On 12 November 2020, the market prices of call option on Johnson & Johnson stock is as follows. The expiration date of these option is 16 April 2021. Calls for April 16, 2021 Contract Name Last Trade Date Strike Last Price Bid Ask Change % Change Volume Open Interest Implied Volatility JNJ210416C00095000 2020-11-09 2:04PM EST 95.00 52.55 0.00 0.00 0.00 1 0 0.00% JNJ210416C00100000 2020-11-09 2:04PM EST 100.00 47.59 0.00 0.00 0.00 1 0 0.00% JNJ210416C00105000 2020-11-02 11:31AM EST 105.00 33.90 0.00 0.00 0.00 1 0 0.00% JNJ210416C00110000 2020-10-04 11:00PM EST 110.00 38.80 29.30 30.50 0.00 8 0.00% JNJ210416C00115000 2020-10-28 9:26AM EST 115.00 26.00 0.00 0.00 0.00 4 0 0.00% JNJ210416C00120000 2020-11-09 10:07AM EST 120.00 29.35 0.00 0.00 0.00 1 0 0.00% JNJ210416C00125000 2020-11-10 2:07PM EST 125.00 24.15 0.00 0.00 0.00 2 0 0.00% JNJ210416C00130000 2020-11-09 1:08PM EST 130.00 20.00 0.00 0.00 0.00 2 0 0.00% JNJ210416C00135000 2020-11-11 1:01PM EST 135.00 16.04 0.00 0.00 0.00 1 0 0.00% JNJ210416C00140000 2020-11-11 3:33PM EST 140.00 12.20 0.00 0.00 0.00 6 0 0.00% JNJ210416C00145000 2020-11-11 3:57PM EST 145.00 9.20 0.00 0.00 0.00 6 0 0.00% JNJ210416C00150000 2020-11-11 3:57PM EST 150.00 6.69 0.00 0.00 0.00 21 0 0.78% JNJ210416C00155000 2020-11-11 2:25PM EST 155.00 4.60 0.00 0.00 0.00 28 0 1.56% JNJ210416C00160000 2020-11-11 3:30PM EST 160.00 3.20 0.00 0.00 0.00 103 0 3.13% JNJ210416C00165000 2020-11-10 1:24PM EST 165.00 2.25 0.00 0.00 0.00 9 0 3.13% JNJ210416C00170000 2020-11-10 9:30AM EST 170.00 1.41 0.00 0.00 0.00 2 0 6.25% (source: Yahoo finance] We find that the market price of call with strike price of $130.00 is $20.00. Assuming that this option is European, we want to compare the market price to the theoretical price that we can obtain from a binomial tree. The current stock price is $147.80 and the risk-free interest rate is 0.5% per annum. In constructing the binomial tree, we make the llowing assumptions: We approximate that the time to expiration is five months. We use a 10-step binomial tree. (a) To find u and d in the binomial tree, we need the volatility of the stock return. Use historical data on Johnson & Johnson stock price in " JNJ.xlsx" on Canvas. There, you can find 1-year long observations of daily stock prices. What is the standard deviation of one-day return? What is the volatility (the standard deviation of one-year return)? (b) What are u and d in the binomial tree? Use the estimated volatility. (c) Construct the table showing the possible stock prices and option payoff at the option maturity and their probabilities as follows: Number of down Stock price at maturity Probability Option Payoff Number of up 10 0 9 1 (d) What is the option price? (e) Compare the theoretical price to the market price. Are they close to each other? What can we do to better estimate the option price? Date Adj Close Date Adj Close Date Adj Close Date Adj Close Date Adj Close 12/11/2019 127.8762 24/12/2019 143.0549 7/2/2020 148.8975 23/3/2020 109.6451 5/5/2020 147.4892 13/11/2019 127.7984 26/12/2019 142.9569 10/2/2020 148.8681 24/3/2020 117.577 6/5/2020 146.0882 14/11/2019 127.4966 27/12/2019 142.8785 11/2/2020 148.9956 25/3/2020 117.794 7/5/2020 145.6048 15/11/2019 131.3713 30/12/2019 142.4374 12/2/2020 148.1133 26/3/2020 124.8676 8/5/2020 146.6999 18/11/2019 131.2642 31/12/2019 142.9961 13/2/2020 147.133 27/3/2020 121.5034 11/5/2020 147.1044 19/11/2019 131.2545 2/1/2020 143.0942 14/2/2020 147.1722 30/3/2020 131.2209 12/5/2020 145.1609 20/11/2019 132.3449 3/1/2020 141.4375 18/2/2020 146.2017 31/3/2020 129.3662 13/5/2020 145.151 21/11/2019 132.8316 6/1/2020 141.261 19/2/2020 146.0056 1/4/2020 127.0774 14/5/2020 145.6542 22/11/2019 134.4185 7/1/2020 142.1237 20/2/2020 145.4469 2/4/2020 131.3591 15/5/2020 148.4165 25/11/2019 134.4773 8/1/2020 142.1041 21/2/2020 146.9761 3/4/2020 132.3653 18/5/2020 148.4954 26/11/2019 134.4675 9/1/2020 142.5256 24/2/2020 143.9474 6/4/2020 137.8801 19/5/2020 147.0156 27/11/2019 135.0361 10/1/2020 142.2021 25/2/2020 142.7044 7/4/2020 135.6308 20/5/2020 145.6936 29/11/2019 134.7812 13/1/2020 142.8295 26/2/2020 141.7474 8/4/2020 141.3331 21/5/2020 144.7367 2/12/2019 134.6538 14/1/2020 143.6333 27/2/2020 137.2488 9/4/2020 139.3304 22/5/2020 143.4155 3/12/2019 134.4577 15/1/2020 144.1137 28/2/2020 132.6712 13/4/2020 137.89 26/5/2020 143.6042 4/12/2019 136.634 16/1/2020 145.2802 2/3/2020 138.1367 14/4/2020 144.0658 27/5/2020 143.932 5/12/2019 136.8104 17/1/2020 146.2311 3/3/2020 133.7663 15/4/2020 145.6739 28/5/2020 145.9983 6/12/2019 137.6143 21/1/2020 146.3291 4/3/2020 141.5501 16/4/2020 147.6569 29/5/2020 147.7665 9/12/2019 137.7319 22/1/2020 145.3293 5/3/2020 140.0999 17/4/2020 149.9753 1/6/2020 146.2168 10/12/2019 137.232 23/1/2020 145.6037 6/3/2020 140.1196 20/4/2020 149.63 2/6/2020 147.2698 11/12/2019 138.2025 24/1/2020 145.3979 9/3/2020 134.6048 21/4/2020 147.6667 3/6/2020 147.6672 12/12/2019 138.5554 27/1/2020 145.8194 10/3/2020 139.7349 22/4/2020 150.9322 4/6/2020 145.7599 13/12/2019 138.5946 28/1/2020 146.5546 11/3/2020 130.0272 23/4/2020 153.4183 5/6/2020 146.3261 16/12/2019 138.9965 29/1/2020 147.5741 12/3/2020 123.7232 24/4/2020 152.7771 8/6/2020 145.7996 17/12/2019 140.7316 30/1/2020 147.3977 13/3/2020 132.4837 27/4/2020 152.2147 9/6/2020 144.995 18/12/2019 140.3689 31/1/2020 145.937 16/3/2020 125.42 28/4/2020 149.3537 10/6/2020 146.8228 19/12/2019 142.4864 3/2/2020 147.2114 17/3/2020 134.7528 29/4/2020 148.2192 11/6/2020 139.9386 20/12/2019 143.1824 4/2/2020 148.6133 18/3/2020 133.1447 30/4/2020 148.0219 12/6/2020 141.2101 23/12/2019 143.5549 5/2/2020 150.9562 19/3/2020 125.3411 1/5/2020 146.2954 15/6/2020 140.3161 24/12/2019 143.0549 6/2/2020 150.5052 20/3/2020 118.2774 4/5/2020 146.2757 16/6/2020 143.5049 Date Adj Close Date Adj Close Date Adj Close Date Adj Close 17/6/2020 143.0678 30/7/2020 145.8691 11/9/2020 147.78 23/10/2020 145.24 18/6/2020 142.4618 31/7/2020 144.7963 14/9/2020 148.35 26/10/2020 143.97 19/6/2020 142.879 3/8/2020 146.3758 15/9/2020 148.89 27/10/2020 143.15 22/6/2020 142.4419 4/8/2020 146.2466 16/9/2020 148.4 28/10/2020 138.36 23/6/2020 141.9155 5/8/2020 147.4188 17/9/2020 147.17 29/10/2020 137.19 24/6/2020 138.8956 6/8/2020 146.5744 18/9/2020 149.18 30/10/2020 137.11 25/6/2020 138.7465 7/8/2020 147.6175 21/9/2020 145.1 2/11/2020 138.69 26/6/2020 136.8988 10/8/2020 147.0513 22/9/2020 144.21 3/11/2020 138.5 29/6/2020 138.1207 11/8/2020 145.9983 23/9/2020 144.44 4/11/2020 139.4 30/6/2020 139.7002 12/8/2020 148.6705 24/9/2020 144.67 5/11/2020 139.76 1/7/2020 139.4519 13/8/2020 147.0215 25/9/2020 145.66 6/11/2020 142.25 2/7/2020 140.0379 14/8/2020 147.2599 28/9/2020 147.11 9/11/2020 146.08 6/7/2020 142.0347 17/8/2020 148.0049 29/9/2020 147.06 10/11/2020 148.28 7/7/2020 141.9055 18/8/2020 149.0976 30/9/2020 148.88 11/11/2020 147.8 8/7/2020 142.3227 19/8/2020 149.3957 1/10/2020 147.32 9/7/2020 141.5479 20/8/2020 150.4189 2/10/2020 146.24 10/7/2020 141.4287 21/8/2020 151.75 5/10/2020 148.23 13/7/2020 144.2499 24/8/2020 152.15 6/10/2020 146.26 14/7/2020 146.942 25/8/2020 152.06 7/10/2020 147.88 15/7/2020 147.2797 26/8/2020 152.3 8/10/2020 148.89 16/7/2020 148.2632 27/8/2020 152.98 9/10/2020 150.97 17/7/2020 148.3625 28/8/2020 153.64 12/10/2020 151.84 20/7/2020 148.6109 31/8/2020 153.41 13/10/2020 148.36 21/7/2020 148.75 1/9/2020 151.52 14/10/2020 148.1 22/7/2020 149.0182 2/9/2020 153.83 15/10/2020 147.19 23/7/2020 148.6208 3/9/2020 149.55 16/10/2020 148.1 24/7/2020 147.1407 4/9/2020 148.59 19/10/2020 144.32 27/7/2020 146.2069 8/9/2020 147.26 20/10/2020 144.55 28/7/2020 145.8592 9/9/2020 149.7 21/10/2020 143.93 29/7/2020 145.5711 10/9/2020 146.91 22/10/2020 145.08 On 12 November 2020, the market prices of call option on Johnson & Johnson stock is as follows. The expiration date of these option is 16 April 2021. Calls for April 16, 2021 Contract Name Last Trade Date Strike Last Price Bid Ask Change % Change Volume Open Interest Implied Volatility JNJ210416C00095000 2020-11-09 2:04PM EST 95.00 52.55 0.00 0.00 0.00 1 0 0.00% JNJ210416C00100000 2020-11-09 2:04PM EST 100.00 47.59 0.00 0.00 0.00 1 0 0.00% JNJ210416C00105000 2020-11-02 11:31AM EST 105.00 33.90 0.00 0.00 0.00 1 0 0.00% JNJ210416C00110000 2020-10-04 11:00PM EST 110.00 38.80 29.30 30.50 0.00 8 0.00% JNJ210416C00115000 2020-10-28 9:26AM EST 115.00 26.00 0.00 0.00 0.00 4 0 0.00% JNJ210416C00120000 2020-11-09 10:07AM EST 120.00 29.35 0.00 0.00 0.00 1 0 0.00% JNJ210416C00125000 2020-11-10 2:07PM EST 125.00 24.15 0.00 0.00 0.00 2 0 0.00% JNJ210416C00130000 2020-11-09 1:08PM EST 130.00 20.00 0.00 0.00 0.00 2 0 0.00% JNJ210416C00135000 2020-11-11 1:01PM EST 135.00 16.04 0.00 0.00 0.00 1 0 0.00% JNJ210416C00140000 2020-11-11 3:33PM EST 140.00 12.20 0.00 0.00 0.00 6 0 0.00% JNJ210416C00145000 2020-11-11 3:57PM EST 145.00 9.20 0.00 0.00 0.00 6 0 0.00% JNJ210416C00150000 2020-11-11 3:57PM EST 150.00 6.69 0.00 0.00 0.00 21 0 0.78% JNJ210416C00155000 2020-11-11 2:25PM EST 155.00 4.60 0.00 0.00 0.00 28 0 1.56% JNJ210416C00160000 2020-11-11 3:30PM EST 160.00 3.20 0.00 0.00 0.00 103 0 3.13% JNJ210416C00165000 2020-11-10 1:24PM EST 165.00 2.25 0.00 0.00 0.00 9 0 3.13% JNJ210416C00170000 2020-11-10 9:30AM EST 170.00 1.41 0.00 0.00 0.00 2 0 6.25% (source: Yahoo finance] We find that the market price of call with strike price of $130.00 is $20.00. Assuming that this option is European, we want to compare the market price to the theoretical price that we can obtain from a binomial tree. The current stock price is $147.80 and the risk-free interest rate is 0.5% per annum. In constructing the binomial tree, we make the llowing assumptions: We approximate that the time to expiration is five months. We use a 10-step binomial tree. (a) To find u and d in the binomial tree, we need the volatility of the stock return. Use historical data on Johnson & Johnson stock price in " JNJ.xlsx" on Canvas. There, you can find 1-year long observations of daily stock prices. What is the standard deviation of one-day return? What is the volatility (the standard deviation of one-year return)? (b) What are u and d in the binomial tree? Use the estimated volatility. (c) Construct the table showing the possible stock prices and option payoff at the option maturity and their probabilities as follows: Number of down Stock price at maturity Probability Option Payoff Number of up 10 0 9 1 (d) What is the option price? (e) Compare the theoretical price to the market price. Are they close to each other? What can we do to better estimate the option price? Date Adj Close Date Adj Close Date Adj Close Date Adj Close Date Adj Close 12/11/2019 127.8762 24/12/2019 143.0549 7/2/2020 148.8975 23/3/2020 109.6451 5/5/2020 147.4892 13/11/2019 127.7984 26/12/2019 142.9569 10/2/2020 148.8681 24/3/2020 117.577 6/5/2020 146.0882 14/11/2019 127.4966 27/12/2019 142.8785 11/2/2020 148.9956 25/3/2020 117.794 7/5/2020 145.6048 15/11/2019 131.3713 30/12/2019 142.4374 12/2/2020 148.1133 26/3/2020 124.8676 8/5/2020 146.6999 18/11/2019 131.2642 31/12/2019 142.9961 13/2/2020 147.133 27/3/2020 121.5034 11/5/2020 147.1044 19/11/2019 131.2545 2/1/2020 143.0942 14/2/2020 147.1722 30/3/2020 131.2209 12/5/2020 145.1609 20/11/2019 132.3449 3/1/2020 141.4375 18/2/2020 146.2017 31/3/2020 129.3662 13/5/2020 145.151 21/11/2019 132.8316 6/1/2020 141.261 19/2/2020 146.0056 1/4/2020 127.0774 14/5/2020 145.6542 22/11/2019 134.4185 7/1/2020 142.1237 20/2/2020 145.4469 2/4/2020 131.3591 15/5/2020 148.4165 25/11/2019 134.4773 8/1/2020 142.1041 21/2/2020 146.9761 3/4/2020 132.3653 18/5/2020 148.4954 26/11/2019 134.4675 9/1/2020 142.5256 24/2/2020 143.9474 6/4/2020 137.8801 19/5/2020 147.0156 27/11/2019 135.0361 10/1/2020 142.2021 25/2/2020 142.7044 7/4/2020 135.6308 20/5/2020 145.6936 29/11/2019 134.7812 13/1/2020 142.8295 26/2/2020 141.7474 8/4/2020 141.3331 21/5/2020 144.7367 2/12/2019 134.6538 14/1/2020 143.6333 27/2/2020 137.2488 9/4/2020 139.3304 22/5/2020 143.4155 3/12/2019 134.4577 15/1/2020 144.1137 28/2/2020 132.6712 13/4/2020 137.89 26/5/2020 143.6042 4/12/2019 136.634 16/1/2020 145.2802 2/3/2020 138.1367 14/4/2020 144.0658 27/5/2020 143.932 5/12/2019 136.8104 17/1/2020 146.2311 3/3/2020 133.7663 15/4/2020 145.6739 28/5/2020 145.9983 6/12/2019 137.6143 21/1/2020 146.3291 4/3/2020 141.5501 16/4/2020 147.6569 29/5/2020 147.7665 9/12/2019 137.7319 22/1/2020 145.3293 5/3/2020 140.0999 17/4/2020 149.9753 1/6/2020 146.2168 10/12/2019 137.232 23/1/2020 145.6037 6/3/2020 140.1196 20/4/2020 149.63 2/6/2020 147.2698 11/12/2019 138.2025 24/1/2020 145.3979 9/3/2020 134.6048 21/4/2020 147.6667 3/6/2020 147.6672 12/12/2019 138.5554 27/1/2020 145.8194 10/3/2020 139.7349 22/4/2020 150.9322 4/6/2020 145.7599 13/12/2019 138.5946 28/1/2020 146.5546 11/3/2020 130.0272 23/4/2020 153.4183 5/6/2020 146.3261 16/12/2019 138.9965 29/1/2020 147.5741 12/3/2020 123.7232 24/4/2020 152.7771 8/6/2020 145.7996 17/12/2019 140.7316 30/1/2020 147.3977 13/3/2020 132.4837 27/4/2020 152.2147 9/6/2020 144.995 18/12/2019 140.3689 31/1/2020 145.937 16/3/2020 125.42 28/4/2020 149.3537 10/6/2020 146.8228 19/12/2019 142.4864 3/2/2020 147.2114 17/3/2020 134.7528 29/4/2020 148.2192 11/6/2020 139.9386 20/12/2019 143.1824 4/2/2020 148.6133 18/3/2020 133.1447 30/4/2020 148.0219 12/6/2020 141.2101 23/12/2019 143.5549 5/2/2020 150.9562 19/3/2020 125.3411 1/5/2020 146.2954 15/6/2020 140.3161 24/12/2019 143.0549 6/2/2020 150.5052 20/3/2020 118.2774 4/5/2020 146.2757 16/6/2020 143.5049 Date Adj Close Date Adj Close Date Adj Close Date Adj Close 17/6/2020 143.0678 30/7/2020 145.8691 11/9/2020 147.78 23/10/2020 145.24 18/6/2020 142.4618 31/7/2020 144.7963 14/9/2020 148.35 26/10/2020 143.97 19/6/2020 142.879 3/8/2020 146.3758 15/9/2020 148.89 27/10/2020 143.15 22/6/2020 142.4419 4/8/2020 146.2466 16/9/2020 148.4 28/10/2020 138.36 23/6/2020 141.9155 5/8/2020 147.4188 17/9/2020 147.17 29/10/2020 137.19 24/6/2020 138.8956 6/8/2020 146.5744 18/9/2020 149.18 30/10/2020 137.11 25/6/2020 138.7465 7/8/2020 147.6175 21/9/2020 145.1 2/11/2020 138.69 26/6/2020 136.8988 10/8/2020 147.0513 22/9/2020 144.21 3/11/2020 138.5 29/6/2020 138.1207 11/8/2020 145.9983 23/9/2020 144.44 4/11/2020 139.4 30/6/2020 139.7002 12/8/2020 148.6705 24/9/2020 144.67 5/11/2020 139.76 1/7/2020 139.4519 13/8/2020 147.0215 25/9/2020 145.66 6/11/2020 142.25 2/7/2020 140.0379 14/8/2020 147.2599 28/9/2020 147.11 9/11/2020 146.08 6/7/2020 142.0347 17/8/2020 148.0049 29/9/2020 147.06 10/11/2020 148.28 7/7/2020 141.9055 18/8/2020 149.0976 30/9/2020 148.88 11/11/2020 147.8 8/7/2020 142.3227 19/8/2020 149.3957 1/10/2020 147.32 9/7/2020 141.5479 20/8/2020 150.4189 2/10/2020 146.24 10/7/2020 141.4287 21/8/2020 151.75 5/10/2020 148.23 13/7/2020 144.2499 24/8/2020 152.15 6/10/2020 146.26 14/7/2020 146.942 25/8/2020 152.06 7/10/2020 147.88 15/7/2020 147.2797 26/8/2020 152.3 8/10/2020 148.89 16/7/2020 148.2632 27/8/2020 152.98 9/10/2020 150.97 17/7/2020 148.3625 28/8/2020 153.64 12/10/2020 151.84 20/7/2020 148.6109 31/8/2020 153.41 13/10/2020 148.36 21/7/2020 148.75 1/9/2020 151.52 14/10/2020 148.1 22/7/2020 149.0182 2/9/2020 153.83 15/10/2020 147.19 23/7/2020 148.6208 3/9/2020 149.55 16/10/2020 148.1 24/7/2020 147.1407 4/9/2020 148.59 19/10/2020 144.32 27/7/2020 146.2069 8/9/2020 147.26 20/10/2020 144.55 28/7/2020 145.8592 9/9/2020 149.7 21/10/2020 143.93 29/7/2020 145.5711 10/9/2020 146.91 22/10/2020 145.08