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Improvement of binomial trees model and Black-Scholes model in option pricing
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Improvement of binomial trees model and Black-Scholes model in option pricing
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IMPROVEMENT OF BINOMIAL TREES MODEL AND BLACK-SCHOLES MODEL IN OPTION PRICING By Hao Zhang A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (Applied Mathematics) June, 2014 II Acknowledgments Myriad of thanks to my thesis advisor, Professor Sergey Lototsky and Professor Ricardo Mancera, for their constructive criticism and helpful mathematical insights. Contents Acknowledgments ................................................................................................. II Abstract ..................................................................................................................iv Chapter One Introduction..................................................................................... 1 Chapter Two Binomial Trees Model ..................................................................... 6 2.1 Time period effect on stock price ............................................................ 6 2.1.1 Data collection and analysis............................................................ 6 2.1.2 Using binomial trees to price ........................................................ 15 2.2 Political effect on stock price ................................................................... 17 2.2.1 The Strategy of Managing Political Risks .................................... 18 Chapter Three Continuous Condition ............................................................... 21 3.1 When RA> R ............................................................................................ 21 3.2 When RA<R ............................................................................................. 21 3.3 More General Condition .......................................................................... 24 Chapter Four Application and Summarization ................................................ 26 References ............................................................................................................. 27 Appendix ............................................................................................................... 29 Appendix A: List of Select Stocks ................................................................. 29 Appendix B: Total Result of O, Y , and X....................................................... 32 Appendix C: Matlab Code for Binomial Trees model ................................... 37 iv Abstract Black-Scholes formula is a common tool for people to price a European option, and it can be derived from binomial trees model by using infinite steps. However Black-Scholes model needs several assumptions which are not possible in real world. In this article, the underlying stock log return is not a random walk. The seasonal and political effect on the stock price will be summarized and applied to the model, therefore to increase the accuracy. And most important by the difference between improved model and original model, people will get a measure about certain risks in option investment. Key Word: Option pricing, Binomial Trees model, Black-Scholes Model Chapter One Introduction Characters of Options Options consist of two type, which are a call option and a put option. A call option give the holder the right to buy the underlying asset at a certain price at a certain time or within a certain time period in the future, while a put option conveys the right to sell. Noticed that American options can be exercised any time before the expiration date, while European option can only be exercised at the expiration data, therefore there may be a different in price between American and European option with the same condition. Also noticed that options give the right to sell or buy not the obligation which is the main difference between options and futures. Options have been used for quite a long time in human history. Supposedly ancient Greek mathematician and philosopher Thales of Miletus was the first option buyer. He predicted the harvest of olive and used options to get a higher pay. (Sander) Modern options appeared in the 1690s, people in London first introduced put and call options during the reign of William and Mary. (Smith) In America, options appeared in nineteenth century and soon becomes an important financial instrument all over the world. Binomial Options Pricing Model As options are widely used in the world to hedge, speculate or arbitrage, how to set the price option become quite critical. A very useful method for option pricing is binomial options pricing model. It was first introduced by Cox, Ross and Rubinstein in 1979. (Cox, Ross and Rubinstein) Bionimal trees model is a discrete model for option pricing. 2 The first step of this model is to create a price tree for the asset price. Given the asset price S at a certain point, the model assume that the underlying asset price will move up or down by a percentage u or d each step. Therefore in the next time period, the asset price will be S*u or S*d. Assuming there is no arbitrage chance and the portfolio is riskless, that we can calculte p = 𝑒 𝑟𝑇 −𝑑 𝑢 −𝑑 . Then continue this to the next time node to create a binomial tree, which can be shown in the following picture. Second step of binomial tree is to find option price at final node. The option value is calculate easily at the expiration time using the following formula. Call Option : Max[(Sn − K), 0] Put Option : Max[(K − Sn), 0] Third step is to track back option value at earlier nodes. As the probability of underlying asset moves up and down is already known, the expectation of option price of earlier point can be easily calculate by using probability knowledge and eliminate the time effect of money. Here is the formula. 3 Earlier Node Price = [p ∗ option price of up node+ (1 − p) ∗ option price of down node ] ∗ e −r∗t As different type of options have different characters, a few more comparison will be added if it is an American option or Bermudan option. European option cannot be exercised early, therefore no adjustment is needed for the price of option in earlier nodes. American option can be exercised any time before the expiration time. Thus after the calculation, a comparison must be applied to each node to check if early exercise will ensure more return. Bermudan option can be exercised at some certain time period before expiration date. So comparison is needed at the nodes when it can be exercised and not needed otherwise. After iteration several times to original point, the option price is measured. The accuracy can be improved by adding more generation to the iteration. Black-Scholes Model In 1973, Fischer Black and Myron Scholes published their result of option pricing model by using a PDE. In fact it is the continuous version of binomial trees model. To give the estimation of option price, they gave several assumptions. (Black and Scholes) The return of a riskless asset is a constant The log returns of underlying asset is a geometric Brownian motion The volatility of underlying asset is constant. The underlying asset pays no dividend. 4 There is no arbitrage opportunity Cash can be borrowed and lent at riskless rate Underlying asset can be shorted or longed at any amount No transition cost Under these ideal condition, the price of option can be valued by the following formula. And put-call parity: Where N() is the cumulative distribution function of the standard normal distribution S is the spot price of the underlying asset K is the strike price r is the risk free rate σ is the volatility of returns of the underlying asset (Black and Scholes) Black-Scholes model is a quite effective way to give an estimation price of option. However as it has so many assumptions, in some certain condition it will lead to quite unreliable results. Most famous example is LTCM (Long Term Capital Management). They 5 use dynamic hedging and large leverage to make money. However in 1997 and 1998, Asian and Russian financial Crisis exceeded the model and lead to a bailout with tremendous lost. Many people have done a lot to improve Black-Scholes model. Paul Wilmott made a lot effect take transaction costs into consideration. (Wilmott). R. Company, A.L. Gonzales, L.Jodar added discrete dividend into Black-Scholes model, (R. Company) etc. All this made Black-Scholes model more accuracy to use. In this article, I will try to change the assumption that underlying asset price is a geometric Brownian motion to improve the accuracy. 6 Chapter Two Binomial Trees Model 2.1 Time period effect on stock price First I start with discrete condition as it will be much easier to research. Time is always connected with money, especially in stock market. There are a number of anomalies in current stock market, for example, January Anomaly, weekend anomaly, holiday anomaly and so on. Here I try to find the relationship between stock price of different sectors and time. Using historical data to predict the seasonal trend of a set of stocks within the same industry. As database is extremely big, I only make a research on the seasonal impact on stock price and more detailed results can be drawn with the same method. 2.1.1 Data collection and analysis Stock price of different companied may have different seasonal trend as they are in different fields of industries. NYSE have concluded the listed stocks into 10 industries and several hundred subsectors. (NYSE) I only take the biggest class into consideration to simplify the work (still 100 stocks with 5 year trading records). Super sectors, sectors and subsectors data can be summarized by the same tool. The list of 10 industries are showing in the below table. Basic Materials Consumer Goods Consumer Services Financials Health Care Industrials Oil & Gas Technology Telecommunications Utilities 7 In each industry, ten stocks are chosen by 4 from large cap companies (Over $10 billion capitalization), 3 from middle cap ($2 billion–$10 billion) and 3 from small cap ($250 million–$2 billion). The total list of 100 stocks are shown in the appendix. Then for each company trading records are collect from Jan 1 st , 2009 to Jan 1 st , 2014 to analyze the seasonal trend. As for seasonal trend and discrete condition, I start with quarter compounding increase rate and then use more complicated continuous compounding. The key point here is to get the relationship between certain season performance and whole year performance. Noticed here I choose 5 year record to summarize the relationship mainly because a former research showed that stock market have a mean reversion character, (Balvers, Wu and Gilliland) which although not fully convinced but generally admitted. Thus 5 year or even longer data would show some characters of stocks price. However I did not use these 5 year data to predict the stock price, as from some researches of behavioral finance. Individual stocks may have momentum, which means they are likely to continue their performance of the past few months. (Jegadeesh and Titman). Therefore I use the relationship and recently one year performance to add the seasonal trend to bionmial trees model to improve the accuracy. 2.1.1.1 Quarter Compounding When using quarter compounding, first I calculate the increase rate of each quarter and the whole year and then take the average of 5 years. Then for each industry also take the average of 10 companies to generate the raw data. 8 Quarter Compounding Annual Rate First Qu Second Qu Third Qu Last Qu Basic Materials Large Cap 0.1502 06 0.0354 01 - 0.0673 3 0.0897 35 0.1379 95 Middle Cap 0.3011 42 0.0822 93 0.0787 67 0.0441 73 0.1648 77 Small Cap 0.3609 47 0.1656 19 - 0.0737 8 0.1547 32 0.3258 47 Total 0.2587 09 0.0885 34 - 0.0254 3 0.0955 65 0.2024 15 Consumer Goods Large Cap 0.2153 7 0.0603 32 0.0285 67 0.0308 13 0.1284 34 Middle Cap 0.4175 03 0.1433 97 0.0627 19 0.1362 32 0.1369 39 Small Cap 0.2431 06 0.0481 31 0.1136 03 0.1374 34 0.0756 24 Total 0.2843 31 0.0815 91 0.0643 24 0.0944 25 0.1151 42 Consumer Services Large Cap 0.1514 31 0.0217 88 - 0.0004 2 0.0544 53 0.1069 49 Middle Cap 0.1029 03 0.1524 58 - 0.0245 8 0.0359 08 0.0254 57 Small Cap 0.3963 04 0.1573 1 0.1533 22 0.1146 04 0.0764 08 Total 0.2103 35 0.1016 46 0.0384 56 0.0669 35 0.0733 39 Financials Large Cap 0.0743 27 0.0030 06 - 0.0184 4 0.1229 32 0.0494 29 Middle Cap 0.1916 41 0.0708 88 0.0152 4 0.0361 18 0.1138 09 Small Cap 0.0032 27 - 0.0521 7 0.0161 84 - 0.0107 6 0.0965 03 Total 0.0881 91 0.0068 17 0.0020 53 0.0567 8 0.0828 65 Health Care Large Cap 0.1644 32 0.0376 61 - 0.0051 2 0.0793 23 0.0743 81 Middle Cap 0.3000 43 0.0940 34 0.0781 69 0.0735 18 0.0800 05 Small Cap 0.0933 07 0.0381 35 0.0960 17 - 0.0305 7 0.0246 2 9 Total 0.1837 78 0.0547 15 0.0502 07 0.0446 13 0.0611 4 Industrials Large Cap 0.1657 82 0.0393 4 0.0063 03 0.0618 55 0.0838 96 Middle Cap 0.0455 74 - 0.0241 6 0.0175 77 0.0523 23 0.0752 61 Small Cap 0.4999 77 0.1680 42 0.1247 61 0.1553 06 0.0940 55 Total 0.2299 78 0.0589 01 0.0452 22 0.0870 31 0.0843 53 Oil & Gas Large Cap 0.0980 81 0.0016 92 - 0.0132 4 0.0400 71 0.0967 9 Middle Cap 0.2219 23 0.0764 37 0.0243 32 0.0546 39 0.0834 04 Small Cap 0.1026 81 - 0.0014 3 - 0.1009 5 0.1285 04 0.1820 23 Total 0.1366 14 0.0231 8 - 0.0282 8 0.0709 71 0.1183 44 Technology Large Cap 0.1738 74 0.0500 93 - 0.0029 2 0.0816 54 0.0736 53 Middle Cap 0.1168 43 0.0045 06 - 0.0378 4 0.0677 97 0.1321 54 Small Cap 0.3763 1 0.1662 87 0.0726 21 0.0624 73 0.1557 38 Total 0.2174 96 0.0712 75 0.0092 66 0.0717 42 0.1158 29 Telecommunicati ons Large Cap 0.2118 77 0.0254 86 0.0594 94 0.0754 29 0.0819 54 Middle Cap 0.0418 86 - 0.0444 5 0.0010 01 0.0808 76 0.0261 28 Small Cap 0.0753 94 - 0.0277 6 0.0487 03 0.1377 76 - 0.0274 9 Total 0.1199 35 - 0.0114 7 0.0387 09 0.0957 67 0.0323 72 Utilities Large Cap 0.1286 5 0.0109 02 0.0300 53 0.0417 46 0.0547 43 Middle Cap 0.1815 22 0.0400 83 0.0365 58 0.0449 36 0.0746 61 Small Cap 0.0884 96 -0.0083 0.0296 28 0.0291 71 0.0455 59 10 Total 0.1324 65 0.0138 95 0.0318 77 0.0389 31 0.0579 63 Now after first step data collection and analysis, some data processions are needed to reach how seasonal performance contribute to the whole year performance. In order to calculate the average, I scale the whole year increase rate to 1 or -1, and then calculate the seasonal contribution. Then with the same absolute scaling I can calculate the average. Let r denote average quarter compounding annual increase rate, and r1, r2, r3 and r4 denote average quarter compounding increase rate of first (Jan 1 st to Mar 31 st ), second (Apr 1 st to Jun 30 th ), third (July 1 st to Sep 30 th ) and last (Oct 1 st to Dec 31 st ) season respectively in the table. Then from basic accounting knowledge I get the formula. S1 = S1 S0 ∗ (1 + r 4 ) 4 = 𝑆 0 ∗ (1 + r1 ) ∗ (1 + r2 ) ∗ (1 + r3 ) ∗ (1 + r4 ) 1 = (1 + r1 ) ∗ (1 + r2 ) ∗ (1 + r3 ) ∗ (1 + r4 ) (1 + 𝑟 4 ) 4 ⑴ Therefore let o1 = 1+r1 1+ r 4 − 1 , then o1 will show the contribution of first season to the whole year. After calculate the average of o1, given the quarter compounding annual increase rate of target stock of the most recent year, reversing the formula gives the prediction of seasonal trend of the target stock. 𝑟 1 𝑡𝑎𝑟𝑔𝑒𝑡 = (𝑜 1 ̅̅̅ + 1) ∗ (1 + r 4 ) − 1 ⑵ 11 By the same method, I generate o2, o3 and o4. Here I take Oil & Gas industry as example and the whole result will also be shown in the appendix. Oil & Gas O1 O2 O3 O4 Test Large Cap - 0.02159 - 0.0396 5 0.01758 4 0.07133 6 1.02221 4 Middle Cap 0.02227 2 - 0.0309 1 - 0.00259 0.02781 5 1.01333 2 Small Cap - 0.02296 - 0.1129 3 0.09647 8 0.13663 1 1.07518 3 Total - 0.00884 - 0.0590 1 0.03519 9 0.07786 8 1.03544 Also noted that I add a test in the table. From formula (1), (1+o1)*(1+o2)*(1+o3)*(1+o4) should equal to 1, however after the average, this property has been changed. Therefore there will be a slight impact on the accuracy. 2.1.1.2 Continuous Compounding When using continuous compounding, same as quarter compounding, I generate the raw result first. Continuous Compounding First Qu Secon d Qu Third Qu Last Qu Annua l 1 Basic Materials Large Cap 0.0434 56 - 0.1368 0.1045 46 0.2187 61 0.0574 92 Middle Cap 0.1130 86 0.0753 15 0.0224 29 0.2598 83 0.1176 78 Small Cap 0.1835 85 - 0.2441 2 0.2150 89 0.2991 91 0.1134 37 Total 0.1063 84 - 0.1053 6 0.1130 74 0.2552 27 0.0923 31 Consumer Goods Large Cap 0.0897 13 0.0140 6 0.0377 09 0.1956 27 0.0842 77 12 Middle Cap 0.2141 64 0.0506 74 0.1874 45 0.2091 4 0.1653 56 Small Cap 0.0602 98 0.1167 41 0.1085 37 0.1025 92 0.0970 42 Total 0.1182 24 0.0558 49 0.1038 78 0.1717 7 0.1124 3 Consumer Services Large Cap 0.0103 24 - 0.0128 1 0.0769 83 0.1699 8 0.0611 19 Middle Cap 0.2002 58 - 0.0743 8 0.0151 42 0.0015 25 0.0356 36 Small Cap 0.2351 02 0.1734 63 0.1040 44 0.1019 81 0.1536 48 Total 0.1347 38 0.0246 0.0665 49 0.0990 44 0.0812 33 Financials Large Cap - 0.0427 1 - 0.0562 6 0.1412 78 0.0635 21 0.0264 58 Middle Cap 0.1042 68 - 0.0054 7 0.0238 88 0.1828 76 0.0763 9 Small Cap - 0.1465 7 - 0.0058 7 - 0.0460 8 0.1233 11 - 0.0188 Total - 0.0297 7 - 0.0259 0.0498 54 0.1172 64 0.0278 6 Health Care Large Cap 0.0491 77 - 0.0190 7 0.1188 91 0.1211 24 0.0675 31 Middle Cap 0.1464 52 0.1071 28 0.0953 04 0.1293 69 0.1195 64 Small Cap 0.0529 74 0.1446 26 - 0.0729 3 0.0323 39 0.0392 51 Total 0.0794 98 0.0678 99 0.0542 67 0.0969 62 0.0746 57 Industrials Large Cap 0.0603 96 0.0011 15 0.0784 72 0.1365 28 0.0691 28 Middle Cap - 0.0658 4 - 0.0069 9 0.0394 78 0.0792 73 0.0114 82 Small Cap 0.2551 89 0.1708 61 0.1892 6 0.1327 65 0.1870 19 13 Total 0.0809 65 0.0496 08 0.1000 1 0.1182 23 0.0872 01 Oil & Gas Large Cap - 0.0037 9 - 0.0461 7 0.0553 11 0.1565 44 0.0404 73 Middle Cap 0.1205 1 0.0295 83 0.0755 69 0.1352 86 0.0902 37 Small Cap - 0.0227 3 - 0.2154 1 0.1582 08 0.2135 61 0.0334 05 Total 0.0278 18 - 0.0742 2 0.0922 58 0.1672 72 0.0532 82 Technology Large Cap 0.0663 18 - 0.0225 5 0.1245 99 0.1152 51 0.0709 04 Middle Cap - 0.0128 - 0.0786 4 0.0775 3 0.2057 4 0.0479 57 Small Cap 0.2039 22 0.0668 37 0.0828 48 0.2267 26 0.1450 83 Total 0.0838 63 - 0.0125 6 0.0979 53 0.1758 4 0.0862 73 Telecommunica tions Large Cap 0.0217 25 0.0894 76 0.1103 36 0.1316 29 0.0882 92 Middle Cap - 0.0831 5 - 0.0059 8 0.1136 45 0.0342 69 0.0146 97 Small Cap - 0.0656 1 0.0508 0.1965 79 - 0.0897 4 0.0230 06 Total - 0.0359 4 0.0492 38 0.1372 02 0.0360 09 0.0466 27 Utilities Large Cap 0.0118 33 0.0477 13 0.0682 49 0.0901 41 0.0544 84 Middle Cap 0.0554 04 0.0548 35 0.0654 78 0.1209 28 0.0741 61 Small Cap - 0.0186 1 0.0449 9 0.0481 28 0.0749 4 0.0373 61 Total 0.0157 7 0.0490 32 0.0613 81 0.0948 17 0.0552 5 14 Here let r, r1, r2, r3, r4 denoted average continuous compounding annual increase rate and first, second, third and last quarter continuous compounding annualized increase rate in the table. Therefore we have S1 = S1 𝑆 0 ∗ 𝑒 𝑟 = 𝑆 0 ∗ 𝑒 𝑟 1 4 ∗ 𝑒 𝑟 2 4 ∗ 𝑒 𝑟 3 4 ∗ 𝑒 𝑟 4 4 𝑟 = 𝑟 1 4 + 𝑟 2 4 + 𝑟 3 4 + 𝑟 4 4 1 = 𝑟 1 4𝑟 + 𝑟 2 4𝑟 + 𝑟 3 4𝑟 + 𝑟 4 4𝑟 Denote 𝑦 1 = 𝑟 1 4𝑟 , then y1+y2+y3+y4 =1. However when r is small on a specific year, y will be too big to effect the average. Therefore, I use another modification. That is x1 2 + 𝑥 2 2 + 𝑥 3 2 + 𝑥 4 2 = 1. By x1 = r1 √r1 2 + 𝑟 2 2 + 𝑟 3 2 + 𝑟 4 2 x2 = r2 √r1 2 + 𝑟 2 2 + 𝑟 3 2 + 𝑟 4 2 x3 = r3 √r1 2 + 𝑟 2 2 + 𝑟 3 2 + 𝑟 4 2 x4 = r4 √r1 2 + 𝑟 2 2 + 𝑟 3 2 + 𝑟 4 2 When using this, one should reverse the modification by getting the target r1 to r4 first. The result of using these two modifications are showing in the below two table. Modification one Y1 Y2 Y3 Y4 Large Cap 1.142968 - 0.54576 -1.20895 1.111738 15 Middle Cap 0.284625 - 0.57417 -0.21613 1.105677 Small Cap -0.25227 - 0.55014 0.616734 0.252348 Total 0.466893 -0.5556 -0.3634 0.852103 Modification Two X1 X2 X3 X4 Large Cap -0.00181 -0.04883 0.156102 0.421373 Middle Cap 0.262404 0.098388 0.201979 0.307179 Small Cap -0.12952 -0.2375 0.2513 0.150696 Total 0.039139 -0.06126 0.198424 0.305912 Also I collect the data of several Oil & Gas ETFs and S&P Oil&Gas index, the results of x and y are quite similar. Second quarter always has the worst performance, and last quarter are usually the best. Therefore it will be a good method to predict options of ETFs as well. x1 x2 x3 x4 Large Cap -0.00181 -0.04883 0.156102 0.421373 Middle Cap 0.262404 0.098388 0.201979 0.307179 Small Cap -0.12952 -0.2375 0.2513 0.150696 Total 0.039139 -0.06126 0.198424 0.305912 Oil & Gas ETF XOP 0.125418 -0.20569 0.431555 0.377349 Oil & Gas ETF DIG 0.020465 -0.20008 0.314185 0.430714 Oil & Gas ETF OIH 0.158572 -0.10119 0.359653 0.295153 S&P Oil&Gas Index 0.217161 -0.12191 0.273045 0.349737 2.1.2 Using binomial trees to price From the above steps, I get the expected return of underlying asset called “ra”. Then in binomial trees model, if ra > r, then the discount rate will be ‘ra’ each step as it will become 16 new risk-free asset. When ra < r, then discount rate will still be r, however the expected stock price are effected by ‘ra’. For example, I chose Chevron to be the target stock. Then I try to calculate its 3 month option price expired at Jun 14 th . Then first I get one year trading data of Chevron. Then calculate its r with quarter compounding and continuous compounding. Here are the results. Quarter r -0.01697 Continuous r r1 r2 r3 r4 0.067702 0.152683 0.00635 0.046605 0.06517 Then reverse it, I get expect r2 in the following table. R2 LC mean reverse with large capital data and r2 Total means reverse with r2 total average data. Quarter r2 LC r2 Total - 0.04372 -0.063 Continuous mod 1 mod 2 r2 LC r2 Total r2 LC r2 Total -0.1478 -0.15046 - 0.00145 - 0.00182 Also calculate u, d from u = e σ√∆𝑡 , d = e −σ√ ∆𝑡 , p = 𝑒 𝑟 2∆𝑡 −𝑑 𝑢 −𝑑 . Then using Matlab to iterate for results. Here is the table of results with a strike price K = 120 and steps n = 3000. Quarter r2 LC r2 Total B-S Model Result Ame C Eur C Ame C Eur C 10.1027 8.979 4 8.344 7 8.869 2 7.843 0 Continuo us mod 1 mod 2 r2 LC r2 Total r2 LC r2 Total Ame C Eur C Ame C Eur C Ame C Eur C Ame C Eur C 17 8.830 5.822 8.830 5.576 9.520 7 9.494 6 9.513 5 9.484 2 Compared with real option price American call option has a value of 8.85. The result is quite acceptable. Also I try to predict the option price of ETFs, but unfortunate these options have so few open interests that there is almost no trade data. However considering ETFs’ character, the prediction should be good as ETF reflect almost the same property my statistic result shows. 2.2 Political effect on stock price Political effect also will have a strong effect on stock price. In some certain field of industry, political event will dominate the investment option, which is also the reason most giant insurance companies are now providing political risk insurance. Also I take Oil and Gas industry as an example. In this field, political risks often play a fatal role in decision making. Quite a lot wars, riots and coups are connected with oil interest. The main reason for this is that firstly oil and gas provide most energy for today’s society and secondly almost more than 80% of proved reserves of oil are located in most political uncertain areas. Here is the data from US Energy Information Administration. (USEIA) Crude Oil Proved Reserves (Billion Barrels) 2010 2011 2012 2013 Venezuela 99.377 211.17 211.17 297.57 Saudi Arabia 262.4 262.6 267.02 267.91 Canada 175.214 175.214 173.6252 173.1052 Iran 137.62 137.01 151.17 154.58 Iraq 115 115 143.1 141.35 Kuwait 104 104 104 104 United Arab Emirates 97.8 97.8 97.8 97.8 Russia 60 60 60 80 Libya 44.27 46.42 47.1 48.01 18 Nigeria 37.2 37.2 37.2 37.2 Kazakhstan 30 30 30 30 Qatar 25.41 25.38 25.38 25.38 World 1355.743 1473.761 1525.957 1645.984 According to a former research, political events in oil and gas industry can be mainly concluded into three classes, regulatory framework, government behavior and political conflict. (Nordal) Most oil companies now have already established systems to manage political risks, therefore the impact on the stocks are not that serious today. However when you invest with options, especially with large leverage, this effect can lead to bankrupt easily. 2.2.1 The Strategy of Managing Political Risks Here in order to pricing options, the relationship between political risks and stock price must be quantified. Therefore I take three steps to achieve the purposes. First thing to be done is to measure the effect is positive or negative. For example, in 2003 from March 19 th to May 1 st , the war on Iraq raise the stock price of most American oil companies, however this may not be a good information for the companies in other countries. Secondly measurement must be made on how large the effect is. I take 10 companies which have the same positive or negative reflections on the stock prices to a certain political event, into a group. Collecting the stock prices data before, during and after this event and also the stock prices of the same time period from joint years when no big political event happens. Then calculating and comparing the increase rate or decrease rate when the certain kind of political event happens. The following table shows the result I get about Iraq War (all the rate are continuous compounding annualized rate) 19 R of the year r during war period R after war in 3 months r before war in 3 mouths BP 0.023263912 -0.05493 0.130445 -0.0759 EPD 0.103241098 0.254576 -0.00471 0.138079 XOM 0.047413336 0.02296 0.005068 0.020821 CVX 0.05963202 -0.15123 0.226764 -0.03902 NE -0.009011376 -0.47772 0.10726 -0.10299 APU 0.054146896 0.346127 0.065048 0.04576 GEL 0.075463656 0.331251 0.446352 -0.28739 CIR 0.119147805 0.373826 0.255615 -0.22269 PZE 0.07888551 0.244815 0.036151 0.145698 ERHE 0.786434047 0.18909 0.584512 1.408666 Average 0.13386169 0.107876 0.185251 0.103105 Also from the database of USEIA I get the proved reserves of oil of Iraq and the entire world are 112.500 and 1,213.112 Billion Barrels respectively in 2003. Therefore 9.27% proved reserves are direct effected. Finally applying this rate into binomial trees model. Noticed that here a probability of how likely this event will happen before the option expiration date is required, which can be found on some report from think tank. Also I take where the war happens into consideration as comparing the ratio of proved reserves. So I made a formula to calculate the impact of war on different places. R 𝑤𝑎𝑟 = ∑ 𝑃 𝑡𝑟𝑎𝑔𝑒𝑡 ∗ 𝑅 Iraq ∗ 𝑅𝑎𝑡𝑖 𝑜 𝑡𝑎𝑟𝑔𝑒𝑡 𝑅𝑎𝑡𝑖 𝑜 𝐼𝑟𝑎𝑞 Where R war means the adjustment of increase rate of political event: war. P 𝑡𝑟𝑎𝑔𝑒𝑡 means the probability of war happens before the option expiration date in target areas. R 𝐼𝑟𝑎𝑞 denotes the average result from the table 20 Ratio target 𝑎𝑛𝑑 𝑅𝑎𝑡𝑖 𝑜 𝐼𝑟𝑎𝑞 denote the ratio of proved reserves in target areas and 2003 Iraq respectively In fact, if we get more data of different wars’ impact on oil stocks and get an average, the result will be more accuracy. However it requires too much time, so I just show the method I use here. Also other political event can be measured in the same method. After R war ′𝑠 calculation, we can set the real increase rate of underlying stocks to be ra = R war + 𝑅 𝑞𝑢𝑎𝑟𝑡𝑒𝑟 𝑒𝑓𝑓𝑒𝑐𝑡 . Then put ‘ra’ into the binomial tree model discussed in chapter 2.1, we can get the modified option price considering both effect of time period and war. 21 Chapter Three Continuous Condition All the above are discussing the discrete condition, in this section. I will try to generalize the trend effected by seasonal and political factors into a PDE formula. From the all the above, what I change is the assumption that underlying asset price is a geometric Brownian motion. The expect return of the stock is thus not risk-free rate, but a adjust rate, say ‘ra’. 3.1 When RA> R Therefore, if other assumption keeps unchanged. When ra bigger than risk-free return, ra will replace the risk-free rate. And then option price becomes, 𝑐 = 𝑆 0 𝑁 (𝑑 1 ) − 𝐾 𝑒 −𝑟𝑎 ∗𝑇 𝑁 (𝑑 2 ) d 1 = ln( 𝑆 0 𝐾 ) + (𝑟𝑎 + 𝜎 2 2 ) 𝑇 𝜎 √𝑇 d 2 = 𝑑 1 − 𝜎 √𝑇 3.2 When RA<R When ‘ra’ smaller than risk-free return, Δ should equal 0 as no one will long this stock. Replace risk free rate to ‘ra’ will give an upper bound of this option price. In fact, each step, the discount rate should still be risk-free rate as it is higher than the stock returns. So under this condition, keeping discount rate unchanged, but change the expect return on underlying assets, we will get a new probability p. E(S 1 ) = 𝑆 0 ∗ 𝑢 ∗ 𝑝 + 𝑆 0 ∗ 𝑑 ∗ (1 − 𝑝 ) = 𝑆 0 ∗ 𝑒 𝑟𝑎 ∗𝑇 22 p = 𝑒 𝑟𝑎 ∗𝑇 − 𝑑 𝑢 − 𝑑 Then assume using an n steps binomial tree model to value a Euro call option, which has a strike price K and life T. Also other parameters are the same with original binomial tree model and Black-Scholes model. For a final node that have experienced j up movements and n-j down movements. The probability of the appearance of this node can be calculate with binomial distribution, and that is 𝐶 𝑛 𝑛 −𝑗 ∗ 𝑝 𝑗 ∗ (1 − 𝑝 ) 𝑛 −𝑗 And for this node, the payoff of European call option is max (S 0 𝑢 𝑗 𝑑 𝑛 −𝑗 − 𝐾 , 0) Thus, the expected payoff of the call option is ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ 𝑝 𝑗 ∗ (1 − 𝑝 ) 𝑛 −𝑗 ∗ 𝑛 𝑗 =0 max (S 0 𝑢 𝑗 𝑑 𝑛 −𝑗 − 𝐾 , 0) Then as I analyzed before, I use risk-free rate r as the discount rate. Therefore the present value of this option is 𝑐 = 𝑒 −𝑟 ∗𝑇 ∗ ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ 𝑝 𝑗 ∗ (1 − 𝑝 ) 𝑛 −𝑗 ∗ 𝑛 𝑗 =0 𝑚𝑎𝑥 (𝑆 0 𝑢 𝑗 𝑑 𝑛 −𝑗 − 𝐾 , 0) 𝑐 = 𝑒 −𝑟 ∗𝑇 ∗ ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ 𝑝 𝑗 ∗ (1 − 𝑝 ) 𝑛 −𝑗 ∗ (𝑆 0 𝑢 𝑗 𝑑 𝑛 −𝑗 − 𝐾 ) 𝑆 0 𝑢 𝑗 𝑑 𝑛 −𝑗 >0 𝑐 = 𝑒 −𝑟 ∗𝑇 ∗ ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ 𝑝 𝑗 ∗ (1 − 𝑝 ) 𝑛 −𝑗 ∗ 𝑛 𝑗 > 𝑛 2 − ln( 𝑆 0 𝑘 ) 2𝜎 √ 𝑇 𝑛 (𝑆 0 𝑢 𝑗 𝑑 𝑛 −𝑗 − 𝐾 ) ⑶ Denoting 23 𝑈 1 = ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ 𝑝 𝑗 ∗ (1 − 𝑝 ) 𝑛 −𝑗 ∗ 𝑗 > 𝑛 2 − 𝑙𝑛 ( 𝑆 0 𝑘 ) 2𝜎 √ 𝑇 𝑛 𝑢 𝑗 𝑑 𝑛 −𝑗 𝑈 2 = ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ 𝑝 𝑗 ∗ (1 − 𝑝 ) 𝑛 −𝑗 𝑗 > 𝑛 2 − 𝑙𝑛 ( 𝑆 0 𝑘 ) 2𝜎 √ 𝑇 𝑛 Then call option price c can be written as 𝑐 = 𝑒 −𝑟 ∗𝑇 ∗ (𝑆 0 𝑈 1 − 𝐾 𝑈 2 ) Now thinking of continuous condition, i.e. let n go to infinite. Then 𝑈 1 = ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ (𝑝𝑢 ) 𝑗 ∗ [𝑑 (1 − 𝑝 )] 𝑛 −𝑗 𝑗 > 𝑛 2 − 𝑙𝑛 ( 𝑆 0 𝑘 ) 2𝜎 √ 𝑇 𝑛 Define 𝑃 ∗ = 𝑝𝑢 𝑝𝑢 + (1 − 𝑝 )𝑑 𝑎𝑛𝑑 1 − 𝑃 ∗ = (1 − 𝑝 )𝑑 𝑝𝑢 + (1 − 𝑝 )𝑑 Therefore 𝑈 1 = [𝑝𝑢 + (1 − 𝑝 )𝑑 ] 𝑛 ∗ ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ (𝑝 ∗ ) 𝑗 ∗ (1 − 𝑝 ∗ ) 𝑛 −𝑗 𝑗 > 𝑛 2 − 𝑙𝑛 ( 𝑆 0 𝑘 ) 2𝜎 √ 𝑇 𝑛 𝑈 1 = 𝑒 𝑟𝑎 ∗𝑇 ∑ 𝐶 𝑛 𝑛 −𝑗 ∗ (𝑝 ∗ ) 𝑗 ∗ (1 − 𝑝 ∗ ) 𝑛 −𝑗 𝑗 > 𝑛 2 − 𝑙𝑛 ( 𝑆 0 𝑘 ) 2𝜎 √ 𝑇 𝑛 As binomial distribution approaches to be a normal distribution when n goes to infinite 𝑈 1 = 𝑒 𝑟𝑎 ∗𝑇 ∗ 𝑁 ( 𝑛 𝑝 ∗ − 𝑛 2 + 𝑙𝑛 ( 𝑆 0 𝑘 ) 2𝜎 √ 𝑇 𝑛 √𝑛 𝑝 ∗ (1 − 𝑝 ∗ ) ) 24 Where N() is the cumulative normal distribution function. Also from the original model 𝑢 = 𝑒 𝜎 √ 𝑇 𝑛 𝑎𝑛𝑑 𝑑 = 𝑒 −𝜎 √ 𝑇 𝑛 substituting for u and d in 𝑝 ∗ , 𝑝 ∗ = 𝑒 𝑟𝑎 ∗𝑇 𝑛 − 𝑒 −𝜎 √ 𝑇 𝑛 𝑒 𝜎 √ 𝑇 𝑛 − 𝑒 −𝜎 √ 𝑇 𝑛 ∗ 𝑒 𝜎 √ 𝑇 𝑛 𝑒 𝑟𝑎 ∗𝑇 𝑛 Expanding exponential functions using Taylor expansion, I get lim 𝑛 →∞ 𝑝 ∗ (1 − 𝑝 ∗ ) = 1 4 , 𝑎𝑛𝑑 lim 𝑛 →∞ √ 𝑛 (𝑝 ∗ − 1 2 ) = (𝑟𝑎 + 𝜎 2 2 )√𝑇 2𝜎 Thus, 𝑈 1 = 𝑒 𝑟𝑎 ∗𝑇 ∗ 𝑁 ( ln( 𝑆 0 𝐾 ) + (𝑟𝑎 + 𝜎 2 2 )𝑇 𝜎 √𝑇 ) Then I do the same process to U2, 𝑈 2 = 𝑁 ( ln( 𝑆 0 𝐾 ) + (𝑟𝑎 − 𝜎 2 2 )𝑇 𝜎 √𝑇 ) Sum it up, the final formula is, 𝑐 = 𝑆 0 ∗ 𝑒 𝑟𝑎 −𝑟 𝑁 (𝑑 1 ) − 𝐾 𝑒 −𝑟𝑇 𝑁 (𝑑 2 ) Where 𝑑 1 = ln( 𝑆 0 𝐾 ) + (𝑟𝑎 + 𝜎 2 2 ) 𝑇 𝜎 √𝑇 𝑑 2 = ln( 𝑆 0 𝐾 ) + (𝑟𝑎 − 𝜎 2 2 )𝑇 𝜎 √𝑇 = 𝑑 1 − 𝜎 √𝑇 3.3 More General Condition 25 To a more general condition, when ‘ra’ changes with time. In this situation, each step, the discount rate should be max (ra, r), and also notice that whenever ‘ra’ smaller than r, Δ should equal zero. The hedge ratio also change dramatically. If the time periods when ra>r and when ra<r are exactly known, using the method in 3.2 can get a formula. Noticed that discount rate only applied once in the whole process, just in formula (3). Combining 3.1 and 3.2 together to different time periods can generate the result. However, when the time periods are not exactly known or ‘ra’ change too many times, it is quite difficult to calculate so many times, therefore numerical method may be more realistic. Numerical methods like binomial trees or Monte-Carlo simulation can be used to estimate the results. 26 Chapter Four Application and Summarization After all the discussion, I give an estimation of option price considering the price trend of underlying assets. However they are all under certain assumptions and comparing to real option price there is still lot more factors to be considered. Therefore I think the main application of this method is not to calculate the option price, but with comparison to original Black-Scholes models, if there is a big difference in the result of option price, people should be aware that there may be a great political or quarter risk. Thus I define a parameter H. H = | 𝑜𝑝𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒 𝑐𝑜𝑛𝑐𝑒𝑟𝑛𝑖𝑛𝑔 𝑝𝑜𝑙𝑖𝑡𝑖𝑐𝑎𝑙 𝑜𝑟 𝑞𝑢𝑎𝑟𝑡𝑒𝑟 𝑟𝑖𝑠𝑘 𝑜𝑟𝑖𝑔𝑛𝑎𝑙 𝐵 − 𝑆 𝑜𝑝𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒 − 1| H can be calculated mostly by numerical method as a reference when making decision. If H is close to 0, showing that political risks or quarter risks are not playing main part on the underlying assets, otherwise, more considerations need to be taken before going to the position. Other factors that will have an impact on the underlying assets price can be analyzed in the same way. And One can use H to measure the how serious the risk is. 27 References Balvers, Ronald, Yangru Wu and Erik Gilliland. "Mean Reversion across National Stock Markets and Parametric Contrarian Investment Strategies." THE JOURNAL OF FINANCE (2000): 754-769. Black, Fischer and Myron Scholes. ""The Pricing of Options and Corporate Liabilities"." Journal of Political Economy (1973): 637–654. Cox, J. C., S. A. Ross and M Rubinstein. ""Option pricing: A simplified approach"." Journal of Financial Economics (1979): 229. Hull, John C. Options,Futures, and Other Derivatives. Boston: Pearson Education, Inc., 2012. Jegadeesh, Narasimhan and Sheridan Titman. "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency." The Journal of Finance (Mar., 1993): 65-91. Nordal, Kjell Bjorn. Political Uncertainty: Valuation and Decision Making with a Focus on Oil Investments. 1998. NYSE. http://www.nyse.com/about/listed/lc_all_industry.html. 2014. R. Company, A.L. Gonz´alez, L. J´odar. "Numerical solution of modified Black–Scholes equation pricing stock options with discrete dividend." Mathematical and Computer Modelling (2006): 1058-1068. Sander, Mattias. Bondesson's Representation of the Variance Gamma Model and Monte Carlo Option Pricing. Lunds Tekniska Högskola, 2008. Smith, B. Mark. History of the Global Stock Market from Ancient Rome to Silicon Valley. University of Chicago Press, 2003. ISBN 0-226-76404-4. USEIA. Crude Oil Proved Reserves. 2014. 2014. <http://www.eia.gov/countries/index.cfm?view=reserves#cabs>. 28 Wilmott, Paul. "Discrete Charms." Risk (1994): 48-51. World Bank. World Investment and Political Risk. Washington, DC: MIGA, World Bank Group, 2013. 29 Appendix Appendix A: List of Select Stocks Basic Materials Large Cap APD Air Products and Chemicals, Inc. NYSE RIO Rio Pinto plc NYSE VALE Vale S.A. NYSE BBL BHP Billiton PLC. NYSE Middle Cap MEO H Methanex Corporation NASDAQ ASH Ashland Inc. (ASH) NYSE ATI Allegheny Technologies Inc. NYSE Small Cap GPRE Green Plains Renewable Energy Inc. NASDAQ GPL Great Panther Silver Ltd. NYSE USEG US Energy Corporation NASDAQ Consumer Goods Large Cap F Ford Motor Company NYSE TM Toyota Motor Corporation NYSE LOW Lowe's Companies Inc. NYSE BTI British American Tobacco PLC. NYSE Middle Cap FOSL Fossil Inc. NYSE NUS Nu Skin Enterprises, Inc. NYSE IFF International Flavors and Fragrances Inc. NYSE Small Cap ROX Castle Brands Inc. NYSE SPAR Spartan Motors Inc. NASDAQ ANDE The Andersons Inc. NASDAQ Consumer Services Large Cap CMCS A Comcast Corporation NASDAQ WMT Wal-Mart Stores Inc. NYSE DAL Delta Air Lines Inc. NYSE MCD McDonald's Corporation NYSE Middle Cap SHLD Sears Holdings Corporation NASDAQ SWY Safeway Inc. NYSE CVC Cablevision Systems Corporation NYSE 30 Small Cap CTCM CTC Media Inc. NASD AQ KKD Krispy Kreme Doughnuts Inc. NYSE PLKI Popeyes Louisiana Kitchen Inc. NASDAQ Financials Large Cap JPM JPMorgan Chase & Co. NYSE C Citigroup Inc. NYSE ING ING Groep N.V. NYSE AMT American Tower Corp. NYSE Middle Cap BOKF BOK Financial Corporation NASDAQ PSEC Prospect Capital Corporation NASDAQ CBG CB Richard Ellis Group Inc. NYSE Small Cap FBIZ First Business Financial Services Inc. NASDAQ EIG Employers Holdings Inc. NYSE CPF Central Pacific Financial Corporation NYSE Health Care Large Cap GILD Gilead Sciences Inc. NASDAQ COV Covidien plc. NYSE PFE Pfizer Inc. NYSE NVS Novartis A.G. NYSE Middle Cap XRAY DENTSPLY International Inc. NASDAQ RDY Dr Reddys Laboratories Ltd. NYSE CTRX Catamaran Corporation NASDAQ Small Cap ICUI ICU Medical Inc. NASDAQ ACOR Acorda Therapeutics Inc. NASDAQ HAE Haemonetics Corporation NYSE Industrials Large Cap ABB ABB Ltd. NYSE EMR Emerson Electric Company NYSE LMT Lockheed Martin Corporation NYSE ECL Ecolab Inc. NYSE FSLR First Solar Inc. NASDAQ 31 Middle Cap ENR Energizer Holdings Inc. NYSE CTAS Cintas Corporation NASDAQ Small Cap POWR PowerSecure International Inc. NYSE AZZ AZZ inc. NYSE CATM Cardtronics Inc. NASDAQ Oil & Gas Large Cap EPD Enterprise Products Partners LP NYSE BP BP Plc. NYSE XOM Exxon Mobil Corporation NYSE PTR PetroChina Company Ltd. NYSE Middle Cap GEL Genesis Energy LP NYSE NE Noble Corporation NYSE APU AmeriGas Partners LP NYSE Small Cap CIR Circor International Inc. NYSE AREX Approach Resources Inc. NASDAQ PZE Petrobras Energia Participaciones S.A. NYSE Technology Large Cap AAPL Apple Inc. NASDAQ MSFT Microsoft Corporation NASDAQ CSCO Cisco Systems Inc. NASDAQ QCO M Qualcomm Inc. NASDAQ Middle Cap HRS Harris Corporation NYSE ARW Arrow Electronics Inc. NYSE FLIR FLIR Systems Inc. NASDAQ Small Cap IDCC InterDigital Inc. NASDAQ LORL Loral Space and Communications Inc. NASDAQ CRAY Cray Inc. NASDAQ Telecommunicati ons Large Cap T AT&T Inc. NYSE TWC Time Warner Cable Inc. NYSE BT BT Group PLC. NYSE VZ Verizon Communications Inc. NYSE Middle Cap WIN Windstream Corporation NASDAQ 32 FTR Frontier Communications Corporation NASDAQ PT Portugal Telecom SGPS S.A. NYSE Small Cap ALSK Alaska Communications Systems Group Inc. NASDAQ ATNI Atlantic Tele- Network Inc. NASDAQ CNSL Consolidated Communications Holdings Inc. NASDAQ Utilities Large Cap NGG National Grid Plc. NYSE DUK Duke Energy Corporation NYSE NEE NextEra Energy Inc. NYSE SO Southern Company NYSE Middle Cap RGP Regency Energy Partners LP NYSE LNT Alliant Energy Corporation NYSE MDU MDU Resources Group Inc. NYSE Small Cap CWT California Water Service Group NYSE ARTN A Artesian Resources Corporation NASDAQ EDE Empire District Electric Company NYSE Appendix B: Total Result of O, Y , and X o1 o2 o3 o4 Test Basic Materials Large Cap - 0.00322 -0.0977 0.04672 4 0.09944 4 1.03347 8 Middl e Cap 0.01206 6 -0.0069 - 0.03522 0.09402 1.05601 Small Cap 0.09209 2 - 0.16928 0.08611 4 0.17631 9 1.15312 Total 0.02995 8 - 0.09194 0.03395 6 0.12088 1.07613 Consumer Goods Large Cap 0.00653 4 - 0.02792 - 0.02106 0.07333 5 1.02457 6 Middl e Cap 0.04756 5 - 0.04108 0.02405 8 0.02678 2 1.05472 5 33 Small Cap - 0.00799 0.04891 6 0.05501 2 0.02521 6 1.10303 4 Total 0.01448 7 - 0.00882 0.01529 7 0.04493 3 1.05715 8 Consumer Services Large Cap -0.018 - 0.03781 0.01670 3 0.07026 1.02637 5 Middl e Cap 0.12567 - 0.04697 0.00816 1 - 0.00176 1.05747 3 Small Cap 0.05896 0.05284 8 0.00133 7 - 0.01283 1.09912 8 Total 0.04818 8 - 0.01336 0.00953 0.02372 7 1.05753 Financials Large Cap - 0.01736 - 0.03055 0.10518 3 0.02987 9 1.08057 Middl e Cap 0.02768 6 - 0.03623 - 0.01507 0.06710 4 1.03931 4 Small Cap - 0.05824 0.01457 3 - 0.00892 0.10158 2 1.04195 Total - 0.01611 - 0.01872 0.03487 5 0.06255 7 1.05660 7 Health Care Large Cap - 0.00361 -0.0447 0.03633 1 0.03319 7 1.01778 6 Middl e Cap 0.02104 7 - 0.00057 - 0.00579 0.00775 4 1.01975 6 Small Cap 0.01363 5 0.07306 9 - 0.05373 0.00192 7 1.03019 4 Total 0.00896 3 0.00387 1 - 0.00333 0.01618 3 1.0221 Industrials Large Cap - 0.00084 - 0.03266 0.01685 0.04185 5 1.02178 7 Middl e Cap - 0.02018 0.00579 1 0.03681 7 0.05666 4 1.07781 Small Cap 0.04402 2 0.00216 6 0.01320 7 - 0.02135 1.03551 8 Total 0.00681 5 - 0.01068 0.02174 7 0.02733 5 1.04271 3 Oil & Gas Large Cap - 0.02159 - 0.03965 0.01758 4 0.07133 6 1.02221 4 Middl e Cap 0.02227 2 - 0.03091 - 0.00259 0.02781 5 1.01333 2 Small Cap - 0.02296 - 0.11293 0.09647 8 0.13663 1 1.07518 3 Total - 0.00884 - 0.05901 0.03519 9 0.07786 8 1.03544 Technology Large Cap 0.00578 1 - 0.04626 0.03728 0.03101 1.02368 7 Middl e Cap - 0.02131 - 0.06512 0.03286 3 0.10234 1.03980 3 34 Small Cap 0.05446 6 - 0.02975 - 0.02194 0.06545 2 1.05488 Total 0.01225 8 - 0.04697 0.01818 9 0.06274 1 1.03788 Telecommunicatio ns Large Cap - 0.02667 0.00733 7 0.02012 4 0.02840 9 1.02587 3 Middl e Cap - 0.05071 - 0.00768 0.06671 8 0.01334 6 1.01692 5 Small Cap - 0.03504 0.02628 7 0.11746 5 - 0.05284 1.04100 1 Total - 0.03639 0.00851 7 0.06330 4 - 0.00048 1.02772 7 Utilities Large Cap - 0.02091 - 0.00213 0.00975 6 0.02187 2 1.00751 8 Middl e Cap - 0.00598 - 0.00942 - 0.00059 0.03035 1 1.01329 5 Small Cap - 0.02958 0.00665 4 0.00731 5 0.02298 4 1.00598 3 Total - 0.01903 - 0.00168 0.00592 0.02474 9 1.00879 1 y1 y2 y3 y4 Basic Materials Large Cap -3.23158 -4.9359 0.90687 7.560613 Middle Cap 0.300442 -0.0741 -0.44378 0.817431 Small Cap 0.541267 -0.68696 -0.01693 -0.03737 Total -1.04012 -2.20268 0.224536 3.258262 Consumer Goods Large Cap 0.110425 -0.24169 0.133704 0.697563 Middle Cap 0.790658 -0.05039 -0.14281 0.135872 Small Cap 0.208547 0.042029 -0.25582 0.605243 Total 0.343932 -0.09919 -0.06611 0.50136 Consumer Services Large Cap -0.04396 -1.6949 0.845403 1.393457 Middle Cap 0.590684 -0.36665 -0.59532 0.704613 Small Cap 0.351298 1.231819 -2.25448 1.138031 Total 0.265012 -0.41841 -0.51678 1.110176 Financials Large Cap -0.14204 -0.27111 0.636101 0.077054 Middle Cap 0.331108 -0.51463 -0.21171 0.995232 35 Small Cap -0.91346 0.475695 0.230489 0.673944 Total -0.23152 -0.12013 0.260074 0.531575 Health Care Large Cap 0.430463 -0.24757 -0.38992 0.707029 Middle Cap 0.555021 -0.13733 -0.76036 0.942667 Small Cap -0.16145 1.302377 -0.42806 -0.11286 Total 0.290255 0.250486 -0.5125 0.531755 Industrials Large Cap 0.221308 -0.14586 0.108019 0.416535 Middle Cap -0.93944 -0.58262 1.116561 0.872168 Small Cap 0.230311 0.727213 0.118358 -0.47588 Total -0.12421 -0.01497 0.413683 0.2855 Oil & Gas Large Cap 1.142968 -0.54576 -1.20895 1.111738 Middle Cap 0.284625 -0.57417 -0.21613 1.105677 Small Cap -0.25227 -0.55014 0.616734 0.252348 Total 0.466893 -0.5556 -0.3634 0.852103 Technology Large Cap -0.10931 -0.94343 1.009089 0.543652 Middle Cap -0.64122 -0.82229 0.392933 1.403914 Small Cap -0.09316 -0.40119 0.10971 0.717972 Total -0.26404 -0.74442 0.554428 0.854027 Telecommunications Large Cap -0.30613 0.15554 -0.04475 1.09534 Middle Cap -1.62076 -1.25545 3.099677 -0.02347 Small Cap -2.10924 1.268477 4.476224 -3.43546 Total -1.24145 0.066124 2.254871 -0.59954 Utilities Large Cap -1.75002 -3.23502 4.077598 1.607439 Middle Cap 0.038998 0.059039 -0.03198 0.667277 Small Cap 0.123534 -0.52835 -0.09497 1.233124 Total -0.65125 -1.4348 1.592954 1.213096 36 x1 x2 x3 x4 Basic Materials Large Cap 0.065236 -0.23298 0.215632 0.392205 Middle Cap 0.128241 0.041025 0.077438 0.409826 Small Cap 0.1727 -0.28904 0.186234 0.134219 Total 0.116377 -0.16759 0.165354 0.320095 Consumer Goods Large Cap 0.23493 0.003809 0.10341 0.344373 Middle Cap 0.328126 0.11159 0.292415 0.302329 Small Cap 0.130452 0.145224 0.073703 0.177274 Total 0.231545 0.078568 0.151199 0.28163 Consumer Services Large Cap 0.124919 -0.05391 0.150049 0.395556 Middle Cap 0.161967 -0.09983 0.053785 0.127575 Small Cap 0.318226 0.191815 0.2043 0.11265 Total 0.194025 0.006031 0.137445 0.23029 Financials Large Cap 0.069862 -0.04612 0.228808 0.269443 Middle Cap 0.187388 0.021116 0.117979 0.354915 Small Cap -0.09227 0.164651 0.099161 0.315207 Total 0.05648 0.037282 0.156665 0.308814 Health Care Large Cap 0.167231 -0.05012 0.263537 0.273566 Middle Cap 0.315222 0.105321 0.165268 0.30478 Small Cap 0.150889 0.307734 -0.03788 0.159221 Total 0.206726 0.103867 0.143631 0.248627 Industrials Large Cap 0.173977 0.01663 0.272138 0.333637 Middle Cap -0.05747 0.033508 0.19447 0.217431 Small Cap 0.370501 0.244333 0.226435 0.138834 Total 0.1635 0.090004 0.235127 0.240334 Oil & Gas Large Cap -0.00181 -0.04883 0.156102 0.421373 Middle Cap 0.262404 0.098388 0.201979 0.307179 Small Cap -0.12952 -0.2375 0.2513 0.150696 37 Total 0.039139 -0.06126 0.198424 0.305912 Technology Large Cap 0.133411 -0.02863 0.238532 0.276993 Middle Cap 0.007874 -0.14981 0.207256 0.332992 Small Cap 0.215441 -0.05012 0.12417 0.314144 Total 0.120359 -0.07143 0.194841 0.304938 Telecommunications Large Cap 0.159851 0.149161 0.183497 0.339447 Middle Cap -0.25944 -0.052 0.340187 0.149404 Small Cap -0.02692 0.030116 0.327937 0.015539 Total -0.02197 0.053099 0.273836 0.185262 Utilities Large Cap 0.156742 0.244219 0.243676 0.304809 Middle Cap 0.275269 0.152905 0.155948 0.29866 Small Cap -0.01647 0.221331 0.155076 0.358803 Total 0.140336 0.209958 0.190777 0.319163 Appendix C: Matlab Code for Binomial Trees model clc clear n = 3000; ra = -0.00182; r = 0.02; K = 120; s0 = 128.83; t = 65/n; ct = t/252; a = exp(ra*ct); ac = zeros(n,n+1); uc = zeros(n,n+1); x = xlsread('data.xlsx','table'); for i = 1:251 x1(i) = log(x(i)/x(i+1)); end m = mean(x1); 38 stdev = sqrt((1/251)*sum((x1-m).^2)); r1 = stdev/(sqrt(1/252)); u = exp(r1*sqrt(ct)); d = 1/u; p = (a-d)/(u-d); for j=1:n+1 ac(n+1,j)=s0*d^(n-j+1)*u^(j-1)-K; if ac(n+1,j)<0 ac(n+1,j)=0; end end for j=1:n+1 uc(n+1,j)=s0*d^(n-j+1)*u^(j-1)-K; if uc(n+1,j)<0 uc(n+1,j)=0; end end if(ra>r) for i=n:-1:1 for j=1:i ac(i,j)=max((p*ac(i+1,j+1)+(1-p)*ac(i+1,j))*exp(-ra*ct),s0*d^(i-j)*u^(j-1)-K); end end rame=ac(1,1); for i=n:-1:1 for j=1:i uc(i,j)=(p*uc(i+1,j+1)+(1-p)*uc(i+1,j))*exp(-ra*ct); end end reur=uc(1,1); else for i=n:-1:1 for j=1:i ac(i,j)=max((p*ac(i+1,j+1)+(1-p)*ac(i+1,j))*exp(-r*ct),s0*d^(i-j)*u^(j-1)-K); end end rame=ac(1,1); for i=n:-1:1 for j=1:i uc(i,j)=(p*uc(i+1,j+1)+(1-p)*uc(i+1,j))*exp(-r*ct); end end 39 reur=uc(1,1); end T = 65/252; d1 = (log(s0/K)+(r+0.5*r1^2)*T)/(r1*sqrt(T)); d2 = d1-r1*sqrt(T); r5a = s0*normcdf(d1,0,1)-exp(-r*T)*K*normcdf(d2,0,1); r5b = r5a-s0+K*exp(r*T);%price of European put option by bs model delta=normcdf(d1,0,1); vega=exp(-d1^2/2)*s0*sqrt(T)/sqrt(2*pi); gamma=exp(-d1^2/2)/(s0*sqrt(T)*r1)/sqrt(2*pi);
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Asset Metadata
Creator
Zhang, Hao
(author)
Core Title
Improvement of binomial trees model and Black-Scholes model in option pricing
School
College of Letters, Arts and Sciences
Degree
Master of Science
Degree Program
Applied Mathematics
Publication Date
07/24/2014
Defense Date
06/23/2014
Publisher
University of Southern California
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Tag
binomial trees model,Black-Scholes model,OAI-PMH Harvest,option pricing
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Lototsky, Sergey V. (
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), Mancera, Ricardo (
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), Sacker, Robert (
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zh9810@gmail.com,zhan217@usc.edu
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Tags
binomial trees model
Black-Scholes model
option pricing