1. S TOCHASTIC M ODELS L ECTURE 4 P ART II B ROWNIAN M OTIONS Nan Chen MSc Program in Financial Engineering The Chinese University of Hong Kong (Shenzhen) Nov 18, 2015

2. Outline 1.Variations on Brownian motion 2.Maximum variables of Drifted Brownian motion

3. 4.3 V ARIATIONS ON B ROWNIAN M OTION

4. 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

5. 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

6. 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

7. Geometric Brownian Motion If is a Brownian motion process with drift and volatility, then the process defined by is called a geometric Brownian motion.

8. 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

9. 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.

10. 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.

11. How Good is the Geometric Brownian Motion as a Financial Model? As a first step to perform statistical analysis, we estimate – Mean: – Standard deviation

12. 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.

13. 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.

14. 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

15. 4.4 M AXIMUM OF B ROWNIAN M OTION WITH D RIFTS

16. 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.

17. Exponential Martingales Defined by Drifted Brownian Motion For any real defines a martingale.

18. 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

19. 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

20. Laplace Transform of Hitting Time When, It is possible to invert the above transform to obtain an explicit expression for the pdf of

21. Maximum Variable of Drifted Brownian Motion A key observation relating and is that Therefore,

22. Maximum Variable of Drifted Brownian Motion After some algebraic operations, we have where with being a standard normal random variable.

23. 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