This assignment has a data gathering component to it. On ThursdayFebru- ary 4 and on Friday February 5 at 2:30pm ET please repeat thefollowing steps: Go to GOOGLE web page on yahoo : GOOGLE on Yahoo. Click on theop- tions link in the upper left corner. Select options which expire inFeb 10, Mar 10 and Jun 10 (by clicking on the respective link). Please save (orrecord) these numbers (for both put and call options). Also write down the GOOGLEstock price at the time when you save the data. You just gathered optiondata on an equity. The most studied data index is the S&P500, but for some reasonYahoo! does not carry its option data anymore (or at least I cannot ndit). This data is available on CBOE. You will need to gure out how to download,clean and identify each option from the le downloaded. We will refer to the data sets gathered in the two consecutive daysas DATA1 (for the rst day) and DATA2 (for the second day) respectivelythroughout this document. Whoever posts comma separated les (or excel les) containing bothdata (Google and S&P500 - cleaned) by 10:00pm February 6 will get 25bonus points counting toward the rst assignment. Please note that in addition to the data you just downloaded youwill also need the following: Recorded underlying equity or index price at the exact moment whenthe rest of the data is downloaded. The short-term interest rate which may be obtained here: http://www.federalreserve.gov/releases/H15/Current/. There are alot of rates posted on the site - they are all yearly, they are inpercents and need to be converted to numbers. There is no theoretical recommendationon which to use, I used to use 3-months Treasury bills which are notavailable anymore. Since then I have been using the \Federal funds(eective)" rate but you can go ahead and try others. You should remember tobe consistent in your choice. Also, make sure that the interest ratethat you use is for the same day when the data you use for the impliedvolatility was gathered. The same site has a link to past (historical)interest rates. 1 Time to Maturity. As you all know options mature in the thirdFriday of the respective month. Note that Feb 10 for example refers to thefact that the options expire in February 2010 not on the 10th of that month.This is important because it can make a dierence when calculating thetime to maturity T ??t. The time to maturity is expressed in years andmay be calculated using calendar days (365) or trading days (295). Eitherway is ne but do remember to be consistent in your choice. (1) Using your choice of computer programminglanguage implement the Black- Scholes formula as a function of current stock price S0, volatility, time to expiration T ?? t (in years), strike price K and short-terminterest rate r (annual). (2) With the function you just constructed, use the bisectionmethod (or any other method you wish) to calculate the implied volatility on therst day you downloaded (DATA1). For this purpose use as the option valuethe average of bid and ask price if they both exist (and if theircorresponding volume is nonzero). Report the implied volatility at the money (forthe option with strike price closest to the traded stock price). Youneed to do it for both index and stock data you have. Then average all theimplied volatilities for the options between in-the-money andout-of-the-money. There is no clearly dened boundary between options at-the-moneyand out-of-the-money or in-the-money options. If we dene moneyness asthe ratio between S0 the stock price today and K the strike price ofthe option some people use values of moneyness between 0.95 and 1.05 to denethe options at the money. Yet, other authors use between 0.9 and 1.1.Use these guidelines if you wish to determine which options' impliedvolatilities should be averaged. (3) Please report the values obtained for every maturity, optiontype and stock. Also compile the average volatilities as described in theprevious point. Comment on what is the dierence in values obtained forGOOG and S&P500. Comment on what happens when the maturityincreases. Comment on what happen when the options become in the moneyrespec- tively out of the money. (4) Construct a binomial tree to calculate the values of theEuropean Call option and one for the European Put option. (5) Next we will use the second dataset DATA2. For each strikeprice in the data use the Stock price for the same day, the implied volatilityyou calcu- lated from DATA1 and the current short-term interest rate(corresponding to the day on which DATA2 was gathered). First use theBlack-Scholes formula, to calculate the option price. Second use the binomialtree to calculate the same values. (6) Implement a binomial tree to calculate the American option,both Call and Put. Repeat the steps in the previous problem and calculatethe respective option prices as if they are American. 2 (7) Create a table which contains the following columns: Bid andAsk values, Black Scholes price, European and American prices calculated usingthe binomial tree. Please write a paragraph detailing the dierencesthat you see between the various option valuations and how they compare withthe bid/ask values. (8) BONUS. For the bonus problem we will use a synthetic example toil- lustrate the power of the binomial tree model. Read Section 2.10 inyour textbook, then construct a binomial tree to calculate the price ofan Amer- ican Up-and-In call option. This is an option that can be exercisedat any time; but it will expire worthless unless by the expiration timethe stock price would have hit the barrier level. Use S0 = 100, strike K =100, maturity T = 0:5 (half an year), volatility = 0:3, short rater = 0:11, and barrier H = 110. Use as many steps in your tree as you thinkare necessary. In all of the applications of the trees to price options use at least 200 steps in your constructions. Anything less will lose points. Just a warning, to my knowledge Excel cannot do 200 steps. 3 **** I realize this is a long question, the portion in RED is whats most important to me...everything elsetips or hints would be much appreciated