Options and Economics
harrykrish
SampleAssignment4.xlsxQuestion 1
| ST Range | Payoff for Long Put Option | Payoff for Short Put Option | Net Profit | ||||||||
| ST ≥ 410 | 0 | 0 | -10.45 | ||||||||
| 400 < ST < 410 | 410 - ST | 0 | 410 - ST -10.45 = 399.55 - ST | ||||||||
| ST ≤ 400 | 410 - ST | ST - 400 | 410-400-10.45 = -0.45 | ||||||||
| ST Range | Payoff for Long Call Option | Payoff for Short Call Option | Net Profit | ||||||||
| ST ≥ 210 | ST -200 | 210 -ST | 210-200-34.82 = -24.82 | ||||||||
| 200 < ST < 210 | ST -200 | 0 | ST - 200 - 34.82 = ST -234.82 | ||||||||
| ST ≤ 200 | 0 | 0 | -34.82 | ||||||||
| ST Range | Payoff from Call | Payoff from Put | Total Payoff | Profit | |||||||
| ST ≤ 365 | 0 | 365 -ST | 365 -ST | 365 -ST -34.05= 330.95 - ST | |||||||
| ST > 365 | ST -365 | 0 | ST -365 | ST -365 -34.05 = ST -399.05 | |||||||
| ST Range | Payoff from Call | Payoff from Put | Total Payoff | Profit | |||||||
| ST ≤ 365 | 0 | 365 -ST | 365 -ST | 365 -ST-33.80 = 331.20 - ST | |||||||
| 365 < ST < 370 | 0 | 0 | 0 | 0-33.80 = -33.80 | |||||||
| ST ≥ 370 | ST - 370 | 0 | ST -370 | ST -370 -33.80= ST - 403.80 |
I looked up Netflix (NFLX) options on the 1st of April 2019. The share price at the time I did the assignment was 363.76 USD.
(i) On the above date a put option for April 18th 2019 with a strike price of $400 costs $46.25 and a put option with a strike price of $410 costs $56.70. A Bear spread using puts is created by selling the $400 put and buying the $410 put. This strategy initially costs $10.45. (56.70 - 46.25). Below is the tabular payoff matrix for this strategy.
(ii) On the above date a call option for April 18th 2019 with a strike price of $200 costs $175.0 and a call option with a strike price of $210 costs $140.18. The rule to create a Bull spread is to buy the lower strike price option and sell the higher strike price option. So a Bull spread using calls is created when an investor buys a call with strike price $200 and sells a call with strike price $210. This strategy initially costs $34.82. (175.0 - 140.18). Below is the tabular payoff matrix for this strategy.
(iii) A straddle is a strategy that involves buying both a put and a call option with the same strike price and expiration date. A trader will profit when the price of the security rises or falls from the strike price by more than the total cost of the premium paid. Looking at put and call options for April 18th 2019 with a strike price of $365, the call costs $15.80 and the put costs $18.25. The straddle is created by buying both the put and the call and costs $15.80 + $18.25 = $34.05. The following is the payoff matrix for the above straddle.
(iv) A strangle is a strategy where the investor holds a position in both a call and put with different strike prices, but with the same expiration date. This strategy is profitable only if the underlying asset has a large price movement. Again taking April 18th 2019 as our expiration date we have a call with a strike price of $370 costing $15.55 and a put with a strike price of $365 costing $18.25. The strangle is created by buying both and costs the investor $15.55 + $18.25 = $33.80. The following is the payoff matrix for the above strangle.
Question 2
| VIX Index | Bid | Ask | ||||||||||
| VIX April 13 Call | 1.40 | 2.70 | ||||||||||
| VIX April 14 Call | 0.85 | 2.00 | ||||||||||
| VIX April 15 Call | 0.50 | 1.45 | ||||||||||
| VIX April 16 Call | 0.65 | 1.10 | ||||||||||
| VIX April 17 Call | 0.60 | 0.90 |
**As I did this question on 08/04/2019 I'm taking the view that the VIX will increas by the end of April 2019 rather than march 2019.**
A Long Call If we are forecasting a rise in expected market volatility, then buying a VIX call option may be an appropriate strategy. The VIX index is currently trading at 13.15 and say you believe that this will rise to $17 by the end of the month. In the table below are the call options as of 08/04/2019 for expiry on the 24/04/2019.
Vertical Call Spread Say that you are bullish on the VIX Index and expect it to rise the a vertical call spread would be a possible strtegy. For this strategy we buy a call option with a lower strike price and sell a call option with a higher strike price. The maximum amount of profit we can gain from a vertical call spread is: Maximum Profit = Difference in Strike Prices - Net Cost For example, an investor would buy an April call option with a strike price of $15 costing $1.45 and sell an April call option with a strike price of $17 for $0.60. The initial cost to the investor is $1.45-$0.60 = $0.85. (This information is correct as of 08/04/2019 and is based on an option with expiry on the 24/04/2019.) Now we can calculate our maximum profit using the formula above, Maximum Profit = Difference in Strike Prices - Net Cost Maximum Profit = (17-15) - 0.85 = $1.15. We can also calculate the breakeven amount and the maximum loss from our vertical call spread. Breakeven = Lower Strike Price + Net Cost Breakeven: 15 + $0.85 = $15.85. Maximum Loss = Net Cost Maximum Loss: $0.85. Looking at all the information above we can see that in order for us to make a profit we will need the VIX price to rise above the breakeven price of $15.85. The maximum amount of profit we can reach is capped at $1.15, which is achieved once the VIX Index reaches a price of $17. (i.e. our higher strike price)
When a trader is designing any of the two above strategies with the hope of benefiting form a rising VIX they should consider both cost and risk. A trader will need to account for not just the cost of the option but also different costs involved in the strategies such as trading and commission costs. If a trader wants to make a profit from either of the strategies above he/she will have to ensure that they take all the cost into consideration before adopting the strategy. The trader should also take risk into account. Risk management will help them to cut down losses. It can also help to protect a trader's account from losing all of his or her money.
In his articles Wigglesworth states that volatility has evolved from an academic idea into a risk management tool and it is now something investors can trade, the same way they can trade a bond or a stock. It now even has its own index, known as the VIX. In these articles he discusses the risks involved in using volatility trades. He discusses how some investors are worried about feedback loop, saying there are risks inherent in having a major input into risk-management models. It is believed that there was over $2tn in strategies that are concerned with stock market volatility. It is being described as a cycle because lower volatility is leading to even lower volatility, in a ‘self-perpetuating cycle’. It is causing alarm because even though it can appear to be stabilizing it can very easily turn into chaos. The hazard and worry is that the multi-trillion dollar short volatility trade, could contribute to a vicious cycle of higher volatility and could end up resulting in a hyper-crash. If this happened, then the cycle would break, and volatility could rise exponentially. There would be no limit to how high it could go. Although in early February a rapid volatility increase resulted in the collapse of several inverse VIX ETPs, markets recovered quickly, and measures were taken o limit damage. Regardless there are still concerns as links still exist between volatility as a tradable asset and as a risk-management tool.
If the VIX were to rise to $17 by 24/04/2019, as predicted, then the calls with strike prices of $13, $14 and $15 would be profitable for the investor to exercise. However the investor would not exercise calls with strike prices of $16 or $17 as they would lead to an overall loss.
Question 3
| STOCK PRICE | PROFIT | ||||||
| 170 | 2.94 | ||||||
| 165 | 2.94 | ||||||
| 164 | 1.94 | ||||||
| 163 | 0.94 | ||||||
| 162 | -0.06 | ||||||
| 160 | -2.06 | ||||||
| 155 | -2.06 | ||||||
| ST Range | Payoff for Long Call Option | Payoff for Short Call Option | Net Profit | ||||
| ST ≥ 165 | ST -160 | 165 -ST | 165-160-2.06= 2.94 | ||||
| 160 < ST < 165 | ST -160 | 0 | ST - 160 - 2.06 = ST -162.06 | ||||
| ST ≤ 160 | 0 | 0 | -2.06 | ||||
| STOCK PRICE | PROFIT | ||||||
| 170 | -1.67 | ||||||
| 165 | -1.67 | ||||||
| 163 | 0.33 | ||||||
| 160 | 3.33 | ||||||
| 158 | 1.33 | ||||||
| 155 | -1.67 | ||||||
| 150 | -1.67 | ||||||
| ST Range | Payoff for first long call | Payoff for second long call | Payoff from short calls | Net Profit | |||
| ST ≤ 155 | 0 | 0 | 0 | -1.67 | |||
| 155< ST ≤ 160 | ST -155 | 0 | 0 | ST -155-1.67= ST -156.67 | |||
| 160 < ST < 165 | ST -155 | 0 | -2(ST -160) | 165- ST -1.67= 163.33-ST | |||
| ST ≥ 165 | ST -155 | ST -165 | -2(ST -160) | -1.67 | |||
| STOCK PRICE | PROFIT | ||||||
| 155 | 1.9 | ||||||
| 160 | -3.1 | ||||||
| 162 | -3.1 | ||||||
| 163 | -3.1 | ||||||
| 164 | -3.1 | ||||||
| 165 | -3.1 | ||||||
| 170 | 1.9 | ||||||
| ST Range | Payoff from Call | Payoff from Put | Total Payoff | Profit | |||
| ST ≤ 160 | 0 | 160 -ST | 160 -ST | 160 -ST-3.10= 156.9 - ST | |||
| 160 < ST < 165 | 0 | 0 | 0 | 0 - 3.10 = -3.10 | |||
| ST ≥ 165 | ST - 165 | 0 | ST -165 | ST -165-3.10= ST - 168.1 |
(i) Bullish means that an investor believes that the stock or the overall market will go up. If we are bullish about Visa, then a bull spread using calls would be an appropriate options strategy that should allow us to profit if we are correct but also protects us from large downside risk if we are incorrect. The rule to create a Bull spread using calls is to buy the lower strike price option and sell the higher strike price option. The investor can only lose their initial cost. Looking at options on Visa with expiry on 26/04/2019 we have a call with a strike price of $160 costing $2.87 and a call with a strike price of $165 costing 0.81. To create the bull spread we buy the $160 call option for $2.87 and we sell the $165 call option. The initial cost is $2.87-$0.81 = $2.06. The table below shows the tabular payoff matrix for this strategy. The graph to the right is the profit diagram.
BULL SPREAD FOR VISA
170 165 164 163 162 160 155 2.94 2.94 1.94 0.94 -0.06 -2.06 -2.06
STOCK PRICE
PROFIT
BUTTERFLY SPREAD FOR VISA
170 165 163 160 158 155 150 -1.67 -1.67 0.33 3.33 1.33 -1.67 -1.67
STOCK PRICE
PROFIT
STRANGLE FOR VISA
155 160 162 163 164 165 170 1.9 -3.1 -3.1 -3.1 -3.1 -3.1 1.9
STOCK PRICE
PROFIT
(ii) If we expect prices to remain stable for the period covered above, then a butterfly spread is a speculative option strategy that would allow us to profit. This option strategy combines both bull and bear spread and has a capped profit and fixed risk. When creating a butterfly spread we must use four option contracts with the same maturity date and with three different strike prices. We purchase two calls, one with a high strike price and one with a low strike price. Then we sell two calls which have a strike price halfway between the high and low strike prices of the options that we purchased. So we would purchase one call with a $155 strike price costing $6.60 and one call with a $165 strike price costing $0.81, and we sell two calls with a strike price of $160 for $2.87 each. The initial cost to the investor is $6.60 + $0.81 - 2(2.87) = $1.67. The table below shows the tabular payoff matrix for this butterfly spread. The graph to the right is the profit diagram. (The above data is correct as of 18/04/2019 for options with expiry on 26/04/2019.)
(iii) If, instead we believe that this market will be very volatile in the near future, but we don’t know whether the price will have a big move up or down then a possible speculative option strategy that would allow us to profit from this would be a strangle. A strangle is an options strategy where the investor buys both a call and a put with different strike prices, but with the same expiration date. This is an unlimited profit, low risk strategy and is profitable only if there is a large price movement. (The following data is correct as of 18/04/2019 for options with expiry on 26/04/2019.) We have a call with strike price $165 costing $0.81 and a put with strike price $160 costing $2.29. The initial cost to the investor is $2.29 + $0.81 = $3.10. The table below shows the tabular payoff matrix for this strangle. The graph to the right below is the profit diagram.
At the time of writing (i.e. 18/04/2019) Visa's share price is $160.45. AS I did this question in April I used April 2019 calls and puts data.
Question 4
| Date | Close | Sigma: | 19.8558721276 | STDEV.S(B2:B252) | |||
| 3/15/18 | 199.059998 | 10-Year US government bond: | 2.59% | Risk-free rate | |||
| 3/16/18 | 200.279999 | S0 (initial stock price): | $171.76 | ||||
| 3/19/18 | 194.529999 | Time to maturity (T) | 36 | from 21/02/2019 to 29/03/2019 | |||
| 3/20/18 | 198.949997 | ||||||
| 3/21/18 | 195.300003 | ||||||
| 3/22/18 | 184.649994 | ||||||
| 3/23/18 | 181.199997 | ||||||
| 3/26/18 | 190.5 | ||||||
| 3/27/18 | 181.889999 | ||||||
| 3/28/18 | 178.910004 | ||||||
| 3/29/18 | 183.539993 | ||||||
| 4/2/18 | 177.610001 | ||||||
| 4/3/18 | 174.669998 | ||||||
| 4/4/18 | 172.070007 | ||||||
| 4/5/18 | 172.570007 | ||||||
| 4/6/18 | 167.520004 | ||||||
| 4/9/18 | 169.869995 | ||||||
| 4/10/18 | 177.100006 | ||||||
| 4/11/18 | 175.360001 | ||||||
| 4/12/18 | 175.919998 | ||||||
| 4/13/18 | 172.039993 | S0 (initial stock price): | 171.76 | ||||
| 4/16/18 | 174.699997 | K (Strike Price): | 180 | ||||
| 4/17/18 | 178.699997 | r (risk-free rate): | 2.59% | ||||
| 4/18/18 | 182.679993 | Sigma (σ)(Volatility): | 19.86% | ||||
| 4/19/18 | 181.389999 | T (time to maturity) | 0.098630137 | 36/365 | |||
| 4/20/18 | 179.110001 | ln(S0 / K) = ln(171.76/180) | -0.0468586973 | ||||
| 4/23/18 | 175.570007 | 0.3140543536 | |||||
| 4/24/18 | 173.089996 | ||||||
| 4/25/18 | 170.220001 | d1 | -0.6791451418 | ||||
| 4/26/18 | 173.899994 | d2 | -0.7415163364 | ||||
| 4/27/18 | 177.160004 | ||||||
| 4/30/18 | 178.539993 | ||||||
| 5/1/18 | 179.5 | The Formula for the price of a Call is as follows: | |||||
| 5/2/18 | 181.449997 | ||||||
| 5/3/18 | 182.449997 | ||||||
| 5/4/18 | 188.889999 | ||||||
| 5/7/18 | 195.350006 | ||||||
| 5/8/18 | 196.309998 | N(d1) | 0.2485229511 | NORMSDIST(F30) | |||
| 5/9/18 | 195.429993 | N(d2) | 0.2291902151 | NORMSDIST(F31) | |||
| 5/10/18 | 195.960007 | e-rT | 0.9974487395 | ||||
| 5/11/18 | 194.360001 | ||||||
| 5/14/18 | 198.639999 | ||||||
| 5/15/18 | 196.610001 | Price of Call with $180 strike: | 1.5373136762 | ||||
| 5/16/18 | 198.110001 | ||||||
| 5/17/18 | 196.020004 | ||||||
| 5/18/18 | 195 | ||||||
| 5/21/18 | 197.639999 | ||||||
| 5/22/18 | 195.869995 | ||||||
| 5/23/18 | 196.800003 | ||||||
| 5/24/18 | 197.369995 | ||||||
| 5/25/18 | 199.199997 | ||||||
| 5/29/18 | 198 | ||||||
| 5/30/18 | 197.979996 | ||||||
| 5/31/18 | 198.009995 | ||||||
| 6/1/18 | 204.339996 | ||||||
| 6/4/18 | 208.949997 | ||||||
| 6/5/18 | 208.369995 | ||||||
| 6/6/18 | 208.300003 | S0 (initial stock price): | 171.76 | ||||
| 6/7/18 | 203.619995 | K (Strike Price): | 165 | ||||
| 6/8/18 | 205.070007 | r (risk-free rate): | 2.59% | ||||
| 6/11/18 | 205.699997 | Sigma (σ)(Volatility): | 19.86% | ||||
| 6/12/18 | 209.080002 | T (time to maturity) | 0.098630137 | ||||
| 6/13/18 | 206.619995 | ln(S0 / K) = ln(171.76/165) | 0.0401526797 | ||||
| 6/14/18 | 210.860001 | 0.3140543536 | |||||
| 6/15/18 | 208 | ||||||
| 6/18/18 | 208.570007 | d1 | 0.7159119438 | ||||
| 6/19/18 | 204.429993 | d2 | 0.6535407492 | ||||
| 6/20/18 | 206.229996 | ||||||
| 6/21/18 | 202.210007 | The Formula for the price of a Put is as follows: | |||||
| 6/22/18 | 202.009995 | ||||||
| 6/25/18 | 191.25 | ||||||
| 6/26/18 | 191.419998 | ||||||
| 6/27/18 | 185.020004 | ||||||
| 6/28/18 | 188.380005 | ||||||
| 6/29/18 | 185.529999 | N(-d1) | 0.2370228615 | ||||
| 7/2/18 | 186.360001 | N(-d2) | 0.256703864 | ||||
| 7/3/18 | 184.75 | e-rT | 0.9974487395 | ||||
| 7/5/18 | 186.880005 | ||||||
| 7/6/18 | 192.270004 | ||||||
| 7/9/18 | 192.75 | Price of Put with strike $165: | 1.5370293297 | ||||
| 7/10/18 | 192.550003 | ||||||
| 7/11/18 | 187.419998 | ||||||
| 7/12/18 | 190.169998 | ||||||
| 7/13/18 | 190.039993 | ||||||
| 7/16/18 | 190.350006 | ||||||
| 7/17/18 | 192.660004 | ||||||
| 7/18/18 | 190.789993 | ||||||
| 7/19/18 | 187.339996 | ||||||
| 7/20/18 | 187.25 | ||||||
| 7/23/18 | 187.039993 | ||||||
| 7/24/18 | 189 | ||||||
| 7/25/18 | 197.979996 | ||||||
| 7/26/18 | 194.179993 | ||||||
| 7/27/18 | 189.419998 | ||||||
| 7/30/18 | 184.820007 | ||||||
| 7/31/18 | 187.229996 | ||||||
| 8/1/18 | 185.270004 | ||||||
| 8/2/18 | 182.600006 | ||||||
| 8/3/18 | 180.839996 | ||||||
| 8/6/18 | 178.619995 | ||||||
| 8/7/18 | 179.919998 | ||||||
| 8/8/18 | 177.520004 | ||||||
| 8/9/18 | 177.190002 | ||||||
| 8/10/18 | 180.009995 | ||||||
| 8/13/18 | 177.679993 | ||||||
| 8/14/18 | 172.529999 | ||||||
| 8/15/18 | 169.830002 | ||||||
| 8/16/18 | 171.990005 | ||||||
| 8/17/18 | 172.779999 | ||||||
| 8/20/18 | 176.289993 | ||||||
| 8/21/18 | 177.919998 | ||||||
| 8/22/18 | 177.850006 | ||||||
| 8/23/18 | 172.229996 | ||||||
| 8/24/18 | 174.229996 | ||||||
| 8/27/18 | 180.649994 | ||||||
| 8/28/18 | 178.190002 | ||||||
| 8/29/18 | 178.5 | ||||||
| 8/30/18 | 174.600006 | ||||||
| 8/31/18 | 175.009995 | ||||||
| 9/4/18 | 170.440002 | ||||||
| 9/5/18 | 164.229996 | ||||||
| 9/6/18 | 159.869995 | ||||||
| 9/7/18 | 162.369995 | ||||||
| 9/10/18 | 156.360001 | ||||||
| 9/11/18 | 157.460007 | ||||||
| 9/12/18 | 161.460007 | ||||||
| 9/13/18 | 165.529999 | ||||||
| 9/14/18 | 164.740005 | ||||||
| 9/17/18 | 158.889999 | ||||||
| 9/18/18 | 156.649994 | ||||||
| 9/19/18 | 162.630005 | ||||||
| 9/20/18 | 165.880005 | ||||||
| 9/21/18 | 164.630005 | ||||||
| 9/24/18 | 163.160004 | ||||||
| 9/25/18 | 164.25 | ||||||
| 9/26/18 | 165.399994 | ||||||
| 9/27/18 | 166.320007 | ||||||
| 9/28/18 | 164.759995 | ||||||
| 10/1/18 | 162 | ||||||
| 10/2/18 | 160.229996 | ||||||
| 10/3/18 | 162.369995 | ||||||
| 10/4/18 | 156.130005 | ||||||
| 10/5/18 | 154.630005 | ||||||
| 10/8/18 | 151.139999 | ||||||
| 10/9/18 | 146.940002 | ||||||
| 10/10/18 | 138.289993 | ||||||
| 10/11/18 | 141.899994 | ||||||
| 10/12/18 | 147.289993 | ||||||
| 10/15/18 | 144.160004 | ||||||
| 10/16/18 | 149.600006 | ||||||
| 10/17/18 | 148.139999 | ||||||
| 10/18/18 | 142.020004 | ||||||
| 10/19/18 | 142.929993 | ||||||
| 10/22/18 | 148.800003 | ||||||
| 10/23/18 | 146.649994 | ||||||
| 10/24/18 | 139.610001 | ||||||
| 10/25/18 | 144.600006 | ||||||
| 10/26/18 | 142.869995 | ||||||
| 10/29/18 | 133.380005 | ||||||
| 10/30/18 | 136.330002 | ||||||
| 10/31/18 | 142.279999 | ||||||
| 11/1/18 | 151.25 | ||||||
| 11/2/18 | 147.589996 | ||||||
| 11/5/18 | 144.639999 | ||||||
| 11/6/18 | 147.440002 | ||||||
| 11/7/18 | 152.5 | ||||||
| 11/8/18 | 148.990005 | ||||||
| 11/9/18 | 144.850006 | ||||||
| 11/12/18 | 142.820007 | ||||||
| 11/13/18 | 146.979996 | ||||||
| 11/14/18 | 150.440002 | ||||||
| 11/15/18 | 156.220001 | ||||||
| 11/16/18 | 154.100006 | ||||||
| 11/19/18 | 149.529999 | ||||||
| 11/20/18 | 145.979996 | ||||||
| 11/21/18 | 149.410004 | ||||||
| 11/23/18 | 150.330002 | ||||||
| 11/26/18 | 156.009995 | ||||||
| 11/27/18 | 156.460007 | ||||||
| 11/28/18 | 159.339996 | ||||||
| 11/29/18 | 156.279999 | ||||||
| 11/30/18 | 160.860001 | ||||||
| 12/3/18 | 163.740005 | ||||||
| 12/4/18 | 158.339996 | ||||||
| 12/6/18 | 155.830002 | ||||||
| 12/7/18 | 153.059998 | ||||||
| 12/10/18 | 151.429993 | ||||||
| 12/11/18 | 151.830002 | ||||||
| 12/12/18 | 151.5 | ||||||
| 12/13/18 | 151.479996 | ||||||
| 12/14/18 | 149 | ||||||
| 12/17/18 | 143.979996 | ||||||
| 12/18/18 | 140.820007 | ||||||
| 12/19/18 | 137.139999 | ||||||
| 12/20/18 | 135.110001 | ||||||
| 12/21/18 | 132 | ||||||
| 12/24/18 | 131.889999 | ||||||
| 12/26/18 | 138 | ||||||
| 12/27/18 | 138.449997 | ||||||
| 12/28/18 | 139.089996 | ||||||
| 12/31/18 | 137.070007 | ||||||
| 1/2/19 | 136.699997 | ||||||
| 1/3/19 | 130.600006 | ||||||
| 1/4/19 | 139.75 | ||||||
| 1/7/19 | 143.100006 | ||||||
| 1/8/19 | 146.789993 | ||||||
| 1/9/19 | 151.919998 | ||||||
| 1/10/19 | 151.690002 | ||||||
| 1/11/19 | 151.320007 | ||||||
| 1/14/19 | 149.270004 | ||||||
| 1/15/19 | 150.880005 | ||||||
| 1/16/19 | 154.839996 | ||||||
| 1/17/19 | 155.970001 | ||||||
| 1/18/19 | 157.020004 | ||||||
| 1/22/19 | 152.149994 | ||||||
| 1/23/19 | 152.029999 | ||||||
| 1/24/19 | 155.860001 | ||||||
| 1/25/19 | 159.210007 | ||||||
| 1/28/19 | 158.919998 | ||||||
| 1/29/19 | 156.880005 | ||||||
| 1/30/19 | 166.820007 | ||||||
| 1/31/19 | 168.490005 | ||||||
| 2/1/19 | 167.970001 | ||||||
| 2/4/19 | 166.699997 | ||||||
| 2/5/19 | 171.830002 | ||||||
| 2/6/19 | 171.520004 | ||||||
| 2/7/19 | 166.960007 | ||||||
| 2/8/19 | 167.360001 | ||||||
| 2/11/19 | 167.449997 | ||||||
| 2/12/19 | 168.710007 | ||||||
| 2/13/19 | 169.399994 | ||||||
| 2/14/19 | 168.380005 | ||||||
| 2/15/19 | 166.149994 | ||||||
| 2/19/19 | 170.179993 | ||||||
| 2/20/19 | 170.710007 | ||||||
| 2/21/19 | 171.660004 | ||||||
| 2/22/19 | 176.919998 | ||||||
| 2/25/19 | 183.25 | ||||||
| 2/26/19 | 183.539993 | ||||||
| 2/27/19 | 184.580002 | ||||||
| 2/28/19 | 183.029999 | ||||||
| 3/1/19 | 183.880005 | ||||||
| 3/4/19 | 187.25 | ||||||
| 3/5/19 | 185 | ||||||
| 3/6/19 | 184.169998 | ||||||
| 3/7/19 | 177.320007 | ||||||
| 3/8/19 | 175.029999 | ||||||
| 3/11/19 | 180.410004 | ||||||
| 3/12/19 | 180.630005 | ||||||
| 3/13/19 | 180.699997 | ||||||
| 3/14/19 | 180.360001 |
The data on the left hand side shows the closing price for Alibaba Group Holding Limited (BABA) over a one year span from 15/03/2018 to 15/03/2019. I used this to get an estimate for sigma, which is shown above to be 19.86%. I also used the 10-Year US government bond on the 15/03/2019 as the risk-free rate. Starting from 21/02/2019, with an initial stock price at time zero of $171.76.
(i) We want to calculate the price of March 29 2019 Call options with a $180 strike. First we need to calculate d1 and d2 using the formulas below.
(ii) We want to calculate the price of March 29 2019 Put options with a $165 strike. Using the same formulas as above and the figures below to calculate d1 and d2 with K = 165.
From the above calculations I found the price of March 29 2019 Call options with a $180 strike to be $1.54.
From the above calculations I found the price of March 29 2019 Put options with a $165 strike to be $1.54.
The actual traded price of the above call option on the 15th of March is $1.58 and for the put option is $1.56. The variation in prices could be due to the fact that the Black-Scholes model assumes that the volatility and risk-free rate are known and are constant. As I assumed the volatility to be the historical volatility in daily prices over the past year and the risk-free rate to be the 10-year government bond rate, these valuse might not be 100% accurate and would resut in the variation of the prices of the Put and Call opions shown above.
Question 5
| PARAMETERS | ||||||||||
| Facebook Spot Price | $160.64 | |||||||||
| Strike Price | $170 | |||||||||
| Risk-free rate | 2.59% | |||||||||
| Time to maturity | 57 days | 21/02/2019-18/04/2019 | ||||||||
| T = 57/365 | 0.1561643836 | |||||||||
| Dividend Yield | 0 | |||||||||
| Implied Volatility | ?? | |||||||||
| April 18 Call option | $2.52 | |||||||||
| PARAMETERS | ||||||||||
| Facebook Spot Price | 160.64 | d1 | -0.5450126987 | |||||||
| Strike Price | 170 | d2 | -0.6342040111 | |||||||
| Risk-free rate | 2.59% | N(d1) | 0.292872392 | |||||||
| T = 57/365 | 0.1561643836 | N(d2) | 0.2629738451 | |||||||
| Implied Volatility | 0.2257 | |||||||||
| Call price | 2.5219208493 | |||||||||
| PARAMETERS | ||||||||||
| Facebook Spot Price | 160.64 | d1 | -0.5453647396 | |||||||
| Strike Price | 170 | d2 | -0.6345065683 | |||||||
| Risk-free rate | 2.59% | N(d1) | 0.2927513432 | |||||||
| T = 57/365 | 0.1561643836 | N(d2) | 0.2628751408 | |||||||
| Implied Volatility | 0.2255747806 | |||||||||
| Call price | 2.5191875728 | |||||||||
| Volatility |
Using the above parameters I want to figure out the volatility that was used to price the April 18 Call option. Firstly I will enter an initial guess value for the volatility, and calculate out d1, d2, N(d1), N(d2) and the call price for the initial guess. I made my initial guess for volatility 20% (0.2). I put it into the volatility cell below (highlighted in yellow). I got a call price of $1.97. So I know that the implied volatility for our $2.52 call option will be higher than 20%. From here there are two ways to find the volatility we are looking for, firstly we could use a trial and error basis. Using this method I can see that an implied volatility of 22.57% gives us a Call price of $2.521921.
The other method is to use the goal seek method in excel.To use this I Go to Data>What If Analysis>Goal Seek on my excel sheet. I then set the Call price to 2.52 (cell F36 in my spreadsheet) by changing the implied volatility (cell B35 in this spreadsheet). This tells me that implied volatility of 0.225575 (shown in the blue cell below) gives a Call price of $2.519188 = $2.52. This is shown below.
So using a trial an error method I arrived at a volatility of 22.57%. Using the Goal Seek method on excel I arrived at a value of 22.56%. Then using the CBOE Options Calculator to calculate the implied volatility and the same parameters as above I arrived at a volatility of 22.55%, this can be seen in the screenshot below. We can see that all three value for volatility are extremely close.
Question 6
| Stock Price | Profit | |||||||
| 31.76 | 1 | |||||||
| 32.76 | 0 | |||||||
| 33.5 | -0.74 | |||||||
| 34 | -0.74 | |||||||
| 34.74 | 0 | |||||||
| 35.74 | 1 | |||||||
| Stock Price | Profit | |||||||
| 30.31 | 2 | |||||||
| 32.31 | 0 | |||||||
| 34.00 | -1.69 | |||||||
| 34.85 | 0 | |||||||
| 35.85 | 2 | |||||||
| Stock Price | Profit | |||||||
| 29.81 | 2 | |||||||
| 31.81 | 0 | |||||||
| 33.00 | -1.19 | |||||||
| 34.19 | 0 | |||||||
| 36.19 | 2 |
(iii) A straddle is a strategy that involves buying both a put and a call option with the same strike price and expiration date. A trader will profit when the price of the security rises or falls from the strike price by more than the total cost of the premium paid. Looking at put and call options for April 5th 2019 with a strike price of $33.00, the call costs $0.99 and the put costs $0.20. The initial cost to the trader is $1.19.
Strangle
31.76 32.76 33.5 34 34.74 35.74 1 0 -0.74 -0.74 0 1
Stock Price
Profit
Combination of 2 Calls and 1 Put
30.31 32.31 34 34.85 35.85 2 0 -1.69 0 2
Stock Price
Profit
Straddle
29.81 31.81 33 34.19 36.19 2 0 -1.19 0 2
Stock Price
Profit
(i) A strangle is a strategy where by the investor holds a position in both a call and put with different strike prices, but with the same expiration date. This strategy is profitable only if the underlying asset has a large price movement. Looking at put and call options for Twitter shares on April 5th 2019. The call costs $0.38 and has a strike price of $34 the put costs $0.36 and has a strike price of $33.50. This strategy initially costs the investor $0.74.
I looked up Twitter (TWTR) options on the 2nd of April 2019. The share price at the time I did the assignment was 33.76 USD.
(ii) A combination of 2 calls and 1 put with the same strike price and expiration is known as a strap. Looking at put and call options for Twitter shares on April 5th 2019. We have a two call costing $0.38 each that have a strike price of $34 and the put costs $0.55 and has a strike price of $34. The initial cost is: 2(0.38)+0.55 = $1.69.
Question 7
| Call Option Value | ||||||||||
| Implied Volatility |
Using the CBOE Options Calculator to find the theoretical BSM price of an April 5th European call option for Coca Cola (KO) with a strike of $51, I found the theoretical call price to be $0.00. This can be seen in the picture below.The price of the option is essentially zero because the stock prcie is extremely unlikely to hit the strike price of $51 in the next two days as it is currently only $46.57. In this case holding the option would be worthless.
The actual price on Yahoo finance of an April 5th European call option for Coca Cola (KO) with a strike of $50 on the day I did this assignment (03/04/2019) was $0.01. (As there was no information available on yahoo finance for an option with a strike price of $51). (Changing the strike price to $50 in the calculator still gives us a result of $0.00). The market price of $0.01 is slightly different to the result from the CBOE options calculator. This could be due to the volatility rate that the CBOE calculator is using.
Delta is the ratio of the change in the price of a stock option to the change in the price of the underlying stock. It is an important parameter used in pricing & hedging of options It usually is represented as a number between minus one and one, and this number indicates how much the value of an option should change when the price of the underlying stock rises by one dollar. In regard to call options, a delta of 0.6 means that for every $1 the underlying stock increases, the call option will increase by $0.60. On the other hand, the delta of a put option would be negative, because as the underlying security increases, the value of the option will decrease. Gamma measures the responsiveness of delta to unit changes in the value of the underlying asset. It is valuable in helping traders predict changes in the delta of an option. Gamma is always positive for both puts and calls and is larger for at-the-money options and is lower for both in- and out-of-the-money options. Vega is the sensitivity of an option value to a change in volatility. Each different option has its own Vega. Even though Vega affects calls and puts similarly, it appears to affect calls a slight bit more than puts. The effect of volatility changes is larger for at-the-money options than it is for the in-the-money options and the out-of-the-money options.
**I found the price for an April 5th call option as I did the quesion on the 3rd April and therefore couldnt get data for March 29 call option.**
In the picure below it shows the implied volatility based on the market price of this call option as of the CBOE calculator. The implied volatility is 55.8%.
