A Time Series Analysis of the Adjusted Closing Stock Prices

Table of Contents

1. Introduction

2. literature review

3. Introduction

4. Methodology

INTRODUCTION
Google Inc. is an American multinational corporation which provides Internet-related products and services, including Internet search, cloud computing, software 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 September 4, 1998, and its initial public offering followed on August 19, 2004. The company is now listed on the NASDAQ stock exchange under the ticker symbol . The company's mission statement from the outset was "to organize the world's information and make it universally accessible and useful”, and the company's unofficial slogan is "Don’t be evil”. In 2006, the company moved to its current headquarters in Mountain View, California.
Objectives
1. To fit a multiple regression model to a data set comprising the put, call and strike prices of a stock belonging to a company listed on a known index. 2. To use the BSM Model to which provides a mathematical science for the pricing and hedging of European Call and Put options as the American Options market 3. We wanted to analyze the data for Google option prices from the S&P index over the past and present time periods in order to be able to forecast the future.

Literature Review 1. Put call parity
In financial mathematics, put–call 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 underlying being a liquid asset. In the absence of liquidity, the existence of a forward contract suffices. Put–call parity requires minimal assumptions and thus does not require assumptions such as those of Black–Scholes or other commonly used financial models. 2. Black-Scholes Model
The Black–Scholes model or Black–Scholes-Merton is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black–Scholes formula, 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. lt is widely used by options market participants
Methodology
The data being analyzed consisted of daily past prices of silver traded on the S&P index since 14th May to 22 September 2012.The group was required to obtain data sets containing put, call and strike prices the data set of option expiring in more than 30 days but less than 100
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 values 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 Close | 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 | 0.475 | 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 | 606.07 | 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 | 475 | 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 | 614.25 | 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 | 17.8 | 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 | 640 | 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 | 119.25 | 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 | 619.54 | 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 | 900 | 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 | 0.225 | 363.25 | 940900 | 580.94 |
Fitting a Multiple regression Model

From the results shownβ0=605.997, β1=0.995, β2= -0.9979. The value of the stock at that point in time wasSt=605.23.
If significant, the estimate β2 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.

Since all the estimates were significant, β2= -0.9979=-e-r(94) r=-ln0.997994=2.236391*10-5 Annualizing r; r=2.236391*10-5*250=0.05592275=5.92275%, 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 β2 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= β0+β1KSt+β2K2St+ νt main purpose is so as to determine the values of β0,β1 & β2

Procedure

From the results shown, we get β0=1.313950 , β1=-1.959886, β2= 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 ANALYSIS
The theoretical model for a time financial time series 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 investigation * 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=d^2
> 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 linear 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: