29:null 2a:["$","title",null,{"children":"Black Scholes Calculator | oscar6echo | Observable"}] 2f:Tb94,viewof form1 = { let name = "S"; let key = (i) => ["", "_min", "_max"].map((e) => name + e)[i]; const S = [115, 80, 150] .map((e, i) => Inputs.number({ label: i === 0 ? "Spot (S), min, max" : null, step: 0.001, value: e, width: i === 0 ? 120 : -20 }) ) .reduce((acc, v, i) => { const k = key(i); acc[k] = v; return acc; }, {}); name = "K"; key = (i) => ["", "_min", "_max"].map((e) => name + e)[i]; const K = [115, 0.0, 200] .map((e, i) => Inputs.number({ label: i === 0 ? "Strike (K), min, max" : null, step: 0.001, value: e, width: i === 0 ? 120 : -20 }) ) .reduce((acc, v, i) => { const k = key(i); acc[k] = v; return acc; }, {}); name = "T"; key = (i) => ["", "_min", "_max"].map((e) => name + e)[i]; const T = [3, 0.01, 10] .map((e, i) => Inputs.number({ label: i === 0 ? "Mat (T), min, max" : null, step: 0.001, value: e, width: i === 0 ? 120 : -20 }) ) .reduce((acc, v, i) => { const k = key(i); acc[k] = v; return acc; }, {}); name = "v"; key = (i) => ["", "_min", "_max"].map((e) => name + e)[i]; const v = [0.03, 0.001, 0.8] .map((e, i) => Inputs.number({ label: i === 0 ? "Vol (v), min, max" : null, step: 0.001, value: e, width: i === 0 ? 120 : -20 }) ) .reduce((acc, v, i) => { const k = key(i); acc[k] = v; return acc; }, {}); name = "r"; key = (i) => ["", "_min", "_max"].map((e) => name + e)[i]; const r = [0.03, -0.02, 0.1] .map((e, i) => Inputs.number({ label: i === 0 ? "Rate (r), min, max" : null, step: 0.001, value: e, width: i === 0 ? 120 : -20 }) ) .reduce((acc, v, i) => { const k = key(i); acc[k] = v; return acc; }, {}); name = "q"; key = (i) => ["", "_min", "_max"].map((e) => name + e)[i]; const q = [0.01, -0.02, 0.05] .map((e, i) => Inputs.number({ label: i === 0 ? "Div (q), min, max" : null, step: 0.001, value: e, width: i === 0 ? 120 : -20 }) ) .reduce((acc, v, i) => { const k = key(i); acc[k] = v; return acc; }, {}); const inputs = { ...S, ...K, ...T, ...v, ...r, ...q }; return Inputs.form(inputs, { template: (obj) => htl.html`

${obj.S} ${obj.S_min} ${obj.S_max}

${obj.K} ${obj.K_min} ${obj.K_max}

${obj.T} ${obj.T_min} ${obj.T_max}

${obj.v} ${obj.v_min} ${obj.v_max}

${obj.r} ${obj.r_min} ${obj.r_max}

${obj.q} ${obj.q_min} ${obj.q_max}

` }); }30:T855,build_chart_2d = () => { const [width, height] = [800, 400]; const { x, y: _ys, y_ref, points, points_full } = data; const ys = [y_ref, ..._ys.filter((e) => e !== y_ref)]; const points_y_ref = points.filter((e) => e.field == y_ref); const scale_x = d3.scaleLinear( [0, width], d3.extent(points_y_ref, (d) => d[x]) ); const scale_y = d3.scaleLinear( [0, height], d3.extent(points_y_ref, (d) => d[y_ref]) ); // const colors = d3["schemeTableau10"]; const colors = d3["schemeCategory10"]; const color = Object.fromEntries(d3.zip(ys, colors)); const curves = ys.map((y) => { if (y === y_ref) { const line = Plot.lineY(points_y_ref, { x, y, stroke: color[y] }); return line; } else { const points_y = points.filter((e) => e.field == y); const scale = d3.scaleLinear( d3.extent(points_y, (d) => d[y]), d3.extent(points_y_ref, (d) => d[y_ref]) ); const line = Plot.lineY( points_y, Plot.mapY((d) => d.map(scale), { x, y, stroke: color[y] }) ); const legend = Plot.text(ys, { x: scale_x(30), y: (d, i) => scale_y(height - (20 + 15 * i)), text: (d) => `--- ${d}`, fill: (d) => color[d] }); return [line, legend]; } }); const pointer_ruleX = Plot.ruleX( points, Plot.pointerX({ x, py: y_ref, stroke: "gray", strokeDasharray: "3,5" }) ); const pointer_dot = Plot.dot( points, Plot.pointerX({ x, y: y_ref, stroke: "black" }) ); const pointer_text = Plot.text( points_full, Plot.pointerX({ px: x, py: y_ref, dy: -17, frameAnchor: "top-right", fontVariant: "tabular-nums", // fill: (d) => "gray", text: (d) => ys.map((y) => `${y} ${d[y].toFixed(3)}`).join(" ") }) ); const pointer_elts = [pointer_ruleX, pointer_dot, pointer_text]; const chart = Plot.plot({ marks: [...curves, ...pointer_elts], x: { nice: false }, y: { axis: "left", label: y_ref, nice: false }, grid: true, width, height }); return chart; }31:Td92,bs = { const erf = (x) => { // constants const a1 = 0.254829592; const a2 = -0.284496736; const a3 = 1.421413741; const a4 = -1.453152027; const a5 = 1.061405429; const p = 0.3275911; const sign = x >= 0 ? 1 : -1; x = Math.abs(x); // A&S formula 7.1.26 const t = 1.0 / (1.0 + p * x); const y = 1.0 - ((((a5 * t + a4) * t + a3) * t + a2) * t + a1) * t * Math.exp(-x * x); return sign * y; }; const cdf = (x, mean, variance) => { if ((x - mean) / variance < -1e4) { return 0; } else if ((x - mean) / variance > 1e4) { return 1; } else { return 0.5 * (1 + erf((x - mean) / Math.sqrt(2 * variance))); } }; // standard normal CDF const N = (z) => { return cdf(z, 0, 1); }; // standard normal PDF const Nprime = (z) => { return (1.0 / Math.sqrt(2 * Math.PI)) * Math.exp(-0.5 * z * z); }; const price_shared = ({ S, K, T, v, r, q }) => { const sqrt_T = Math.sqrt(T); const d1 = (1 / (v * sqrt_T)) * (Math.log(S / K) + (r - q + (v * v) / 2) * T); const d2 = d1 - v * sqrt_T; const PV = Math.exp(-r * T); const PV_K = K * PV; const D = Math.exp(-q * T); const payoff = Math.max(0, S - K); const PV_payoff = PV * payoff; return { sqrt_T, d1, d2, PV, PV_K, D, payoff, PV_payoff }; }; const price_call = (input) => { const { S, K, T, v, r, q } = input; const { sqrt_T, d1, d2, PV, PV_K, D, payoff, PV_payoff } = price_shared(input); const N_d1 = N(d1); const N_minus_d1 = N(-d1); const N_prime_d1 = Nprime(d1); const N_d2 = N(d2); const N_minus_d2 = N(-d2); const N_prime_d2 = Nprime(d2); const price = N_d1 * S * D - N_d2 * PV_K; const delta = D * N_d1; const gamma = (D * N_prime_d1) / (S * v * sqrt_T); const vega = S * D * N_prime_d1 * sqrt_T; const theta = (-D * S * N_prime_d1 * v) / (2 * sqrt_T) - r * PV_K * N_d2 + q * S * D * N_d1; const rho = PV_K * T * N_d2; const voma = (S * D * N_prime_d1 * sqrt_T * d1 * d2) / v; return { d1, d2, N_d1, N_minus_d1, N_prime_d1, N_d2, N_minus_d2, N_prime_d2, PV, PV_K, D, price, delta, gamma, vega, theta, rho, voma, payoff, PV_payoff }; }; const price_put = (input) => { const { S, K, T, v, r, q } = input; const { sqrt_T, d1, d2, PV, PV_K, D, payoff, PV_payoff } = price_shared(input); const N_d1 = N(d1); const N_minus_d1 = N(-d1); const N_prime_d1 = Nprime(d1); const N_d2 = N(d2); const N_minus_d2 = N(-d2); const N_prime_d2 = Nprime(d2); const price = N_minus_d2 * PV_K - N_minus_d1 * S * D; const delta = -D * N_minus_d1; const gamma = (D * N_prime_d1) / (S * v * sqrt_T); const vega = S * D * N_prime_d1 * sqrt_T; const theta = (-D * S * N_prime_d1 * v) / (2 * sqrt_T) + r * PV_K * N_minus_d2 - q * S * D * N_minus_d1; const rho = -PV_K * T * N_minus_d2; const voma = (S * D * N_prime_d1 * sqrt_T * d1 * d2) / v; return { d1, d2, N_d1, N_minus_d1, N_prime_d1, N_d2, N_minus_d2, N_prime_d2, PV, PV_K, D, price, delta, gamma, vega, theta, rho, voma, payoff, PV_payoff }; }; return { price_call, price_put }; }32:T9cd,definitions = [ { name: "equa-diff", formula: tex`\frac{\partial V}{\partial t} + (r-q) S \frac{\partial V}{\partial S} + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} \sigma^2S_t^2 =rV ` }, { name: "condition-limit", formula: tex` \forall S \; V(S, T)=Max(0, S-K)` }, { name: "payoff", type: "greek", symbol: "P/O", formula_call: tex`Max(0, S-K)`, formula_put: tex`Max(0, K-S)` }, { name: "price", type: "greek", symbol: tex`V`, formula_call: tex`Se^{-qT}N(d_1)-Ke^{-rT}N(d_2)`, formula_put: tex`Ke^{-rT}N(-d_2)-Se^{-qT}N(-d_1)` }, { name: "delta", type: "greek", symbol: tex`\Delta=\frac{\partial V}{\partial S}`, formula_call: tex`e^{-qT}N(d_1)`, formula_put: tex`-e^{-qT}N(-d_1)` }, { name: "gamma", type: "greek", symbol: tex`\Gamma=\frac{\partial \Delta}{\partial S}=\frac{\partial^2 V}{\partial S^2}`, formula_call: tex`e^{-qT}\frac{N'(d_1)}{\sigma \sqrt{T}}`, formula_put: "idem" }, { name: "vega", type: "greek", symbol: tex`vega=\frac{\partial V}{\partial \sigma}`, formula_call: tex`Se^{-qT}N'(d_1) \sqrt{T}`, formula_put: tex`Ke^{-rT}N'(d_2) \sqrt{T}` }, { name: "theta", type: "greek", symbol: tex`\Theta=-\frac{\partial V}{\partial T}`, formula_call: tex`-e^{qT}\frac{SN'(d_1) \sigma}{2 \sqrt{T}}-rKe^{-rT}N(d_2)+qSe^{-qT}N(d_1)`, formula_put: tex`-e^{qT}\frac{SN'(d_1) \sigma}{2 \sqrt{T}}+rKe^{-rT}N(-d_2)-qSe^{-qT}N(-d_1)` }, { name: "rho", type: "greek", symbol: tex`\rho=\frac{\partial V}{\partial r}`, formula_call: tex`KTe^{-rT}N(d_2)`, formula_put: tex`-KTe^{-rT}N(-d_2)` }, { name: "voma", type: "greek", symbol: tex`voma=\frac{\partial vega}{\partial \sigma}=\frac{\partial ^2 V}{\partial \sigma ^2}`, formula_call: tex`Se^{-qT}N'(d_1) \sqrt{T}\frac{d_1 d_2}{\sigma}`, formula_put: "idem" }, { name: "d1", type: "math", symbol: tex`d_1`, formula: tex`\frac{log(S/K) + (r-q +\sigma^2/2)T}{\sigma \sqrt{T}}` }, { name: "d2", type: "math", symbol: tex`d_2`, formula: tex`\frac{log(S/K) + (r-q -\sigma^2/2)T}{\sigma \sqrt{T}}=d_1 - \sigma \sqrt{T - t}` }, { name: "cdf", type: "math", symbol: tex`N(x)`, formula: tex`\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{t^{2}}{2}}dt` }, { name: "pdf", type: "math", symbol: tex`N'(x)`, formula: tex`\frac{1}{\sqrt{2 \pi}}e^{-\frac{x^{2}}{2}}` } ]33:[] 2d:{"id":"bd7ab0919ec35fb3","slug":"black-scholes-calculator","title":"Black Scholes Calculator","thumbnail":"61f3612583407adf7bd1a1ecada85b11d43607b8dd247c14ce081d8daa2297e6","access_level":"public","create_time":"$D2024-02-08T13:09:03.203Z","update_time":"$D2024-02-09T15:53:30.408Z","view_time":null,"first_public_time":"$D2024-02-08T13:09:03.203Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"caa24eaf0dba3b8c","login":"oscar6echo","name":"","avatar_url":"https://avatars-next.observableusercontent.com/b9c6f536a16e403a7143c5cf9ef43970ec3b956fd7f1fb31d74113cc741e32ac","tier":"starter_2024"},"version":1175,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"375c413046669a50","login":"oscar6echo","name":"","avatar_url":"https://avatars-next.observableusercontent.com/b9c6f536a16e403a7143c5cf9ef43970ec3b956fd7f1fb31d74113cc741e32ac","tier":null}],"members":[{"type":"individual","id":"375c413046669a50","login":"oscar6echo","name":"","avatar_url":"https://avatars-next.observableusercontent.com/b9c6f536a16e403a7143c5cf9ef43970ec3b956fd7f1fb31d74113cc741e32ac","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Black Scholes Calculator","cells":[{"id":0,"value":"# Black Scholes Calculator\n\nThe [Black Scholes model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model) gives the price of European options under theoretical market conditions. It was introduced by Fischer Black and Myron Scholes in their 1973 paper [The pricing of options and corporate liabilites](https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":279,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":356,"value":"viewof form2 = Inputs.form({\n dim: Inputs.radio([\"2D\", \"3D\"], { label: \"Dimension\", value: \"2D\" }),\n type: Inputs.radio([\"Call\", \"Put\"], { label: \"Type\", value: \"Call\" }),\n n_step: Inputs.number({\n label: \"# steps\",\n width: 100,\n value: 100,\n min: 5,\n max: 1000,\n step: 1\n })\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":554,"value":"viewof form4 = {\n const inputs = {\n y_ref: Inputs.select(form3.y, {\n label: \"y axis\",\n width: 100,\n disabled: form2.dim === \"3D\"\n })\n };\n return Inputs.form(inputs);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":370,"value":"viewof form3 = {\n const ins = [\"S\", \"K\", \"T\", \"v\", \"r\", \"q\"];\n const outs = [\n \"d1\",\n \"d2\",\n \"N_d1\",\n \"N_d2\",\n \"N_minus_d1\",\n \"N_minus_d2\",\n \"N_prime_d1\",\n \"N_prime_d2\",\n \"PV\",\n \"PV_K\",\n \"D\",\n \"price\",\n \"delta\",\n \"gamma\",\n \"vega\",\n \"theta\",\n \"rho\",\n \"voma\",\n \"payoff\",\n \"PV_payoff\"\n ];\n\n const inputs_2d = {\n x: Inputs.select(ins, { label: \"x\", width: 100, value: \"S\" }),\n y: Inputs.select(outs, {\n label: \"y\",\n width: 100,\n value: [\"price\", \"delta\", \"vega\"],\n multiple: true\n })\n };\n const inputs_3d = {\n x: Inputs.select(ins, { label: \"x\", width: 100, value: \"S\" }),\n y: Inputs.select(ins, { label: \"y\", width: 100, value: \"T\" }),\n z: Inputs.select(outs, { label: \"z\", width: 100, value: \"price\" })\n };\n\n const inputs = form2.dim === \"3D\" ? inputs_3d : inputs_2d;\n\n return Inputs.form(inputs);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":510,"value":"{\n if (form2.dim === \"2D\") return build_chart_2d();\n if (form2.dim === \"3D\") return build_chart_3d();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":368,"value":"data = {\n if (form2.dim === \"2D\") return build_data_2d();\n if (form2.dim === \"3D\") return build_data_3d();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":944,"value":"build_chart_3d = () => {\n const [width, height] = [800, 600];\n const surface = [\n {\n z: data.points_vis,\n type: \"surface\",\n showscale: false,\n hidesurface: false,\n contours: {\n z: {\n show: true,\n usecolormap: true,\n highlightcolor: \"red\",\n project: { z: true }\n }\n }\n }\n ];\n const layout = {\n scene: { camera: { eye: { x: -1.25, y: -1.25, z: 0.1 } } },\n width: width,\n height: height,\n margin: {\n l: 0,\n r: 0,\n b: 0,\n t: 0\n }\n };\n\n let div = d3\n .create(\"div\")\n .style(\"width\", `${width}px`)\n .style(\"height\", `${height}px`)\n .style(\"border\", `1px solid gray`);\n\n const plot = plotly.newPlot(div.node(), surface, layout);\n\n return plot;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":941,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":916,"value":"build_data_3d = () => {\n const price_option = form2.type === \"Call\" ? bs.price_call : bs.price_put;\n\n const keys = Object.keys(form1).filter((e) => e.startsWith(form3.x));\n const x_min = form1[form3.x + \"_min\"];\n const x_max = form1[form3.x + \"_max\"];\n const y_min = form1[form3.y + \"_min\"];\n const y_max = form1[form3.y + \"_max\"];\n\n const { S, K, T, v, r, q } = form1;\n\n const points_vis = d3\n .range(x_min, x_max, (x_max - x_min) / form2.n_step)\n .map((x, i) => {\n return d3\n .range(y_min, y_max, (y_max - y_min) / form2.n_step)\n .map((y, j) => {\n const input = {\n ...{ S, K, T, v, r, q },\n ...{ [form3.x]: x, [form3.y]: y }\n };\n const priced = price_option(input);\n const point = { ...input, ...priced };\n return point[form3.z];\n });\n });\n\n console.log({ points_vis });\n\n return {\n points_vis,\n x: form3.x,\n y: form3.y,\n y_ref: form4.y_ref\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":904,"value":"build_data_2d = () => {\n const price_option = form2.type === \"Call\" ? bs.price_call : bs.price_put;\n\n const keys = Object.keys(form1).filter((e) => e.startsWith(form3.x));\n const x_min = form1[form3.x + \"_min\"];\n const x_max = form1[form3.x + \"_max\"];\n\n const { S, K, T, v, r, q } = form1;\n\n const points_full = d3\n .range(x_min, x_max, (x_max - x_min) / form2.n_step)\n .map((x) => {\n const input = { ...{ S, K, T, v, r, q }, ...{ [form3.x]: x } };\n const priced = price_option(input);\n const point = { ...input, ...priced };\n return point;\n });\n\n const points = points_full\n .map((e) => {\n const arr = form3.y.map((y) => ({\n [form3.x]: e[form3.x],\n [y]: e[y],\n field: y\n }));\n return arr;\n })\n .flat();\n\n return {\n points_full,\n points,\n x: form3.x,\n y: form3.y,\n y_ref: form4.y_ref\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":188,"value":"---\n\n## Theory\n\n\n### Differential Equation\n\n\n\nThe price and their greeks i.e. partial derivatives are deducted from the following [differential equation](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation) and limit condition: \n\n+ ${formula('equa-diff')}\n+ ${formula('condition-limit')}\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":194,"value":"### Solution\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":67,"value":"{\n const data = definitions.filter((e) => e.type === \"greek\");\n const katex = (d) => htl.html`${d}`;\n return Inputs.table(data, {\n format: { symbol: katex, formula_call: katex, formula_put: katex },\n columns: [\"name\", \"symbol\", \"formula_call\", \"formula_put\"],\n width: {\n name: 70,\n symbol: 170\n }\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":250,"value":"{\n const data = definitions.filter((e) => e.type === \"math\");\n const katex = (d) => htl.html`${d}`;\n return Inputs.table(data, {\n format: { symbol: katex, formula: katex },\n columns: [\"name\", \"symbol\", \"formula\"],\n width: {\n name: 70,\n symbol: 170\n }\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1154,"value":"---\n## Annex","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":265,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":132,"value":"formula = (name) => {\n const obj = definitions.find((e) => e.name === name);\n return obj.formula || obj.formula_call;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":892,"value":"plotly = require(\"plotly.js-dist-min@2.29.0\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":712,"value":"d3 = require(\"d3@7\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2e:"$2d:body"