Quant from Scratch by QuantWright

Options and Derivatives · Lesson 1

Forwards and futures: the price of waiting

Agree today to buy a share in a year: what price is fair? Not a guess about next year, but the one price that leaves nobody a sure profit, worked out from interest and the costs of holding.

Know the basics? Read the lesson notes

A price for next year

A made-up share (a small piece of ownership in a company) sells for €100 today. You want to own one in a year. You could buy it now. Or you could sign a forward: a promise to buy later at a price agreed today, on a set delivery day (the day the share is handed over).

Money can be borrowed or lent at 5% a year, the interest rate: borrow €100 today and you repay €105 in a year. Every number in this lesson is made up.

What forward price, for delivery in one year, is fair to both sides?

Keep your guess in mind. The next screen works it out from two ways of owning the share.

Buy now, or agree now and pay later

Two plans both end the year with one share. Plan A: borrow €100 at 5% today and buy the share; in a year, repay the loan. Plan B: sign a forward at a price we call F; pay nothing today, and pay F in a year when the share is delivered. Press each moment to see the cash.

The cash in each plan (made-up numbers)

Plus: money you receive. Minus: money you pay.

PlanTodayIn a yearYou end with
A: buy now???
B: forward???

Both plans cost nothing today and end with one share. Plan A then costs €105: the €100 back, plus €5 interest. Two plans that cost nothing today and end in the same place must cost the same in a year, or someone profits for free (the next two steps show how). So the plans match only if F = €105: the answer to the guess is B.

What it costs to hold the thing a contract is on (the underlying, here the share) until delivery is the cost of carry; here, a year's interest. With €100 of your own, buying now gives up the €5 it could earn: €105 again. No guess about next year's price went into this.

When the forward price is too high

Say a made-up trading firm offers to buy the share from you in a year at €108, agreed today. You own no share and have no money. Try this: borrow €100 at 5%, buy the share, and sign the forward to sell it at €108. Fill in the last line.

WhenWhat you doCash
TodayBorrow €100 at 5%+€100
TodayBuy the share−€100
TodaySign to sell it at €108 in a year€0
In a yearDeliver the share, receive €108+€108
In a yearRepay the loan with interest−€105
In a yearNet?

What is the net in a year, in euros?

Hint: add the two cash lines in a year, +108 and −105.

108 − 105 = €3, whatever the share sells for in a year: you never sell it in the market, you deliver it. You put in no money of your own and took no price risk.
In a year you receive €108 for the share and repay €100 + €5 = €105, so the net is 108 − 105 = €3, whatever the share's price then.

A profit locked in with no money of your own and no risk is an arbitrage. This one, buying the underlying and holding it while selling it forward, is cash-and-carry. Anyone offered €108 would profit by repeating it, which in this made-up market pushes the price back toward €105. A price that leaves no such trade is a no-arbitrage price.

When the forward price is too low

Now the firm offers to sell you the share in a year at €102. The profit needs the opposite plan, and it uses short selling: borrowing a share from a broker (made up) that holds shares for others, selling it now, and giving a share back later. Which plan locks in a profit?

In a year the loan pays back €105; you pay €102 for a share through the forward and give it back to its lender. You keep 105 − 102 = €3, whatever the share's price then.
That plan owns two shares in a year and owes €105 + €102 for them, so its result depends on the share's price: nothing is locked in. Plan A works: 105 − 102 = €3 at any price.
The forward's buyer can lock in a profit: sell a borrowed share, lend the €100, and in a year get €105, pay €102 and return the share, keeping €3 at any price.

This is reverse cash-and-carry. Above €105, cash-and-carry locks in the forward price minus 105; below it, the reverse locks in 105 minus the forward price. Drag the forward price.

The profit locked in at each forward price (made-up numbers)

Across: the forward price agreed today. Up: the sure profit in a year, from whichever trade works.

The trade that workscash-and-carry
Sure profit in a year€3.00

Only at €105 does neither trade pay. That is the no-arbitrage forward price.

A sure profit, or a bet?

Back to the forward at €108. Here are 10,000 made-up prices of the share in a year. Each starts at €100; on each of 250 trading days a fair coin moves it €1 up or down (a random walk: past moves say nothing about which way the next goes), and a steady rise adds €5 over the year. So in this made-up world the share grows on average at the 5% interest rate.

On how many of the 10,000 does the cash-and-carry (borrow, buy, sell forward at €108) end with a profit?

Keep your guess in mind. The 10,000 prices are below.

10,000 simulated prices of the share in a year

Each bar counts paths ending at that price; they ran from €39 to €171, spread about evenly either side of €105 (with "Grows 12%", €7 higher). Dashed: the forward price. Orange: paths ending above it, where the long side gains. Grey: at or below it.

Paths so far0
Long side's average payoff–
Cash-and-carry's result–

What the share you hold gains or loses, the forward you sold gives back, so the price at the end never matters. The cash-and-carry made exactly €3 on 10,000 of 10,000 paths in this simulation: the answer is C. Guess A counts the forward alone and forgets the share.

The forward's buyer alone, the long side, gets the price in a year minus €108 (its payoff), which averaged −€2.98: a bet that loses about €3 on average. Switch to €105: the average is €0.02, about zero, so here neither side gains on average, before costs.

Now switch to a world where the share grows 12% a year. At €105 the long side averages +€7.02: buying forward is like borrowing €100 to buy the share, and the share now outgrows the loan. The cash-and-carry still makes exactly €0 at €105 and €3 at €108 on every path. The price comes from carry, not from a forecast. This can't show real prices, which don't move in €1 steps, or costs.

Theory

The forward price: today's price, carried

When holding the underlying pays nothing and costs nothing to store, the forward price that leaves no arbitrage is today's price grown at the interest rate until delivery. This is the no-arbitrage forward price. If interest is added continuously rather than once a year (continuous compounding), the growth factor is erT, a little more than (1 + r)T for any rate above zero.

F = S × (1 + r)T
F = S × erT   (interest added continuously)
F
the forward price, agreed today, for delivery at time T
S
the underlying's price today
r
the interest rate per year, as a decimal (5% is 0.05), the same for borrowing and lending
T
the years until delivery (half a year is 0.5)
(1 + r)T
(1 + r) multiplied by itself T times (T can be a fraction)
e
a fixed number, about 2.718
Worked example: this lesson's share

S = €100, r = 0.05. One year: 100 × 1.05 = €105. Two years: 100 × 1.052 = 100 × 1.1025 = €110.25. Half a year: 100 × 1.050.5 = 100 × 1.0247 = €102.47; 1.050.5 is the square root of 1.05, about 1.0247 (a calculator does it).

Continuous, one year: 100 × e0.05 = 100 × 1.0513 = €105.13.

Holds when
  • you can borrow and lend at the same rate r;
  • trades are free;
  • the underlying can be bought, held at no cost and sold short;
  • holding it pays nothing before delivery;
  • both sides pay what they owe.
Breaks when

Holding pays income or costs money to store: the next two screens. Rates differ, trades cost money or short selling is limited: a band instead of one price, at the end of this lesson.

The situationA made-up share sells for €80. Money can be borrowed or lent at 4% a year. Holding the share pays nothing.

What is the no-arbitrage forward price for delivery in 2 years, in euros?

Hint: 1.042 = 1.04 × 1.04 = 1.0816. Multiply by €80.

80 × 1.042 = 80 × 1.0816 = €86.53.
Grow €80 at 4% for two years: 80 × 1.04 × 1.04 = 80 × 1.0816 = €86.53. 80 × 1.04 = €83.20 is one year only; 80 × 1.08 = €86.40 leaves out the interest on the first year's interest.

A share that pays you to hold it

Many companies pay a dividend: cash paid out for each share, to whoever holds it. Say the share pays €3 in half a year. Plan A's holder gets it; the forward's buyer, who doesn't hold the share yet, does not. Interest is still 5%.

Is the no-arbitrage forward price for delivery in one year above, at, or below €105?

Buying now brings the €3; the forward's buyer misses it. So owning through the forward must be cheaper, not dearer: below €105.
Plan A now comes with €3 that plan B misses, so the plans match only if the forward price is lower: below €105.
The forward's buyer misses the €3, so the forward is cheaper by what that €3 is worth.

Follow plan A's loan of €100 at 5%. After half a year it has grown to 100 × 1.0247 = €102.47 (1.050.5 = 1.0247). The €3 dividend repays part of it, leaving €99.47, which grows to 99.47 × 1.0247 = €101.93 by delivery. Buying now costs that, so the forward must too. The answer is C.

The price falls by €3.07, a little more than €3, because the €3 arrives half a year early and earns interest until delivery.

A metal that costs money to keep

Now a made-up metal at €100 a bar. Keeping a bar in a vault for a year costs €2, paid at delivery. Holding it pays nothing, and interest is 5%. What forward price for delivery in one year leaves no arbitrage, in euros?

Hint: buying now has two costs by delivery: the loan, 100 × 1.05, and the €2 vault fee.

Buying now costs 100 × 1.05 = €105 by delivery, plus the €2 fee: €107.
Carry it as plan A: repay 100 × 1.05 = €105, then pay the €2 vault fee: €107. Storage is a cost of holding, so it raises the forward price; a dividend lowers it.

In present values, the way the dividend was handled: the fee is worth 2 ÷ 1.05 = €1.905 today, so F = (100 + 1.905) × 1.05 = €107.00. Income lowers the forward price; storage raises it.

Theory

Income and storage shift the carry

The general rule: take the present value (what a later amount is worth today) of any income the holder receives out of today's price, add the present value of any storage cost, and carry the rest to delivery at the interest rate.

F = (S − I + U) × (1 + r)T
F
the forward price, agreed today, for delivery at time T
S
the underlying's price today
I
the present value today of income paid to the holder before delivery, such as dividends (0 if none): I = each payment ÷ (1 + r)(years until paid), added up
U
the present value today of storage costs paid up to delivery (0 if none), worked out the same way
r
the interest rate per year, as a decimal, the same for borrowing and lending
T
the years until delivery
Worked example: your two cases

The share with a €3 dividend in half a year: I = 3 ÷ 1.050.5 = 2.928, U = 0, so F = (100 − 2.928) × 1.05 = €101.93.

The metal with a €2 fee at delivery: I = 0, U = 2 ÷ 1.05 = 1.905, so F = (100 + 1.905) × 1.05 = €107.00.

Holds when
  • the income and storage amounts and dates are known today;
  • the first formula's conditions hold: one rate for borrowing and lending, free trades, short selling possible.
Breaks when

A dividend is cut or changed, so I was wrong; or holders of a stored good value having it on hand and won't sell or lend it for the reverse trade, so the forward price can sit below the formula (for stored goods this is called a convenience yield, not covered here).

The situationA made-up share sells for €60 and will pay a €1.50 dividend in half a year. Money can be borrowed or lent at 4% a year.

What is the no-arbitrage forward price for delivery in one year, in euros?

Hint: I = 1.50 ÷ 1.040.5 = 1.50 ÷ 1.0198. Then (60 − I) × 1.04.

I = 1.50 ÷ 1.0198 = 1.471. F = (60 − 1.471) × 1.04 = 58.529 × 1.04 = €60.87.
I = 1.50 ÷ 1.0198 = 1.471, so F = (60 − 1.471) × 1.04 = 58.529 × 1.04 = €60.87. With no dividend it would be 60 × 1.04 = €62.40; taking off €1.50 undiscounted gives €60.84.

Futures: the same price, paid day by day

A futures contract is a forward with standard terms, traded through an exchange that stands between buyer and seller. Each side keeps a deposit there (money set aside as a guarantee), and each day's gain or loss moves from one side's deposit to the other's: daily settlement. A forward pays everything on delivery day. So futures gains arrive early and can earn interest, and losses must be paid early.

With a rate known in advance, the extra interest from the daily timing can be offset exactly (not derived here), and the futures price equals the forward price (Cox, Ingersoll and Ross, 1981). That is a condition, not a law.

Which change could pull a futures price away from the forward price for the same delivery day?

With a fixed rate, bigger swings mean bigger daily payments, but the extra interest can still be offset, so the prices stay equal. A rate that moves with the price breaks the match: B.
Then the futures buyer's early gains tend to come when rates are high, so they earn more, and early losses when rates are low, so they cost less. That favours the futures buyer, so the futures price sits a little above the forward's (Hull, chapter 5).
Daily settlement alone doesn't change the price while the rate is fixed: the extra interest can be offset exactly. A rate that moves with the price breaks the match: B.

When rates move and tend to rise as the underlying rises, the futures price sits a little above the forward's (Hull, chapter 5). With a fixed rate, every formula in this lesson gives the futures price too.

With costs, one price becomes a band

So far we assumed one interest rate, free trades and easy short selling. Now drop them. Cash-and-carry pays a cost c on top of the share's price, so it borrows S + c, at the higher rate. A year later its profit is F − (S + c) × (1 + rborrow), above zero only above an upper edge. The reverse gets S − c from its short sale and lends it at the lower rate; its profit, (S − c) × (1 + rlend) − F, is above zero only below a lower edge. For a one-year forward, with no dividend or storage:

(S − c) × (1 + rlend) ≤ F ≤ (S + c) × (1 + rborrow)
c
the cost of buying or selling one share today
rborrow
the yearly rate you pay when you borrow money
rlend
the yearly rate you earn when you lend money

Borrowing at 6%, lending at 4%, c = €0.50: (100 − 0.50) × 1.04 = €103.48 up to (100 + 0.50) × 1.06 = €106.53. Move both sliders for the €100 share, then switch short selling off.

Forward prices at which a trade locks in a profit, one year (made-up numbers)

Across: the forward price. Orange: the band where neither trade pays. Grey: prices where one trade locks in a profit. Dashed: lower edge. Solid: upper edge. Dot: the one-rate formula, €105.

Borrow / lend at5% / 5%
Lower edge€105.00
Upper edge€105.00

Between the two edges neither trade pays, so the forward price can sit anywhere in that range: the no-arbitrage band. Push both sliders to zero and it shrinks back to €105.

With short selling off, only someone who already owns the share can run the reverse trade, by selling it; without such owners, nothing holds the price up from below. The one-rate formula is the band shrunk to a line.

Practice 1 of 3

A dealer's offer for stored metal

The situationA made-up metal sells for €40 today. Storing it for a year costs €1, paid at delivery. Money can be borrowed or lent at 3% a year, and trades are free. A dealer offers to buy it from you in one year at €43.00.

How many euros a unit does cash-and-carry lock in, at delivery?

Hint: the no-arbitrage price is 40 × 1.03 + 1. The trade locks in the dealer's price minus that.

Borrow €40, buy, store, deliver: repay 40 × 1.03 = €41.20, pay the €1 fee, €42.20 in all, and receive €43.00: €0.80 a unit, at any price of the metal then.
Carrying costs 40 × 1.03 = €41.20 plus the €1 fee = €42.20; the dealer pays €43.00, so the trade locks in 43.00 − 42.20 = €0.80. Leaving out the fee gives €1.80.
Practice 2 of 3

"The market expects a rise"

The situationA friend sees a made-up share at €100 and its one-year forward at €105, with interest at 5% and no dividends. "So the market expects the share to rise 5% this year," they say.

What does the €105 tell you?

The €105 came from the loan, not from anyone's view: at any other price, cash-and-carry or its reverse locks in a profit. In the lesson's simulation, €105 held just as well in the world where the share grew 12% a year on average.
Nothing about direction went into it: 100 × 1.05 is set by interest. A forward above today's price is what carry gives whenever the rate is above zero and holding pays nothing.
100 × 1.05 = €105: today's price carried at 5% for a year. The same price holds whatever traders expect, because any other price lets someone lock in a profit.
Practice 3 of 3

Inside the band or outside?

The situationA made-up share sells for €50 and pays no dividend. You can borrow at 7% and lend at 3% a year; trades are free and short selling is allowed. A one-year forward is offered at €52.80.

Can anyone lock in a riskless profit?

Cash-and-carry pays only above 50 × 1.07 = €53.50; the reverse only below 50 × 1.03 = €51.50. €52.80 is between them, so neither trade pays.
There is no single 5% rate here. Cash-and-carry borrows at 7%, and repaying 50 × 1.07 = €53.50 costs more than the €52.80 it receives.
The reverse lends at 3%, so it gets back only 50 × 1.03 = €51.50, less than the €52.80 it pays through the forward.
✓

Lesson done

A forward price is not a forecast. When holding the underlying pays nothing and costs nothing, and one interest rate serves for borrowing and lending, it is today's price carried to delivery, because at any other price cash-and-carry or its reverse locks in a profit with no money and no risk. Income lowers it and storage raises it. A futures price equals it when the interest rate is fixed. Different rates, trading costs and limits on short selling widen the single price into a band.

Key ideas

Cost of carry
what it costs to hold the underlying until delivery: interest, plus storage, minus income
Arbitrage
a profit locked in with no money of your own and no risk
No-arbitrage forward price
the forward price at which no combination of trades locks in an arbitrage; with one rate, free trades and short selling, a single price
Cash-and-carry
borrow, buy the underlying and sell it forward; locks in a profit when the forward price is above the no-arbitrage price
Reverse cash-and-carry
sell the underlying short, lend the money and buy it forward; locks in a profit when the forward price is below the no-arbitrage price
Short selling
borrowing an asset, selling it now and giving one back later
Dividend
cash a company pays out for each share to whoever holds it
Continuous compounding
interest added continuously, so money grows by erT instead of (1 + r)T
Daily settlement
a futures contract's gains and losses paid every day between the two sides' deposits; with a fixed interest rate the futures price equals the forward price
No-arbitrage band
the range of forward prices at which neither cash-and-carry nor its reverse pays, once rates differ or trades cost money

F = S × (1 + r)T  ·  F = S × erT   (no income, no storage, one rate)

F = (S − I + U) × (1 + r)T

(S − c) × (1 + rlend) ≤ F ≤ (S + c) × (1 + rborrow)   (one year, no income, no storage)

Lesson notes

A forward is a promise to buy or sell later at a price agreed today. Its fair price is not a forecast of the later price: when holding the underlying pays nothing and costs nothing, one interest rate r serves for borrowing and lending, trades are free and short selling is possible, the only forward price that leaves nobody a riskless profit is today's price carried to delivery, F = S × (1 + r)T. Income paid to the holder lowers it and storage costs raise it. A futures price equals it when the interest rate is fixed; different rates, costs and limits on short selling turn the single price into a band.

Two ways to own the share in a year

A made-up share sells for €100 and money can be borrowed or lent at 5% a year (all numbers made up). To own one share in a year, plan A borrows €100 today and buys it, then repays €105 in a year; plan B signs a forward at a price F and pays F on the delivery day, a year away (the day the share is handed over). Both cost nothing today and end with one share; two plans like that must cost the same in a year, or someone profits for free, so they match only if F = €105. Holding the underlying (the thing the contract is on) until delivery costs a year's interest: the cost of carry. No guess about next year's price enters.

Too high: cash-and-carry

If a made-up trading firm offers €108 for delivery in a year, borrow €100, buy the share and sign to sell it at €108. In a year you deliver the share, receive €108 and repay €105: a net of €3, whatever the share's price then, with no money of your own and no price risk. That is an arbitrage, and this trade is cash-and-carry; anyone offered €108 would profit by repeating it, which in this made-up market pushes the price back toward €105. A price that leaves no such trade is a no-arbitrage price.

Too low: reverse cash-and-carry

If the firm offers to sell at €102, use short selling (borrowing a share from a broker, made up, that holds shares for others, selling it now and giving one back later): sell a borrowed share for €100, lend the €100 at 5% and sign to buy at €102. In a year the loan pays back €105, you pay €102 for a share and return it, keeping €3 at any price. This is reverse cash-and-carry. Above €105 cash-and-carry locks in the forward price minus 105; below, the reverse locks in 105 minus the forward price; only at €105 does neither pay.

A sure profit, or a bet: the simulation

The lesson drew 10,000 made-up prices of the share a year ahead: each started at €100 and took 250 daily fair-coin steps of €1 up or down (a random walk: past moves say nothing about which way the next goes) plus a steady rise of €5 over the year, so the share grew on average at the 5% interest rate. In this simulation the cash-and-carry at €108 made exactly €3 on 10,000 of 10,000 paths: the share's gain or loss is given back by the forward sold. The forward's buyer alone (the long side), whose payoff is the price in a year minus the forward price, averaged −€2.98 at €108 and €0.02 at €105, about zero, before costs. With the same coin tosses in a world where the share grew 12% a year on average (€12 on €100), the long side averaged +€7.02 at €105, the same position as borrowing €100 to buy a share that outgrows the loan, while the cash-and-carry still made exactly €0 at €105 and €3 at €108 on every path. The forward price comes from carry, not from a forecast. Real prices don't move in €1 steps and real trades cost money; the simulation shows neither.

Income and storage

A dividend is cash a company pays out for each share to whoever holds it. If the share pays €3 in half a year, plan A's holder gets it and the forward's buyer does not, so the forward must be cheaper. Follow plan A's loan of €100 at 5%: after half a year it has grown to 100 × 1.0247 = €102.47 (1.050.5, the square root of 1.05, is 1.0247); the €3 dividend repays part of it, leaving €99.47, which grows to 99.47 × 1.0247 = €101.93 by delivery. So F = €101.93, €3.07 below 105 because the €3 arrives early and earns interest until delivery. The same answer in present values (what a later amount is worth today): the €3 is worth 3 ÷ 1.0247 = €2.928 today, and (100 − 2.928) × 1.05 = 97.072 × 1.05 = €101.93. A made-up metal at €100 a bar that costs €2 a year to store, paid at delivery, works the other way: buying now costs 100 × 1.05 + 2 = €107 by delivery, so F = €107. Income lowers the forward price; storage raises it.

Futures and forwards

A futures contract is a forward with standard terms traded through an exchange that stands between buyer and seller. Each side keeps a deposit there (money set aside as a guarantee), and each day's gain or loss moves from one side's deposit to the other's: daily settlement. Gains therefore arrive early and can earn interest, and losses must be paid early. With a rate known in advance, the extra interest from the daily timing can be offset exactly (not derived here), and the futures price equals the forward price (Cox, Ingersoll and Ross, 1981). If rates move and tend to rise when the underlying rises, the futures buyer's early gains earn more and its early losses cost less, so the futures price sits a little above the forward's (Hull, chapter 5). Equality is a condition, not a law.

A band, not a line

For a one-year forward on a share with no dividend or storage cost, with a cost c paid on top of the share's price whenever it is bought or sold today and different rates, cash-and-carry borrows S + c and its profit a year later is F − (S + c) × (1 + borrowing rate), so it pays only above (S + c) × (1 + borrowing rate); the reverse lends the S − c it gets from the short sale, its profit is (S − c) × (1 + lending rate) − F, and it pays only below (S − c) × (1 + lending rate). Borrowing at 6%, lending at 4% and a cost of €0.50 give a band from (100 − 0.50) × 1.04 = €103.48 to (100 + 0.50) × 1.06 = €106.53, the no-arbitrage band: inside it neither trade pays, so the forward price can sit anywhere in it. If short selling is not allowed, only someone who already owns the share can run the reverse trade (by selling it); without such owners, nothing holds the price up from below. The one-rate formula is the band shrunk to a line, when its conditions hold.

Formula: the forward price, today's price carried

When holding the underlying pays nothing and costs nothing to store, the no-arbitrage forward price is today's price grown at the interest rate until delivery (the cost-of-carry formula). With interest added continuously (continuous compounding), the growth factor is erT.

F = S × (1 + r)T
F = S × erT   (interest added continuously)
F
the forward price, agreed today, for delivery at time T
S
the underlying's price today
r
the interest rate per year, as a decimal (5% is 0.05), the same for borrowing and lending
T
the years until delivery (half a year is 0.5)
(1 + r)T
(1 + r) multiplied by itself T times (T can be a fraction)
e
a fixed number, about 2.718
Worked example

S = €100, r = 0.05. One year: 100 × 1.05 = €105. Two years: 100 × 1.052 = 100 × 1.1025 = €110.25. Half a year: 100 × 1.050.5 = 100 × 1.0247 = €102.47. Continuous, one year: 100 × e0.05 = 100 × 1.0513 = €105.13.

Holds whenBreaks when

Holding pays income or costs money to store (the next formula). Rates differ, trades cost money or short selling is limited: a band instead of one price.

Formula: income and storage shift the carry

Take the present value of any income out of today's price, add the present value of any storage cost, and carry the rest to delivery.

F = (S − I + U) × (1 + r)T
F
the forward price, agreed today, for delivery at time T
S
the underlying's price today
I
the present value today of income paid to the holder before delivery, such as dividends (0 if none): I = each payment ÷ (1 + r)(years until paid), added up
U
the present value today of storage costs paid up to delivery (0 if none), worked out the same way
r
the interest rate per year, as a decimal, the same for borrowing and lending
T
the years until delivery
Worked example

The share with a €3 dividend in half a year: I = 3 ÷ 1.050.5 = 2.928, U = 0, so F = (100 − 2.928) × 1.05 = €101.93. The metal with a €2 fee at delivery: I = 0, U = 2 ÷ 1.05 = 1.905, so F = (100 + 1.905) × 1.05 = €107.00.

Holds whenBreaks when

A dividend is cut or changed, so I was wrong; or holders of a stored good value having it on hand and won't sell or lend it for the reverse trade, so the forward price can sit below the formula (for stored goods this is called a convenience yield, not covered here).

One more worked example

New made-up numbers

A made-up share sells for €50, pays a €1 dividend in half a year, and money costs 3% a year to borrow or earns 3% to lend. What is the no-arbitrage forward price for delivery in one year?

Without the dividend: 50 × 1.03 = €51.50.

The dividend's present value: I = 1 ÷ 1.030.5 = 1 ÷ 1.0149 = 0.985.

F = (50 − 0.985) × 1.03 = 49.015 × 1.03 = €50.49. Check with cash-and-carry: borrow €50 and buy; in half a year put the €1 dividend on deposit (it grows to 1 × 1.030.5 = €1.0149); in a year repay €51.50 and receive F. The trade breaks even when F + 1.0149 = 51.50, so F = €50.49.

Common mistakes

Check yourself

  1. A made-up share sells for €200, pays nothing before delivery, and money costs 3% a year. What is the no-arbitrage forward price for delivery in one year?
    Answer

    200 × 1.03 = €206.

  2. For the same share, a dealer offers to buy in a year at €210. Which trade locks in a profit, and how much?
    Answer

    Cash-and-carry: borrow €200, buy the share, sign to sell at €210. In a year receive €210 and repay €206: €4, at any price of the share then.

  3. A made-up metal sells for €200 a bar and costs €3 a year to store, paid at delivery; money costs 3% a year. What is the one-year forward price?
    Answer

    Buying now costs 200 × 1.03 = €206 by delivery, plus the €3 fee: €209. In the formula: U = 3 ÷ 1.03 = 2.913, and (200 + 2.913) × 1.03 = €209.00.

  4. Why does a futures price equal the forward price for the same day only when the interest rate is fixed?
    Answer

    Daily settlement pays gains and losses early, which earns or costs interest. With a fixed, known rate that timing can be undone exactly by holding a little more or less of the contract each day. If rates move with the price, the early payments earn or cost different amounts and the two prices can differ.

  5. A €100 share pays nothing; you can borrow at 5% and lend at 2%, trades are free and short selling is allowed. A one-year forward is offered at €101. Is there an arbitrage?
    Answer

    Yes. The band runs from 100 × 1.02 = €102 to 100 × 1.05 = €105, and €101 is below it: sell a borrowed share for €100, lend it at 2%, sign to buy at €101. In a year you get €102, pay €101 and return the share: €1.

Try it yourself

Run the simulation. The 10,000 prices and every number in this lesson come from one Python script with a fixed random seed (below). Change the steady rise, the number of days or the forward price, and see which numbers move and which stay fixed.

Work it out by hand. For any forward price, write both plans as a cash table: today, and on delivery day. If one plan costs nothing today and leaves a sure amount on delivery day, that amount is the arbitrage; the forward price that makes it zero is the no-arbitrage price.

The code behind every number (Python, seed 2026111201)
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For the precise: the simulationSeed 2026111201; 10,000 paths × 250 fair-coin steps of €1, drawn as one block of uniform numbers (a day is up when its number is below 0.5), plus a steady rise of €5 (world 1) or €12 (world 2) over the year; world 2 uses the same tosses, so each of its final prices is world 1's plus €7. This script draws its own coins; it does not use the course's shared simulated market. World 1's final prices ran from €39 to €171, averaging €105.02, with a standard deviation (the typical distance from the average) of €15.83, against €1 × √250 = €15.81 for the coin part. At a forward price of €105 in world 1, the long side gained on 4,738 paths and lost on 4,732 (the rest ended at exactly €105). At €108, the forward's seller alone, holding no share, lost on 4,261 of 10,000: the sure €3 belongs to the whole cash-and-carry, not to the seller's side alone. The hedged result is (F − ST) + (ST − 105) = F − 105 on every path, before costs.

Honest notes

Where this comes from