1. Wiener Processes and Itô’s Lemma

2. A Wiener Process
We consider a variable z whose value changes continuously
Define f(m,v) as a normal distribution with mean m and variance v
The change in a small interval of time Dt is Dz
The variable follows a Wiener process if
The values of Dz for any 2 different (non-overlapping) periods of time are independent

3. Properties of a Wiener Process
Mean of Δz=0
Standard deviation of Δz=
Variance of Δz= Δt
z follows a Markov process.
Mean of [z (T ) – z (0)] is 0
Variance of [z (T ) – z (0)] is T
Standard deviation of [z (T ) – z (0)] is

4. Wiener process

5. Generalized Wiener Processes (See page 284-86)
A Wiener process has a drift rate (i.e. average change per unit time) of 0 and a variance rate of 1
In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants

6. Generalized Wiener Processes (continued)
Mean change in x per unit time is a
Variance of change in x per unit time is b2
term add noise or variability to the path followed by x.

7. Generalized Winner Process

8. Generalized Wiener Processes (continued)
Mean of [x (T ) – x (0)] is aT
Variance of [x (T ) – x (0)] is b2T
Standard deviation of [x(T ) – x (0)] b*sqrt(T)

9. Taking Limits . . . What does an expression involving dz and dt mean?
It should be interpreted as meaning that the corresponding expression involving Dz and Dt is true in the limit as Dt tends to zero
In this respect, stochastic calculus is analogous to ordinary calculus
Generalized Winner Process:
dx=adt+bdz

10. Generalized Wiener Processes
Mean of [x (T ) – x (0)] is aT
Variance of [x (T ) – x (0)] is b2T
Standard deviation of [x(T ) – x (0)] b*sqrt(T)

11. Itô Process
In an Itô process the drift rate and the variance rate are functions of time
dx=a(x,t) dt+b(x,t) dz
The discrete time equivalent
is only true in the limit as Dt tends to
zero
Ito process is also Markov.

12. An Ito Process for Stock Prices (See pages 286-89)
It is reasonable to assume that the percentage return in a short period of time is random, but its variability is the same regardless of the stock price. Thus we have
. where m is the expected return s is the volatility.
The discrete time equivalent is
The process is known as geometric Brownian motion

13. Monte Carlo Simulation
Fro the process of stock price, we have
We can sample random paths for the stock price by sampling values for e
Suppose m= 0.15, s= 0.30, and Dt = 1 week (=1/52 years), then

14. Monte Carlo Simulation – One Path Excel code : NORMINV(RAND(),0,1)

15. Monte Carlo Simulation
Monte Carlo methods use statistics gathered from random sampling to model and simulate a complicated distribution.
“If “the future” is the random selection of one sequence of events from a probability space, our focus should revolve around modeling the distribution more so than identifying the ultimate future path.”

16. Itô’s Lemma
If we know the stochastic process followed by x, Itô’s lemma tells us the stochastic process followed by some function G (x, t )
Since a derivative is a function of the price of the underlying and time, Itô’s lemma plays an important part in the analysis of derivative securities

17. Itô’s Lemma-formal argument

18. Itô’s Lemma-formal argument

19. Itô’s Lemma-formal argument

20. Application of Ito’s Lemma to a Stock Price Process
We argued that under a reasonable assumption:
Both S and G are affected by dz - the noise in the system.

21. Examples