Introduction to Quant Finance · Lesson 23
Forwards, futures and options
Contracts whose value depends on another price: agree a price today and pay the difference later, or buy a right without the duty.
Know the basics? Read the lesson notesA price agreed today
A farmer will harvest wheat in six months. A miller will need wheat then. Neither knows what wheat will sell for in six months, so today they sign a promise: on that day the miller buys the farmer's wheat at €200 a tonne, whatever wheat sells for then. Every price in this lesson is made up.
A promise to buy or sell later at a price agreed today is a forward contract, or a forward. It is the simplest derivative: a contract whose value depends on another price. The thing whose price it depends on, here wheat, is the underlying.
Delivery day comes, and wheat is selling at €240 a tonne. Who is glad they signed?
Draw the miller's line
On delivery day the contract is worth, to the miller, the price then minus the agreed price: at €240, 240 − 200 = +€40 a tonne. What a contract pays at its end is its payoff. Drag the price at delivery from one end to the other; the dot draws the miller's payoff at every price.
The forward's payoff on delivery day (made-up prices)
Across: the price of wheat on delivery day. Up: the payoff, in euros a tonne. Dashed: the agreed price, €200.
A straight line through €0 at the agreed price: each €1 on the price is €1 of payoff, gain above €200 and loss below.
The side that agreed to buy is called long; the side that agreed to sell is short. The farmer's payoff is the mirror image, 200 minus the price (the grey line), so the two always add up to zero: one side's gain is the other side's loss.
Why sign it
The farmer grows wheat, so a fall in its price hurts. Under the forward, the farmer gets €200 a tonne at any price: when wheat sells for less, the contract pays the farmer the difference; when it sells for more, the farmer pays it. Drag the price and compare the farmer's income with and without the contract.
The farmer's income per tonne on delivery day (made-up prices)
Grey: without the forward, the market price. Orange: with it, the market price plus the forward's payoff (200 minus the price).
With the forward, the farmer's income is €200 at every price. The price risk is gone, and so is any gain from a rise.
Taking on a contract that cancels a price risk you already have is hedging. An importer that must pay a supplier in a foreign currency later can hedge the other way, with the long side of a forward on that currency: if the currency gets dearer, the forward pays the difference. Taking the long side with no wheat, only to gain if the price rises, is speculating.
Futures: a forward with a referee
A forward is a private promise, and one side may fail to pay. A futures contract is a forward with standard terms (here, 100 tonnes per contract) traded on an exchange, a market with set rules that stands between buyer and seller. Each side deposits money with the exchange, called margin: here €2,000. Every day the exchange sets a closing price (the day's last price) and moves that day's gain or loss from one side's margin to the other's. This is daily settlement.
You bought one contract at €200. Press Next day.
One futures contract, five days (made-up prices)
Top: the day's closing price. Bottom: your margin after that day's payment.
The five payments add up to +€600, the same as (206 − 200) × 100: what a forward agreed at €200 pays when the price ends at €206. A futures contract pays the same total, day by day.
On day 3 the price was €191, 4.5% below €200, and €900 of your €2,000 margin, 45%, had gone to the other side.
Leverage: a small move, a big swing
Your €2,000 margin carries wheat worth 100 × €200 = €20,000, ten times as much. Controlling more than the money you put down is leverage: it makes every move in the price ten times bigger as a share of your money, and that holds for losses exactly as for gains.
Wheat falls 5%, from €200 to €190. What happens to your €2,000 margin?
Your €2,000 margin after a move in the price (made-up prices)
The bar is your margin; the move is from €200.
A 10% fall takes the whole €2,000; past that, the loss is more than you deposited. If the margin runs out, the exchange asks for more money or closes the position. A 5% rise adds €1,000 the same way.
Options: the right, not the duty
A forward binds both sides. An option gives its buyer a right without a duty. A call is the right to buy the underlying at a fixed price, the strike, on a set last day.
Take a call on one tonne of wheat with a strike of €200. On the last day, if wheat sells for more than €200, the buyer uses the right and gains the difference. At €200 or less, the buyer simply doesn't use it, and the payoff is 0. Drag the price on the last day.
A call's payoff on its last day, strike €200 (made-up prices)
Across: the price of wheat on the last day. Up: the payoff, in euros a tonne. Grey dashed: the forward's line, for comparison.
Above €200 the call pays like the forward: at €240, +€40. Below €200 it pays nothing instead of a loss. That bend at the strike is what makes an option different.
The premium, and the capped loss
A right without a duty isn't free. The buyer pays the seller up front, when they agree, a price called the premium: say €12 a tonne for this call. The option's last day is its expiry. The buyer's result is the payoff minus the premium.
At expiry wheat sells for €150. How many euros a tonne did the call's buyer lose, in all?
Hint: at €150, is the right to buy at €200 worth using? Then subtract the premium, and type the loss as a positive number.
The call buyer's result: payoff minus the €12 premium (made-up prices)
Orange: the call buyer. Grey dashed: the forward's buyer, who paid no premium.
However low wheat goes, the call's buyer loses at most the €12 premium. To come out ahead, wheat must end above 200 + 12 = €212. The seller of the call holds the other side: it keeps the €12, but owes the buyer every euro above €200, with no limit.
The put: the right to sell
A put is the right to sell the underlying at the strike on expiry. It pays when the price ends below the strike: at €170, the buyer of a put with a strike of €200 sells for €200 what sells for €170 elsewhere, a payoff of €30. Drag the price.
A put's payoff at expiry, strike €200 (made-up prices)
Across: the price of wheat at expiry. Up: the payoff, in euros a tonne.
The put's line is the call's turned around: it rises as the price falls, and is 0 at or above the strike.
How often does a bought call pay?
Take 10,000 made-up prices, each starting at €200. Every day a fair coin (heads and tails equally likely) moves each one: heads up €1, tails down €1, for 100 days. A running total of random steps like this is a random walk; each toss is new, so past moves say nothing about which way the next one goes. On every walk, a call with a strike of €200 expires on day 100.
Out of the 10,000 calls, how many expire worthless, with a payoff of 0?
Where 10,000 simulated prices ended on day 100
Each bar counts walks ending at that price. Orange: the call pays. Grey: it expires worthless.
With the coin fair, about as many walks ended above €200 as below: 4,546 above and 4,606 below. Another 848 ended at exactly €200, where the right to buy at €200 is worth nothing too. So in this simulation 5,454 of 10,000 calls, a little over half, expired worthless.
Averaged over all 10,000, the payoff was €3.97 a tonne, before any trading costs (fees, and the gap between buying and selling prices). Move the strike: a higher one leaves more calls worthless.
Stack a call and a put
Payoffs add up: holding two contracts pays the sum of their payoffs (payoffs only, before premiums). Take a bought call and a sold put, both with a strike of €200. Selling the put means taking the other side of it: you receive its premium and pay its payoff, so its line is the put's turned upside down. Add each one, then drag the price.
A bought call plus a sold put, both strike €200 (made-up prices)
Payoffs only, before premiums. Solid black: the call. Dotted black: the sold put. Wide orange: the two added up. Grey dashed: the forward's line, price minus €200. Lines that share a stretch are drawn a little apart.
The sum is the forward's straight line: call payoff minus put payoff = price − 200 at every price.
Above €200 the call pays and the put pays nothing; below €200 the call pays nothing and the sold put costs the difference. This identity is put–call parity, here in its payoff form, on the last day.
Payoffs at the end, and put–call parity
A derivative's payoff is what it pays at its end, per unit of the underlying (per tonne of wheat, say), worked out from the price then. Three payoffs cover this lesson, and put–call parity links them: a call's payoff minus a put's, same strike, is the payoff of a forward agreed at the strike. The function max(a, b) means the larger of a and b, so max(S − K, 0) is S − K when that is above 0, and 0 otherwise.
- S
- the price of the underlying on delivery day (forward) or at expiry (option)
- F
- the forward's agreed price
- K
- the option's strike: the price at which it lets its buyer buy (call) or sell (put)
- premium
- what the option's buyer paid the seller up front
- max(a, b)
- the larger of the two numbers
Strike €200, price €240: call max(40, 0) = 40, put max(−40, 0) = 0, call − put = 40 = 240 − 200. Price €170: call 0, put max(30, 0) = 30, call − put = −30 = 170 − 200.
The call bought for €12, price €150: 0 − 12 = −€12. The futures contract: (206 − 200) × 100 tonnes = €600.
- the options can be used only at expiry, as here (some can be used earlier);
- both sides pay what they owe (futures margin is there for this);
- results are before trading costs (fees, and the gap between buying and selling prices), and before interest on money paid up front;
- amounts are per unit: multiply by the number of units.
You ask what a contract is worth before its end. Payoffs don't give that price: pricing needs no-arbitrage arguments (a fair price leaves nobody a sure profit from combining contracts) and models (rules for how prices might move), taught in a later course. Parity's full version, with prices today instead of payoffs, adds interest, and comes with them.
What is the call's payoff minus the put's payoff, in euros?
Hint: work out max(S − K, 0) and max(K − S, 0) with S = 58 and K = 50, then subtract.
What this lesson does not do
Three things payoff lines don't show. Turn them over one at a time.
- A payoff line is not a price.
What an option is worth today, before anyone knows the final price, is pricing. It needs the no-arbitrage arguments and price models of a later course.
- The simulation's counts belong to its own world.
The 5,454 worthless calls and the €3.97 average came from fair-coin walks with €1 steps, before costs. Prices that drift, jump or swing more give other counts. The average is not what the €12 premium should be: that needs pricing, in a later course.
- The risk comes with the gain.
Leverage turned a 4.5% fall into 45% of the margin, and it works the same way up. The seller of a call has no cap on its loss. Before costs, the two sides' payoffs add up to zero: one side's gain is the other's loss.
The importer's hedge
Was the hedge a mistake?
A put, all in
What is the put buyer's result, payoff minus premium, in euros per unit?
Hint: payoff = max(K − S, 0) with K = 80 and S = 68. Then subtract the premium.
A smaller margin
What share of the margin is lost, in percent?
Hint: the loss is the fall per unit times the number of units. Then divide by the margin.
Lesson done
A forward fixes a price today and pays the difference later; a futures contract does it day by day, on a small deposit (margin) that makes your money swing many times faster than the price, both ways (leverage). An option is a right without a duty; at expiry a bought call plus a sold put pays what a forward at the strike pays. Payoffs are not prices.
Key ideas
- Derivative
- a contract whose value depends on the price of something else, its underlying
- Forward contract
- a promise to buy or sell later at a price agreed today
- Long / short
- the side that agreed to buy / to sell
- Payoff
- what a contract pays at its end, worked out from the price then
- Hedging / speculating
- taking a contract that cancels a price risk you already have / taking the price risk on purpose
- Futures contract
- a forward with standard terms, traded on an exchange, settled every day
- Margin
- money each side of a futures contract deposits with the exchange; the daily gains and losses are paid from it
- Leverage
- controlling more than the money put down, so gains and losses on that money are a multiple of the price's move
- Call / put
- the right, not the duty, to buy / to sell the underlying at the strike at expiry
- Strike / expiry / premium
- an option's fixed price / its last day / what its buyer pays up front
- Put–call parity (payoff form)
- at expiry, a call's payoff minus a put's, same strike, equals the price minus the strike
forward: S − F · call: max(S − K, 0) · put: max(K − S, 0)
max(S − K, 0) − max(K − S, 0) = S − K