STOCHASTIC MODELS LECTURE 4 PART II

BROWNIAN MOTIONS

Nan ChenMSc Program in Financial EngineeringThe Chinese University of Hong Kong

(Shenzhen)Nov 18, 2015

Outline1. Variations on Brownian motion2. Maximum variables of Drifted

Brownian motion

4.3 VARIATIONS ON BROWNIAN MOTION

Brownian Motion with Drift

• Let be a standard Brownian motion process. If we attach it with a deterministic drift, i.e., let

we say that is a Brownian motion with drift and volatility

Properties of Drifted Brownian Motion• The above drifted Brownian motion has the

following properties:– – it has stationary and independent increments;– is normally distributed with mean and

variance

Drifted Brownian Motion as a Limit of Scaled Random Walk• As the standard Brownian motion, the drifted

Brownian motion can also be obtained through taking limits on a sequence of scaled random walks.– Consider a random walk that in each time unit

either goes up or down the amount with respective probabilities

and– Let

Geometric Brownian Motion

• If is a Brownian motion process with drift and volatility , then the process defined by

is called a geometric Brownian motion.

Properties of Geometric Brownian Motion• Recall that the moment generating function

of a normally distributed random variable is given by

• From this, we can derive that

Geometric Brownian Motion as a Useful Financial Model• Geometric Brownian motion is useful in the

modeling of stock price over time. By taking this model, you implicitly assume that – the (log-)return of stock price is normally

distributed;– the daily returns are independent and identically

distributed from day to day.

How Good is the Geometric Brownian Motion as a Financial Model? • We take the stock of IBM to examine the

goodness of fit. The data source is in the attached excel file, containing daily closing prices from Jan. 2, 2001 to Dec. 31, 2010.

• The daily returns of IBM stock price demonstrate randomness. The return is defined as follows.

How Good is the Geometric Brownian Motion as a Financial Model? • As a first step to perform statistical analysis, we

estimate– Mean:

– Standard deviation

How Good is the Geometric Brownian Motion as a Financial Model?

• Normalize the data by

and compare its distribution against the standard normal distribution.

• We find that the daily price returns behave in a similar manner to normally distributed samples, except at the extreme of the range.

How Good is the Geometric Brownian Motion as a Financial Model? • The distribution from real stock price data

demonstrates the following leptokurtic features, compared with the normal distribution:– Fat tail– Higher peak

• Black swans in financial markets.

Timescale Invariance

• The normal approximation works over a range of different timescales. What we need to change are only the mean and standard deviation for the distribution. – For mean, we have

– For standard deviation, we have

4.4 MAXIMUM OF BROWNIAN MOTION WITH DRIFTS

Maximum of Brownian Motion with Drifts• For being a Brownian motion with

drift and volatility define

We will determine the distribution of in this section.

Exponential Martingales Defined by Drifted Brownian Motion • For any real

defines a martingale.

Hitting Probability

• Fix two positive constants and Let

In the exponential martingale in the last slide, we take

and apply the martingale stopping theorem. We will have

Laplace Transform of Hitting Time

• Consider a constant Let

In words, it is the first time, if any, the process reaches the level

• Set

Using the exponential martingale, we have

Laplace Transform of Hitting Time• When ,

• It is possible to invert the above transform to obtain an explicit expression for the pdf of

Maximum Variable of Drifted Brownian Motion• A key observation relating and is that

• Therefore,

Maximum Variable of Drifted Brownian Motion• After some algebraic operations, we have

where

with being a standard normal random variable.

Homework Assignments (Due on Dec. 2)• Read the material about martingales, and

Sec. 10.1-10.3, 10.5 of Ross’s textbook.• Exercises:– Exercises 7 and 9, p. 639 of Ross– Exercises 18, 21, 22, p. 641 of Ross– Exercises 29, 31, p. 643 of Ross– (Optional) Exercises 26, 27, 28, p. 642-643 of

Ross