{"id":4659,"date":"2022-11-13T16:35:38","date_gmt":"2022-11-13T08:35:38","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=4659"},"modified":"2022-11-14T09:46:37","modified_gmt":"2022-11-14T01:46:37","slug":"risk-neutral-pricing","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=4659","title":{"rendered":"Risk-Neutral Pricing"},"content":{"rendered":"\n<p>Our basic logic is, <\/p>\n\n\n\n<p>$$ \\text{Option Value} =\\text{Discounted Expectation of the Payoff} $$<\/p>\n\n\n\n<p>However, there is not a perfect correlation between the <strong>Option<\/strong> and the <strong>Underlying Asset<\/strong>. That fact encourage us to do the risk-netural method for valuing an option.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Geometric Brownian Motion<\/h4>\n\n\n\n<ul><li>Non-dividend paying stock, (<span class=\"katex math inline\">S_0&gt;0<\/span>)<\/li><\/ul>\n\n\n\n<p>$$ dS_t = \\mu S_t\\ dt+\\sigma S_t \\ dW_t $$<\/p>\n\n\n\n<ul><li>Money market, or bank account, (<span class=\"katex math inline\">B_0 = 1<\/span>)<\/li><\/ul>\n\n\n\n<p>$$ dB_t = r B_t\\ d_t $$<\/p>\n\n\n\n<p>and,<\/p>\n\n\n\n<p>$$ \\{ W_t \\}_{t\\in [0,T]} : \\mathbb{P} \\sim Brownian \\ Motion $$<\/p>\n\n\n\n<p>$ \\{S_t\\} $ is not a <strong>martingals <\/strong>under <span class=\"katex math inline\">\\mathbb{P}<\/span> because <span class=\"katex math inline\">\\mu S_t<\/span> is not zero.<\/p>\n\n\n\n<p>So, we apply a transformation to <span class=\"katex math inline\">S_t<\/span> and to make it be a martingal (eliminate the drift term).<\/p>\n\n\n\n<p>We here look at <span class=\"katex math inline\">\\{ \\frac{S_t}{B_t} \\}<\/span>,<\/p>\n\n\n\n<p>$$ d\\big( \\frac{S_t}{B_t} \\big)=-r\\frac{S_t}{B_t}dt +\\frac{1}{B_t} dS_t +\\frac{1}{2}dS_t (-r\\frac{1}{B_t}dt) $$ <\/p>\n\n\n\n<p>$$ d\\big( \\frac{S_t}{B_t} \\big)=-r\\frac{S_t}{B_t}dt   +\\bigg(  \\frac{1}{B_t}-\\frac{1}{2}r\\frac{1}{B_t}dt \\bigg)\\bigg( \\mu S_t\\ dt +\\sigma S_t \\ dW_t \\bigg)  $$<\/p>\n\n\n\n<p>The cross terms dt &amp; dW would decay to zero quickly in the stochastic integration. We therefore would get,<\/p>\n\n\n\n<p>$$ d\\big( \\frac{S_t}{B_t} \\big) = \\sigma ( \\frac{S_t}{B_t} )\\underbrace{\\bigg( \\frac{\\mu &#8211; r}{\\sigma} dt + dW_t \\bigg)}_{d\\tilde{W_t}} $$<\/p>\n\n\n\n<p>The drift disappears, instead we get <span class=\"katex math inline\">d\\big( \\frac{S_t}{B_t} \\big)d\\tilde{W_t}<\/span><\/p>\n\n\n\n<p>$$  d\\big( \\frac{S_t}{B_t} \\big)d\\tilde{W_t} $$<\/p>\n\n\n\n<p>$$ d\\tilde{W_t} = \\frac{\\mu -r}{\\sigma} dt +dW_t $$<\/p>\n\n\n\n<p>by Girsanov&#8217;s Theorem that <span class=\"katex math inline\">\\{a_t \\}_{t\\in [0,T]}<\/span> be and adapted <span class=\"katex math inline\">\\{ \\mathcal{F} \\}_{t\\in [0,T]}<\/span> Ito Process, so that <span class=\"katex math inline\">\\mathbb{Q}<\/span> is an equivalent measure on <span class=\"katex math inline\">\\mathbb{P}<\/span> such that <span class=\"katex math inline\">\\tilde{W_t}<\/span> is a <strong>Brownian Motion<\/strong>, <span class=\"katex math inline\">\\tilde{W_t} = W_t +\\int_0^t a_u du<\/span>.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>$$ dS_t = rS_t \\ d_t +\\sigma S_t \\ d\\tilde{W_t} $$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Implication<\/h4>\n\n\n\n<p>The <strong>implication <\/strong>is that,<\/p>\n\n\n\n<p>We can construct a relicating portfolio <span class=\"katex math inline\">(S_t, B_t)<\/span> with value process <span class=\"katex math inline\">\\{ V_t \\}_{t\\in[0,T]}<\/span>. Under <span class=\"katex math inline\">\\mathbb{Q}<\/span> the discounted value process exists that is a martingale <span class=\"katex math inline\">\\{ e^{-rt}V_t \\}<\/span>.<\/p>\n\n\n\n<p>$$ e^{-rt} C_t =\\mathbb{E}_{\\mathbb{Q}}[ e^{-rT} C_T | \\mathcal{F}_t ] $$<\/p>\n\n\n\n<p>$$ =\\mathbb{E}_{\\mathbb{Q}}[ e^{-rT} V_T | \\mathcal{F}_t ] = e^{-rt} V_t $$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Risk Neutral Expectation Pricing Formula<\/h4>\n\n\n\n<p>We therefore get,<\/p>\n\n\n\n<p>$$ \\frac{C_t}{B_t} = \\mathbb{E}_{\\mathbb{Q}}\\bigg[  \\frac{C_T}{B_t} \\bigg| \\mathcal{F}_t  \\bigg]  $$<\/p>\n\n\n\n<p>$$ B_T=1,\\quad and \\quad B_t=exp\\{ \\int_0^t r_s ds \\} $$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Our basic logic is, $$ \\text{Option Value} =\\text{Discounted Expectation of the Payoff} $$ However, there is not a perfect correlation between the Option and the Underlying Asset. That fact encourage us to do the risk-netural method for valuing an option. Geometric Brownian Motion Non-dividend paying stock, (S_0&gt;0) $$ dS_t = \\mu S_t\\ dt+\\sigma S_t \\ &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=4659\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Risk-Neutral Pricing<\/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":[6,18,26],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4659"}],"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=4659"}],"version-history":[{"count":44,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4659\/revisions"}],"predecessor-version":[{"id":4721,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4659\/revisions\/4721"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4659"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4659"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4659"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}