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# Financial Economics

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### Financial Economics

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 Sbe the price of the stock and build a 12-period 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 one-month T-Bills for each month in the past 10 years and use this to estimate the risk-free interest rate.

E.Find the actual price of the call option with exercise price.

Compare the actual price to the price I computed.

 ECON 4751

project 2 3

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 1996-2015, 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 − ℎ ℎ)

ℎ ℎ

For example, to get the yearly stock return of 2015-12-1, I use the adjusted close price in 201512-1 minus the adjusted close price in 2014-12-1, and then divided by the adjusted close price in 2014-12-1. 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 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.

 ECON 4751

project 2 5

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 12-period 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,

 ECON 4751 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 12-period tree for the price of the stock (Let Sbe 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 Dec-1-15 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 Jan-1-16, Feb-1-16, etc. up to Dec-1-16.

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 Dec-1-15 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 12-period 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 risk-free interest rate.

In order to use the binomial model, I need a risk-free interest rate. So we are going to find the returns to one-month T-Bills for each month in the past 10 years and estimate the

 ECON 4751

project 2 9

average monthly return of T-Bills. Using this estimate as our risk-free rate for each

month. Below is the table showing the historical data.

Then I use the data above from historical data of one-month T-Bills 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 R=1.3565 %

 ECON 4751

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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 12-period 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 DEC-1-16.

Before we are going to build the tree, the parameters we have known is shown in the table below.

 ECON 4751

project 2 11

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=uS0-X

Cd=dS0-X

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 two-state problems:

 ECON 4751

project 2 12

Cu=uS0-X

Cd=dS0-X (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*U12S– 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*U12S– C12)/(1+r) before. Hence (H*U12S– C12)/(1+r) = H*U11S– Cu11and then we can

get the payoff Cu11.

If we rearranging the equation H, and plugging this into (H*U12S– C12)/(1+r) = H*U11S– Cu11

We can find that at last Cu11=((Cu12-Cu11d)/(u-d)) – ((d*Cu12-u*Cu11d)/((u-d)*(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.

 ECON 4751

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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 January-20-2017 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.

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• Title: Financial Economics
• Price: £ 79
• Post Date: 2018-11-10T06:41:58+00:00
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