1. S TOCHASTIC M ODELS L ECTURE 5 S TOCHASTIC C ALCULUS Nan Chen MSc Program in Financial Engineering The Chinese University of Hong Kong (Shenzhen) Nov 25, 2015

2. Outline 1.Simple Processes and Ito Integral 2.Properties of Ito integral Linearity Martingale Quadratic variation 3.Ito Formula

3. 5.1 I TO I NTEGRAL

4. Motivating Examples Suppose that you buy 2 shares of IBM stock today and decide to sell them tomorrow. The stock price today is $40 per share, and it increases to $50 by tomorrow. How much will you earn through these transactions? Answer:

5. Motivating Examples Suppose that your trading strategy over the next three days is that – Buy 20 shares today; – Increase your holding to 40 shares tomorrow; – Reduce the number of shares down to 10 the day after tomorrow. The stock price change over the three days is – $20 today – $25 tomorrow – $18 the day after tomorrow How much can you earn?

6. An Integral Representation Consider a finite set of time points (partition): – At each, an investor will changes his position in a stock to shares and hold until the next time points. – The stock price process is given by.

7. An Integral Representation Following the above strategy, at any given time the investor’s wealth is for

8. Ito Integral from Simple Processes In general, consider a Brownian motion and an (adaptive) stochastic process such that there exists a partition on : is a constant on each subinterval An Ito integral is defined to be

9. Ito Integral for General Integrands For a general stochastic process, we always can find a sequence of simple processes such that If the limit of the sequence of Ito integrals exists, then we define

10. Example I: Ito Integral Calculate

11. 5.2 P ROPERTIES OF I TO I NTEGRAL

12. Linearity of Ito Integrals Consider two simple processes over a finite number of time points: – Compute

13. Linearity of Ito Integrals (Continued)

14. Martingale Property The Ito integral is a martingale, i.e., for any we have

15. Quadratic Variation of Ito Integral If we are given a time horizon, we choose a time step size for some, and compute We can show that –

16. Quadratic Variation of Ito Integral Therefore, we have We define the quadratic variation of a Brownian motion as

17. Quadratic Variation of Ito Integral Let Its quadratic variation can be shown to be

18. 5.3 I TO F ORMULA FOR B ROWNIAN M OTION

19. Taylor’s Expansion For a function we have For a multivariate function

20. Ito Formula for Brownian Motion Let be a function for which the partial derivatives,, and are defined and continuous. Then,