{"id":4735,"date":"2022-11-15T22:08:41","date_gmt":"2022-11-15T14:08:41","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=4735"},"modified":"2022-11-15T22:54:09","modified_gmt":"2022-11-15T14:54:09","slug":"black-and-scholes-1973","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=4735","title":{"rendered":"Black and Scholes, 1973"},"content":{"rendered":"\n<p><strong>See <a href=\"https:\/\/fanyuzhao.com\/?p=4597\">A bit Stochastic Calculus <\/a>.<\/strong><\/p>\n\n\n\n<p>For,<\/p>\n\n\n\n<p>$$ dS_t = \\mu S_t \\ dt +\\sigma S_t \\ dW_t $$<\/p>\n\n\n\n<ul><li>In calculating <span class=\"katex math inline\">d f(S_t)<\/span>, we would get, (by Taylor Expansion)<\/li><\/ul>\n\n\n\n<p>$$ df(S_t) = \\bigg( \\frac{\\partial f}{\\partial t} +  \\frac{\\partial f}{\\partial S_t}\\mu S_t +\\frac{1}{2}\\frac{\\partial^2 f}{\\partial S_t^2}\\sigma^2 S_t^2  \\bigg)dt + \\frac{\\partial f}{\\partial S_t}\\sigma S_t \\ dW_t  $$<\/p>\n\n\n\n<ul><li>A Special Form of <span class=\"katex math inline\">f(\\cdot)<\/span> is <span class=\"katex math inline\">f(S) = log(S)<\/span>,<\/li><\/ul>\n\n\n\n<p>$$ d\\ log(S_t) = \\bigg( \\mu &#8211; \\frac{1}{2}\\sigma^2 \\bigg)dt + \\sigma \\ dW_t $$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>We get <span class=\"katex math inline\">Y_t = log S_t<\/span> is the price of a derivative security with respect to <span class=\"katex math inline\">S_t<\/span> and <span class=\"katex math inline\">t<\/span> and then,<\/p>\n\n\n\n<p>$$ dY_t= \\bigg( \\frac{\\partial Y_t}{\\partial t}  +   \\frac{\\partial Y_t}{\\partial S_t}\\mu S_t +\\frac{1}{2}\\frac{\\partial^2 Y_t}{\\partial S_t^2}\\sigma^2 S_t^2  \\bigg)dt + \\frac{\\partial Y_t}{\\partial S_t}\\sigma S_t \\ dW_t  $$<\/p>\n\n\n\n<p>Consider a portfolio <span class=\"katex math inline\">\\Pi<\/span> is constructed with (1) short one derivative, and (2) long some fraction of stocks, <span class=\"katex math inline\">\\Delta<\/span>, such that the portfolio is risk natural. (<span class=\"katex math inline\">\\Delta = \\frac{\\partial Y}{\\partial S}<\/span>)<\/p>\n\n\n\n<p>$$ \\Pi_t = -Y +\\Delta \\ S_t $$<\/p>\n\n\n\n<p>Differentiate it,<\/p>\n\n\n\n<p>$$ d\\Pi_t =    -dY_t +\\frac{dY}{dS}dS_t $$<\/p>\n\n\n\n<p>Subtitute <span class=\"katex math inline\">dY_t<\/span> and <span class=\"katex math inline\">dS_t<\/span> into the above equation, we would then get the stochastic process of portfolio, by Ito&#8217;s Lemma.<\/p>\n\n\n\n<p>$$ d\\Pi_t =-\\bigg( \\frac{\\partial Y}{\\partial t} +  \\frac{1}{2}\\frac{\\partial^2 Y}{\\partial S^2} \\sigma^2 S_t^2 \\bigg)  dt $$<\/p>\n\n\n\n<p>The diffusion term <span class=\"katex math inline\">dW_t<\/span> disappears, and that means the portfolio is riskless during the interval <span class=\"katex math inline\">dt<\/span>. Under a no arbitrage assumption, this portfolio can only earn the riskless return, <span class=\"katex math inline\">r<\/span>.<\/p>\n\n\n\n<p>$$ d\\Pi_t =r\\Pi_t \\ dt $$<\/p>\n\n\n\n<ul><li>Subtitute <span class=\"katex math inline\">d\\Pi_t<\/span> and <span class=\"katex math inline\">\\Pi_t<\/span> into, we would get the Partial Differential Equation (PDE) \/ Black-Scholes equation,<\/li><\/ul>\n\n\n\n<p>$$    &#8211; \\bigg( \\frac{\\partial Y}{\\partial t}  +  \\frac{1}{2}\\frac{\\partial^2 Y}{\\partial S^2} \\sigma^2 S_t^2 \\bigg) dt = r\\bigg(- Y_t + \\frac{\\partial Y}{\\partial S}S_t \\bigg)dt  $$<\/p>\n\n\n\n<p>$$ rY_t = \\frac{\\partial Y}{\\partial t} +  \\frac{1}{2}\\frac{\\partial^2 Y}{\\partial S^2} \\sigma^2 S_t^2 +   \\frac{\\partial Y}{\\partial S}S_t $$<\/p>\n\n\n\n<p>Then, we guess (where <span class=\"katex math inline\">U(.)<\/span> is a function of <span class=\"katex math inline\">S_t<\/span> at time t=T),<\/p>\n\n\n\n<p>$$ Y_t = e^{-r(T-t)} U(S_T) $$<\/p>\n\n\n\n<p>For a European call with strike price, K, <span class=\"katex math inline\">U(S_T)<\/span> would be the payoff at maturity,<\/p>\n\n\n\n<p>$$ U(S_T) = max( S &#8211; K , 0 )  $$<\/p>\n\n\n\n<p>Finally, through a series of process to find a specific solution of the PDE, we can solve the value of call, (<span class=\"katex math inline\">\\Phi(\\cdot)<\/span> is the cumulative standard normal distribtion)<\/p>\n\n\n\n<p>$$ c = S_t\\Phi(d_1) &#8211; K e^{-r (T-t)}\\Phi(d_2) $$<\/p>\n\n\n\n<p>with,<\/p>\n\n\n\n<p>$$  d_1 =\\frac{ log(S_t\/K) + (r-\\frac{1}{2}\\sigma^2)(T-t)}{\\sigma \\sqrt{T-t}}  $$<\/p>\n\n\n\n<p>$$ d_1 = \\sigma \\sqrt{T-t} $$<\/p>\n\n\n\n<div data-wp-interactive=\"\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!selectors.core.file.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/11\/42-2010_Handbook.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Embed of 42-2010_Handbook..\"><\/object><a id=\"wp-block-file--media-6ee9dfd2-51e6-4469-b211-b22cd1a32cc7\" href=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/11\/42-2010_Handbook.pdf\">42-2010_Handbook<\/a><a href=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/11\/42-2010_Handbook.pdf\" class=\"wp-block-file__button\" download aria-describedby=\"wp-block-file--media-6ee9dfd2-51e6-4469-b211-b22cd1a32cc7\">Download<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>See A bit Stochastic Calculus . For, $$ dS_t = \\mu S_t \\ dt +\\sigma S_t \\ dW_t $$ In calculating d f(S_t), we would get, (by Taylor Expansion) $$ df(S_t) = \\bigg( \\frac{\\partial f}{\\partial t} + \\frac{\\partial f}{\\partial S_t}\\mu S_t +\\frac{1}{2}\\frac{\\partial^2 f}{\\partial S_t^2}\\sigma^2 S_t^2 \\bigg)dt + \\frac{\\partial f}{\\partial S_t}\\sigma S_t \\ dW_t $$ A &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=4735\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Black and Scholes, 1973<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,6,18,26],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4735"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4735"}],"version-history":[{"count":53,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4735\/revisions"}],"predecessor-version":[{"id":4792,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4735\/revisions\/4792"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4735"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4735"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4735"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}