S TOCHASTIC M ODELS L ECTURE 4 P ART II B ROWNIAN M OTIONS Nan Chen MSc Program in Financial...
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Transcript of S TOCHASTIC M ODELS L ECTURE 4 P ART II B ROWNIAN M OTIONS Nan Chen MSc Program in Financial...
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