{"id":17846,"date":"2023-05-10T13:28:00","date_gmt":"2023-05-10T05:28:00","guid":{"rendered":"https:\/\/www.tejwin.com\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/"},"modified":"2026-04-10T14:33:13","modified_gmt":"2026-04-10T06:33:13","slug":"black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks","status":"publish","type":"insight","link":"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/","title":{"rendered":"\u3010Quant\u3011Black Scholes model and Greeks"},"content":{"rendered":"\n<figure class=\"wp-block-image aligncenter size-large\"><img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"683\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/0_ROirY-0kG2mR9el6-1024x683.jpg\" alt=\"\u3010Quant\u3011Black Scholes model and Greeks\" class=\"wp-image-8461\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/0_ROirY-0kG2mR9el6-1024x683.jpg 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/0_ROirY-0kG2mR9el6-300x200.jpg 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/0_ROirY-0kG2mR9el6-150x100.jpg 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/0_ROirY-0kG2mR9el6-768x512.jpg 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/0_ROirY-0kG2mR9el6.jpg 1400w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">Photo by&nbsp;<a href=\"https:\/\/unsplash.com\/de\/@homajob?utm_source=medium&amp;utm_medium=referral\" target=\"_blank\" rel=\"noreferrer noopener\">Scott Graham<\/a>&nbsp;on&nbsp;<a href=\"https:\/\/unsplash.com\/?utm_source=medium&amp;utm_medium=referral\" target=\"_blank\" rel=\"noreferrer noopener\">Unsplash<\/a><\/figcaption><\/figure>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_81 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69da4ef413892\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"ez-toc-cssicon\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69da4ef413892\"  aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#key_takeaways\" >key takeaways!<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Keywords_Black_Scholes_model_options_call_put\" >Keywords: Black Scholes model, options, call, put<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Highlight\" >Highlight:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Preface\" >Preface<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Programming_environment_and_Module_required\" >Programming environment and Module required<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Database\" >Database<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Import_data\" >Import data<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Data_processing\" >Data processing<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Model_introduction\" >Model introduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Programming\" >Programming<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Theoretical_price\" >Theoretical price<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Delta\" >Delta<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Gamma\" >Gamma<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Vega\" >Vega<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Theta\" >Theta<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Rho\" >Rho<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Real_data_exercise\" >Real data exercise<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Conclusion\" >Conclusion<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/www.tejwin.com\/en\/insight\/black-scholes-%e6%a8%a1%e5%9e%8b%e8%88%87-greeks\/#Further_Reading\" >Further Reading<\/a><\/li><\/ul><\/nav><\/div>\n<h2 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"key_takeaways\"><\/span><span style=\"color: #000000;\"><strong>key takeaways!<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol>\n<li><span style=\"color: #3366ff;\"><strong>Pythonic BS Model:<\/strong> <\/span>Code the Nobel Prize-winning formula into a reusable module.<\/li>\n<li><span style=\"color: #3366ff;\"><strong style=\"font-size: revert;\">Greeks Visualized:<\/strong><\/span><span style=\"font-size: revert; color: initial;\"> Master the 5\u00a0Greeks Indicators:\u00a0through intuitive charts. <\/span><\/li>\n<li><span style=\"color: #3366ff;\"><strong style=\"font-size: revert;\">Pricing Diagnostics:<\/strong><\/span><span style=\"font-size: revert; color: initial;\"> Use <\/span><strong style=\"font-size: revert; color: initial;\">TEJ API<\/strong><span style=\"font-size: revert; color: initial;\"> for TAIEX cases and uncover why theoretical prices deviate from the market\u2014discussed at the end of the article.<\/span><\/li>\n<\/ol>\n<p><!-- notionvc: 97d31d06-dfb7-4ca0-9f1c-009d8ba9b3dc --><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"fe15\"><span class=\"ez-toc-section\" id=\"Keywords_Black_Scholes_model_options_call_put\"><\/span>Keywords: Black Scholes model, options, call, put<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"e457\"><span class=\"ez-toc-section\" id=\"Highlight\"><\/span>Highlight:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Difficulty\uff1a\u2605\u2605\u2605\u2605\u2606<\/li>\n\n\n\n<li>Using transaction data for options pricing.<\/li>\n\n\n\n<li>Advise: The primary focal point for today`s article is to codify Black Scholes formula via Python. Detailed introduction for Black Scholes formula and attributes of options are not included in this article. As a result, previews for options and Black Scholes model is suggested before reading this article.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"1ca3\"><span class=\"ez-toc-section\" id=\"Preface\"><\/span>Preface<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"f417\">In 1997, Robert Merton and Myron Scholes won the Nobel Prize in Economics for their Black-Scholes options pricing formula, beating out many other contenders. The Black-Scholes model is still a widely-used option pricing model in the financial industry and by investors due to its excellent mathematical properties, simplicity, and ease of use. Today, we will focus on programming this model and Greeks derived from Black Scholes model.<\/p>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button has-custom-width wp-block-button__width-100 is-style-fill\"><a class=\"wp-block-button__link has-vivid-cyan-blue-to-vivid-purple-gradient-background has-background has-medium-font-size has-custom-font-size wp-element-button\" href=\"https:\/\/www.tejwin.com\/en\/insight\/three-major-institutional-investors-position-based-trading-strategy-for-taiex-futures\/\" style=\"border-style:none;border-width:0px;border-radius:10px;padding-top:10px;padding-right:10px;padding-bottom:10px;padding-left:10px\" target=\"_blank\" rel=\"noreferrer noopener nofollow\">\u2728<strong>Maybe you&#8217;re interested in: How to Build a Trading Strategy Based on Three Major Institutional Investors&#8217; Positions for TAIEX Futures<\/strong><\/a><\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"c4b4\"><span class=\"ez-toc-section\" id=\"Programming_environment_and_Module_required\"><\/span>Programming environment and Module required<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"dcc5\">Windows 11 and Jupyter Notebook is used as editor<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code># \u8f09\u5165\u6240\u9700\u5957\u4ef6\nimport math \nimport tejapi\nimport pandas as pd \nimport numpy as np \nimport matplotlib.pyplot as plt \nfrom scipy.stats import norm\nplt.style.use('bmh')\nplt.rcParams&#91;'font.sans-serif']=&#91;'Microsoft YaHei']\n\n# \u767b\u5165TEJ API\napi_key = 'YOUR_KEY'\ntejapi.ApiConfig.api_key = api_key\ntejapi.ApiConfig.ignoretz = True<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2968\"><span class=\"ez-toc-section\" id=\"Database\"><\/span>Database<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"60ef\">Stock trading database: Unadjusted daily stock price, database code is (TWN\/APRCD).<br>Derivatives database: Options daily transaction information, database code is (TWN\/AOPTION).<\/p>\n\n\n\n<p>Before running simulations, however, practitioners must collect&nbsp;<strong>accurate and high-frequency market data<\/strong>&nbsp;to serve as input for these models. Whether you\u2019re modeling volatility, interest rate changes, or stock price movements,&nbsp;<strong>reliable historical data is the foundation of any sound pricing strategy<\/strong>.<br>If you\u2019re looking to apply option pricing techniques in practice, start with&nbsp;<a href=\"https:\/\/www.tejwin.com\/en\/databank-solution\/market-data\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">TEJ Market Data<\/mark><\/strong><\/a> \u2014a comprehensive solution for time-series stock prices, implied volatility, and benchmark interest rates that are essential for simulation-based pricing.<\/p>\n\n\n\n<p class=\"has-text-align-left has-medium-font-size\"><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"aa63\"><span class=\"ez-toc-section\" id=\"Import_data\"><\/span>Import data<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"46f7\">Using the unadjusted close prices of the Taiwan Weighted Index (Y9999), with a time period from March 16, 2021 to April 10, 2023. And loading the Taiwan Weighted Index Call Option (TXO202304C15500), which is a European call option, with a start trading date of March 16 and an expiration date of April 19, and a strike price of 15500.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>gte, lte = '2021-03-16', '2023-04-10'\n# \u6a19\u7684\u7269\u50f9\u683c\nstocks = tejapi.get('TWN\/APRCD',\n                   paginate = True,\n                   coid = 'Y9999',\n                   mdate = {'gte':gte, 'lte':lte},\n                   chinese_column_name = True,\n                   opts = {\n                       'columns':&#91; 'mdate','close_d']\n                   }\n                  )\n# \u9078\u64c7\u6b0a\u50f9\u683c\noptions = tejapi.get(\n    'TWN\/AOPTION',\n    paginate = True,\n    coid = 'TXO202304C15500',\n    mdate = {'gte':gte, 'lte':lte},\n    chinese_column_name = True,\n    opts = {\n        'columns':&#91;'mdate', 'coid','settle', 'kk', 'theoremp', 'acls', 'ex_price', 'td1y', 'avolt', 'rtime']\n    }\n)\n# \u91cd\u8a2d\u65e5\u671f\u70baindex\nstocks = stocks.set_index('\u5e74\u6708\u65e5')\noptions = options.set_index('\u65e5\u671f')<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9d24\"><span class=\"ez-toc-section\" id=\"Data_processing\"><\/span>Data processing<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"423d\">Calculating daily return and moving return volatility with a window of 252 days.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>stocks&#91;'\u65e5\u5831\u916c'] = np.log(stocks&#91;'\u6536\u76e4\u50f9(\u5143)']) - np.log(stocks&#91;'\u6536\u76e4\u50f9(\u5143)'].shift(1))\nstocks&#91;'\u79fb\u52d5\u5831\u916c\u6ce2\u52d5\u5ea6'] = stocks&#91;'\u65e5\u5831\u916c'].rolling(252).std()<\/code><\/pre>\n\n\n\n<p id=\"d6b5\">The resultant dataframe after processing is shown below:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/1-41-1024x576.png\" alt=\"\" class=\"wp-image-33817\" style=\"width:817px;height:auto\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/1-41-1024x576.png 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/1-41-300x169.png 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/1-41-150x84.png 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/1-41-768x432.png 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/1-41-1536x864.png 1536w, https:\/\/www.tejwin.com\/wp-content\/uploads\/1-41.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"200e\"><span class=\"ez-toc-section\" id=\"Model_introduction\"><\/span>Model introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"7334\">First, we can take a look at the formula before programming.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/1XBEWwBbve_7gYe9j7A8PDQ.png\" alt=\"\"\/><\/figure>\n\n\n\n<p id=\"1716\">\u25cf C(St, t), P(St, t) : call, put price at day t.<br>\u25cf St, K : price of underlying asset and strike price at day t.<br>\u25cf T, t : Maturity date and day t.<br>\u25cf sigma : historical moving return volatility, in here, we take a window of 252 days.<br>\u25cf r : risk-free rate.<br>\u25cf N() : cumulative density function of standard normal distribution.<\/p>\n\n\n\n<p id=\"e7f8\">By observing above formula, we can see that there are 5 factors affecting options theoretical price: underlying asset price, strike price, time to maturity(T-t), return volatility and risk-free rate. Apart from strike price, other four factors are time-varying. In order to study the relationships between these four factors and options price, Greeks are invented to quantify the relationships. They are&nbsp;<strong>delta, gamma, vega, theta and rho<\/strong>.<\/p>\n\n\n\n<p id=\"48de\">The concepts of Greeks are simple. They are just the partial derivatives of options price with respect to underlying asset price, volatility, time to maturity and risk-free rate. In particular, gamma is the second-order partial derivatives of options price with respect to underlying asset price. Thanks to the effort from those great ancient mathematicians, every single Greeks has an analytic solution, so we don`t have to deal with those nasty partial derivative function\ud83d\ude06.<\/p>\n\n\n\n<p id=\"98e6\">The below graph presents all Greeks analytic solutions. From top to bottom, there are delta of call, delta of put, gamma of call and put, vega of call and put, theta of call, theta of put, rho of call, rho of put.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/1VtfphYZc0exuguPyW1nJzQ.png\" alt=\"\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"392d\"><span class=\"ez-toc-section\" id=\"Programming\"><\/span>Programming<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"b8aa\">Without further ado, let me just show you the code\ud83d\ude0e.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>class BS_formula:\n    def __init__(self, s0, k, r, sigma, T):     \n        self.s0 = s0 # \u6a19\u7684\u7269\u50f9\u683c\n        self.k = k # \u5c65\u7d04\u50f9\u683c\n        self.r = r # \u7121\u98a8\u96aa\u5229\u7387\n        self.sigma = sigma # \u6b77\u53f2\u6ce2\u52d5\u5ea6\n        self.T = T # \u5269\u9918\u5230\u671f\u6642\u9593\n        self.d1 = (np.log(s0\/k)+(r+sigma**2\/2)*T) \/ (sigma * np.sqrt(T))\n        self.d2 = ((np.log(s0\/k)+(r+sigma**2\/2)*T) \/ (sigma * np.sqrt(T))) - sigma*np.sqrt(T)\n        \n    def BS_price(self): # \u8a08\u7b97\u7406\u8ad6\u50f9\u683c\n        c = self.s0*norm.cdf(self.d1) - self.k*np.exp(-self.r*self.T)*norm.cdf(self.d2)\n        p = self.k*np.exp(-self.r*self.T)*norm.cdf(-self.d2) - self.s0*norm.cdf(-self.d1)\n        return c,p\n        \n    def BS_delta(self): # \u8a08\u7b97 delta\n        return norm.cdf(self.d1), norm.cdf(self.d1)-1\n    \n    def BS_gamma(self): # \u8a08\u7b97 gamma\n        return norm.pdf(self.d1)\/(self.s0*self.sigma*np.sqrt(self.T)), norm.pdf(self.d1)\/(self.s0*self.sigma*np.sqrt(self.T))\n    \n    def BS_vega(self): # \u8a08\u7b97 vega\n        return self.s0*np.sqrt(self.T)*norm.pdf(self.d1), self.s0*np.sqrt(self.T)*norm.pdf(self.d1)\n    \n    def BS_theta(self): # \u8a08\u7b97 theta \n        c_theta = -self.s0*norm.pdf(self.d1)*self.sigma \/ (2*np.sqrt(self.T)) - self.r*self.k*np.exp(-self.r*self.T)*norm.cdf(self.d2)\n        p_theta = -self.s0*norm.pdf(self.d1)*self.sigma \/ (2*np.sqrt(self.T)) + self.r*self.k*np.exp(-self.r*self.T)*norm.cdf(-self.d2)\n        return c_theta, p_theta\n    \n    def BS_rho(self): # \u8a08\u7b97 rho  \n        return self.k*self.T*np.exp(-self.r*self.T)*norm.cdf(self.d2), -self.k*self.T*np.exp(-self.r*self.T)*norm.cdf(-self.d2)<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9e50\"><span class=\"ez-toc-section\" id=\"Theoretical_price\"><\/span>Theoretical price<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"b5b9\">From above code, ceteris paribus, we can visualize the relationship between underlying asset price and options price. In the graph down below, we can see that the curves of call and put prices are symmetric, while call price and underlying asset price are positively-correlated, put price and underlying asset price are negatively-correlated. Besides,&nbsp;<strong>a stylized fact that withholding a call has limited risk but unlimited profit can also be discovered via this picture<\/strong>, since you can get infinity profit when deeply in-the-money, and at most no profit when deeply out-of-the-money. Unlike withholding a call, The profit of withholding a put when deeply in-the-money can not be infinity, since the lowest price for an underlying asset is zero.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>s0 = np.linspace(200,800)\nk = 500\nr = 0.00\nsigma = 0.2\nT = 252\/252\n\nmybs = BS_formula(s0, k, r, sigma, T)\nc, p = mybs.BS_price()\n\nfig = plt.figure(figsize = (12,8))\nplt.plot(s0, c, label = '\u8cb7\u6b0a')\nplt.plot(s0, p, label = '\u8ce3\u6b0a')\nplt.axvline(x = 500, color = 'black', linestyle = '--')\nplt.xlabel('\u6a19\u7684\u7269\u50f9\u683c', fontsize = 15)\nplt.ylabel('\u9078\u64c7\u6b0a\u50f9\u683c', fontsize = 15)\nplt.title('\u9078\u64c7\u6b0a\u50f9\u683c VS. \u6a19\u7684\u7269\u50f9\u683c', fontsize = 20)\nplt.legend(fontsize = 14)\nplt.savefig('black scholes put call price.png')\nplt.show()<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/2-39-1024x576.png\" alt=\"\" class=\"wp-image-33819\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/2-39-1024x576.png 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/2-39-300x169.png 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/2-39-150x84.png 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/2-39-768x432.png 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/2-39-1536x864.png 1536w, https:\/\/www.tejwin.com\/wp-content\/uploads\/2-39.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3f4a\"><span class=\"ez-toc-section\" id=\"Delta\"><\/span>Delta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"13de\">Then, ceteris paribus, we can visualize the relationship between underlying asset price and delta. The financial meaning of delta refers to the amount by which the price of an option increases or decreases when the underlying asset price increases by one unit. From picture down below, It can be observed that when the option is deep out-of-the-money, the delta of both call and put tends towards zero. When the option is deep in-the-money, the delta of the call tends towards one while the delta of the put option tends towards negative one.&nbsp;<strong>This indicates that small changes in the underlying asset price have little impact on the options price when it is deep out-of-the-money, while small changes in the underlying asset price can cause significant fluctuations in the option price when it is deep in-the-money and the magnitude of the fluctuations is roughly equal to that of the underlying asset.<\/strong><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>s0 = np.linspace(200,800)\nk = 500\nr = 0.00\nsigma = 0.2\nT = 252\/252\n\nmybs = BS_formula(s0, k, r, sigma, T)\nc, p = mybs.BS_delta()\n\nfig = plt.figure(figsize = (12,8))\nplt.plot(s0, c, label = '\u8cb7\u6b0a')\nplt.plot(s0, p, label = '\u8ce3\u6b0a')\nplt.axvline(x = 500, color = 'black', linestyle = '--')\nplt.axhline(y = 0, color = 'black', linestyle = '--')\nplt.xlabel('\u6a19\u7684\u7269\u50f9\u683c', fontsize = 15)\nplt.ylabel('Delta\u503c', fontsize = 15)\nplt.title('Delta\u503c VS. \u6a19\u7684\u7269\u50f9\u683c', fontsize = 20)\nplt.legend(fontsize = 14)\nplt.savefig('black scholes put call delta.png')\nplt.show()<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/3-34-1024x576.png\" alt=\"\" class=\"wp-image-33821\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/3-34-1024x576.png 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/3-34-300x169.png 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/3-34-150x84.png 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/3-34-768x432.png 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/3-34-1536x864.png 1536w, https:\/\/www.tejwin.com\/wp-content\/uploads\/3-34.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"295a\"><span class=\"ez-toc-section\" id=\"Gamma\"><\/span>Gamma<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"e6a9\">Gamma is the second-order partial derivative of call price with respect to underlying asset price, it can be understood as the slope of delta curve. Ceteris paribus, we can visualize the relationship between underlying asset price and gamma. The gammas of call and put are actually the same, both reach to maximum during at-the-money.&nbsp;<strong>That means the magnitude of incremental of delta first accelerate then decelerate, when the underlying asset price moves from deeply out-of-the-money to deeply in-the-money<\/strong>. Code is shown at the end of article.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/4-33-1024x576.png\" alt=\"\" class=\"wp-image-33823\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/4-33-1024x576.png 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/4-33-300x169.png 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/4-33-150x84.png 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/4-33-768x432.png 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/4-33-1536x864.png 1536w, https:\/\/www.tejwin.com\/wp-content\/uploads\/4-33.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p id=\"3db0\">Moreover, we can observe the change of gamma under different moneyness and time to maturity. A fun fact can be discovered that while in-the-money and approaching to maturity, gamma surges incredibly. And that is the well-known gamma risk. Gamma risk means that while approaching to maturity, the price volatility of options at-the-money usually is usually large. Therefore, most more risk-averting investors try to offset the positions while approaching to maturity. However, some investors might do it inversely, they may long positions before maturity and expect to earn a more decent risk premium.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>s0, s1, s2 = 400, 500, 600 \nk = 500\nr = 0.00\nsigma = 0.2\nT = np.linspace(1, 0.01)\n\nmybs0 = BS_formula(s0, k, r, sigma, T)\nc0, p0 = mybs0.BS_gamma()\n\nmybs1 = BS_formula(s1, k, r, sigma, T)\nc1, p1 = mybs1.BS_gamma()\n\nmybs2 = BS_formula(s2, k, r, sigma, T)\nc2, p2 = mybs2.BS_gamma()\n\nfig = plt.figure(figsize = (12,8))\nplt.plot(T, c0, label = '\u8cb7\u6b0a(\u50f9\u5916)')\nplt.plot(T, c1, label = '\u8cb7\u6b0a(\u50f9\u5e73)')\nplt.plot(T, c2, label = '\u8cb7\u6b0a(\u50f9\u5167)')\nplt.xlabel('\u5269\u9918\u6642\u9593', fontsize = 15)\nplt.ylabel('Gamma\u503c', fontsize = 15)\nplt.title('Gamma\u503c VS. \u5269\u9918\u6642\u9593', fontsize = 20)\nplt.legend(fontsize = 14)\nplt.axis(&#91;1.005, -0, -0.005, .045])\nplt.savefig('black scholes put call gamma2.png')\nplt.show()<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/5-15-1024x576.png\" alt=\"\" class=\"wp-image-33825\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/5-15-1024x576.png 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/5-15-300x169.png 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/5-15-150x84.png 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/5-15-768x432.png 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/5-15-1536x864.png 1536w, https:\/\/www.tejwin.com\/wp-content\/uploads\/5-15.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4e0c\"><span class=\"ez-toc-section\" id=\"Vega\"><\/span>Vega<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"bdf8\">Similar to gamma, the vega of call and put are exactly the same. The meaning of vega refers to the amount by which the price of an option increases or decreases when the volatility of underlying asset price increases by one unit. From below graph, it can be discovered that vega reach to the maximum during at-the-money. That is, the price of options is most sensitive to volatility during at-the-money. Code is shown in the end of article.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/6-15-1024x576.png\" alt=\"\" class=\"wp-image-33827\" style=\"width:840px;height:560px\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/6-15-1024x576.png 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/6-15-300x169.png 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/6-15-150x84.png 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/6-15-768x432.png 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/6-15-1536x864.png 1536w, https:\/\/www.tejwin.com\/wp-content\/uploads\/6-15.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2741\"><span class=\"ez-toc-section\" id=\"Theta\"><\/span>Theta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"1c94\">The meaning of theta is the amount by which the price of an option increases or decreases when the time to maturity increases by one unit. We can visualize the relationship between theta and time to maturity. We can see that during deeply in-the-money and out-of-the-money, theta would approach to zero while approaching to maturity. On the contrary, there is a steep downward movement when at-the-money. Moreover, theta is always negative, because as the time moves, the time value of options gradually declines. Code is shown in the end of article.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/7-9-1024x576.png\" alt=\"\" class=\"wp-image-33829\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/7-9-1024x576.png 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/7-9-300x169.png 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/7-9-150x84.png 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/7-9-768x432.png 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/7-9-1536x864.png 1536w, https:\/\/www.tejwin.com\/wp-content\/uploads\/7-9.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"c278\"><span class=\"ez-toc-section\" id=\"Rho\"><\/span>Rho<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"444c\">Eventually, rho represents the amount by which the price of an option increases or decreases when the risk-free rate increases by one unit. Also we plot the below curve to observe the relationship between rho and underlying asset price. Since the underlying asset price increase, rho becomes larger. Code is shown in the end of article.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/\/8-5-1024x576.png\" alt=\"\" class=\"wp-image-33831\" srcset=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/8-5-1024x576.png 1024w, https:\/\/www.tejwin.com\/wp-content\/uploads\/8-5-300x169.png 300w, https:\/\/www.tejwin.com\/wp-content\/uploads\/8-5-150x84.png 150w, https:\/\/www.tejwin.com\/wp-content\/uploads\/8-5-768x432.png 768w, https:\/\/www.tejwin.com\/wp-content\/uploads\/8-5-1536x864.png 1536w, https:\/\/www.tejwin.com\/wp-content\/uploads\/8-5.png 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"76d0\"><span class=\"ez-toc-section\" id=\"Real_data_exercise\"><\/span>Real data exercise<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p id=\"6640\">Last but not least, we take the TAIEX call options for example, calculating the theoretical call price at a strike price of 15500, time to maturity of 6 days.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>r = 0.012\ns0 = stocks.loc&#91;'2023-04-10']&#91;'\u6536\u76e4\u50f9(\u5143)']\nk = 15500\nsigma = stocks.loc&#91;'2023-04-10']&#91;'\u79fb\u52d5\u5831\u916c\u6ce2\u52d5\u5ea6']*np.sqrt(252)\nT = 6\/252\n\nmybs = BS_formula(s0, k, r, sigma, T)\nc, p = mybs.BS_price()\nc_delta, p_delta = mybs.BS_delta()\nc_gamma, p_gamma = mybs.BS_gamma()\nc_vega, p_vega = mybs.BS_vega()\nc_theta, p_theta = mybs.BS_theta()\nc_rho, p_rho = mybs.BS_rho()\n\nprint('==2023-04-10\u5c65\u7d04\u50f9\u70ba525\u7684\u53f0\u7a4d\u96fb\u8cb7\u6b0a==')\nprint('\u7576\u524d\u6a19\u7684\u7269\u50f9\u683c\u70ba %.3f\uff0c \u5e74\u5316\u6ce2\u52d5\u5ea6\u70ba %.3f\uff0c \u5269\u9918\u671f\u9593\u70ba %.3f'%(s0, sigma, T*252))\nprint('\u8cb7\u6b0a\u7406\u8ad6\u50f9\u683c: %.4f\uff0c \u8ce3\u6b0a\u7406\u8ad6\u50f9\u683c: %.4f' %(c,p))\nprint('\u8cb7\u6b0adelta: %.4f\uff0c \u8ce3\u6b0adelta: %.4f' %(c_delta,p_delta))\nprint('\u8cb7\u6b0agamma: %.4f\uff0c \u8ce3\u6b0agamma: %.4f' %(c_gamma,p_gamma))\nprint('\u8cb7\u6b0avega: %.4f\uff0c \u8ce3\u6b0avega: %.4f' %(c_vega,p_vega))\nprint('\u8cb7\u6b0atheta: %.4f\uff0c \u8ce3\u6b0atheta: %.4f' %(c_theta,p_theta))\nprint('\u8cb7\u6b0arho: %.4f\uff0c \u8ce3\u6b0arho: %.4f' %(c_rho,p_rho))<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/1c1MrKkmB_jlAsniEBKc1wg.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>options.loc&#91;'2023-04-10'] # \u5be6\u969b\u8cb7\u6b0a\u50f9\u683c\n<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.tejwin.com\/wp-content\/uploads\/1y_Mtnf-WKiDzoDvA2O8srQ.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p id=\"392f\">Comparing the theoretical price we calculated with the strike price, we can see that there is a significant difference (435.54\u2013385), indicating that the call option may be undervalued at this moment. In addition, TEJ API also provides a theoretical pricing service in the derivatives financial product database. The theoretical price calculated by TEJ API has a slight difference from ours (440.36 v.s. 435.54), which may be due to different methods of calculating historical volatility or using different risk-free interest rate standards.<\/p>\n\n\n\n<div style=\"height:31px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Conclusion\"><\/span>Conclusion<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>The Black-Scholes model remains a foundational tool in options pricing, offering analytical clarity and practical utility through its associated Greeks. By quantifying how option prices respond to market variables, investors and professionals can better understand and manage risk exposure. However, even the most elegant model relies on accurate inputs \u2014 and that\u2019s where high-quality market data becomes indispensable. Whether for pricing, hedging, or simulation, <a href=\"https:\/\/www.tejwin.com\/en\/solution\/market-data\/\" target=\"_blank\" rel=\"noreferrer noopener\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">TEJ Market Data<\/mark><\/a> offers the historical depth and real-time granularity needed to power your quantitative strategies with confidence.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-container-core-buttons-is-layout-b5994976 wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button has-custom-width wp-block-button__width-100 is-style-fill\"><a class=\"wp-block-button__link has-vivid-cyan-blue-background-color has-background has-text-align-center wp-element-button\" href=\"https:\/\/www.tejwin.com\/en\/databank-solution\/market-data\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Predict Market Pricing with thorough data and investment model!<br><strong>Get Started with TEJ Database Now<\/strong>!<\/strong><\/a><\/div>\n<\/div>\n\n\n\n<div style=\"height:29px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Further_Reading\"><\/span>Further Reading<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<div class=\"wp-block-group is-horizontal is-content-justification-left is-nowrap is-layout-flex wp-container-core-group-is-layout-e5fbeb3a wp-block-group-is-layout-flex\">\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:100%\">\n<figure class=\"wp-block-embed alignleft is-type-wp-embed is-provider-tej wp-block-embed-tej\"><div class=\"wp-block-embed__wrapper\">\n<blockquote class=\"wp-embedded-content\" data-secret=\"4ed0Sflu5H\"><a href=\"https:\/\/www.tejwin.com\/insight\/%e9%81%b8%e6%93%87%e6%ac%8a%e5%ae%9a%e5%83%b9-%e8%92%99%e5%9c%b0%e5%8d%a1%e7%be%85%e6%a8%a1%e6%93%ac%e6%b3%95\/\">\u9078\u64c7\u6b0a\u5b9a\u50f9-\u8499\u5730\u5361\u7f85\u6a21\u64ec\u6cd5<\/a><\/blockquote><iframe class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"\u9078\u64c7\u6b0a\u5b9a\u50f9-\u8499\u5730\u5361\u7f85\u6a21\u64ec\u6cd5 &#8212; TEJ\" src=\"https:\/\/www.tejwin.com\/insight\/%e9%81%b8%e6%93%87%e6%ac%8a%e5%ae%9a%e5%83%b9-%e8%92%99%e5%9c%b0%e5%8d%a1%e7%be%85%e6%a8%a1%e6%93%ac%e6%b3%95\/embed\/#?secret=SRSBtoYcXZ#?secret=4ed0Sflu5H\" data-secret=\"4ed0Sflu5H\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe>\n<\/div><\/figure>\n<\/div>\n<\/div>\n\n\n\n<figure class=\"wp-block-embed alignright is-type-wp-embed is-provider-tej wp-block-embed-tej\"><div class=\"wp-block-embed__wrapper\">\n<blockquote class=\"wp-embedded-content\" data-secret=\"GTbgzW3aKs\"><a href=\"https:\/\/www.tejwin.com\/en\/insight\/quantitative-data-analysis\/\">Quantitative Data Analysis Explained: Methods &amp; Finance Uses<\/a><\/blockquote><iframe class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"Quantitative Data Analysis Explained: Methods &amp; Finance Uses &#8212; TEJ\" src=\"https:\/\/www.tejwin.com\/en\/insight\/quantitative-data-analysis\/embed\/#?secret=8kZo25w2Jf#?secret=GTbgzW3aKs\" data-secret=\"GTbgzW3aKs\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe>\n<\/div><\/figure>\n<\/div>\n\n\n\n<p><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In 1997, Robert Merton and Myron Scholes won the Nobel Prize in Economics for their Black-Scholes options pricing formula, beating out many other contenders. The Black-Scholes model is still a widely-used option pricing model in the financial industry and by investors due to its excellent mathematical properties, simplicity, and ease of use. Today, we will focus on programming this model and Greeks derived from Black Scholes model.<\/p>\n","protected":false},"featured_media":8496,"template":"","tags":[2962,2987,3199],"insight-category":[690,50],"class_list":["post-17846","insight","type-insight","status-publish","has-post-thumbnail","hentry","tag-market-data","tag-quant","tag-tej-database","insight-category-data-analysis","insight-category-fintech"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/insight\/17846","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/insight"}],"about":[{"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/types\/insight"}],"version-history":[{"count":15,"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/insight\/17846\/revisions"}],"predecessor-version":[{"id":44979,"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/insight\/17846\/revisions\/44979"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/media\/8496"}],"wp:attachment":[{"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/media?parent=17846"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/tags?post=17846"},{"taxonomy":"insight-category","embeddable":true,"href":"https:\/\/www.tejwin.com\/en\/wp-json\/wp\/v2\/insight-category?post=17846"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}