A Time Series Analysis of the Adjusted Closing Stock Prices

Table of Contents 1. ? inception 2. ?literature review 3. ?Introduction 4. ?methodological analysis INTRODUCTION Google Inc. is an American multinational corporation which provides Internet-related products and services, including Internet search, cloud com adjusting, softw atomic number 18 and advertising technologies. The company was founded by Larry Page and Sergey Brin while both attended Stanford University. Google was first incorporated as a privately held company on September4, 1998, and its initial public offering followed on August19, 2004. The company is right off listed on the NASDAQ stock exchange under the ticker symbol .The companys mission statement from the outset was to organize the worlds information and make it universally accessible and efficacious, and the companys unofficial slogan is Dont be evil. In 2006, the company moved to its current headquarters in Mountain View, California. Objectives 1. To fit a multiple regression illustration to a info set compris ing the put, call and sweetheart prices of a stock belonging to a company listed on a known index. 2. To use the BSM Model to which provides a numerical science for the pricing and hedging of European Call and Put options as the American Options market 3.We treasured to analyze the data for Google option prices from the SP index over the past and present time periods in order to be able to calculate the future. Literature Review 1. Put call parity In financial mathematics, putcall parity defines a relationship between the price of a European call option and European put option in a frictionless market both with the identical strike price and expiry, and the inherent being a liquid asset. In the absence of liquidity, the existence of a forward contract suffices.Putcall parity take aims minimal assumptions and thus does not require assumptions such as those of BlackScholes or other commonly employ financial models. 2. Black-Scholes Model The BlackScholes model or BlackScholes-Me rton is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the BlackScholes design, which gives the price of European-style options. The formula led to a boom in options trading and legitimized scientifically the activities of the Chicago Board Options Exchange and other options markets around the world. t is widely used by options market participants Methodology The data being analyzed consisted of daily past prices of silver traded on the SP index since 14th May to 22 September 2012. The assemblage was required to obtain data sets containing put, call and strike prices the data set of option expiring in more than than 30 days but less than blow The data was obtained from marketwatch. com on 14th May 2012 copied to excel and imported to R, with the stock price at $605. 23. The group chose options expiring on 22th September 2012 for the 1st data set, with 94 days to expiry.An average of the Bid and Ask prices of both the call and put options was then calculated as shown below. The observes in the columns labeled call put were calculated as an average of the corresponding Bid Ask call and put prices respectively. A number of statistical methods were applied to analyze the data on the R program. We first started by importing the data to the R program below is a table showing the data. Strike call put Strikesq Adj about 295 311. 75 0. 45 87025 605. 23 300 306. 45 0. 425 90000 613. 66 305 303. 1 0. 45 93025 609. 15 310 297. 6 0. 75 96100 612. 79 315 291. 4 0. 5 99225 607. 55 320 286. 6 0. 55 102400 596. 97 325 282. 75 0. 6 105625 611. 02 330 277. 85 0. 65 108900 607. 26 335 273. 4 0. 775 112225 604. 43 340 266. 6 0. 7 115600 604. 85 345 262. 1 0. 75 119025 614. 98 350 256. 7 0. 8 122500 615. 47 355 253. 3 0. 875 126025 609. 72 360 248. 35 0. 875 129600 601. 27 365 243. 45 0. 925 133225 597. 6 370 237. 35 1 136900 596. 06 375 232. 4 1. 05 140625 599. 3 380 227. 45 1. 05 144400 607 . 45 385 222. 55 1. 2 148225 609. 57 390 218. 85 1. 325 152100 606. 7 395 212. 45 1. 45 156025 624. 6 400 207. 9 1. 525 160000 651. 01 405 202. 95 1. 6 164025 635. 96 410 198. 15 1. 65 168100 626. 86 415 193. 15 1. 825 172225 630. 84 420 188. 4 2. 025 176400 632. 32 425 183. 6 2. 25 180625 635. 15 430 180 2. 375 184900 642. 62 435 175. 25 2. 55 189225 646. 92 440 170. 3 2. 9 193600 641. 24 445 164. 6 3. 025 198025 648. 41 450 160. 9 3. 3 202500 655. 76 455 155. 15 3. 55 207025 647. 02 460 150. 6 3. 85 211600 649. 33 465 146. 8 4. 05 216225 642. 59 470 141. 15 4. 55 220900 646. 05 75 137. 65 4. 95 225625 639. 98 480 132. 05 5. 35 230400 633. 49 485 128. 5 5. 8 235225 633. 98 490 123. 45 6. 2 240100 625. 04 495 118. 65 6. 75 245025 621. 13 500 114. 1 7. 4 250000 615. 99 505 110. 75 7. 95 255025 617. 78 510 105. 65 8. 5 260100 605. 15 515 101. 35 9. 45 265225 600. 25 520 98 10. 25 270400 607. 14 525 93. 15 11. 1 275625 606. 8 530 89. 55 11. 95 280900 604. 96 535 85. 15 13. 05 286225 61 4. 25 540 80. 6 14. 15 291600 621. 25 545 76. 85 15. 3 297025 622. 4 550 72. 9 16. 35 302500 618. 25 555 69. 5 17. 308025 618. 39 560 66. 05 19. 2 313600 609. 31 565 62. 8 20. 75 319225 609. 9 570 59. 15 22. 45 324900 606. 11 575 56. 5 24. 05 330625 607. 94 580 52. 75 26 336400 614 585 49. 7 27. 85 342225 604. 64 590 46. 7 29. 6 348100 606. 52 595 43. 9 31. 9 354025 605. 56 600 40. 95 34. 35 360000 609. 76 605 38. 45 36. 7 366025 612. 2 610 36. 1 38. 85 372100 605. 91 615 33. 55 41. 35 378225 611. 46 620 31. 05 44. 15 384400 609. 85 625 29. 5 46. 9 390625 606. 77 630 27. 35 49. 75 396900 609. 09 635 25. 3 52. 95 403225 596. 33 40 23. 2 56 409600 585. 11 645 21. 6 59. 2 416025 580. 83 650 19. 95 62. 65 422500 580. 11 655 18. 6 66. 15 429025 577. 69 660 16. 85 70. 1 435600 579. 98 665 15. 6 73. 95 442225 568. 1 670 14. 4 77. 55 448900 569. 49 675 13. 3 81. 35 455625 580. 93 680 12. 25 85. 55 462400 585. 52 685 11. 05 88. 25 469225 585. 99 690 10. 05 93. 4 476100 639. 57 695 9. 55 96. 45 483025 632. 91 700 8. 45 102. 25 490000 628. 58 705 7. 75 105. 25 497025 624. 99 710 7. 1 110. 65 504100 629. 64 715 6. 75 114. 75 511225 625. 96 720 5. 95 119. 5 518400 623. 14 725 5. 65 122. 65 525625 622. 46 730 5. 05 128. 5 532900 650. 02 735 4. 55 131. 95 540225 659. 01 740 4. 25 137. 6 547600 668. 28 745 3. 95 142. 35 555025 665. 41 750 3. 5 147 562500 645. 9 755 3. 25 151. 7 570025 642. 4 760 2. 975 155. 95 577600 639. 7 765 2. 725 161. 4 585225 640. 25 770 2. 525 166. 45 592900 633. 14 775 2. 2 169. 9 600625 629. 7 780 2. 125 174. 75 608400 625. 82 785 1. 975 180. 55 616225 630. 37 790 1. 775 185. 45 624100 621. 83 795 1. 65 190. 35 632025 625. 96 800 1. 525 195. 15 640000 619. 4 810 1. 35 205. 05 656100 618. 07 820 1. 175 214. 95 672400 625. 63 830 0. 975 224. 75 688900 625. 39 840 0. 825 234. 95 705600 627. 42 850 0. 725 244 722500 616. 05 860 0. 65 254. 25 739600 623. 39 870 0. 525 265 756900 623. 77 880 0. 475 274. 55 774400 625. 65 890 0. 425 284. 6 792100 620. 36 9 00 0. 375 293. 45 810000 613. 77 910 0. 375 304. 7 828100 599. 39 920 0. 3 314. 45 846400 582. 93 930 0. 3 323. 3 864900 588. 19 940 0. 275 333. 25 883600 563 950 0. 25 343. 25 902500 570. 11 960 0. 25 353. 25 921600 580 970 0. 25 363. 25 940900 580. 94 Fitting a Multiple regression Model From the results shown? 0=605. 997, ? 1=0. 995, ? 2= -0. 9979. The value of the stock at that point in time wasSt=605. 23. If significant, the see ? 2 was to be equated to -e-r (T-t) and the value for r equated. In this formula, T-t is the time to expiry of the options (94 days in our case) and r is the interest on a daily bottom (short rate), which was then supposed to be annualized. Since all the estimates were significant, ? 2= -0. 9979=-e-r(94) r=-ln0. 997994=2. 236391*10-5 Annualizing r r=2. 236391*10-5*250=0. 05592275=5. 2275%, which is the risk. The formula call(Ct)= 605. 997+0. 995put(Pt)-0. 9979(Strike(Kt)) was the model we used to derive values of call prices in relation to the multiple regression model. A plot of these call and strike options is shown below If significant, the estimate ? 2 was to be equated to -e-r (T-t) and the value for r equated. In this formula, T-t is the time to expiry of the options (94 days in our case) and r is the interest on a daily basis (short rate), which was then supposed to be annualized. PROCEDURE FOR FITTING Finally we drew a graph of Call against Strike and this was the graph obtained.The code and resulting graph are shown below, GRAPH FOR CALL AGAINST STRIKE BSM MODEL METHODS To fit the BSM Model and generate theoretical call prices, we obtained and truncated historical data from finance. yahoo. com as shown in the column labeled Adj. Close The code snapshot below created a function BSM73 We then computed the BSM73 by using the given the data, annualized interest rate (r), stock price, strike price and days to maturity generates the theoretical call prices. The proposed model to be fitted to fit the regression model CtSt= ? 0+ ? 1KSt+? 2K2St+ ? t main purpose is so as to determine the values of ? ,? 1 ? 2 Procedure From the results shown, we get ? 0=1. 313950 , ? 1=-1. 959886, ? 2= 0. 001195. The value of the stock at that point in time was St=605. 23. PLOT BSM CALL PRICE (Yt) AGAINST STRIKE PRICES For data analysis conducted for September 2012 options with T-t=94 days and r=5. 922%, the proposed model can be used in option pricing. It can be concluded from the analysis that for options with a longer time to expiry and a smaller interest rate, the proposed model prices the options more accurately than the BSM model in the price ranges where most options are traded. TIME SERIES ANALYSISThe theoretical model for a time financial time serial publication data is given by Xt = Trend + ARMA + GARCH + WN Where WN is the white noise in the data. We assumed that the GARCH component is equal to 0 We proceeded to investigate whether indeed the data at hand had trend in it. We used the following tools in our investi gation * Box plots * ACF * Histogram * Plotting the data Time series of the data. Summary of strike price data. Box plots ACF OF GARCH NOISE Code y=log(strike) d=diff(y) garch=d2 acf(garch,lag=100,main=ACF of Garch Noise) Histogram Code hist(strike,main=Histogram of ADJ Closing prices)De-trending the data After having confirmed that the data contained unidimensional trend, we proceeded to de-trend the data by 1. Finding the natural logarithm of the data y=log(strike) 2. Differencing the data d=diff(y) We confirmed that the data the data was actually stationery at this point by using the following techniques * Finding the ACF of the de-trended data and Plotting the de-trended data FIT ARMA (p, q) We found that an ARIMA (2, 2, 0) was the best model for our data FORECASTING We used the ARIMA (2,2,0) model to predict the adjusted closing share prices for the next 10 days