Payoffs

All the diagrams below take only into account the asset/contract payoff at maturity as a function of the underlying asset value. They do not represent the profit/loss of holding a position on such assests as they ignore any trading costs.

\(S_T\) is the underlying asset value at maturity \(T\), and \(F_0\) is the futures price at time zero, and \(X\) is the option exercise price.

Underlying asset

Change the value of the asset at maturity \(S_T\) using the slider below, the payoff is represented with a dot:

viewof s = Inputs.range([0, 150], {step: 1, label: "S_T", value:100})

longAssetPayoff = Plot.plot({
  caption: "Payoffs on a Long position on an asset",
  x: {
    domain: [0, 150],
    label: "S_T"
  },
  y: {
    domain: [-150, 150],
    label: "Payoff"
  },
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.line([[0, 0], [150, 150]], {stroke: "blue"}),
    Plot.dot(
      [[s, s]], 
      {
        fill: "red", 
        r: 5,
        title: d => `S_T: ${s}
Payoff: ${s}`,
        tip: true
      })
  ],
  title: "Long position payoff"
})

shortAssetPayoff = Plot.plot({
  caption: "Payoffs on a Short position on an asset",
  x: {
    domain: [0, 150],
    label: "S_T"
  },
  y: {
    domain: [-150, 150],
    label: "Payoff"
  },
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.line([[0, 0], [150, -150]], {stroke: "red"}),
    Plot.dot(
      [[s, -s]], 
      {
        fill: "blue", 
        r: 5,
        title: d => `S_T: ${s}
Payoff: ${s}`,
        tip: true
      })
  ],
  title: "Short position payoff"
})

When you purchase an asset (long position), the payoff is simply the value of the asset at maturity \(S_T\). The payoff function is represented by the straight line passing through the origin with a slope of 1.

When you sell an asset (short position), the payoff is the negative of the value of the asset at maturity, \(-S_T\). The payoff function is represented by the straight line passing through the origin with a slope of -1.

Futures contract

You can change the future price at time zero (\(F_0\)) and observe how it affects the payoff diagram, using the slider below:

viewof f0 = Inputs.range([50, 150], {step: 1, label: "F_0", value:100})
viewof stf = Inputs.range([50, 150], {step: 1, label: "S_T", value:100})

longFuturePayoff = Plot.plot({
  caption: "Payoffs on a Long position on a futures contract",
  x: {
    domain: [0, 150],
    label: "S_T"
  },
  y: {
    domain: [-150, 150],
    label: "Payoff"
  },
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.ruleX([f0], {stroke: "gray", strokeDasharray: "4,4"}),
    Plot.line([[0, -f0], [150, 150 - f0]], {stroke: "blue"}),
    Plot.dot(
      [[stf, stf - f0]], 
      {
        fill: "red", 
        r: 5,
        title: d => `S_T: ${s}
F_0: ${f0}
Payoff: ${s - f0}`,
        tip: true
      })
  ],
  title: "Long futures position payoff"
})

shortFuturePayoff = Plot.plot({
  caption: "Payoffs on a Short position on a futures contract",
  x: {
    domain: [0, 150],
    label: "S_T"
  },
  y: {
    domain: [-150, 150],
    label: "Payoff"
  },
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.ruleX([f0], {stroke: "gray", strokeDasharray: "4,4"}),
    Plot.line([[0, f0], [150, f0 - 150]], {stroke: "red"}),
    Plot.dot(
      [[stf, f0 - stf]], 
      {
        fill: "blue", 
        r: 5,
        title: d => `S_T: ${s}
F_0: ${f0}
Payoff: ${f0 - stf}`,
        tip: true
      })
  ],
  title: "Short futures position payoff"
})

The payoff of a futures contract at maturity is the difference between the spot price of the underlying asset at maturity (\(S_T\)) and the futures price agreed upon at the initiation of the contract (\(F_0\)):

• For a long position (buyer of the futures contract), the payoff is \(S_T - F_0\)
• For a short position (seller of the futures contract), the payoff is \(F_0 - S_T\)

Note that unlike other derivatives, futures contracts have no upfront premium cost as they are obligations to buy/sell at a predetermined price.

Option Contracts

Options give the holder the right, but not the obligation, to buy (call option) or sell (put option) the underlying asset at a predetermined price (exercise price or strike price). You can adjust the parameters below to see how they affect option payoffs.

viewof x = Inputs.range([50, 150], {step: 1, label: "X (Exercise Price)", value:100})
viewof sto = Inputs.range([0, 150], {step: 1, label: "S_T", value:100})

Call Options

A call option gives the holder the right to buy the underlying asset at the exercise price.

longCallPayoff = Plot.plot({
  caption: "Payoffs on a Long position on a Call option at the option maturity",
  x: {
    domain: [0, 200],
    label: "S_T"
  },
  y: {
    domain: [-100, 100],
    label: "Payoff"
  },
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.ruleX([x], {stroke: "gray", strokeDasharray: "4,4"}),
    Plot.line([[0, 0], [x, 0], [200, 200 - x]], {stroke: "blue"}),
    Plot.dot(
      [[sto, Math.max(0, sto - x)]], 
      {
        fill: "red", 
        r: 5,
        title: d => `S_T: ${sto}
X: ${x}
Payoff: ${Math.max(0, sto - x)}`,
        tip: true
      })
  ],
  title: "Long Call position payoff"
})

shortCallPayoff = Plot.plot({
  caption: "Payoffs on a Short position on a Call option at the option maturity",
  x: {
    domain: [0, 200],
    label: "S_T"
  },
  y: {
    domain: [-100, 100],
    label: "Payoff"
  },
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.ruleX([x], {stroke: "gray", strokeDasharray: "4,4"}),
    Plot.line([[0, 0], [x, 0], [200, x - 200]], {stroke: "red"}),
    Plot.dot(
      [[sto, -Math.max(0, sto - x)]], 
      {
        fill: "blue", 
        r: 5,
        title: d => `S_T: ${sto}
X: ${x}
Payoff: ${-Math.max(0, sto - x)}`,
        tip: true
      })
  ],
  title: "Short Call position payoff"
})

Put Options

A put option gives the holder the right to sell the underlying asset at the exercise price.

longPutPayoff = Plot.plot({
  caption: "Payoffs on a Long position on a Put option at the option maturity",
  x: {
    domain: [0, 200],
    label: "S_T"
  },
  y: {
    domain: [-150, 150],
    label: "Payoff"
  },
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.ruleX([x], {stroke: "gray", strokeDasharray: "4,4"}),
    Plot.line([[0, x], [x, 0], [200, 0]], {stroke: "blue"}),
    Plot.dot(
      [[sto, Math.max(0, x - sto)]], 
      {
        fill: "red", 
        r: 5,
        title: d => `S_T: ${sto}
X: ${x}
Payoff: ${Math.max(0, x - sto)}`,
        tip: true
      })
  ],
  title: "Long Put position payoff"
})

shortPutPayoff = Plot.plot({
  caption: "Payoffs on a Short position on a Put option at the option maturity",
  x: {
    domain: [0, 200],
    label: "S_T"
  },
  y: {
    domain: [-150, 150],
    label: "Payoff"
  },
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.ruleX([x], {stroke: "gray", strokeDasharray: "4,4"}),
    Plot.line([[0, -x], [x, 0], [200, 0]], {stroke: "red"}),
    Plot.dot(
      [[sto, -Math.max(0, x - sto)]], 
      {
        fill: "blue", 
        r: 5,
        title: d => `S_T: ${sto}
X: ${x}
Payoff: ${-Math.max(0, x - sto)}`,
        tip: true
      })
  ],
  title: "Short Put position payoff"
})

Option payoffs at maturity:

• Long Call: \(\max(0, S_T - X)\)
• Short Call: \(-\max(0, S_T - X)\)
• Long Put: \(\max(0, X - S_T)\)
• Short Put: \(-\max(0, X - S_T)\)

Remember that these diagrams show only the payoff at maturity and do not account for the premium paid to acquire the options or received for writing them.