Speculation and Leverage

Stock vs. Call Options

This page demonstrates how leverage can amplify outcomes by comparing two speculation strategies using the same initial capital: buying the stock versus buying call options. We use a simple, consistent setup:

European call option, no dividends.
Risk‑free rate: 4% (continuously compounded), volatility: 20%, time to maturity: 1 year.
Options control 100 shares per contract.
Option premium is computed by Black–Scholes (see the Payoffs page for details).

Note: Long calls require only the premium upfront.

Set Capital and Spot Price

viewof capital = Inputs.range([1000, 50000], {step: 100, label: "Investment Capital ($)", value: 1000})
viewof s0 = Inputs.range([50, 150], {step: 1, label: "Current Price S_0", value: 100})

// Strike equals spot (ATM)
x  = s0;
r = 0.04; T = 1; sigma = 0.20; q = 0; // BS parameters

// Formatters
formatUSD = (value, digits = 0) => new Intl.NumberFormat('en-US', {
  style: 'currency', currency: 'USD', maximumFractionDigits: digits, minimumFractionDigits: digits
}).format(value)
formatInt = (n) => new Intl.NumberFormat('en-US', {maximumFractionDigits: 0}).format(n)

Option Premium (Black–Scholes)

erf = x => {
  const sign = Math.sign(x); const ax = Math.abs(x);
  const t = 1 / (1 + 0.3275911 * ax);
  const y = 1 - (((((1.061405429*t - 1.453152027)*t) + 1.421413741)*t - 0.284496736)*t + 0.254829592)*t*Math.exp(-ax*ax);
  return sign * y;
}
N = z => 0.5 * (1 + erf(z / Math.SQRT2));

// Black–Scholes d1, d2 and call premium C0 (no dividends)
d1 = (Math.log(s0 / x) + (r - q + 0.5 * sigma * sigma) * T) / (sigma * Math.sqrt(T));
d2 = d1 - sigma * Math.sqrt(T);
c0 = s0 * Math.exp(-q*T) * N(d1) - x * Math.exp(-r*T) * N(d2);
contractSize = 100;
contractCost = c0 * contractSize;

md`Call premium (Black–Scholes, r = 4% cc, σ = 20%, T = 1): **${c0.toFixed(2)}**`

Side‑by‑Side: What Can You Buy?

Strategy 1: Buy Stock

shares = Math.max(0, Math.floor(capital / s0))
stockCost = shares * s0
stockLeftover = Math.max(0, capital - stockCost)

// P/L at selected S_T
plStockPoint = shares * (st - s0)

html`<div class="profit-loss-table">
<table>
<thead><tr><th colspan="2">Stock Purchase</th></tr></thead>
<tbody>
  <tr><td>S0</td><td>${formatUSD(s0, 2)}</td></tr>
  <tr><td>Shares Purchased</td><td><strong>${formatInt(shares)}</strong></td></tr>
  <tr><td>Cost</td><td>${formatUSD(stockCost, 0)} (leftover ${formatUSD(stockLeftover, 0)})</td></tr>
  <tr><td>P/L at S_T = ${st}</td><td><strong>${formatUSD(plStockPoint, 0)}</strong></td></tr>
 </tbody>
</table>
</div>`

// Visual: capital usage bar
Plot.plot({
  caption: "Capital usage (Stock)",
  x: {label: "USD", tickFormat: "~s"},
  y: {domain: ["Leftover", "Cost"], label: null},
  height: 120,
  marginLeft: 80,
  marks: [
    Plot.barX([
      {k: "Cost", v: stockCost},
      {k: "Leftover", v: stockLeftover}
    ], {y: "k", x: "v", fill: d => d.k === "Cost" ? "steelblue" : "#ddd"}),
    Plot.ruleX([0])
  ]
})

Strategy 2: Buy Call Options

contracts = Math.max(0, Math.floor(capital / contractCost))
contractsCost = contracts * contractCost
contractsLeftover = Math.max(0, capital - contractsCost)
notionalControlled = contracts * contractSize

// P/L at selected S_T
plOptionsPoint = notionalControlled * (Math.max(0, st - x) - c0)

breakeven = x + c0

html`<div class="profit-loss-table">
<table>
<thead><tr><th colspan="2">Long Call Options</th></tr></thead>
<tbody>
  <tr><td>S0, X</td><td>${formatUSD(s0, 2)}, ${formatUSD(x, 2)} (ATM)</td></tr>
  <tr><td>Contracts Purchased</td><td><strong>${formatInt(contracts)}</strong> × ${contractSize} shares</td></tr>
  <tr><td>Premium per Contract</td><td>${formatUSD(contractCost, 2)}</td></tr>
  <tr><td>Total Premium</td><td>${formatUSD(contractsCost, 0)} (leftover ${formatUSD(contractsLeftover, 0)})</td></tr>
  <tr><td>Breakeven at T</td><td>${formatUSD(breakeven, 2)}</td></tr>
  <tr><td>P/L at S_T = ${st}</td><td><strong>${formatUSD(plOptionsPoint, 0)}</strong></td></tr>
</tbody>
</table>
</div>`

// Visual: capital usage bar
Plot.plot({
  caption: "Capital usage (Options)",
  x: {label: "USD", tickFormat: "~s"},
  y: {domain: ["Leftover", "Premium"], label: null},
  height: 120,
  marginLeft: 80,
  marks: [
    Plot.barX([
      {k: "Premium", v: contractsCost},
      {k: "Leftover", v: contractsLeftover}
    ], {y: "k", x: "v", fill: d => d.k === "Premium" ? "seagreen" : "#ddd"}),
    Plot.ruleX([0])
  ]
})

Profit/Loss vs. S_T

// Central slider controlling S_T
viewof st = Inputs.range([0, 200], {step: 1, label: "Underlying Price at Maturity S_T", value: s0})

domainMax = 200
stockSeries = d3.range(0, domainMax + 1).map(p => ({x: p, y: shares * (p - s0)}))
optionSeries = d3.range(0, domainMax + 1).map(p => ({x: p, y: notionalControlled * (Math.max(0, p - x) - c0)}))

Plot.plot({
  caption: "P/L for Stock vs. Call Options",
  x: {domain: [0, domainMax], label: "S_T ($/share)"},
  y: {label: "Profit/Loss ($)", tickFormat: "~s"},
  grid: true,
  pointer: "plot",
  marginLeft: 84,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([s0], {stroke: "gray", strokeDasharray: "4,4"}),
    Plot.line(stockSeries, {x: "x", y: "y", stroke: "steelblue", strokeWidth: 2}),
    Plot.line(optionSeries, {x: "x", y: "y", stroke: "seagreen", strokeWidth: 2}),
    Plot.dot([[st, plStockPoint]], {fill: "steelblue", r: 5, title: d => `Stock\nS_T: ${st}\nP/L: ${formatUSD(plStockPoint, 0)}`, tip: true}),
    Plot.dot([[st, plOptionsPoint]], {fill: "seagreen", r: 5, title: d => `Calls\nS_T: ${st}\nP/L: ${formatUSD(plOptionsPoint, 0)}`, tip: true}),
    // Pointer tooltips and crosshairs
    Plot.tip(Plot.pointerX(stockSeries), {x: "x", y: "y", title: d => `Stock\nS: ${d.x}\nP/L: ${formatUSD(d.y, 0)}`}),
    Plot.tip(Plot.pointerX(optionSeries), {x: "x", y: "y", title: d => `Calls\nS: ${d.x}\nP/L: ${formatUSD(d.y, 0)}`}),
    Plot.crosshair(stockSeries, {x: "x", y: "y", stroke: "#444", opacity: 0.6})
  ],
  title: "Leverage Comparison"
})

html`<div class="plot-legend">
  <span><span class="swatch" style="background: steelblue;"></span>Stock</span>
  <span><span class="swatch" style="background: seagreen;"></span>Call Options</span>
</div>
<div class="plot-legend">
  <span><span class="swatch-line" style="border-top-color: gray; border-top-style: dashed;"></span>S0
    <span class="badge">${s0.toFixed(0)} $/share</span>
  </span>
</div>`

Point‑in‑Time Comparison

plBars = [
  {k: "Stock", v: plStockPoint},
  {k: "Calls", v: plOptionsPoint}
]

Plot.plot({
  caption: `P/L at S_T = ${st}`,
  x: {label: "Strategy"},
  y: {label: "Profit/Loss ($)", tickFormat: "~s"},
  height: 240,
  marks: [
    Plot.ruleY([0]),
    Plot.barY(plBars, {x: "k", y: "v", fill: d => d.k === "Stock" ? "steelblue" : "seagreen"}),
    Plot.text(plBars, {x: "k", y: d => d.v >= 0 ? d.v : 0, text: d => formatUSD(d.v, 0), dy: d => d.v >= 0 ? -6 : 12, fill: "#333"})
  ]
})

Payoffs and P/L Table

payoffStock = shares * (st - s0)
payoffCalls = notionalControlled * Math.max(0, st - x) - notionalControlled * c0
returnStockPct = stockCost > 0 ? (payoffStock / stockCost) * 100 : 0
returnCallsPct = contractsCost > 0 ? (payoffCalls / contractsCost) * 100 : 0

html`<div class="profit-loss-table">
<table>
<thead>
  <tr><th>Item</th><th>Stock</th><th>Call Options</th></tr>
</thead>
<tbody>
  <tr><td>Initial Capital</td><td colspan="2">${formatUSD(capital, 0)}</td></tr>
  <tr><td>Units Purchased</td><td>${formatInt(shares)} shares</td><td>${formatInt(contracts)} contracts × ${contractSize} = ${formatInt(notionalControlled)} shares</td></tr>
  <tr><td>Initial Outlay</td><td>${formatUSD(stockCost, 0)}</td><td>${formatUSD(contractsCost, 0)}</td></tr>
  <tr><td>Option Premium (per share)</td><td>—</td><td>${formatUSD(c0, 2)}</td></tr>
  <tr><td>Terminal Value at S_T</td><td>${formatUSD(shares * st, 0)}</td><td>${formatUSD(notionalControlled * Math.max(0, st - x), 0)}</td></tr>
  <tr><td>P/L at S_T</td><td><strong>${formatUSD(payoffStock, 0)}</strong></td><td><strong>${formatUSD(payoffCalls, 0)}</strong></td></tr>
  <tr><td>Return on Capital</td><td><strong>${returnStockPct.toFixed(1)}%</strong></td><td><strong>${returnCallsPct.toFixed(1)}%</strong></td></tr>
</tbody>
</table>
</div>`

Notes

Stock position P/L: \((S_T − S_0) \times \text{shares}\).
Long call P/L: \([max(0, S_T − X) − C_0] \times \text{contracts} \times 100\), with \(X = S_0\) and \(C_0\) from Black–Scholes.
For the stock position, the maximum loss is the entire initial outlay if the stock price drops to zero.
For options, the initial outlay is the premium; risk is limited to that premium for long calls.

See also: the overview of option premium and payoffs on the Payoffs page.