- The document discusses how to estimate parameters and determine an optimal portfolio allocation given $180 million to invest. - There are two main problems: estimating the drift and volatility of different stocks, and determining the proportion of each stock/security to include in the portfolio. - Two methods are presented for estimating drift: 1) averaging historical returns and 2) using experience to select from a range of possible drift values. - Applying the methods to Apple stock data from 1980-2016, method 1 achieved an annual return of 18.8% by investing proportionally based on the estimated drift and volatility.

1. faculty advisor: Professor Levental
Lingjun Miao

2. A question
Suppose you are given $ 180
million today to make an
investment.
What will you do?
$180
M

4. Contents
Two main problems
Definition Explanation
Problem 1
Problem 2

5. Two main problems
Estimating two
important
parameters
drift & volatility
determining the
proportion of
different stocks
or securities in
the portfolio

6. Definition Explanation
Portfolio a set of securities held by an investor
Drift
a slow and gradual movement or change from one
place or condition to another
Volatility
a statistical measure of the dispersion of returns for
a given security or market index. Commonly, the
higher the volatility, the riskier the security

7. Problem 1
∆𝑠 𝑛∆𝑡
𝑆 𝑛∆𝑡
= 𝛼∆𝑡 + ∆𝑡𝑍 𝑛 𝜎 𝑛∆𝑡(1)
 ∆t —— one day, it equals to (1/250) because we took 1
as one year and there are about 250 trading days in one
year.
 ∆𝑆 𝑛∆𝑡—— the change of a stock’s price during ∆t at time
n
 𝑆 𝑛∆𝑡—— the price of a stock during ∆t at time n
 α —— drift
 𝜎 —— volatility.
 Z —— standard normal distribution

8. Problem 1
𝜎 𝑛∆𝑡 =
2
∆𝑡
∆𝑆 𝑛∆𝑡
𝑆 𝑛∆𝑡
− 𝑙𝑜𝑔
𝑆 𝑛+1 ∆𝑡
𝑆 𝑛∆𝑡
(2)
By ITO formula and
some other
calculation
We can get the
estimation of 𝜎

9. Problem 1
𝜎 𝑛 = 𝑘=0
9
𝜎 𝑛−𝑘 𝑎 𝑘
𝑘=0
9
𝑎 𝑘 (3)
• 0<a<1
• we took a as 0.9

10. Problem 1
∆𝑠 𝑛∆𝑡
𝑆 𝑛∆𝑡
= 𝛼∆𝑡 + ∆𝑡𝑍 𝑛 𝜎 𝑛∆𝑡(1)

11. Problem 2
𝜃 =
𝛼
𝜎2 (4)
At the beginning, suppose our
portfolio is made up of one stock
and money in bank. The proportion
of the stock is 𝜃

12. Problem 1
Method1
𝛼 by averaging
Method2
𝛼 Depending on
experience
Two methods to estimate 𝜶

13. Step 1
We downloaded stock data of
Apple from 1980 to 2016 from
Yahoo finance. By using the
adjusted price, we calculated 𝜎 𝑛
for every day.
Problem 1 Method1
𝜎 𝑛 = 𝑘=0
9
𝜎 𝑛−𝑘 𝑎 𝑘
𝑘=0
9
𝑎 𝑘 (3)

14. Problem 1 Method1
Step 2 Depending on experience, we
restricted α to multiples of 0.01
between 0 to 0.2, and then
given any α from the range, we
calculated the corresponding
𝜃 for every day.

15. Problem 1 Method1
𝑉𝑡+∆𝑡 = 𝑉𝑡 𝜃
∆𝑆 𝑛∆𝑡
𝑆 𝑛∆𝑡
+ 1 (5)
Step 3
we used the formula (5) to
calculate how much money we
will get today if we invest 1
dollar in 1980 into the money
market and Apple by proportion
of 𝜃.

16. Step 4
we calculated 𝑉𝑡 𝑉𝑡−200∆𝑡
everyday, and took as our
estimate for 𝛼 , the 𝛼 that
produces the biggest ratio.
Problem 1 Method1

17. we used the best 𝛼 selected in step 4
to calculate 𝜃 by formula (4)
we invested 1 dollar again into the
portfolio made out of money market
and Apple in 1980 by proportion 𝜃
We calculated 𝑉𝑡+∆𝑡 by formula (5)
74.47 dollars,
which means the
annual return
rate is 18.8%,
pretty
good!
Problem 1 Method1
Test

18. Problem 1 Method2
Step 1
𝛼0 = 250 ∗ 𝐴𝑉𝐸𝑅𝐴𝐺𝐸(
∆𝑆 𝑛
𝑆 𝑛
𝑡𝑜
∆𝑆 𝑛+200
𝑆 𝑛+200
) (6)
we calculated 𝛼0 for every day
by the formula(6):

19. Problem 1 Method2
Step 2
we calculated the corresponding
𝜃 for every day.

20. Problem 1 Method2
Step 3
we used the formula (5) to
calculate how much money we
will get today if we invest 1 dollar
in 1980 into the money market
and Apple by proportion of 𝜃.
𝑉𝑡+∆𝑡 = 𝑉𝑡 𝜃
∆𝑆 𝑛∆𝑡
𝑆 𝑛∆𝑡
+ 1 (5)

21. Problem 1 Method1
Step 2 Depending on experience, we
restricted α to multiples of 0.01
between -0.2 to 0.2, and then
given any α from the range, we
calculated the corresponding
𝜃 for every day.

22. Step 4
we calculated 𝑉𝑡 𝑉𝑡−100∆𝑡( 𝑉𝑡 𝑉𝑡−300∆𝑡)
everyday, and took as our estimate for
𝛼, the 𝛼 that produces the biggest
ratio.
Problem 1 Method1

23. Experiments on SP500
“SP500” is a portfolio of 500 biggest
companies in stock market, we used
“index fund” with symbol “SPY”
If we buy SPY(𝜃 = 1)
If we use method 1 for SPY
If we use method 2 for SPY
$1➡$6.75
$1➡$850
$1➡$54

24. Thank you for the department of
mathematics
Thank for Professor Levental
Thank you for Dr. Wald