:
A process where the drift and variance rate can be a function of both x and time:
dx = a(x, t)dt + b(x,t)dz
The change in x over a short period of time is normally distributed
The gWp and gBm are examples of Itô processes.

Question 8

gBm is an appropriate stochastic process for modelling the behaviour of ??.

Answer 8

gBm is an appropriate stochastic process for modelling the behaviour of stock prices.

Question 9

Ito’s Lemma:
dx = a(x, t)dt + b(x,t)dz
=> dG = ?????
if x is the ?? process, G could be that of the ?.

Answer 9

Ito’s Lemma:
dx = a(x, t)dt + b(x,t)dz
=> dG = [∂G/∂x * a+ ∂G/∂t +1/2 (∂^2 G)/(∂x^2 ) * b^2 ]dt+ ∂G/∂x bdz
if x is the stock price process G could be that of the option.