On December 31, 2021, Shopifys stock closed at $174.169. On the same day, you are asked to value an American call option on the stock with an exercise price of $175 and three months remaining to maturity. The risk-free rate is 5% per annum, compounded semi-annually. Shopify has not paid any dividends at all since its founding. a) The excel file SHOP.TO.csv has the daily closing prices of Shopifys stock in the 2021 calendar year. Using these daily closing prices to estimate Shopifys stock return volatility. b) Estimate the market price of the above American call option on Shopifys stock on December 31, 2021, based on the closing stock price of $174.169. c) Repeat b) except that the daily high price of $178.140 is assumed to be the stock price. d) Repeat b) except that the daily low price of $174.077 is assumed to be the stock price.

FILE #;

Date. Open High. Low Close 1/4/2021 145.25 145.306 137.868 139.619 1/5/2021 140.099 141.757 138.382 141.256 1/6/2021 139.325 140.643 136.857 137.605 1/7/2021 141.041 147.376 140.229 147.09 1/8/2021 149.122 151.672 146.526 151.672 1/11/2021 150.001 155.85 149 150.241 1/12/2021 152.443 153.899 148.9 151.266 1/13/2021 152.009 152.65 149.055 152.356 1/14/2021 156.021 156.022 147.943 148.868 1/15/2021 150.538 151.715 146.516 149.095 1/18/2021 147.001 153.671 147 151.712 1/19/2021 151.184 151.628 146.4 149.439 1/20/2021 149.516 153.566 149.274 151.739 1/21/2021 152.6 153.299 149.121 151.085 1/22/2021 150.7 152.775 149.707 152.254 1/25/2021 153.114 158.647 149 153.664 1/26/2021 153.386 154.369 147 147.212 1/27/2021 146.963 146.963 138.5 139.77 1/28/2021 142.635 145.524 139.86 143.403 1/29/2021 142.849 143.47 138.493 139.429 2/1/2021 142.199 147.659 141 147.245 2/2/2021 149.354 158.53 149.122 158.132 2/3/2021 157.701 159.215 153.32 156.218 2/4/2021 158.624 164.5 158.235 161.446 2/5/2021 162.699 165.7 160.554 164.385 2/8/2021 166.655 171.794 165.5 171.794 2/9/2021 173.921 184.546 173.118 182.299 2/10/2021 182.548 190.058 179.424 185.734 2/11/2021 186.4 187.436 182.862 185.801 2/12/2021 185.951 187.5 182.5 184.72 2/16/2021 186.719 189.229 181.796 185.603 2/17/2021 174.495 185.3 171.95 180.418 2/18/2021 176.093 179.685 175 175.56 2/19/2021 175.714 181.5 173.836 181.058 2/22/2021 177.021 178.616 171.829 174.226 2/23/2021 160.083 166.3 152.787 163.726 2/24/2021 164.55 165.8 158.898 163.244 2/25/2021 161.639 164.738 153.884 157.249 2/26/2021 159.978 164.873 156.8 164.873 3/1/2021 166.574 168.458 162.5 165.541 3/2/2021 166.72 166.72 158.494 165.379 3/3/2021 162.968 165.039 154.104 154.503 3/4/2021 151.966 154.229 141.318 145.599 3/5/2021 145.951 145.951 129.643 143.18 3/8/2021 142.434 147.338 135.77 136.651 3/9/2021 141.175 145.1 139.356 143.106 3/10/2021 148.009 148.009 136.448 137.822 3/11/2021 141.409 145.275 137.296 143.153 3/12/2021 140.513 142.858 137.263 142.291 3/15/2021 142.291 144.614 138.8 144.538 3/16/2021 147.526 148.32 140 141.785 3/17/2021 140 146.269 137.2 145.154 3/18/2021 142.699 143.122 138.649 139.827 3/19/2021 139.767 142.323 138.6 139.998 3/22/2021 139.998 145.285 139.882 145.054 3/23/2021 145.609 147.012 143.622 146.168 3/24/2021 147.541 149 138.61 139.046 3/25/2021 137.779 138.968 132.843 133.752 3/26/2021 132.436 136.344 126.745 133.4 3/29/2021 134.399 135.4 129.581 130.72 3/30/2021 129.52 135.5 128.812 132.456 3/31/2021 135.299 140.582 134.437 138.743 4/1/2021 143.9 148.131 142.636 144.818 4/5/2021 147.4 147.4 142.26 143.438 4/6/2021 144.679 148 143.515 145.34 4/7/2021 144.9 148.156 144.1 146.212 4/8/2021 148.4 153.97 148.4 153.684 4/9/2021 152.081 154.689 149.5 153.703 4/12/2021 153.021 153.677 149.2 150.852 4/13/2021 152.679 156.5 152.679 155.557 4/14/2021 156.4 156.931 147.215 147.606 4/15/2021 150.009 153.21 148.6 151.178 4/16/2021 151.532 152.5 148.65 151.034 4/19/2021 149.695 150.205 143.228 143.228 4/20/2021 142.424 145.01 139.681 141.608 4/21/2021 141.607 143.816 140.435 142.599 4/22/2021 143.991 144.542 136.596 136.672 4/23/2021 136.678 138.344 136.006 137.373 4/26/2021 137.377 144.376 137.377 144.155 4/27/2021 145.104 145.104 140.825 143.045 4/28/2021 150.592 161 147.776 158.947 4/29/2021 159.5 159.821 150.9 151.433 4/30/2021 149 150.155 144.425 145.09 5/3/2021 146.639 146.639 137.5 137.5 5/4/2021 137 140.778 132.531 140.566 5/5/2021 141.105 141.829 136.461 137.605 5/6/2021 136.001 137.015 130.291 132.449 5/7/2021 135.948 137.299 133.193 134.536 5/10/2021 133.861 133.861 128.616 130.328 5/11/2021 125.22 135.529 124.855 134.289 5/12/2021 130.99 133.429 128 131.549 5/13/2021 133.996 134.536 125.174 127.499 5/14/2021 128.5 131.77 126.427 131.447 5/17/2021 130.993 134.287 129.5 132.26 5/18/2021 132.997 138.842 132.145 136.565 5/19/2021 132.523 142 132.523 141.922 5/20/2021 142.581 147.598 141.653 146.434 5/21/2021 147.399 150.5015 143.8 147.378 5/25/2021 152.574 154.2 148.484 149.668 5/26/2021 151.465 153.493 149.2 149.94 5/27/2021 148.999 151.577 146.779 149.282 5/28/2021 151.02 153.632 149.958 150.242 5/31/2021 150.9 150.9 147.9 148.032 6/1/2021 151.159 153.75 149.514 151.072 6/2/2021 152.2 152.3 146.864 147.947 6/3/2021 147.518 148.741 144.742 144.896 6/4/2021 145.704 148.016 145.271 145.715 6/7/2021 146.001 148.785 143.485 148.294 6/8/2021 149.633 151.38 146.5 148.828 6/9/2021 148.704 149.815 147.166 147.202 6/10/2021 147.218 149.559 145 149.325 6/11/2021 149.335 151.89 149.228 151.808 6/14/2021 151.795 159.1 151.622 158.583 6/15/2021 159.755 164.198 156.744 161.211 6/16/2021 161.649 168.5 160.919 167.047 6/17/2021 167.1 180.4715 166.698 179.001 6/18/2021 178.557 184.645 177.5 182.037 6/21/2021 182.951 192 180.1 182.37 6/22/2021 183.222 187.5 181.62 185.036 6/23/2021 185.242 187.995 183.688 184.809 6/24/2021 186 188.121 181.726 182.284 6/25/2021 182.853 182.853 175.684 181.242 6/28/2021 183.112 188 182.5 184.467 6/29/2021 185.579 187.384 182.72 183.877 6/30/2021 182.801 184.382 180.46 181.287 7/2/2021 181.296 185.01 178.682 180.825 7/5/2021 181.283 181.9 180.318 181.418 7/6/2021 181.501 191.841 181.501 191.636 7/7/2021 193.55 197.499 188.628 189.866 7/8/2021 185.021 188.06 182.149 183.803 7/9/2021 183.801 186.424 182.19 186.297 7/12/2021 188.355 191.305 184.084 186.224 7/13/2021 186.999 191.858 186.694 189.986 7/14/2021 191.191 193.032 182.463 183.004 7/15/2021 185.64 185.844 179.105 181.479 7/16/2021 182.164 184 179.982 181.963 7/19/2021 179.923 188.274 177.736 187.512 7/20/2021 188.134 194.388 186.22 193.494 7/21/2021 193.003 197.287 191.965 197.201 7/22/2021 197.499 201.435 197.201 200.351 7/23/2021 202.201 206.979 200.945 206.845 7/26/2021 207.303 207.588 198.005 198.772 7/27/2021 199.495 201.445 190.447 195.83 7/28/2021 198.078 198.8 188.704 193.066 7/29/2021 190.386 193.845 187.829 189.938 7/30/2021 186.7 190.128 185.688 187.3 8/3/2021 193.8 195.634 188.321 191.008 8/4/2021 191.78 195.5 190.327 195.231 8/5/2021 195.479 196.654 192.288 193.167 8/6/2021 192.001 194.332 188.776 191.546 8/9/2021 190.201 195.036 189.629 194.887 8/10/2021 194.887 199 189.815 191.244 8/11/2021 191.815 192.077 185.34 185.75 8/12/2021 186.687 189.5 182.881 188.151 8/13/2021 188.999 188.999 185.192 187.465 8/16/2021 187.174 192.5 185 187.205 8/17/2021 186.064 188.2 184.2 187.085 8/18/2021 187.14 190.95 185.987 188.676 8/19/2021 187.203 191.65 187.1 188.11 8/20/2021 189.253 191.4 185 186.195 8/23/2021 185.06 188.123 184.5 187.631 8/24/2021 190.364 196.7 190.335 193.981 8/25/2021 194.778 196.51 190.8 195.343 8/26/2021 194.851 196.336 192.034 192.043 8/27/2021 192.801 194.406 192.145 193.414 8/30/2021 194 196.525 192.468 195.2 8/31/2021 196.026 196.15 191.9 192.659 9/1/2021 193.404 195.805 191.883 194.256 9/2/2021 194.264 199.3 193.3 193.495 9/3/2021 193.499 196.128 193.2 194.648 9/7/2021 195.701 196.749 192.372 194.478 9/8/2021 194.225 194.741 187.101 187.527 9/9/2021 188.344 192.23 188.344 190.182 9/10/2021 191.419 191.419 187.465 188.18 9/13/2021 189.799 189.799 181.5 186.668 9/14/2021 186.588 187.37 183.118 183.894 9/15/2021 184.995 186.58 181.446 186.141 9/16/2021 186.137 187.474 183.858 187.029 9/17/2021 184.598 189.552 184.405 189.523 9/20/2021 185 187.66 180 183.56 9/21/2021 185.64 185.64 182.7 184.33 9/22/2021 184.954 187.4 183.692 187.192 9/23/2021 186.5 187.567 184.5 186.725 9/24/2021 186.194 187.246 180.928 182.829 9/27/2021 180.966 183.218 179.024 182.268 9/28/2021 179 180.109 172.618 173.49 9/29/2021 175.001 176.892 171.11 171.791 9/30/2021 172.469 174.728 171.337 171.922 10/1/2021 171.361 173.5 169.346 170.707 10/4/2021 167.48 167.6 161.7 165.475 10/5/2021 165.479 170.172 165.479 165.601 10/6/2021 165.42 170.993 165.016 169.849 10/7/2021 172.868 176.322 171.913 175.681 10/8/2021 177.011 178.499 171.369 171.734 ######## 170.898 171.841 168.4 169.092 ######## 171.279 174.067 170.567 174.067 ######## 176 176.396 173.5 174.163 ######## 175.48 176.981 173 176.337 ######## 175.838 181.529 175.697 181.427 ######## 182.9 185.396 181.958 183.682 ######## 185.175 185.62 180.339 183.405 ######## 181.799 186.264 181.5 185.058 ######## 181.895 181.9 173.891 176.433 ######## 176.6 178.185 172.969 176.585 ######## 177.084 177.097 168.2 168.863 ######## 170.838 171.111 165.236 168.167 ######## 166.058 185.06 164.672 180.004 ######## 177.209 184.2 175.595 180.702 11/1/2021 182.251 189.435 181.171 188.983 11/2/2021 188.983 190.259 182.489 183.129 11/3/2021 185.899 185.999 180.748 185.14 11/4/2021 187.042 193.01 186.207 191.964 11/5/2021 192.451 192.879 188.085 190.257 11/8/2021 191.458 194.82 190.199 191.074 11/9/2021 191.674 194.624 189.703 193.53 ######## 191.434 191.607 183.761 184.283 ######## 187.95 190.33 187.41 187.765 ######## 189.8 210.083 189.799 209.501 ######## 206.615 212.86 204.5 207.282 ######## 206.999 214.698 204.535 210.467 ######## 208.801 210.675 206.447 206.599 ######## 207.441 213.7 206.658 211.847 ######## 216.52 222.873 213.179 213.982 ######## 216.511 216.511 200.6 202.767 ######## 202.72 204.665 195 199.394 ######## 197.949 207.915 195.493 206 ######## 208.414 210.668 204.61 207.504 ######## 209.979 212 200.134 200.395 ######## 201.3 201.867 194.827 200 ######## 203.691 207.44 193.64 194.103 12/1/2021 197.999 198.21 186.983 187.325 12/2/2021 187.435 190.223 183.464 185.39 12/3/2021 186.017 187.571 174.271 180.967 12/6/2021 176.744 182.58 171.958 182.353 12/7/2021 186 194.3 185.853 192.447 12/8/2021 189.799 193.98 188.579 192.229 12/9/2021 190.994 194.5 188.594 189.912 ######## 190.809 194.5 183.597 185.722 ######## 184 188.027 175.837 179.215 ######## 176.244 179.77 170.991 174.047 ######## 174.14 176.508 166.457 175.359 ######## 181.521 181.521 166.086 169.369 ######## 167.2 173.171 164.5 170.537 ######## 169.354 171.063 164.958 166.221 ######## 170.304 177.087 168.741 177.016 ######## 176.483 178.679 174.5 177.667 ######## 178.8 184.79 177.287 184 ######## 182.806 184.999 181.237 181.355 ######## 177.018 179.114 173.013 178.147 ######## 177.631 183.345 177.631 178.151 ######## 178.139 178.14 174.077 174.169