faculty advisor: Professor Levental

Lingjun Miao

A question

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

$180M

Contents

Two main problemsDefinition ExplanationProblem 1Problem 2

Two main problems

Estimating two important parameters

drift & volatility

determining the proportion of different stocks or securities in the portfolio

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

Volatilitya statistical measure of the dispersion of returns for a given security or market index. Commonly, the higher the volatility, the riskier the security

Problem 1

(1)

—— one day, it equals to (1/250) because we took 1 as one year and there are about 250 trading days in one year.

—— the change of a stock’s price during at time n

—— the price of a stock during at time n —— drift —— volatility. Z —— standard normal distribution

Problem 1

(2)By ITO formula and some other calculation

We can get the estimation of

Problem 1

(3)

• 0<a<1 • we took a as 0.9

Problem 1

(1)

Problem 2

(4)

At the beginning, suppose our portfolio is made up of one stock and money in bank. The proportion of the stock is

Problem 1

Method 1

by averagingMethod 2

Depending on experience

Two methods to estimate

Step 1

We downloaded stock data of Apple from 1980 to 2016 from Yahoo finance. By using the adjusted price, we calculated for every day.

Problem 1 Method 1

(3)

Problem 1 Method 1

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

Problem 1 Method 1

(5)

Step 3we 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 .

Step 4

we calculated everyday, and took as our estimate for , the that produces the biggest ratio.

Problem 1 Method 1

we used the best selected in step 4to calculate by formula (4)

we invested 1 dollar again into the portfolio made out of money market

and Apple in 1980 by proportion

We calculated by formula (5)

74.47 dollars, which means the annual

return rate is 18.8%,

pretty good!

Problem 1 Method 1

Test

Problem 1 Method 2

Step 1

(6)

we calculated for every day by the formula(6):

Problem 1 Method 2

Step 2

we calculated the corresponding for every day.

Problem 1 Method 2

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 .

(5)

Problem 1 Method 1

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

Step 4

we calculated () everyday, and took as our estimate for , the that produces the biggest ratio.

Problem 1 Method 1

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() If we use method 1 for SPY If we use method 2 for SPY

$1 $6.75➡$1 $850➡$1 $54➡

Thank you for the department of mathematics

Thank for Professor Levental

Thank you for Dr. Wald