Arbitrage: Cash-and-Carry and Reverse

Interactive dashboard illustrating how arbitrageurs enforce price consistency in forward markets (no dividends).

Arbitrage enforces price consistency between the spot market and the forward market. When the observed forward price \(F_0\) deviates from the no-arbitrage benchmark \(S_0 e^{rT}\) (for a non-dividend-paying asset), an arbitrageur can lock in a risk-free profit using cash-and-carry (if the forward is overpriced) or reverse cash-and-carry (if the forward is underpriced).

Market Dashboard

Use the sliders to set the spot price, market forward price, risk-free rate, and time to maturity. The no-arbitrage forward price \(S_0 e^{rT}\) is computed for reference. When mispricing exists, the Execute button becomes active and glows.

import {Inputs} from "@observablehq/inputs"

// Inputs
viewof S0 = Inputs.range([10, 200], {step: 1, value: 100, label: "Spot Price S₀"})
viewof r = Inputs.range([0, 0.2], {step: 0.001, value: 0.05, label: "Risk-free rate r"})
viewof T = Inputs.range([0.0, 3.0], {step: 0.01, value: 1.00, label: "Time to maturity T (years)"})

// No‑arbitrage benchmark used to seed F0
noArb = S0 * Math.exp(r * T)

// Forward price (allow finer decimals, start at no‑arb)
viewof F0 = Inputs.range([10, 300], {step: 0.001, value: noArb, label: "Market Forward Price F₀"})

// Derived quantities
mispricing = F0 - noArb

// Define no-arbitrage precision to 3 decimals
NA_DECIMALS = 3
fairTol = 0.5 * Math.pow(10, -NA_DECIMALS)
isFair = Math.abs(mispricing) <= fairTol
direction = isFair ? "fair" : (mispricing > 0 ? "overpriced" : "underpriced")

fmt = d => d.toLocaleString(undefined, {maximumFractionDigits: 4})

viewof execute = {
  if (isFair) {
    // Hide the button entirely when no arbitrage
    const placeholder = html`<div></div>`
    placeholder.style.display = 'none'
    return placeholder
  }
  const b = Inputs.button("Execute Arbitrage")
  b.classList.add("arb-btn")
  b.classList.toggle("glow", true)
  return b
}

md`**No-arbitrage forward price**: $${fmt(noArb)}  
**Status**: ${direction === "fair" ? "No arbitrage (fairly priced)" : (direction === "overpriced" ? "Forward overpriced → cash-and-carry" : "Forward underpriced → reverse cash-and-carry")}  
**Potential Profit at Maturity**: $${fmt(isFair ? 0 : Math.abs(mispricing))}`

Note

The no-arbitrage forward price for a non-dividend-paying asset is \(S_0 e^{rT}\). This model uses a continuously compounded rate (\(r\)) because it is the standard for pricing derivatives, reflecting interest that accrues constantly. If \(F_0 > S_0 e^{rT}\), sell the forward and carry the asset (cash-and-carry). If \(F_0 < S_0 e^{rT}\), do the reverse (reverse cash-and-carry).

Execute Arbitrage

Click Execute to see the sequence of trades and the risk-free payoff at maturity. The animation reverses depending on whether \(F_0\) is above or below \(S_0 e^{rT}\).

// import {tex} from "@observablehq/tex"
// Wrap the animation in a single reactive cell that depends on `execute`.
viewof arb_steps = {
  // Re-run this cell on every click
  execute;

  const container = html`<div class="arb-container"></div>`

  const overpriced = mispricing > fairTol
  const underpriced = mispricing < -fairTol

  // Simple inline SVG icon set for a professional look
  function iconSVG(name) {
    const wrap = document.createElement("span")
    wrap.className = "icon"
    const common = 'width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="1.8" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"'
    let svg = ""
    switch (name) {
      case "cash": svg = `<svg ${common}><rect x="2" y="6" width="20" height="12" rx="2"></rect><circle cx="12" cy="12" r="3"></circle></svg>`; break
      case "asset": svg = `<svg ${common}><rect x="4" y="6" width="16" height="12" rx="2"></rect><path d="M8 10h8M8 14h5"/></svg>`; break
      case "contract": svg = `<svg ${common}><path d="M14 2H6a2 2 0 0 0-2 2v16a2 2 0 0 0 2 2h12a2 2 0 0 0 2-2V8z"/><path d="M14 2v6h6"/><path d="M9 13h6M9 17h6"/></svg>`; break
      case "ff": svg = `<svg ${common}><polygon points="13 19 22 12 13 5 13 19"></polygon><polygon points="2 19 11 12 2 5 2 19"></polygon></svg>`; break
      case "deliver": svg = `<svg ${common}><path d="M21 15v4a2 2 0 0 1-2 2H5a2 2 0 0 1-2-2v-4"/><polyline points="7 10 12 5 17 10"/><line x1="12" y1="5" x2="12" y2="19"/></svg>`; break
      case "repay": svg = `<svg ${common}><path d="M3 10h11a4 4 0 0 1 0 8H7"/><polyline points="7 14 3 10 7 6"/></svg>`; break
      case "short": svg = `<svg ${common}><path d="M3 3v18h18"/><path d="M7 15l4-4 4 4 4-4"/></svg>`; break
      case "invest": svg = `<svg ${common}><path d="M3 3v18h18"/><path d="M7 17l4-6 4 4 4-8"/></svg>`; break
      case "receive": svg = `<svg ${common}><path d="M21 9V5a2 2 0 0 0-2-2H5a2 2 0 0 0-2 2v4"/><polyline points="7 10 12 15 17 10"/><line x1="12" y1="15" x2="12" y2="3"/></svg>`; break
      default: svg = `<svg ${common}><circle cx="12" cy="12" r="4"/></svg>`
    }
    wrap.innerHTML = svg
    return wrap
  }

  function step(content, icon) {
    const row = document.createElement("div")
    row.className = "step"
    const badge = iconSVG(icon)
    const body = document.createElement("div")
    body.className = "body"
    if (typeof content === 'string') {
      body.textContent = content
    } else {
      body.append(content)
    }
    row.append(badge, body)
    container.append(row)
  }

  function wait(ms) { return new Promise(res => setTimeout(res, ms)) }

  async function run() {
    container.innerHTML = ""
    if (!overpriced && !underpriced) {
      step(md`No arbitrage opportunity: the forward is fairly priced.`, "✅")
      return
    }

    if (overpriced) {
      // Cash-and-carry (F0 > S0 e^{rT})
      step(md`Step 1: Borrow ${tex`S_0`} at rate ${tex`r`}. Receive ${fmt(S0)} in cash.`, "cash")
      await wait(700)
      step(md`Step 2: Buy the asset in the spot market for ${fmt(S0)}.`, "asset")
      await wait(700)
      step(md`Step 3: Sell a forward contract at the high price ${fmt(F0)}.`, "contract")
      await wait(700)
      step(md`Fast forward to maturity ${tex`T`}.`, "ff")
      await wait(700)
      step(md`Step 4: Deliver the asset into the forward and receive ${fmt(F0)}.`, "deliver")
      await wait(700)
      step(md`Step 5: Repay loan principal and interest ${tex`S_0 e^{rT}`} = ${fmt(noArb)}.`, "repay")
    } else {
      // Reverse cash-and-carry (F0 < S0 e^{rT})
      step(md`Step 1: Short the asset in the spot market; receive ${fmt(S0)}.`, "short")
      await wait(700)
      step(md`Step 2: Invest the proceeds at the risk-free rate ${tex`r`} to maturity.`, "invest")
      await wait(700)
      step(md`Step 3: Buy a forward contract at the low price ${fmt(F0)}.`, "contract")
      await wait(700)
      step(md`Fast forward to maturity ${tex`T`}.`, "ff")
      await wait(700)
      step(md`Step 4: Receive the asset via the long forward by paying ${fmt(F0)}.`, "receive")
      await wait(700)
      step(md`Step 5: Your investment matures to ${fmt(noArb)}. Use ${fmt(F0)} of this to settle the forward, receiving the asset, and then return the asset to close the short position.`, "repay")
    }

    await wait(400)
    const profit = Math.abs(mispricing)
    const res = document.createElement("div")
    res.className = "result"
    const heading = document.createElement("div")
    heading.innerHTML = `<strong>Risk‑free Profit at Maturity:</strong> ${fmt(profit)}`
    const explain = document.createElement("div")
    explain.className = "explain"
    const explNode = overpriced
      ? md`Profit = ${tex`F_0 - S_0 e^{rT}`} = ${fmt(F0)} − ${fmt(noArb)} = ${fmt(profit)}`
      : md`Profit = ${tex`S_0 e^{rT} - F_0`} = ${fmt(noArb)} − ${fmt(F0)} = ${fmt(profit)}`
    explain.append(explNode)
    res.append(heading, explain)
    container.append(res)
  }

  run()
  return container
}

Payoff Summary

import {tex} from "@observablehq/tex"
viewof payoff_table = {
  const overpriced = mispricing > fairTol
  const underpriced = mispricing < -fairTol

  function el(tag, className) {
    const e = document.createElement(tag)
    if (className) e.className = className
    return e
  }
  function cellContent(...parts) {
    const span = document.createElement('span')
    for (const p of parts) {
      if (p == null) continue
      if (typeof p === 'string') span.append(document.createTextNode(p))
      else span.append(p)
    }
    return span
  }
  function row(nameNode, t0Node, tTNode, isTotal = false) {
    const tr = el('tr')
    const td1 = el('td')
    const td2 = el('td', 'num')
    const td3 = el('td', 'num')
    td1.append(typeof nameNode === 'string' ? document.createTextNode(nameNode) : nameNode)
    td2.append(typeof t0Node === 'string' ? document.createTextNode(t0Node) : t0Node)
    td3.append(typeof tTNode === 'string' ? document.createTextNode(tTNode) : tTNode)
    if (isTotal) { td1.style.fontWeight = td2.style.fontWeight = td3.style.fontWeight = '700' }
    tr.append(td1, td2, td3)
    return tr
  }

  const tbl = el('table', 'payoff-table')
  const thead = el('thead')
  const thr = el('tr')
  const th1 = el('th'); th1.textContent = 'Component'
  const th2 = el('th'); th2.textContent = 't = 0'
  const th3 = el('th'); th3.append(document.createTextNode('t = '), tex`T`, (()=>{const s=document.createElement('sup'); s.textContent='*'; return s})())
  thr.append(th1, th2, th3)
  thead.append(thr)
  tbl.append(thead)
  const tb = el('tbody')

  if (!overpriced && !underpriced) {
    tb.append(row('No arbitrage (fair price)', '0', '0'))
  } else if (overpriced) {
    tb.append(row('Borrow (loan)', `+${fmt(S0)}`, `−${fmt(noArb)}`))
    tb.append(row('Buy asset (spot)', `−${fmt(S0)}`, cellContent('+', tex`S_T`)))
    tb.append(row('Short forward', '0', cellContent(`${fmt(F0)} − `, tex`S_T`)))
    tb.append(row('Total', '0', `${fmt(F0 - noArb)}`, true))
  } else {
    tb.append(row('Short asset (spot)', `+${fmt(S0)}`, cellContent('−', tex`S_T`)))
    tb.append(row(cellContent('Invest at ', tex`r`), `−${fmt(S0)}`, `+${fmt(noArb)}`))
    tb.append(row('Long forward', '0', cellContent(tex`S_T`, ` − ${fmt(F0)}`)))
    tb.append(row('Total', '0', `${fmt(noArb - F0)}`, true))
  }
  tbl.append(tb)
  
  const footer = document.createElement('div')
  const footP = document.createElement('p')
  footP.style.cssText = 'font-size: 0.8rem; text-align: right; margin-top: 4px;'
  const sup = document.createElement('sup'); sup.textContent = '*'; footP.append(sup)
  footP.append(document.createTextNode(' At maturity, '))
  footP.append(tex`S_T`)
  footP.append(document.createTextNode(' is the spot price of the asset. The positions are structured to hedge against its value, making the final profit risk-free.'))
  footer.append(footP)

  const wrapper = document.createElement('div')
  wrapper.append(tbl, footer)

  return wrapper
}

What’s Going On?

• Overpriced forward (\(F_0 > S_0 e^{rT}\)): cash-and-carry.
  • Borrow cash, buy the asset, sell the forward; at maturity deliver the asset, receive \(F_0\), and repay \(S_0 e^{rT}\). Profit: \(F_0 − S_0 e^{rT}\).
• Underpriced forward (\(F_0 < S_0 e^{rT}\)): reverse cash-and-carry.
  • Short the asset, invest proceeds, buy the forward; at maturity receive the asset via the forward for \(F_0\), return the asset, and keep \(S_0 e^{rT} − F_0\).

Tip

This demo assumes no dividends, no carry costs, and frictionless borrowing/lending at the same continuously compounded rate \(r\) for horizon \(T\).

style = html`<style>
.arb-container { border: 1px solid #e5e7eb; border-radius: 8px; padding: 12px; background: #fafafa; }
.step { display: grid; grid-template-columns: 28px 1fr; gap: 8px; align-items: start; padding: 6px 0; border-bottom: 1px dashed #e5e7eb; }
.step:last-child { border-bottom: none; }
.icon { display: inline-grid; place-items: center; width: 28px; height: 28px; color: #0f766e; }
.body { font-size: 0.98rem; }
.result { margin-top: 10px; padding: 10px; border-radius: 6px; background: #f0fdf4; color: #14532d; border: 1px solid #86efac; }
.result .explain { font-size: 0.9rem; margin-top: 4px; color: #065f46; }
/* Execute button styling */
.arb-btn { appearance: none; background: #16a34a !important; color: #ffffff !important; border: 1px solid #15803d !important; padding: 8px 14px; border-radius: 8px; font-weight: 700; letter-spacing: .2px; }
.arb-btn:hover { background: #159347 !important; }
.arb-btn:disabled { background: #e5e7eb !important; color: #6b7280 !important; border-color: #e5e7eb !important; cursor: not-allowed; }
button.glow { box-shadow: 0 0 0 0 rgba(52,211,153,0.7); animation: glow 1.8s infinite; border-color: #10b981 !important; }
button.glow:hover { background: #bbf7d0; }
@keyframes glow { 0% { box-shadow: 0 0 0 0 rgba(34,197,94,0.7); } 70% { box-shadow: 0 0 0 12px rgba(34,197,94,0); } 100% { box-shadow: 0 0 0 0 rgba(34,197,94,0); } }

/* Payoff table */
.payoff-table { width: 100%; border-collapse: collapse; margin: 10px 0; font-size: 0.95rem; }
.payoff-table thead th { text-align: left; border-bottom: 2px solid #e5e7eb; padding: 6px; }
.payoff-table thead th:nth-child(2),
.payoff-table thead th:nth-child(3) { text-align: right; }
.payoff-table td { border-bottom: 1px solid #f1f5f9; padding: 6px; }
.payoff-table td:nth-child(2),
.payoff-table td:nth-child(3) { text-align: right; }
.payoff-table td.num { text-align: right; font-variant-numeric: tabular-nums; }
</style>`