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4751 







project 







2 







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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project 

2 

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.