Financial Economics

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.

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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)

C. Forecast the future possible price of the stock.

•Build a 12-period 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 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

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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.

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 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 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 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.

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Then we need to use these parameters to calculate the payoff of the call, build tree like below:

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.

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:

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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.

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=((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.

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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.