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Introduction:
The stock I choose is AT&T, INC.to generate this optimal portfolio construction. AT&T Inc. is an American multinational telecommunications corporation, headquartered at Whitacre Tower in downtown Dallas, Texas. AT&T is the second largest provider of mobile telephone and the largest provider of fixed telephone in the United States, and also provides broadband subscription television services.
Outline:
A.Choose one stock that has been traded for more than 15 years and analyze it. (AT&T, INC)
•Provide a graph of the yearly stock return over time.
•Provide a table of summary statistics on returns.
B.Choose reasonable values for the parameters u and d.
•Convert the annual returns that we got from excel into continuously compounded annual returns.
•Find the sample standard deviation of the continuously compounded annual returns.
•Use the number of periods (12) and the sample standard deviation to adjust u and d.
C.Forecast the future possible prices of the stock.
•Let S0 be the price of the stock and build a 12period tree for the price of the stock where u and d are used to forecast the future stock’s price.
D.Use the binomial model to price a call option on the stock.
•Build a tree that has the option price at the origin vertex and at the end vertices.
•Find the returns to onemonth TBills for each month in the past 10 years and use this to estimate the riskfree interest rate.
E.Find the actual price of the call option with exercise price.
•Compare the actual price to the price I computed.
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•Summarize what I learned from the experiment.
Data section:
A. Choose one publicly traded stock.
The stock I choose is AT&T, INC.
Before I use the binomial model, I need a way to choose reasonable values for the parameters u and d. So that I need to get some basic statistics of this stock. I got the historical price data from the first trading day of December through 19962015, which contains the last 20 years. And then I use these data through EXCEL to calculate the yearly stock return.
To calculate the yearly stock return, I use the formula
ℎ ℎ
For example, to get the yearly stock return of 2015121, I use the adjusted close price in 2015121 minus the adjusted close price in 2014121, and then divided by the adjusted close price in 2014121. Then I just use the EXCEL automatically to get other years’ yearly return. The table below shows the historical price data from 2015 to 1996 and the calculated yearly return.
Date

Open

High

Low

Close

Volume

Adj Close

yearly return









12/1/15

33.779999

33.970001

33.580002

33.77

33523000

32.893505

0.063051326









12/1/14

35.279999

35.369999

32.07

33.59

27279700

30.942537

0.006594704









12/2/13

35.18

35.299999

33.599998

35.16

21568400

30.739817

0.097220959









12/3/12

34.23

34.689999

33.099998

33.709999

25837000

28.016068

0.174876869









12/1/11

28.93

30.299999

28.51

30.24

24893900

23.845961

0.090239279









12/1/10

28.120001

29.559999

28.030001

29.379999

21588700

21.872227

0.116025153









ECON







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4









12/1/09

27.18

28.610001

26.940001

28.030001

23437800

19.598328

0.046777088









12/1/08

28

30.65

26.57

28.5

34290000

18.722542

0.280535526









12/3/07

38.490002

42.790001

37.709999

41.560001

26775800

26.022886

0.20608776









12/1/06

33.950001

36.209999

33.740002

35.75

25337800

21.576279

0.531621767









12/1/05

25.15

25.6

24.280001

24.49

12611400

14.087211

0.002336996









12/1/04

25.4

26.559999

24.950001

25.77

8332400

14.054366

0.037790113









12/1/03

23.299999

26.15

22.950001

26.07

9426900

13.54259

0.016794701









12/2/02

28.75

29.1

24.85

27.110001

7028100

13.318903

0.283115434









12/3/01

37.599998

40.290001

37.200001

39.169998

7303400

18.578867

0.160866127









12/1/00

54.6875

55

42.625

47.75

8059400

22.140528

0.000705901









12/1/99

52.0625

55.5

47.375

48.75

4802700

22.12491

0.07382148









12/1/98

48.25

54.875

47.25

53.625

2307600

23.888386

0.497099417









12/1/97

73

76.125

69.5625

73.25

2673000

15.956446

0.455790318









12/2/96

52.375

55.25

48.5

51.875

1865400

10.960676

0.06327295









http://finance.yahoo.com/q/hp?s=T&a=11&b=1&c=1995&d=11&e=1&f=2015&g=m
After finishing calculating the yearly stock return, I generate a graph of the yearly stock return of AT&T stock over time. From this graph, we can see the fluctuation of the yearly stock return clearly.
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In addition to the graph, I would like to provide a table of summary statistics on returns, including the mean, variance, skewness, kurtosis, median, interquartile range, and maximum and minimum values.

SUMMARY OF STATISTICS




Mean


0.074070042




Variance


0.049156136




Skewness


0.645230162




Kurtosis


0.453572607




Median


0.0422836




Interquartile range


0.5320568




Maximum values


0.531621767




Minimum values


0.283115434




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B. Choose reasonable values for the parameters u and d.
• Convert the annual returns into continuously compounded annual returns.
I use the formula (textbook, page 162): Continuously compounded rate of return rcc= In (1+effective annual rate)
As we all known, the relationship between APR (annual percentage rate of return) and EAR (effective annual rate of return) is: 1+EAR= (1+T*APR)1/T. If we fix APR and increase the frequency of compounding (make T very small and approaches to zero), (1+EAR) will converge to e APR. If we fix EAR and increase the frequency of compounding (make T very small and approaches to zero), (APR) will converge to In(1+EAR). This is called Continuously compounded rate of return.
Below is the table showing the continuously compounded rate of return. Use the formula Rcc=ln(1+EAR)
•Find the sample standard deviation of the continuously compounded annual returns. After we calculating the continuously compounded annual returns, we can get the variance of returns is σ2=0.04152903, and the standard deviation of the returns is σ
=0.2037867268. This is the unbiased estimate of the standard deviation of the continuously compounded annual returns.
•Use the number of periods together with σ to adjust u and d.
The binomial model we will use for this project is a 12period model. T=12. Then we calculate the parameters u and d. The up and down factors are calculated using underlying volatility, σ, and the time duration of the step, which is t=1/12. (measured in years). From the condition that the variance of the log of the price is σ2t, we have:
u=exp(σ ∆ ) d=exp(σ
∆ ) so that,
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8

u= 1.060592974

Use as u=1.06 d=0.9428687767

Use as d=0.94


C. Forecast the future possible price of the stock.
•Build a 12period tree for the price of the stock (Let S0 be the price of the stock on Dec1 15).
From the original historical data of the AT&T stock, we can see that the price on Dec115 was $32.893505, so we set S0=$32.893505 ≈ $32.89. Then I use S0=$32.89, u=1.06, d=0.94 to build a 12 period tree for the price of the stock to forecast the price on Jan116, Feb116, etc. up to Dec116.
Assume a stock price can take two possible values: The stock will either go up and down. Call the factor by which it goes up u, and factor by which it goes down d.
In this case, we set the AT&T stock price on Dec115 as S0=$32.89, the stock price will either increase by the factor of u=1.06 to $34.8634 (=$32.89*1.06) or fall by a factor of d=0.94 to $30.9166(=$32.89*0.94). In the next period, there would be four possibilities. When the price was already increased to $34.8634, it would either increase by the factor of u=1.06 to $39.075204(=$36.8634*1.06) or decrease by a factor of d=0.94 to $32.771596(=$34.8634*0.94). On the other hand, when the price was already fall to $30.9166. it would still either increase by the factor of u=1.06 to $32.117596 and decrease by the factor of d=0.94 to $29.061604. This is always the case that with the time changing, the stock price would either increase by a factor of u and decrease by a factor of d.
So that we can build a 12period tree for the price of the AT&T stock in EXCEL.
D. Use the binomial model to price a call option on the stock.
•Estimate the riskfree interest rate.
In order to use the binomial model, I need a riskfree interest rate. So we are going to find the returns to onemonth TBills for each month in the past 10 years and estimate the
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average monthly return of TBills. Using this estimate as our riskfree rate for each
month. Below is the table showing the historical data.
The original data below is from https://research.stlouisfed.org/fred2/series/TB4WK/downloaddataUNIT: PERCENT
Then I use the data above from historical data of onemonth TBills for each month in the past ten years to estimate the average monthly return. (UNIT: PERCENT)
So the average monthly risk free return we estimated is Rf =1.3565 %
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•Build a tree that has the option price (C) at the origin vertex and at the end vertices. After we have already built a 12period tree for the price of the AT&T stock, we can use those data to build a tree that has C, the option price at the origin vertex and at the end vertices. Replace the notation for the end vertices with the option payoff given the price forecast on DEC116.
Before we are going to build the tree, the parameters we have known is shown in the table below.
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Then we need to use these parameters to calculate the payoff of the call, build tree like below:



Cu12


Cu4


Cu11d


Cu3


Cu10d2


Cu3d


Cu9d3


Cuu




Cu8d4


2

…….


Cu d


Cu

2 2


Cu7d5

Cu d


C

Cud= Cdu


Cu6d6

Cd

Cud2



Cdd


Cu5d7


Cud3


Cu4d8


Cd3


Cu3d9


Cd4





Cu2d10

Cu1d11




Cd12




Cu=uS0X
Cd=dS0X
At each final node of the tree — i.e. at expiration of the option — the option value is simply its intrinsic, or exercise, value.
Max

[ (), 0 ], for a call option


Max

[ ( –

), 0 ], for a put


option:




Where is the strike price and

is the spot price of the underlying asset at the

period. So

in the period 12, I just use the stock price I calculated minus the exercise price, compared with zero, the bigger one is what we want for the payoff of the call in period 12.
Then we are going to calculate the payoff in the period 11.
We can generate The Hedge Ratio for other twostate problems:
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Cu=uS0X
Cd=dS0X (X is the exercise price)
This ratio allows us to come up with a simple algorithm to follow when pricing call option.
The first H in the period 11 =


6789:6788; , which means (the highest payoff in period

12– the 789
lower payoff in period 12) divided by (the highest stock price in period 12 – the lower stock price in period 12). This ratio represents that the portfolio is composed of H shares and one call
written. So the payoff will be H*U12S0 – C12. And then we need to find the present value of this
portfolio using the risk free risk we estimated. So to get this portfolio today, we need to pay
(H*U12S0 – C12)/(1+r) before. Hence (H*U12S0 – C12)/(1+r) = H*U11S0 – Cu11, and then we can
get the payoff Cu11.
If we rearranging the equation H, and plugging this into (H*U12S0 – C12)/(1+r) = H*U11S0 – Cu11
We can find that at last Cu11=((Cu12Cu11d)/(ud)) – ((d*Cu12u*Cu11d)/((ud)*(1+r))). So we can set the general payoff like this. Then I use EXCEL to do the rest part. The table below is the option price tree for all 12 periods.
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So the call price Icalculated at lastis $7.68488.
E. Comparison & Summary
• Compare the actual price with the price I calculated.
The option price we calculated is for 2016, so I went to the Yahoo Finance to get the option price close to the end of the 2016.The cloestprice is on January202017 and the price is $7.00.
As we can see from the option price tree, the call price I calculated is $7.68488, which is very similar to the actual price.