30:Ta4e, Plots=html`

Plots

Price	${tex`P_C = e^{-rT} \left(S e^{bT} \cdot N(d_1) - K \cdot N(d_2)\right)`} ${tex`P_P = e^{-rT} \left(K \cdot N(-d_2) - S e^{bT} \cdot N(-d_1)\right)`}
Delta	${tex`\Delta_C = e^{(b - r)T} \cdot N(d_1)`} ${tex`\Delta_P = e^{(b - r)T} \cdot N(-d_1)`}
Gamma	${tex`\Gamma_C = \Gamma_P = {e^{(b - r)T} \over S\sigma\sqrt{T} } \cdot P(d_1)`}
Vega	${tex`V_C = V_P = e^{(b - r)T} S\sqrt{T} \cdot P(d_1)`}
Rho	${tex`\rho_{C, P} = {\partial P_{C, P}\over \partial r} + {\partial P_{C, P}\over \partial b}`} ${tex` {\partial P_{C, P}\over \partial r} = -T \cdot P_{C, P} `}
Carry Rho	${tex`\rho^C_{C, P} = {\partial P_{C, P}\over \partial b}`} ${tex` {\partial P_{C, P}\over \partial b} = S\cdot T \cdot \Delta_{C, P} `}
Theta	${tex`\theta^C_{C, P} = {\partial P_{C, P}\over \partial t}`}
Charm	${tex`c_{C, P} = {\partial^2 P_{C, P}\over \partial t\partial S}`}
Theta 2	${tex`\theta^2_{C, P} = {\partial^2 P_{C, P}\over \partial t^2}`}

`31:T85e,function DrawValues(model) { const kMin =parseFloat(MinStrike); const kMax = parseFloat(MaxStrike); const nStrikes = parseInt(NumOfStrikes); const dK = (kMax - kMin) / (nStrikes - 1); var strikes = []; var callPrice = [], putPrice = []; var callDelta = [], putDelta = []; var callGamma = [], putGamma = []; var callVega = [], putVega = []; var callRho = [], putRho = []; var callCarryRho = [], putCarryRho = []; var callTheta = [], putTheta = []; var callCharm = [], putCharm = []; var callTheta2 = [], putTheta2 = []; var tmpC = 0.0, tmpP = 0.0; for (let i = 0; i < nStrikes; ++i) { strikes.push(kMin + i * dK); model.K = strikes[strikes.length - 1]; [tmpC, tmpP] = model.Price(); callPrice.push(tmpC), putPrice.push(tmpP); [tmpC, tmpP] = model.Delta(); callDelta.push(tmpC), putDelta.push(tmpP); [tmpC, tmpP] = model.Gamma(); callGamma.push(tmpC), putGamma.push(tmpP); [tmpC, tmpP] = model.Vega(); callVega.push(tmpC), putVega.push(tmpP); [tmpC, tmpP] = model.Rho(); callRho.push(tmpC), putRho.push(tmpP); [tmpC, tmpP] = model.CarryRho(); callCarryRho.push(tmpC), putCarryRho.push(tmpP); [tmpC, tmpP] = model.Theta(); callTheta.push(tmpC), putTheta.push(tmpP); [tmpC, tmpP] = model.Charm(); callCharm.push(tmpC), putCharm.push(tmpP); [tmpC, tmpP] = model.Theta2(); callTheta2.push(tmpC), putTheta2.push(tmpP); } Plot(strikes, callPrice, putPrice, "Price", "price-chart"); Plot(strikes, callDelta, putDelta, "Delta", "delta-chart"); Plot(strikes, callGamma, putGamma, "Gamma", "gamma-chart"); Plot(strikes, callVega, putVega, "Vega", "vega-chart"); Plot(strikes, callRho, putRho, "Rho", "rho-chart"); Plot(strikes, callCarryRho, putCarryRho, "Carry Rho", "carry-rho-chart"); Plot(strikes, callTheta, putTheta, "Theta", "theta-chart"); Plot(strikes, callCharm, putCharm, "Charm", "charm-chart"); Plot(strikes, callTheta2, putTheta2, "Theta2", "theta2-chart"); } 32:T1093, class PricingModel { constructor(S, K, r, b, sigma, T, dayCountingConvention) { this.S = Math.max(1e-4, Number(S)); if (this.S <= 0.0) throw RangeError("Spot price must be positive"); this.K = Math.max(1e-4, Number(K)); if (this.K <= 0.0) throw RangeError("Strike price must be positive"); this.r = Number(r); this.b = Number(b); this.sigma = Math.max(1e-4, Number(sigma)); if (this.sigma <= 0.0) throw RangeError("Volatility must be positive"); T = Math.max(1e-4, Number(T)); if (this.T <= 0.0) throw RangeError("Strike price must be positive"); dayCountingConvention = Number(dayCountingConvention); if (ALLOWED_DAY_COUNTING_CONVENTION.indexOf(dayCountingConvention) === -1) throw RangeError("Invalid day counting convention"); this.T = T / dayCountingConvention; } Price() { throw TypeError("This method must be overridden"); } __FiniteDifferenceFirstDerivative(paramDelegate) { const dp = 1e-4; paramDelegate(this, dp); const [callPriceUp, putPriceUp] = this.Price(); paramDelegate(this, -2.0 * dp); const [callPriceDown, putPriceDown] = this.Price(); paramDelegate(this, dp); const callDerivative = (callPriceUp - callPriceDown) / (2.0 * dp); const putDerivative = (putPriceUp - putPriceDown) / (2.0 * dp); return [callDerivative, putDerivative]; } __FiniteDifferenceSecondDerivative(paramDelegate) { const dp = 1e-4; paramDelegate(this, dp); const [callPriceUp, putPriceUp] = this.Price(); paramDelegate(this, -2.0 * dp); const [callPriceDown, putPriceDown] = this.Price(); paramDelegate(this, dp); const [callPrice, putPrice] = this.Price(); const callDerivative = (callPriceUp - 2.0 * callPrice + callPriceDown) / (dp * dp); const putDerivative = (putPriceUp - 2.0 * putPrice + putPriceDown) / (dp * dp); return [callDerivative, putDerivative]; } __FiniteDifferenceSecondPartialDerivative(paramDelegate1, paramDelegate2) { const dp = 1e-4; const dq = 1e-4; paramDelegate1(this, dp); paramDelegate1(this, dq); const [callPriceUpUp, putPriceUpUp] = this.Price(); paramDelegate2(this, -2.0 * dq); const [callPriceUpDown, putPriceUpDown] = this.Price(); paramDelegate1(this, -2.0 * dp); const [callPriceDownDown, putPriceDownDown] = this.Price(); paramDelegate2(this, +2.0 * dq); const [callPriceDownUp, putPriceDownUp] = this.Price(); paramDelegate1(this, +dp); paramDelegate2(this, -dq); const callDerivative = (callPriceUpUp - callPriceUpDown - callPriceDownUp + callPriceDownDown) / (4.0 * dp * dq); const putDerivative = (putPriceUpUp - putPriceUpDown - putPriceDownUp + putPriceDownDown) / (4.0 * dp * dq); return [callDerivative, putDerivative]; } Delta() { return this.__FiniteDifferenceFirstDerivative(function (that, dp) { that.S += dp; }); } Gamma() { return this.__FiniteDifferenceSecondDerivative(function (that, dp) { that.S += dp; }); } Vega() { return this.__FiniteDifferenceFirstDerivative(function (that, dp) { that.sigma += dp; }); } Rho() { const rho = this._Rho(); const rhoB = this.CarryRho(); return [rho[0] + rhoB[0], rho[1] + rhoB[1]]; } _Rho() { return this.__FiniteDifferenceFirstDerivative(function (that, dp) { that.r += dp; }); } CarryRho() { return this.__FiniteDifferenceFirstDerivative(function (that, dp) { that.b += dp; }); } Theta() { return this.__FiniteDifferenceFirstDerivative(function (that, dp) { that.T -= dp; }); } Charm() { return this.__FiniteDifferenceSecondPartialDerivative(function (that, dp) { that.T -= dp; }, function (that, dp) { that.S += dp; }); } Theta2() { return this.__FiniteDifferenceSecondDerivative(function (that, dp) { that.T -= dp; }); } }; 33:Ta17,class BlackScholes extends PricingModel { constructor(S, K, r, b, sigma, T, dayCountingConvention) { super(S, K, r, b, sigma, T, dayCountingConvention); } Price() { var [d1, d2] = this.GetD(); var Nd1 = std_cdf(d1), Nd2 = std_cdf(d2); const discountFactor = Math.exp(-this.r * this.T); const growthFactor = Math.exp(this.b * this.T); const callPrice = discountFactor * (this.S * growthFactor * Nd1 - this.K * Nd2); const forwardPrice = discountFactor * (this.S * growthFactor - this.K); const putPrice = callPrice - forwardPrice; // call-put parity return [callPrice, putPrice]; } Delta() { var [d1, _] = this.GetD(); var Nd1 = std_cdf(d1); const growthFactorTimesDiscountFactor = Math.exp((this.b - this.r) * this.T); const callDelta = growthFactorTimesDiscountFactor * Nd1; const putDelta = growthFactorTimesDiscountFactor * (Nd1 - 1.0); return [callDelta, putDelta]; } Gamma() { var [d1, _] = this.GetD(); const Pd1 = std_pdf(d1); const oneOverS = 1.0 / this.S; const oneOverSigmaSqrtT = 1.0 / (this.sigma * Math.sqrt(this.T)); const growthFactorTimesDiscountFactor = Math.exp((this.b - this.r) * this.T); const gamma = oneOverS * Pd1 * oneOverSigmaSqrtT * growthFactorTimesDiscountFactor; return [gamma, gamma]; } Vega() { var [d1, _] = this.GetD(); const Pd1 = std_pdf(d1); const growthFactorTimesDiscountFactor = Math.exp((this.b - this.r) * this.T); const vega = this.S * Math.sqrt(this.T) * growthFactorTimesDiscountFactor * Pd1; return [vega, vega]; } _Rho() { const price = this.Price(); return [-this.T * price[0], -this.T * price[1]]; } CarryRho() { var [d1, _] = this.GetD(); var Nd1 = std_cdf(d1); const growthFactorTimesDiscountFactor = Math.exp((this.b - this.r) * this.T); const callCarryRho = this.S * this.T * Nd1 * growthFactorTimesDiscountFactor; const putCarryRho = -this.S * this.T * (1.0 - Nd1) * growthFactorTimesDiscountFactor; return [callCarryRho, putCarryRho]; } GetD() { const logMoneyness = Math.log(this.S / this.K); const sigmaSqrtT = this.sigma * Math.sqrt(this.T); const tmp = (logMoneyness + this.b * this.T) / sigmaSqrtT; const d1 = tmp + .5 * sigmaSqrtT; const d2 = tmp - .5 * sigmaSqrtT; return [d1, d2]; } }34:[] 2e:{"title":"Option Greeks","cells":[{"id":0,"value":"md`# Option Greeks`","pinned":false,"hidden":false,"mode":"ojs"},{"id":180,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":359,"value":"{\n plotter(\"price-chart\", \"Price\");\n plotter(\"delta-chart\", \"Delta\");\n plotter(\"gamma-chart\", \"Gamma\");\n plotter(\"vega-chart\", \"Vega\");\n plotter(\"rho-chart\", \"Rho\");\n plotter(\"carry-rho-chart\", \"Carry Rho\");\n plotter(\"theta-chart\", \"Theta\");\n plotter(\"charm-chart\", \"Charm\");\n plotter(\"theta2-chart\", \"Theta 2\");\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":343,"value":"bs = new BlackScholes(S, S, r, b, sigma, T, dcc);","pinned":false,"hidden":false,"mode":"ojs"},{"id":351,"value":"DrawValues(bs)","pinned":true,"hidden":false,"mode":"ojs"},{"id":200,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":357,"value":"function plotter(id, analytic) {\n var ctx = document.getElementById(id).getContext('2d');\n\n var chart = new Chart(ctx, {\n type: 'line',\n options: {\n elements: {\n line: {\n tension: 0 // disables bezier curves\n },\n point: {\n radius: 0\n }\n },\n animation: {\n duration: 0 // general animation time\n },\n hover: {\n animationDuration: 0 // duration of animations when hovering an item\n },\n responsiveAnimationDuration: 0 // animation duration after a resize\n }\n });\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":207,"value":"\nfunction Plot(strikes, callData, putData, analytic, id) {\n\n Chart.helpers.each(Chart.instances, function(instance){\n if(instance.chart.canvas.id !== id)\n return;\n instance.chart.config.data.labels = strikes;\n instance.chart.config.data.datasets = [\n {\n fill: false,\n label: \"Call \" + analytic,\n data: callData,\n borderColor: \"blue\",\n },\n {\n fill: false,\n label: \"Put \" + analytic,\n data: putData,\n borderColor: \"red\"\n },\n ];\n\n instance.update();\n });\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":198,"value":"md`## Setup`","pinned":false,"hidden":false,"mode":"ojs"},{"id":320,"value":"maxPrice=500;","pinned":false,"hidden":false,"mode":"ojs"},{"id":339,"value":"viewof MaxStrike = slider({title: \"MAX STRIKE\",description: \"\",min:Spot ,max:maxPrice ,value:Spot+\n 10,step:1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":270,"value":"viewof MinStrike = slider({title: \"Min\",description: \"\",min:1 ,max: Spot,value:Spot-9,step:1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":272,"value":"viewof NumOfStrikes = slider({title: \"\",description: \"\",min: 5,max:50 ,value:20 ,step:1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":279,"value":"viewof Spot = slider({title: \"SPOT\",description: \"\",min:1 ,max:maxPrice ,value:55\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":335,"value":"S=parseFloat(Spot)","pinned":false,"hidden":false,"mode":"ojs"},{"id":281,"value":"viewof RiskFreeRate= slider({title: \"Risk-Free Interest Rate (%):\",description: \"Discounting Rate\",min:-100 ,max: 100,value:-0.01,step:0.01})","pinned":false,"hidden":false,"mode":"ojs"},{"id":325,"value":"r= 0.01 * parseFloat(RiskFreeRate)","pinned":false,"hidden":false,"mode":"ojs"},{"id":285,"value":"viewof CarryRate=slider({title: \"Carry Rate (%):\",description: \"Cost of carrying the option's underlying. If there's no borrow rate, use the same value as the risk-free interest rate\",min:-100 ,max: 100,value:-0.01,step:0.01})","pinned":false,"hidden":false,"mode":"ojs"},{"id":327,"value":"b= 0.01 * parseFloat(CarryRate);","pinned":false,"hidden":false,"mode":"ojs"},{"id":314,"value":"viewof Volatility = slider({title: \"Volatility:\",description: \"\",min:.01,max: Infinity,value:30,step:0.01})","pinned":false,"hidden":false,"mode":"ojs"},{"id":316,"value":"sigma=0.01 * parseFloat(Volatility)","pinned":false,"hidden":false,"mode":"ojs"},{"id":318,"value":"viewof DaysToExpiry=slider({title: \"Days to Expiry\",description: \"\",min:1 ,max:maxPrice ,value:30,step:1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":346,"value":"T=DaysToExpiry","pinned":false,"hidden":false,"mode":"ojs"},{"id":330,"value":"viewof DayCountingConvention = radio({options:[\"CalendarDays\",\"BusinessDays\"],value:\"CalendarDays\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":332,"value":"dcc= DayCountingConvention === \"CalendarDays\"? 252 : 365","pinned":false,"hidden":false,"mode":"ojs"},{"id":115,"value":"ALLOWED_DAY_COUNTING_CONVENTION = [365, 252];\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":419,"value":"md`\n----\n\n## Model`","pinned":false,"hidden":false,"mode":"ojs"},{"id":114,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":148,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":422,"value":"md`### Utility`","pinned":false,"hidden":false,"mode":"ojs"},{"id":103,"value":"\noneOverSqrt2PI = 1.0 / Math.sqrt(2.0 * Math.PI);\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":112,"value":"\nfunction std_cdf(x) \n{\n return cdf(x, 0.0, 1.0);\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":110,"value":"\nfunction std_pdf(x) \n{\n return pdf(x, 0.0, 1.0);\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":108,"value":"\nfunction cdf(x, mean, variance) \n{\n return 0.5 * (1 + erf((x - mean) / (Math.sqrt(2 * variance))));\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":106,"value":"\nfunction pdf(x, mean, variance) \n{\n const z = (x - mean) / variance;\n return Math.exp(-.5 * z * z) * oneOverSqrt2PI;\n}\n ","pinned":false,"hidden":false,"mode":"ojs"},{"id":101,"value":"\nfunction erf(x) \n{\n // save the sign of x\n const sign = (x >= 0) ? 1 : -1;\n x = Math.abs(x);\n \n const a1 = 0.254829592;\n const a2 = -0.284496736;\n const a3 = 1.421413741;\n const a4 = -1.453152027;\n const a5 = 1.061405429;\n const p = 0.3275911;\n \n // Abramovitz-Stegun formula 7.1.26\n const t = 1.0 / (1.0 + p * x);\n const y = 1.0 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-x * x);\n return sign * y;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":265,"value":"md`## ANX`","pinned":false,"hidden":false,"mode":"ojs"},{"id":10,"value":"d3=require('d3','d3-array')","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"import {text,slider,radio} from '@jashkenas/inputs'","pinned":false,"hidden":false,"mode":"ojs"},{"id":213,"value":"Chart = require('https://cdnjs.cloudflare.com/ajax/libs/Chart.js/2.8.0/Chart.bundle.js')","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W34","stdlib":"1","titling":"auto"}