Options and Economics

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Question 1

4th of April with closing price 367.33
a)	Bear spread using Puts
	Stock Price	Payoff	Profit
	St ≥ 370	0	-3.48	
	362.50 ≤ St < 370	370 − St	362.50 − St
	St < 362.50	7.5	4.02
b)	Bull spread using Calls
	Stock Price	Payoff	Profit
	St ≥ 370	0	31.4	
	362.50 ≤ St < 370	St – 370	St − 362.50
	St < 362.50	-7.5	-23.9
c)	Straddle
	Stock Price	Payoff	Profit
	St > 362.50	St − 362.50	St −328.60	
	St ≤ 362.50	362.50 − St	396.40 − St
	This shows that the straddle will lead to a loss if the final stock price is between 328.6 and 396.40
d)	Strangle
	Range of stock price	Payoff from call	Payoff from put	Total payoff	Net payoff
	St ≤ 362.5	0	362.5 − St	362.5 − St	334 − St	
	362.5 < St < 370	0	0	0	-28.5
	St ≥ 370	St – 370	0	St – 370	St – 398.5

An option is a financial derivative that represents a contract sold by one party to another party. The contract offers the buyer the right to call (buy) or put (sell) a financial asset at a strike price (previously decided price) on a specific date. Buying a Call - You have the right to buy a security at a predetermined price. Selling a Call - You have an obligation to deliver the security at a predetermined price to the option buyer. Buying a Put - You have the right to sell a security at a predetermined price. Selling a Put - You have an obligation to buy the security at a predetermined price to the option buyer. Taking the closing price of 367.33 on the 4th April we can create the tabular payoff matrix. The strike prices for 18th April are 362.50 and 370. A bear spread is an option strategy, which aims to maximize profit when the price of the underlying security declines. The put bear spread can be looked at as a limited profit and limited risk options strategy that can be used when the options trader is moderately bearish on the security. The following figure shows the bear spread using puts: A bear spread is created by selling the 362.50 put option and buying 370. The prices are 14.85 and 18.33 respectively. This strategy will have the initial cost of -3.48.

A bull spread is an option strategy designed to profit from a moderate rise in the price of the security. Here, the lower strike price is bought and the higher strike price is sold, both with the same expiry date. The following figure shows the bull spread using calls: A bull spread is created by buying the 362.50 call option and selling the 370. The prices are 17.75 and 13.65 respectively. This strategy gives rise to an initial cash inflow of 31.40.

A straddle is an option strategy in which the investor holds onto a date, paying both premiums. This allows for making a profit regardless of whether the price of the security goes up or down. Here a large movement is the strike price is expected but the direction is unsure of. It combines the benefit of a call and a put. The following figure shows a straddle: A straddle is created by buying both call and put. A call with the strike price 362.50 costs 17.75 while the put costs 16.15. This strategy costs 33.90.

A strangle is a strategy where the investor holds a position in both a call and a put with different strike prices but with same maturity and underlying asset. The call strike price is higher than the put strike price. This is used when there is expected to be a large movement in the price in the near future but the direction of the movement is unsure of. The following figure shows the strangle: A strangle is created by buying a put option with the strike price of 362.50 for 14.85 and buy a call with strike price 370 for 13.65. The total cost of setting up the strangle is 28.5

Question 2

The CBOE Volatility Index (VIX) is a measure of expected price fluctuations in the S&P 500 Index options over the next 30 days. The VIX, often termed as the "fear index," is calculated in real time by the Chicago Board Options Exchange (CBOE). It is a mathematical measure of how much the market thinks the SPX will fluctuate in the next 12 months. On the 22nd of Febuary the VIX index was trading at 14.21. Hence, the index is expected to move by (14.21%/ sqrt12) = 4.10%
Since there is a belief that the VIX will rise by the end of April 2019, the two strategies are a long VIX call and VIX vertical spread. The question stated to use March 2019 calls and puts, however I choose to use April calls and puts as I am doing this question in April
a) Long VIX Call is a strategy used when an investor forecasts an increase in the expected market volatility. Thus purchasing a call for less than the current price would result in a profit (if VIX is increasing) but also allowing the investor to not purchase should there be a fall in the VIX. In this case the index is at 14.21 and an increase in the VIX is forecasted. The following table shows the ranging strike prices from 10 - 14 for April 24th.
										
CALLS	Last	Change	Vol	Bid	Ask	Strike	PUTS	Last	Change	Vol	Bid	Ask
quote	3	-0.35	2	2.9	3.35	10	quote	0	0	0	0	0.02
quote	0	0	0	2.14	2.85	10.5	quote	0	0	0	0	0.03
quote	2	0.22	5	1.65	2.35	11	quote	0.02	-0.01	15	0.02	0.03
quote	1.75	-0.1	13	1.19	1.87	11.5	quote	0.05	0.02	64	0.02	0.09
quote	1.11	-0.29	170	1.05	1.44	12	quote	0.13	0.03	201	0.1	0.14
						12.09	Current
quote	0.8	-0.25	74	0.75	0.95	12.5	quote	0.25	0	62	0.25	0.5
quote	0.55	-0.18	101	0.5	0.8	13	quote	0.52	0.14	30	0.5	0.82
quote	0.35	-0.4	12	0.3	0.61	13.5	quote	0.81	0.06	1	0.68	1.2
quote	0.41	-0.09	398	0.2	0.45	14	quote	1.35	0.17	13	1.05	1.61
quote	0.2	0	124	0.14	0.38	14.5	quote	1.74	0.54	3	1.45	2.05
quote	0.16	-0.09	102	0.11	0.2	15	quote	2.3	0.26	354	2.15	2.51
quote	0.2	0	211	0.06	0.15	16	quote	3.16	0.36	3	2.8	3.46
quote	0.07	-0.08	95	0.05	0.09	17	quote	4.08	0.27	4	3.75	4.42
i) VIX vertical spread is a strategy that can be used when an expectation of an increase in VIX is formed. A higher volatility means that there are larger swings in the option prices. Hence, a good way to trade is with a bullish outlook where you buy a call on one date and sell on another at a higher strike price. Assuming that the VIX increases to 15 in the future we can sell it at 17. The spread is created by buying the 15 call option and selling the 17. The prices are 0.16 and 0.07 respectively. The matrix can be formed as follows:
Stock Price	Long C	Short C	Net Profit
St ≥ 17	St - 15	17 – St	2
15 < St < 17	St - 15	0	St - 15
St ≤ 17	0	0	-0.09
ii) A trader may benefit from an increase in volatility by using the Straddle strategy where, an investor purchases a call option and put option on the same underlying with the same strike price and with the same maturity. For my example I am choosing the call and put strike price to be 15. The price for the call is 0.16 and the price for the put is 2.3. This strategy enables investor to profit from underlying price change. Hence investor expects volatility and he can take advantage of the leverage.
Straddle:
Stock Price	Payoff	Profit
St >15	St − 15	St −12.54
St ≤ 15	15 − St	17.46 − St
This shows that the straddle will lead to a loss if the final stock price is between 12.54 and 17.46
Convergence-Divergence Relationships: Traders should keep a real time VIX on their market screens at all times, comparing the indicator’s trend with price action on the most popular index futures contracts. Convergence-divergence relationships between these instruments will help in trade planning and risk management. Charting The VIX: The VIX daily chart displays vertical spikes that reflect periods of high stress, induced by economic, political or environmental catalysts. Traders wuold observe these levels when trying to interpret patterns. Trading Instruments: VIX futures offer the purest exposure to the indicator’s ups and downs but equity derivatives have gained a strong following with the retail trading crowd in recent years.
Wigglesworth articles highlight a number of risks involved in using volatility trades. As a trader you should be aware of potential losses, complexities, costs, time decay. It is made clear to us that with higher risk, you would expect a higher return. For example, stocks are more volatile than bonds, and therefore you would expect a higher return from them. Markowitz points out that it is best not to 'put all your eggs in one basket' and have a varied portfolio. An investor portfolio can come down to two things - the expected return on the portfolio, and its variance. One of the many reasons that investors choose to trade options is due to the flexibility and versatility they offer, and the wide range of strategies that can be used. In particular, there are a number of strategies that can be used to either limit the risk of taking a position or reduce the upfront costs of taking a position. Linked to the liquidity of some options is the costs involved in trading them. The price of an options contract is quoted on the exchanges giving a bid price and an ask price. Creating an options spread involves entering two or more positions on different options that are based on the same underlying security. There are very good reasons for creating these spreads, but the fact is that taking multiple positions effectively on a single trade does result in higher commissions. The article continues to point out the risks involved when looking at volatility. Investors will often look at the price within the past 30 days, how JP Morgans VaR model spread across the finance industry as it was a popular tool for looking at the history of the price. When look at the time until expirtion of an option, all options have some kind of time value factored in to them, and typically the longer they have until expiration the higher that time value is. Therefore, any options that you own will always be losing some of their value as time goes on. Of course, this doesn’t mean that they always go down in value, but time decay can negatively impact the value of any options that you hold on to.

Question 3

Share price on 10th April
	Bullish investors believe stocks are going up. If the investor is bullish about the share then Strangle is the best strategy to undertake so as to be protected from large downside risk and earn profit if the expectations are correct. A strangle is a strategy where the investor holds a position in both a call and a put with different strike prices but with same maturity and underlying asset. The call strike price is higher than the put strike price. This is used when there is expected to be a large movement in the price in the near future but the direction of the movement is unsure of. The payoff matrix for the share price on 10th april is as follows:
i)
	Stock price	157.57
	Strike Price for call	160
	Price for call	0.15
	Strike price for put	155
	Price for put	0.28
	Initial cost	0.43
	Range of stock price	Payoff from call	Payoff from put	Total payoff	Net payoff
	St ≤ 155	0	155 − St	155 − St	154.57 − St
	155 < St < 160	0	0	0	-0.43
	St ≥ 160	St – 160	0	St – 160	St – 160.43
ii)	If the expectation is for the prices to remain stable then the best option strategy to be adopted is a butterfly spread. A butterfly spread is a neutral option strategy combining bull and bear. It uses four option contracts with the same expiration date but three different strike prices.The trader sells two option contracts at the middle strike price and one option contract at a higher strike price and one at a lower. The pay off matrix for the share price on the 10th April is as follows:
	Buy call with strike price	155	160
	Price of call	2.75	0.51
	Sell 2 calls with	157
	Price of the	0.91
	Initial Investment	1.44
	Stock Price	Profit
	St < 155	− 1.44
	155 < St < 157	(St - 155 )– 1.44
	157 < St < 160	(160− St) – 1.44
	St > 160	− 1.44
	There is a loss borne if the price moves either ways
iii)	If there is an expectation for the market to be very volatile in the near future, you think there will be a change in price without being sure of the direction then the best strategy is a straddle. A straddle is an option strategy in which the investor holds onto a date, paying both premiums. This allows for making a profit regardless of whether the price of the security goes up or down. Here a large movement is the strike price is expected but the direction is unsure of. It combines the benefit of a call and a put. The Payoff matrix for 10th April share prics is as follows:
	Stock price	157.57
	Strike Price for call	155
	Price for call	2.75
	Strike price for put	155
	Price for put	0.28
	Initial cost	3.03
	Stock Price	Payoff	Profit
	St > 155	St − 155	St − 151.97
	St ≤ 155	155 − St	158.03 − St
	This shows that the straddle will lead to a loss if the final stock price is between 151.97 and 158.03. Thus a jump of 3.03 euros in each direction would be negative in order to secure a profit.

155

160

Profit

157.57

Strangle

Butterfly

Straddle

Question 4

Febuary 21st			20th March (the call and put prices for 29th march)
Standard Deviation	0.412		Standard Deviation	0.412
Stock Price	171.76		Stock Price	179.11
Stock Price Date	2/21/19		Stock Price Date	3/20/19				Date	AdjClose	Log return
								Feb 20, 2019	171
Exercise Price for Call	180		Exercise Price for Call	180				Feb 19, 2019	166.98	-0.0237895117
Exercise Price for Put	165		Exercise Price for Put	165		Standard Deviation	0.025951313	Feb 15, 2019	168.61	0.009714311
Exercise Price Date	3/29/19		Exercise Price Date	3/29/19		Annualised Historical Value	0.411964323	Feb 14, 2019	167.64	-0.0057695327
								Feb 13, 2019	169.91	0.013450062
Maturity (in days)	36		Maturity (in days)	9				Feb 12, 2019	169.6	-0.0018261617
Maturity (in years)	0.143		Maturity (in years)	0.036				Feb 11, 2019	168.85	-0.0044319765
								Feb 08, 2019	163.83	-0.030181442
Dividend Yield	0		Dividend Yield	0				Feb 07, 2019	169.16	0.0320157078
Risk Free Rate	0.0256		Risk Free Rate	0.0256				Feb 06, 2019	171.86	0.0158351793
								Feb 05, 2019	168.55	-0.0194477502
								Feb 04, 2019	166.32	-0.0133187981
								Feb 01, 2019	168	0.0100503359
	Strike Price 180	Strike Price 165		Strike Price 180	Strike Price 165			Jan 31, 2019	167.8	-0.0011911854
d1	-0.1996	0.3592	d1	-0.0130	1.1046			Jan 30, 2019	161.29	-0.0395688071
d2	-0.3553	0.2035	d2	-0.0909	1.0268			Jan 29, 2019	159.04	-0.014048244
N(d1)	0.4209	0.6403	N(d1)	0.4948	0.8653			Jan 28, 2019	157.8	-0.0078273345
N(d2)	0.3612	0.5806	N(d2)	0.4638	0.8477			Jan 25, 2019	158.91	0.0070095958
								Jan 24, 2019	151.47	-0.0479504187
Call Price	7.52	14.52	Call Price	5.22	15.24			Jan 23, 2019	154.65	0.0207769136
								Jan 22, 2019	154.4	-0.0016178615
Put Price	15.10	7.16	Put Price	5.94	0.98			Jan 18, 2019	158.45	0.0258924485
								Jan 17, 2019	152.11	-0.0408351428
								Jan 16, 2019	152.81	0.0045913763
								Jan 15, 2019	150.68	-0.0140369368
								Jan 14, 2019	148.5	-0.0145734246
		The Black Schole model gives us a lower value for the call and a higher value for put as compared to the Yahoo page.This difference can arise as the Black-Schole model takes a constant risk free ineterest rate but in the actual market the inetrest rates can change rapidly in certain periods.						Jan 11, 2019	151.83	0.0221765157
								Jan 10, 2019	149.81	-0.0133936494
								Jan 09, 2019	149.89	0.0005338672
								Jan 08, 2019	145	-0.0331679493
								Jan 07, 2019	140.55	-0.0311704451
								Jan 04, 2019	134.26	-0.0457850788
								Jan 03, 2019	134.27	0.0000744796
								Jan 02, 2019	134.13	-0.0010432192
								Dec 31, 2018	141.83	0.0558196784
								Dec 28, 2018	139.2	-0.0187174094
								Dec 27, 2018	135.05	-0.0302666677
								Dec 26, 2018	132.87	-0.0162738736
								Dec 24, 2018	130	-0.0218367561
								Dec 21, 2018	137.08	0.0530302466
								Dec 20, 2018	135.83	-0.0091605932
								Dec 19, 2018	141.04	0.0376394343
								Dec 18, 2018	144.28	0.0227123179
								Dec 17, 2018	146.5	0.0152695724
								Dec 14, 2018	147.71	0.0082254636
								Dec 13, 2018	153.05	0.0355137733
								Dec 12, 2018	155.24	0.0142076411
								Dec 11, 2018	155.26	0.0001288245
								Dec 10, 2018	150.39	-0.031869211
								Dec 07, 2018	155.4	0.032770518
								Dec 06, 2018	153	-0.0155645165
								Dec 04, 2018	164.88	0.0747800152
								Dec 03, 2018	168.64	0.0225483288
								Nov 30, 2018	157.9	-0.0658043441
								Nov 29, 2018	158.08	0.0011393127
								Nov 28, 2018	159.01	0.0058658593
								Nov 27, 2018	154.64	-0.0278672584
								Nov 26, 2018	153.21	-0.0092903056
								Nov 23, 2018	147.3	-0.0393382059
								Nov 21, 2018	149.06	0.011877586
								Nov 20, 2018	144.48	-0.0312078198
								Nov 19, 2018	152.02	0.0508710015
								Nov 16, 2018	155.04	0.019671057
								Nov 15, 2018	152.9	-0.0138990352
								Nov 14, 2018	150.21	-0.0177497979
								Nov 13, 2018	144.83	-0.0364736742
								Nov 12, 2018	145.01	0.0012420647
								Nov 09, 2018	145.57	0.003854365
								Nov 08, 2018	150.99	0.0365565389
								Nov 07, 2018	150.77	-0.0014581127
								Nov 06, 2018	145.87	-0.0330396827
								Nov 05, 2018	146.22	0.0023965228
								Nov 02, 2018	152.56	0.0424456244
								Nov 01, 2018	144.98	-0.0509621594
								Oct 31, 2018	141.35	-0.0253567177
								Oct 30, 2018	132.28	-0.066318196
								Oct 29, 2018	142.42	0.0738595504
								Oct 26, 2018	139	-0.0243065054
								Oct 25, 2018	142.5	0.0248680666
								Oct 24, 2018	145.18	0.0186323521
								Oct 23, 2018	143.22	-0.0135924423
								Oct 22, 2018	148.99	0.03949728
								Oct 19, 2018	145.34	-0.0248033644
								Oct 18, 2018	145.85	0.0035028711
								Oct 17, 2018	150.68	0.0325796865
								Oct 16, 2018	145.71	-0.0335400378
								Oct 15, 2018	144.77	-0.0064720689
								Oct 12, 2018	148.62	0.0262464366
								Oct 11, 2018	135.53	-0.0921996947
								Oct 10, 2018	142.5	0.0501489817
								Oct 09, 2018	147.97	0.0376675508
								Oct 08, 2018	150.2	0.0149581888
								Oct 05, 2018	156.14	0.0387853014
								Oct 04, 2018	160.06	0.0247957042
								Oct 03, 2018	163.15	0.0191212781
								Oct 02, 2018	159.79	-0.0208095699
								Oct 01, 2018	165.92	0.0376452913
								Sep 28, 2018	164.9	-0.0061665149
								Sep 27, 2018	166.39	0.0089952009
								Sep 26, 2018	165.52	-0.005242397
								Sep 25, 2018	164.05	-0.008920774
								Sep 24, 2018	162.81	-0.0075873827
								Sep 21, 2018	169.46	0.040033034
								Sep 20, 2018	166.89	-0.015281998
								Sep 19, 2018	158.82	-0.0495634273
								Sep 18, 2018	156.88	-0.0122903036
								Sep 17, 2018	161.5	0.0290239608
								Sep 14, 2018	167.88	0.0387442958
								Sep 13, 2018	165.41	-0.0148221982
								Sep 12, 2018	158.2	-0.0445671849
								Sep 11, 2018	153.18	-0.0322463549
								Sep 10, 2018	158.59	0.034708555
								Sep 07, 2018	159.95	0.0085390109
								Sep 06, 2018	164.16	0.0259802956
								Sep 05, 2018	167.48	0.0200223792
								Sep 04, 2018	173.5	0.0353136582
								Aug 31, 2018	173.11	-0.0022503688
								Aug 30, 2018	177.33	0.0240851729
								Aug 29, 2018	179.35	0.0113268005
								Aug 28, 2018	182.15	0.0154913195
								Aug 27, 2018	177.1	-0.0281159787
								Aug 24, 2018	175	-0.0119285709
								Aug 23, 2018	184.97	0.0554076758
								Aug 22, 2018	178.15	-0.0375677577
								Aug 21, 2018	177.63	-0.0029231568
								Aug 20, 2018	175.22	-0.013660408
								Aug 17, 2018	172.52	-0.0155291555
								Aug 16, 2018	172.33	-0.0011019285
								Aug 15, 2018	167.11	-0.0307589651
								Aug 14, 2018	175.14	0.0469333759
								Aug 13, 2018	179.65	0.0254248595
								Aug 10, 2018	175.57	-0.0229726898
								Aug 09, 2018	179.31	0.0210783277
								Aug 08, 2018	180	0.0038406994
								Aug 07, 2018	180.29	0.0016098147
								Aug 06, 2018	180.7	0.002271532
								Aug 03, 2018	184.15	0.0189124453
								Aug 02, 2018	181.54	-0.0142746278
								Aug 01, 2018	186	0.0242706586
								Jul 31, 2018	186.4	0.0021482285
								Jul 30, 2018	190.22	0.0202863948
								Jul 27, 2018	196.1	0.0304434361
								Jul 26, 2018	193.21	-0.0148470529
								Jul 25, 2018	190.61	-0.0135482245
								Jul 24, 2018	190.19	-0.0022058832
								Jul 23, 2018	187.18	-0.0159528518
								Jul 20, 2018	189.49	0.012265532
								Jul 19, 2018	188.68	-0.0042837943
								Jul 18, 2018	192.45	0.0197839213
								Jul 17, 2018	188.65	-0.0199429333
								Jul 16, 2018	189.57	0.004864903
								Jul 13, 2018	191.61	0.0107037068
								Jul 12, 2018	190.77	-0.0043935423
								Jul 11, 2018	188.6	-0.0114401437
								Jul 10, 2018	192.89	0.022491708
								Jul 09, 2018	194.45	0.0080549824
								Jul 06, 2018	186.01	-0.0443746249
								Jul 05, 2018	187.17	0.0062168591
								Jul 03, 2018	187.88	0.0037861664
								Jul 02, 2018	181.66	-0.0336666531
								Jun 29, 2018	185.36	0.0201630721
								Jun 28, 2018	183.2	-0.0117214279
								Jun 27, 2018	193.46	0.0544923205
								Jun 26, 2018	193.34	-0.0006204757
								Jun 25, 2018	197.85	0.0230588708
								Jun 22, 2018	203.38	0.0275669825
								Jun 21, 2018	205.84	0.0120230176
								Jun 20, 2018	205.05	-0.0038453161
								Jun 19, 2018	203.53	-0.0074404377
								Jun 18, 2018	205.16	0.0079767484
								Jun 15, 2018	207.49	0.0112929832
								Jun 14, 2018	207.72	0.0011078732
								Jun 13, 2018	209.44	0.0082462832
								Jun 12, 2018	206.95	-0.011960084
								Jun 11, 2018	206.5	-0.0021768058
								Jun 08, 2018	201.11	-0.0264483904
								Jun 07, 2018	207.46	0.031086528
								Jun 06, 2018	209.86	0.0115020917
								Jun 05, 2018	209.95	0.0004287654
								Jun 04, 2018	205.12	-0.0232742334
								Jun 01, 2018	199.5	-0.0277809374
								May 31, 2018	198	-0.0075472056
								May 30, 2018	199.7	0.0085492097
								May 29, 2018	197.94	-0.008852286
								May 25, 2018	197.57	-0.0018710025
								May 24, 2018	198.12	0.0027799558
								May 23, 2018	193.88	-0.0216334967
								May 22, 2018	198.3	0.0225416245
								May 21, 2018	197.8	-0.0025246163
								May 18, 2018	196.43	-0.0069502854
								May 17, 2018	196	-0.0021914745
								May 16, 2018	196.75	0.0038192281
								May 15, 2018	195.23	-0.0077555366
								May 14, 2018	195.9	0.0034259743
								May 11, 2018	196.4	0.002549071
								May 10, 2018	196.3	-0.0005092946
								May 09, 2018	195.84	-0.002346102
								May 08, 2018	194.2	-0.0084094435
								May 07, 2018	190.41	-0.0197089139
								May 04, 2018	180.4	-0.0540030344
								May 03, 2018	183.5	0.0170380599
								May 02, 2018	180.8	-0.0148232195
								May 01, 2018	177.58	-0.0179702364
								Apr 30, 2018	178.09	0.0028678289
								Apr 27, 2018	177.11	-0.005518032
								Apr 26, 2018	173.25	-0.0220353704
								Apr 25, 2018	170.52	-0.0158830462
								Apr 24, 2018	177.63	0.0408501433
								Apr 23, 2018	178.63	0.0056138922
								Apr 20, 2018	179.36	0.0040783319
								Apr 19, 2018	183.26	0.0215109502
								Apr 18, 2018	178.9	-0.024078919
								Apr 17, 2018	174.83	-0.0230129173
								Apr 16, 2018	172.01	-0.0162614586
								Apr 13, 2018	176.72	0.0270139445
								Apr 12, 2018	175.93	-0.0044803705
								Apr 11, 2018	176.48	0.0031213669
								Apr 10, 2018	175.1	-0.0078503162
								Apr 09, 2018	169.75	-0.0310304729
								Apr 06, 2018	169.84	0.000530051
								Apr 05, 2018	175.48	0.0326682589
								Apr 04, 2018	166.88	-0.050250085
								Apr 03, 2018	179.26	0.0715622747
								Apr 02, 2018	182.81	0.0196100962
								Mar 29, 2018	180.88	-0.0106135341
								Mar 28, 2018	180.73	-0.0008296231
								Mar 27, 2018	192.24	0.0617403866
								Mar 26, 2018	187.89	-0.0228879062
								Mar 23, 2018	186.85	-0.0055505293
								Mar 22, 2018	190.75	0.0206575142
								Mar 21, 2018	198.8	0.0413356241
								Mar 20, 2018	194.95	-0.0195561788
								Mar 19, 2018	198	0.0155239153
								Mar 16, 2018	198.4	0.0020181642
								Mar 15, 2018	198.44	0.0002015926
								Mar 14, 2018	190.29	-0.0419375631
								Mar 13, 2018	193.88	0.0186901867
								Mar 12, 2018	192.3	-0.0081827584
								Mar 09, 2018	189.64	-0.0139291146
								Mar 08, 2018	189.05	-0.0031160077
								Mar 07, 2018	184.37	-0.0250669223
								Mar 06, 2018	185.19	0.004437717
								Mar 05, 2018	179.41	-0.0317086356
								Mar 02, 2018	178.01	-0.0078339609
								Mar 01, 2018	186.18	0.0448742192
								Feb 28, 2018	187.25	0.0057306747
								Feb 27, 2018	192.59	0.0281189545
								Feb 26, 2018	194.46	0.0096629094
								Feb 23, 2018	190.18	-0.0222554943
								Feb 22, 2018	190.2	0.000105158
								Feb 21, 2018	189.37	-0.0043733768

Question5

Stock price	160.64
Stock price date	2/21/18
Strike price	170
Strike price date	4/18/18
closing price	2.52
Expiration in days	56
Expiration in years	0.2222222222
Dividend Yield	0
Risk free rate	0.0256
Implied standard deviation	0.1863104335	(Goal Seek method)
Assumed standard deviation	0.7	(at random)
d1	-0.5361285073
d2	-0.6239560879
N(d1)	0.2959348697
			The implied volatility calculated by the Black-Schole's Method is lower (18.63) than the one calculated using the CBOE Options Calclulator. This differential can arise because the implied volatility is that value of the volatility which when put in an option pricing model will return a theoretical value equal to the current market price of the option however, the CBOE calculator measures the expectation of market volatility using the S&P 500 options as an index.
N(d2)	0.266328212
Call Price	2.5200191483

Question 6

Trading strategies on 20th April:
i) Strangle
Price	34.35
Strike price of Put	32
Price for Put	0.03
Strike price for call	35
Price of call	0.09
Initial Cost	0.12
Range of stock price	Payoff from call	Payoff from put	Total payoff	Net payoff
St ≤ 32	0	32 − St	32 − St	31.88 − St
32 < St < 35	0	0	0	-0.12
St ≥ 35	St – 35	0	St – 35	St – 35.12
ii) Two calls & one put (Strap)
Strap options offer unlimited profit potential on upward price movement and limited profit potential on downward price movement. In order to construct a strap we buy 2 ATM call options and buy 1 ATM put option.
Strike price of call	34
Price for call	0.62
Strike Price of Put	34
Price for put	0.1
Initial cost	0.72
Range of stock price	Payoff from call	Payoff from put	Total payoff	Net payoff
St ≤ 34	0	34 − St	34 − St	33.28 − St
St ≥ 34	St – 68	0	St – 68	St – 68.72
iii) Straddle
Strike price of call	34.5
Price for call	0.62
Strike Price of Put	34.5
Price for put	0.1
Initial cost	0.72
Stock Price	Payoff	Profit
St > 34.50	St − 34.50	St −33.78
St ≤ 34.50	34.50 − St	35.22 − St
This shows that the straddle will lead to a loss if the final stock price is between 33.78 and 35.22

Strap

Profit

34.35

Strangle

Straddle

Question 7

Using the option Calculator:				Call	Put
Style:	European		Symbol	KO 190426C00048000	KO 190426P00048000
Price	47.48		Option Value	0.3927	0.8894
Strike	48		Delta	0.3785	-0.6215
Exp date	April 26, 2019		Gamma	0.2504	0.2504
Days to expiration	7		Theta	-0.0425	-0.0392
Volatility %	23.1		Vega	0.025	0.025
Int Rate %	2.4874		Rho	0.0034	-0.0058
Dividends Date	03/14/19
Dividends Amount	0.4
Frequency	Quarterly
Implied Volatility	Option Price	Volatility %
Call	0.41	23.8
On the 19th of April, the share price of Coca Cola is 47.48. The call for the strike price 51 is not trading in the market as it is indeed out-of-the-money. For that reason I decided to use a strike price of 48. The call price was 0.41. The theoritical BSM price is 0.3927 with volatility 23.1%, the implied volatility was 23.80% for a price of 0.41. As can be seen, implied volatility is higher than normal volatility which will lead to higher option premiums for both calls and puts.
Delta: Delta is one of the four major risk measures used by option traders. It measures the degree to which an option is exposed to shifts in the price of the underlying asset (i.e. stock) or commodity (i.e. futures contract). Values range from 1.0 to –1.0 (or 100 to –100, depending on the convention employed) Gamma: Measures the rate of change of delta, it is therefore the first derivative of delta and is used when trying to gauge the price movement of an option, relative to the amount it is in or out of the money. It is an significant measure of the convexity of a derivative's value, in relation to the underlying. Vega: This is the measurement of an option's price sensitivity to changes in the volatility of the underlying asset. Vega represents the amount that an option contract's price changes in reaction to a 1% change in the implied volatility of the underlying asset.