Option payoffs and strategies · RiskAtoms Academy

By the end you will be able to

1Draw payoff diagrams for spreads, straddles and collars.
2Match a strategy to the market view it expresses.
3Explain how a collar or a protective put changes the risk of owning a share.
4Build any payoff diagram by adding the payoffs of its legs.

Why it matters

Calls and puts are like building bricks. Put two or three together and you can build almost any shape of risk: insurance with a deductible, a bet on calm, a bet on chaos, a cap and a floor. Payoff diagrams let you see the shape at a glance.

Core explanation

Adding up the legs

A strategy is a set of legs: options, and sometimes the share itself. At expiry each leg pays what you saw in L1.7.06, so the whole strategy pays the sum of its legs. On a payoff diagram that means adding the lines, one price at a time.

Reading a diagram in three checks

Where does it bend? Only at the strikes. Between strikes every line is straight. These bends are the kinks.
Slope on the far right. Each long call or share adds +1, each short call −1. A positive slope means unlimited upside; a negative one means unlimited risk.
Slope on the far left. Each long put adds −1 (it gains as the price falls), each short put +1, and each share +1.

The strategies, by market view

Strategy	Legs	View
Bull call spread	Buy call K₁, sell call K₂ (K₂ > K₁)	Moderate rise
Bear put spread	Buy put K₂, sell put K₁ (K₁ < K₂)	Moderate fall
Long straddle	Buy call and put, same K	Big move, either way
Long strangle	Buy OTM call and OTM put	Very big move
Short straddle	Sell call and put, same K	Calm
Protective put	Own share, buy put	Insure a holding
Covered call	Own share, sell call	Earn income, give up upside
Collar	Own share, buy put, sell call	Floor and cap

Spreads trade some upside for a lower cost. A bull call spread buys a call and sells a higher-strike call: the sold call pays part of the cost but caps the gain. Straddles and strangles are bets on how much the price moves, not which way. A long straddle pays off from a big move either way, while a strangle is cheaper but needs a bigger move. Sell them instead and you win if the market stays calm, with large losses if it does not.

Strategies with the share change the risk of a holding. A protective put is insurance: the strike acts like the level of cover and the premium like the insurance cost. A covered call earns a premium in return for giving up gains above the strike. A collar does both: the sold call pays for the bought put, and the holding is boxed between a floor and a cap.

Name that shape

Profit at expiry; share price 70 to 130

The formal version

A strategy's payoff is a function of $S_T$ built by adding functions (see the Functions and graphs Math Atom). Take the bull call spread with strikes $K_1 < K_2$ :

\text{Payoff} = \max(S_T - K_1, 0) - \max(S_T - K_2, 0)

Split the price range at the two strikes and look at each piece:

\text{Payoff} = \begin{cases} 0 - 0 = 0 & S_T \le K_1 \\ (S_T - K_1) - 0 = S_T - K_1 & K_1 < S_T \le K_2 \\ (S_T - K_1) - (S_T - K_2) = K_2 - K_1 & S_T > K_2 \end{cases}

Flat, then rising, then flat at $K_2 - K_1$ . The straddle works the same way and collapses to a single absolute value:

\max(S_T - K, 0) + \max(K - S_T, 0) = |S_T - K|

Profit, as before, is the payoff minus the net premium paid (or plus the net premium received).

Try it

Pick a strategy to load its legs, then change them. Flip a leg between long and short, move a strike, or add a leg of your own, and watch the shape change. Open “Leg by leg” to see how each piece adds up at the marked price.

Payoff diagramStrategy builder

Value option strategies →

Legs

callKpremium

Tap or drag the chart to move the line

Price of the underlying at expiry$110

At an expiry price of $110

Payoff10.00

Profit+6.00

Leg by leg

Long call, strike 100, premium 6.00payoff 10.00, profit +4.00
Short call, strike 110, premium 2.00payoff 0.00, profit +2.00

Net cost today$4.00

Most you can make$6.00

Most you can lose$4.00

Breakeven104.00

Worked example

A collar, then a straddle

An investor holds a euro-area share bought at €50. She buys a put with strike €45 for €1.20 and sells a call with strike €58 for €1.10. All options expire on the same date; prices are illustrative.

The collar

Step 1 of 4

1
Net cost of the options
1.20 paid − 1.10 received = €0.10 per share. Nearly a zero-cost collar.

Regulatory lens · looking ahead

Clearing houses margin a strategy as a whole: a spread whose loss is capped needs less margin than its two legs taken separately. Bank capital rules also look at options as portfolios; the FRTB curvature charge (L4.3) exists because option payoffs bend.

Sparks

SparkA straddle is a bet on volatility

A long straddle does not care which way the price moves, only how far. That makes it a way to trade volatility itself, and its price tells you how big a move the market expects. L1.7.09 shows why volatility drives option prices.

SparkThe short straddle that sank a bank

Nick Leeson, the trader whose losses brought down Barings in 1995, had sold large numbers of straddles on the Nikkei 225 index: a bet that Japanese shares would stay calm. When they fell sharply, the losses on the puts kept growing. L1.7.12 tells the story.

Key takeaways

A strategy's payoff is the sum of its legs' payoffs. Lines bend only at the strikes.
Spreads cut the cost of an option by selling another, at the price of a capped gain.
Straddles and strangles bet on the size of a move, not its direction. Selling them bets on calm, with large risk.
Protective puts insure a holding; covered calls sell its upside; a collar sets a floor and a cap.
Check the slopes at the far left and right to spot unlimited gains or losses.

Reaction

Check what stuck

Question 1 · objective 1

You buy a call with strike 100 for 6 and sell a call with strike 110 for 2. What is the most you can make at expiry, per share?

Question 2 · objective 2

You expect a share to make a big move after a court ruling, but you do not know which way. Which strategy fits?

ABull call spreadBCovered callCLong straddleDShort straddle

Question 3 · objective 1

A straddle buys a call and a put, both with strike 50, for 3 and 2.50. What is the upper breakeven at expiry?

Question 4 · objective 3

What does a collar do for an investor who owns a share?

ADoubles the gain if the share risesBSets a floor and a cap on the value of the holdingCRemoves all costs of holding the shareDTurns the share into a bond paying a fixed coupon

Question 5 · objective 3

You own a share bought at 40, a put with strike 36 bought for 1, and a call with strike 46 sold for 1. At expiry the share is at 30. What is your total profit per share? (Enter a loss as a negative number.)

Glossary

Leg: One of the contracts that make up a strategy, such as a long call or a short put.
Payoff diagram: A chart of what a position is worth at expiry against the price of the underlying then.
Kink: A point where a payoff line changes slope. In option strategies the kinks sit at the strikes.
Spread: A strategy that buys one option and sells another of the same type with a different strike.
Straddle: Buying a call and a put with the same strike and expiry. It gains from a big move in either direction.
Strangle: Buying an out-of-the-money call and an out-of-the-money put. Cheaper than a straddle, but it needs a bigger move.
Protective put: Owning a share and buying a put on it, so the value cannot fall below the strike.
Covered call: Owning a share and selling a call on it, earning the premium but giving up gains above the strike.
Collar: Owning a share, buying a put below the price and selling a call above it: a floor and a cap.

Deep Dives & Origins

Deep Dives

John C. Hull, Options, Futures, and Other Derivatives, 11th edition (Pearson, 2021), the chapter “Trading Strategies Involving Options”. Beginner.
Sheldon Natenberg, Option Volatility and Pricing, 2nd edition (McGraw-Hill, 2015). Intermediate.

Origins

John C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021.
Sheldon Natenberg, Option Volatility and Pricing, 2nd edition, McGraw-Hill, 2015.

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