{"pageProps":{"post":{"_id":"1071b830-efd9-4603-b955-5a799ede0770","title":"Black Scholes Model with Stock Simulation","date":"2021-07-17T16:28:00.000Z","excerpt":"The article explores the use of geometric Brownian Motion (GBM) to simulate the price of stocks with extension to the Black-Scholes-Merton (BSM) model.","coverImage":{"_type":"image","asset":{"_ref":"image-4681d383e20f2b026034aac48e7ad1dcda2a68d2-640x480-png","_type":"reference"}},"slug":"black-scholes","content":{"compiledSource":"/*@jsxRuntime automatic @jsxImportSource react*/\nconst {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0];\nconst {useMDXComponents: _provideComponents} = arguments[0];\nfunction _createMdxContent(props) {\n const _components = Object.assign({\n p: \"p\",\n h2: \"h2\",\n a: \"a\",\n span: \"span\",\n h3: \"h3\",\n math: \"math\",\n semantics: \"semantics\",\n mrow: \"mrow\",\n msub: \"msub\",\n mi: \"mi\",\n annotation: \"annotation\",\n div: \"div\",\n mo: \"mo\",\n mn: \"mn\",\n mfrac: \"mfrac\",\n msup: \"msup\",\n pre: \"pre\",\n code: \"code\",\n img: \"img\",\n mtable: \"mtable\",\n mtr: \"mtr\",\n mtd: \"mtd\",\n mstyle: \"mstyle\",\n munderover: \"munderover\",\n mspace: \"mspace\",\n msqrt: \"msqrt\",\n svg: \"svg\",\n path: \"path\",\n strong: \"strong\"\n }, _provideComponents(), props.components);\n return _jsxs(_Fragment, {\n children: [_jsx(_components.p, {\n children: \"The article explores the use of geometric Brownian Motion (GBM) to simulate the price of stocks. The concept naturally extends to the Black-Scholes-Merton (BSM) model which is widely used to estimate the pricing variation of financial instruments such as an options contract. 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_jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"σ\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\sigma\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"σ\"\n })]\n })\n })]\n })\n }), \" are constants. We also assume the stock does not pay any dividends, there are no transaction costs and the continuously compounding interest rate is \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n children: \"r\"\n }), _jsx(_components.mo, {\n children: \">\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"r > 0\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.5782em\",\n verticalAlign: \"-0.0391em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.02778em\"\n },\n children: \"r\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \">\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6444em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: \"0\"\n })]\n })]\n })]\n })\n }), \". Note that \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"W\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"W_t\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.13889em\"\n },\n children: \"W\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2806em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.1389em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \" is normally distributed with mean \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mn, {\n children: \"0\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"0\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6444em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: \"0\"\n })]\n })\n })]\n })\n }), \" and variance \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"t\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"t\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6151em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"t\"\n })]\n })\n })]\n })\n }), \". For an arbitrary initial value \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"S_0\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \" the above SDE has the analytic solution:\"]\n }), \"\\n\", _jsx(_components.div, {\n className: \"math math-display\",\n children: _jsx(_components.span, {\n className: \"katex-display\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n display: \"block\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.mi, {\n children: \"exp\"\n }), _jsx(_components.mo, {\n children: \"⁡\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mo, {\n fence: \"true\",\n children: \"[\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mo, {\n fence: \"true\",\n children: \"(\"\n }), _jsx(_components.mi, {\n children: \"μ\"\n }), _jsx(_components.mo, {\n children: \"−\"\n }), _jsxs(_components.mfrac, {\n children: [_jsxs(_components.msup, {\n children: [_jsx(_components.mi, {\n children: \"σ\"\n }), _jsx(_components.mn, {\n children: \"2\"\n })]\n }), _jsx(_components.mn, {\n children: \"2\"\n })]\n }), _jsx(_components.mo, {\n fence: \"true\",\n children: \")\"\n })]\n }), _jsx(_components.mi, {\n children: \"t\"\n }), _jsx(_components.mo, {\n children: \"+\"\n }), _jsx(_components.mi, {\n children: \"σ\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"W\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n }), _jsx(_components.mo, {\n fence: \"true\",\n children: \"]\"\n })]\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"S_t = S_0 \\\\exp \\\\left[ \\\\left( \\\\mu - \\\\frac{\\\\sigma^2}{2} \\\\right) t + \\\\sigma W_t \\\\right]\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2806em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"2.4411em\",\n verticalAlign: \"-0.95em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.1667em\"\n }\n }), _jsx(_components.span, {\n className: \"mop\",\n children: \"exp\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.1667em\"\n }\n }), _jsxs(_components.span, {\n className: \"minner\",\n children: [_jsx(_components.span, {\n className: \"mopen delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size3\",\n children: \"[\"\n })\n }), _jsxs(_components.span, {\n className: \"minner\",\n children: [_jsx(_components.span, {\n className: \"mopen delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size3\",\n children: \"(\"\n })\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"μ\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsx(_components.span, {\n className: \"mbin\",\n children: \"−\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mopen nulldelimiter\"\n }), _jsx(_components.span, {\n className: \"mfrac\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"1.4911em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-2.314em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsx(_components.span, {\n className: \"mord\",\n children: \"2\"\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3.23em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"frac-line\",\n style: {\n borderBottomWidth: \"0.04em\"\n }\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3.677em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"mord\",\n children: _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"σ\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsx(_components.span, {\n className: \"vlist-t\",\n children: _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.8141em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-3.063em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"2\"\n })\n })]\n })\n })\n })\n })\n })]\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.686em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"mclose nulldelimiter\"\n })]\n }), _jsx(_components.span, {\n className: \"mclose delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size3\",\n children: \")\"\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.1667em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"t\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsx(_components.span, {\n className: \"mbin\",\n children: \"+\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"σ\"\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.13889em\"\n },\n children: \"W\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2806em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.1389em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mclose delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size3\",\n children: \"]\"\n })\n })]\n })]\n })]\n })]\n })\n })\n }), \"\\n\", _jsx(_components.p, {\n children: \"The GBM is technically a Markov process. The stock price hence follows a random walk and is consistent with the weak form of the efficient market hypothesis (EMH).\"\n }), \"\\n\", _jsx(_components.pre, {\n className: \"language-python\",\n children: _jsxs(_components.code, {\n className: \"language-python code-highlight\",\n children: [_jsxs(_components.span, {\n className: \"code-line\",\n children: [_jsx(_components.span, {\n className: \"token comment\",\n children: \"# Calculate stock price at time t\"\n }), \"\\n\"]\n }), _jsxs(_components.span, {\n className: \"code-line\",\n children: [\"S \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"=\"\n }), \" S0 \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"*\"\n }), \" np\", _jsx(_components.span, {\n className: \"token punctuation\",\n children: \".\"\n }), \"exp\", _jsx(_components.span, {\n className: \"token punctuation\",\n children: \"(\"\n }), \"r \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"*\"\n }), \" t \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"-\"\n }), \" \", _jsx(_components.span, {\n className: \"token number\",\n children: \"1\"\n }), _jsx(_components.span, {\n className: \"token operator\",\n children: \"/\"\n }), _jsx(_components.span, {\n className: \"token number\",\n children: \"2\"\n }), \" \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"*\"\n }), \" sigma\", _jsx(_components.span, {\n className: \"token operator\",\n children: \"**\"\n }), _jsx(_components.span, {\n className: \"token number\",\n children: \"2\"\n }), \" \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"*\"\n }), \" t \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"+\"\n }), \" sigma \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"*\"\n }), \" np\", _jsx(_components.span, {\n className: \"token punctuation\",\n children: \".\"\n }), \"random\", _jsx(_components.span, {\n className: \"token punctuation\",\n children: \".\"\n }), \"randn\", _jsx(_components.span, {\n className: \"token punctuation\",\n children: \"(\"\n }), _jsx(_components.span, {\n className: \"token punctuation\",\n children: \")\"\n }), \" \", _jsx(_components.span, {\n className: \"token operator\",\n children: \"*\"\n }), \" np\", _jsx(_components.span, {\n className: \"token punctuation\",\n children: \".\"\n }), \"sqrt\", _jsx(_components.span, {\n className: \"token punctuation\",\n children: \"(\"\n }), \"t\", _jsx(_components.span, {\n className: \"token punctuation\",\n children: \")\"\n }), _jsx(_components.span, {\n className: \"token punctuation\",\n children: \")\"\n }), \"\\n\"]\n })]\n })\n }), \"\\n\", _jsx(_components.p, {\n children: _jsx(_components.img, {\n src: \"https://cdn.sanity.io/images/xj3nsfqa/production/b313d4ff114ad65fcee36eedb493792174d5810a-640x480.png\",\n alt: \"image\"\n })\n }), \"\\n\", _jsxs(_components.h3, {\n id: \"probability-density-function\",\n children: [_jsx(_components.a, {\n className: \"anchor\",\n href: \"#probability-density-function\",\n children: _jsx(_components.span, {\n className: \"icon icon-link\"\n })\n }), \"Probability density function\"]\n }), \"\\n\", _jsxs(_components.p, {\n children: [\"Since the exponent of \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"S_t\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2806em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \" is normally distributed, we can use the change of variable formula to calculate the probability distribution of \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"S_t\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2806em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \". Note that for \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"S_t\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2806em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \" the exponent is normally distributed with mean \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n children: \"ln\"\n }), _jsx(_components.mo, {\n children: \"⁡\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mn, {\n children: \"0\"\n })]\n }), _jsx(_components.mo, {\n children: \"+\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mo, {\n fence: \"true\",\n children: \"(\"\n }), _jsx(_components.mi, {\n children: \"μ\"\n }), _jsx(_components.mo, {\n children: \"−\"\n }), _jsxs(_components.mfrac, {\n children: [_jsxs(_components.msup, {\n children: [_jsx(_components.mi, {\n children: \"σ\"\n }), _jsx(_components.mn, {\n children: \"2\"\n })]\n }), _jsx(_components.mn, {\n children: \"2\"\n })]\n }), _jsx(_components.mo, {\n fence: \"true\",\n children: \")\"\n })]\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\ln S_0 + \\\\left( \\\\mu - \\\\frac{\\\\sigma^2}{2} \\\\right) t\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8444em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsx(_components.span, {\n className: \"mop\",\n children: \"ln\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.1667em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"0\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsx(_components.span, {\n className: \"mbin\",\n children: \"+\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1.8em\",\n verticalAlign: \"-0.65em\"\n }\n }), _jsxs(_components.span, {\n className: \"minner\",\n children: [_jsx(_components.span, {\n className: \"mopen delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size2\",\n children: \"(\"\n })\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"μ\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsx(_components.span, {\n className: \"mbin\",\n children: \"−\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mopen nulldelimiter\"\n }), _jsx(_components.span, {\n className: \"mfrac\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsxs(_components.span, {\n className: \"vlist\",\n style: {\n height: \"1.0179em\"\n },\n children: [_jsxs(_components.span, {\n style: {\n top: \"-2.655em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"2\"\n })\n })\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3.23em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"frac-line\",\n style: {\n borderBottomWidth: \"0.04em\"\n }\n })]\n }), _jsxs(_components.span, {\n style: {\n top: \"-3.394em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"3em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: _jsxs(_components.span, {\n className: \"mord mtight\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"σ\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsx(_components.span, {\n className: \"vlist-t\",\n children: _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.8913em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.931em\",\n marginRight: \"0.0714em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.5em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size3 size1 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"2\"\n })\n })]\n })\n })\n })\n })\n })]\n })\n })\n })]\n })]\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.345em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"mclose nulldelimiter\"\n })]\n }), _jsx(_components.span, {\n className: \"mclose delimcenter\",\n style: {\n top: \"0em\"\n },\n children: _jsx(_components.span, {\n className: \"delimsizing size2\",\n children: \")\"\n })\n })]\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.1667em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"t\"\n })]\n })]\n })]\n })\n }), \" and variance \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msup, {\n children: [_jsx(_components.mi, {\n children: \"σ\"\n }), _jsx(_components.mn, {\n children: \"2\"\n })]\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"\\\\sigma^2 t\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8141em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.03588em\"\n },\n children: \"σ\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsx(_components.span, {\n className: \"vlist-t\",\n children: _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.8141em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-3.063em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"2\"\n })\n })]\n })\n })\n })\n })\n })]\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"t\"\n })]\n })\n })]\n })\n }), \". Hence, \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"S_t\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.8333em\",\n verticalAlign: \"-0.15em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2806em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n })]\n })\n })]\n })\n }), \" follows a log-normal distribution:\"]\n }), \"\\n\", _jsx(_components.div, {\n className: \"math math-display\",\n children: _jsx(_components.span, {\n className: \"katex-display\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n display: \"block\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n }), _jsx(_components.mo, {\n children: \"∼\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n mathvariant: \"script\",\n children: \"L\"\n }), _jsx(_components.mi, {\n 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Thereafter, the net present value (NPV) of the strike price \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"K\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"K\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6833em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07153em\"\n },\n children: \"K\"\n })]\n })\n })]\n })\n }), \" with risk-free interest rate \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"r\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"r\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.4306em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.02778em\"\n },\n children: \"r\"\n })]\n })\n })]\n })\n }), \" and time to maturity \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"t\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"t\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6151em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"t\"\n })]\n })\n })]\n })\n }), \" is multiplied by the cumulative standard normal distribution and subtracted from the resulting value. The call option price \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsx(_components.mrow, {\n children: _jsx(_components.mi, {\n children: \"C\"\n })\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"C\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6833em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07153em\"\n },\n children: \"C\"\n })]\n })\n })]\n })\n }), \" is therefore given by:\"]\n }), \"\\n\", _jsx(_components.div, {\n className: \"math math-display\",\n children: _jsx(_components.span, {\n className: \"katex-display\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n display: \"block\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n children: \"C\"\n }), _jsx(_components.mo, {\n children: \"=\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"S\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n }), _jsx(_components.mi, {\n children: \"N\"\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \"(\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"d\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \")\"\n }), _jsx(_components.mo, {\n children: \"−\"\n }), _jsx(_components.mi, {\n children: \"K\"\n }), _jsxs(_components.msup, {\n children: [_jsx(_components.mi, {\n children: \"e\"\n }), _jsxs(_components.mrow, {\n children: [_jsx(_components.mo, {\n children: \"−\"\n }), _jsx(_components.mi, {\n children: \"r\"\n }), _jsx(_components.mi, {\n children: \"t\"\n })]\n })]\n }), _jsx(_components.mi, {\n children: \"N\"\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \"(\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"d\"\n }), _jsx(_components.mn, {\n children: \"2\"\n })]\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \")\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"C = S_t N(d_1) - K e^{-rt} N(d_2)\"\n })]\n })\n })\n }), _jsxs(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: [_jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"0.6833em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07153em\"\n },\n children: \"C\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n }), _jsx(_components.span, {\n className: \"mrel\",\n children: \"=\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2778em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1em\",\n verticalAlign: \"-0.25em\"\n }\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.05764em\"\n },\n children: \"S\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.2806em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"-0.0576em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.10903em\"\n },\n children: \"N\"\n }), _jsx(_components.span, {\n className: \"mopen\",\n children: \"(\"\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"d\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"0em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mclose\",\n children: \")\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n }), _jsx(_components.span, {\n className: \"mbin\",\n children: \"−\"\n }), _jsx(_components.span, {\n className: \"mspace\",\n style: {\n marginRight: \"0.2222em\"\n }\n })]\n }), _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1.0936em\",\n verticalAlign: \"-0.25em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.07153em\"\n },\n children: \"K\"\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"e\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsx(_components.span, {\n className: \"vlist-t\",\n children: _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.8436em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-3.113em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsxs(_components.span, {\n className: \"mord mtight\",\n children: [_jsx(_components.span, {\n className: \"mord mtight\",\n children: \"−\"\n }), _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n style: {\n marginRight: \"0.02778em\"\n },\n children: \"r\"\n }), _jsx(_components.span, {\n className: \"mord mathnormal mtight\",\n children: \"t\"\n })]\n })\n })]\n })\n })\n })\n })\n })]\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.10903em\"\n },\n children: \"N\"\n }), _jsx(_components.span, {\n className: \"mopen\",\n children: \"(\"\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"d\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"0em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 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_jsx(_components.mo, {\n stretchy: \"false\",\n children: \"(\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"d\"\n }), _jsx(_components.mn, {\n children: \"2\"\n })]\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \")\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"N(d_2)\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1em\",\n verticalAlign: \"-0.25em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.10903em\"\n },\n children: \"N\"\n }), _jsx(_components.span, {\n className: \"mopen\",\n children: \"(\"\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"d\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"0em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"2\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mclose\",\n children: \")\"\n })]\n })\n })]\n })\n }), \" is the risk-adjusted probability that the option will be exercised. \", _jsx(_components.span, {\n className: \"math math-inline\",\n children: _jsxs(_components.span, {\n className: \"katex\",\n children: [_jsx(_components.span, {\n className: \"katex-mathml\",\n children: _jsx(_components.math, {\n xmlns: \"http://www.w3.org/1998/Math/MathML\",\n children: _jsxs(_components.semantics, {\n children: [_jsxs(_components.mrow, {\n children: [_jsx(_components.mi, {\n children: \"N\"\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \"(\"\n }), _jsxs(_components.msub, {\n children: [_jsx(_components.mi, {\n children: \"d\"\n }), _jsx(_components.mn, {\n children: \"1\"\n })]\n }), _jsx(_components.mo, {\n stretchy: \"false\",\n children: \")\"\n })]\n }), _jsx(_components.annotation, {\n encoding: \"application/x-tex\",\n children: \"N(d_1)\"\n })]\n })\n })\n }), _jsx(_components.span, {\n className: \"katex-html\",\n \"aria-hidden\": \"true\",\n children: _jsxs(_components.span, {\n className: \"base\",\n children: [_jsx(_components.span, {\n className: \"strut\",\n style: {\n height: \"1em\",\n verticalAlign: \"-0.25em\"\n }\n }), _jsx(_components.span, {\n className: \"mord mathnormal\",\n style: {\n marginRight: \"0.10903em\"\n },\n children: \"N\"\n }), _jsx(_components.span, {\n className: \"mopen\",\n children: \"(\"\n }), _jsxs(_components.span, {\n className: \"mord\",\n children: [_jsx(_components.span, {\n className: \"mord mathnormal\",\n children: \"d\"\n }), _jsx(_components.span, {\n className: \"msupsub\",\n children: _jsxs(_components.span, {\n className: \"vlist-t vlist-t2\",\n children: [_jsxs(_components.span, {\n className: \"vlist-r\",\n children: [_jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.3011em\"\n },\n children: _jsxs(_components.span, {\n style: {\n top: \"-2.55em\",\n marginLeft: \"0em\",\n marginRight: \"0.05em\"\n },\n children: [_jsx(_components.span, {\n className: \"pstrut\",\n style: {\n height: \"2.7em\"\n }\n }), _jsx(_components.span, {\n className: \"sizing reset-size6 size3 mtight\",\n children: _jsx(_components.span, {\n className: \"mord mtight\",\n children: \"1\"\n })\n })]\n })\n }), _jsx(_components.span, {\n className: \"vlist-s\",\n children: \"​\"\n })]\n }), _jsx(_components.span, {\n className: \"vlist-r\",\n children: _jsx(_components.span, {\n className: \"vlist\",\n style: {\n height: \"0.15em\"\n },\n children: _jsx(_components.span, {})\n })\n })]\n })\n })]\n }), _jsx(_components.span, {\n className: \"mclose\",\n children: \")\"\n })]\n })\n })]\n })\n }), \" is the factor by which the present value of contingent receipt of the stock exceeds the current stock price.\"]\n }), \"\\n\", _jsx(_components.div, {\n className: \"math math-display\",\n children: _jsx(_components.span, {\n className: \"katex-display\",\n 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