A baker in Monterrey needs wheat in six months and a farmer in the Bajío has it. Each can wait and take whatever the price turns out to be, or they can sign a paper today fixing the price of delivery on a future date. One of them will be unhappy with the number when the day comes. Both may still prefer the paper, because for each the uncertainty was the problem, not the price. That paper is a forward contract, the oldest derivative, and the simplest answer to the question this course asks: how do you reshape a risk you cannot remove?
The same idea runs through an airline's fuel bill, a government's oil budget, a company swapping a floating-rate loan for a fixed one, and a homeowner's insurance policy, which is secretly an option. Done for protection it is hedging; done for a bet, speculation. The contract is the same; intention and size decide which.
Plain statement: this piece describes how derivatives work and are priced. It is not investment, tax or legal advice, and it recommends no trade, no position and no product. The assets, prices and firms in the examples are invented; real episodes named below are history, not guidance. Words such as long, short, holder and writer name the two sides of a contract.
What should the agreed price for delivery in six months be? Not anyone's guess about the future. Suppose Marlow wheat (an invented grade) costs 300 per tonne today, money earns 4% a year and storing a tonne costs 2% of its value a year. A dealer can borrow 300, buy a tonne, store it and deliver it in six months, which costs 300 × e(0.04+0.02)×0.5 = 309.14. So the fair forward price is 309.14. If the market quotes 315, a dealer who agrees to deliver at 315 and does the borrowing and storing earns 5.86 per tonne whatever the wheat price does; if the quote is lower, holders of inventory profit from the reverse. Competition pushes the quote to the cost of carry. This is no-arbitrage pricing, the engine of the whole subject.
The convenience yield explains why some commodity futures sit below spot when goods are scarce: holding the real thing is worth something. For a financial asset, y is its payout.
Invented grade of wheat. The trade described works for holders of storage and borrowing at the stated rate; real frictions (bid-ask spreads, borrowing premiums, quality differences) widen the no-arbitrage band.
A futures contract is a forward that has been standardised, traded on an exchange and settled every day. Each evening the exchange's clearing house moves cash from losers to winners according to the day's price change, and each side has posted a deposit, the margin. If the deposit falls below a maintenance level, a margin call demands a top-up that day. Daily settling lets strangers trade without trusting each other, since credit risk resets nightly. Its cost is liquidity: a hedger can hold exactly the right position and still be forced to find cash at the worst moment, as Metallgesellschaft was, below.
Take a contract on 50 tonnes at 300 (a value of 15,000) with a 10% initial margin of 1,500 and a maintenance level of 75% of it, 1,125. On the path in the tool below, the long side faces two margin calls totalling 950 although the price ends only 1 lower, a 50 loss. A 10% deposit means a 1% price move is a 10% move on your money.
Invented nine-day price path, 50 tonnes. Red dots mark margin calls; maintenance is 75% of initial margin. Interest on margin is ignored.
A forward binds both sides. An option gives one side a right and the other an obligation. A call gives its holder the right, not the duty, to take an asset at a fixed strike price on or before an expiry date; a put gives the right to deliver it at the strike. The holder pays the writer a premium up front, because a right is worth money and a duty costs it. Insurance has the same shape: the premium buys a payout in the bad case and nothing otherwise. The shapes of the payoffs at expiry are the vocabulary of the field: a call pays max(S − K, 0), a put pays max(K − S, 0).
The history is old. Aristotle tells in his Politics (book 1) how Thales, forecasting a good olive harvest, paid a small deposit for the use of the presses in Miletus and rented them out at the peak; whether that was an option or a futures contract is still argued. Rice trading in Osaka's Dōjima district, licensed in 1697, had futures-like contracts by around 1710. Exchange-listed options date from 26 April 1973, when the Chicago Board Options Exchange opened.
Combining options builds new shapes. Choose a strategy below and see the profit at expiry once the cost of the options is included. A protective put (holding the asset plus a put) is a floor, like insurance; a covered call (holding the asset and having written a call) gives up the upside above the strike for income now; a straddle (a call and a put at one strike) pays if the price moves a lot in either direction; a bull spread is a call at one strike held against a call at a higher strike written, which caps both loss and gain.
Invented share at 100, six months to expiry, interest rate 3%. Premiums come from the Black–Scholes formula below. Profit ignores interest on the premium. A bull spread uses a second strike 10 higher.
How can an option have a price without a forecast? Because a trader can replicate it. In 1973 Fischer Black and Myron Scholes, and independently Robert Merton, showed how; the discrete version is the binomial tree of John Cox, Stephen Ross and Mark Rubinstein (1979). Start small. A share is at 100; in a year it will be either 122.14 or 81.87 (a 20% volatility, as the tree is built below). The interest rate is 5%. A call with strike 100 pays 22.14 in the up case and nothing in the down case. Hold 0.55 shares and borrow 42.82: in the up case the shares are worth 67.2, the loan is 45.0, and you net 22.14; in the down case you get 45.0 − 45.0 = 0. The portfolio copies the option exactly, so the option must cost what the portfolio costs: 0.55 × 100 − 42.82 = 12.16. If it were any different, there would be a free lunch.
Notice what never entered: the chance that the share goes up. Rewrite the same price as a discounted average using the risk-neutral probability q = (erT − d) ÷ (u − d), here 0.577: price = e−rT × [q × 22.14 + (1−q) × 0]. That q is not a forecast. It is the weight that makes the replication arithmetic work, and it is the same for the optimist and the pessimist. Believing the share will rise 30% does not move the price. What beliefs change is the payoff you expect to collect, not what the replicating portfolio costs. A second consequence is put-call parity: a call minus a put with the same strike and date equals the share minus the discounted strike, C − P = S − Ke−rT, exactly, since the two sides produce identical payoffs.
Add more steps and the tree's price converges on the Black–Scholes price. Slide the number of steps from 1 to 200.
Share 100, strike 100, one year. Up and down factors e±σ√Δt. Change your belief and watch the price stay still while the expected payoff moves. Beliefs that imply a probability outside 0 to 1 are capped.
Let the tree's steps shrink to nothing and you get the Black–Scholes–Merton formula. Their 1973 papers, Black and Scholes in the Journal of Political Economy and Merton in the Bell Journal of Economics, produced a closed formula whose derivation reduces to a rate-of-change equation of the kind in Differential Equations (34/03), related to the heat equation. Scholes and Merton shared the 1997 Nobel prize in economics; Black had died in 1995. Louis Bachelier had modelled random price movements in his 1900 thesis, long before.
Four of the five inputs can be looked up. The fifth, volatility, cannot, so market prices are usually quoted by the volatility that matches them: the implied volatility. In an option market, one is literally trading views on how much the share will move, not which way. For our share (100, strike 100, six months, 3%, volatility 30%) the call costs 9.15 and the put 7.66, and parity holds: 9.15 − 7.66 = 1.49 = 100 − 100e−0.015.
The sensitivities of the price to each input are called the Greeks, and a risk manager lives by them. Delta is the change in option price for a 1-unit move in the share: 0.57 for this call, so the option behaves like 57 shares per 100 options. Gamma is how fast delta itself changes (0.0185 per unit): large near the strike and near expiry. Vega is the sensitivity to volatility (0.28 for each volatility point), and theta is the loss of value as time passes (about −2.7 cents a day, all else equal). Time itself is a cost to the holder: a right that is running out is worth less.
Interest rate fixed at 3%, no payouts on the share, European-style options. Illustrative invented share. Theta is per calendar day, vega per volatility point.
The model assumes smooth random paths, constant volatility, no jumps and costless trading. None of that is true, and the market knows it: options at different strikes imply different volatilities (the "smile" or "skew"). The formula survives as a language: traders use it to translate a price into a volatility and back, then argue about the volatility.
The replicating portfolio is a recipe as well as a pricing trick. A bank that has written a call hedges it by holding delta shares and adjusting as the price moves. With continuous trading and the model's assumptions, the hedge cancels all the risk. In practice trading is discrete, so the hedge is only approximately right between adjustments, and volatility is not what the model assumed.
Run it. For a one-year, at-the-money call written at an implied volatility of 30%, with the actual future volatility on the slider, the tool simulates 300 random paths and shows the hedger's final profit or loss. Two results appear: the spread falls roughly with the square root of the number of adjustments (from about 8.2 with one adjustment to 0.7 with 250), and the average profit is about the difference between the option's value at the volatility that actually happened and the one it was sold at. A hedged writer has not removed risk; it has swapped price-direction risk for volatility risk.
Option sold at 30% volatility (price about 13.3 on a share at 100, strike 100, 3% rate). 300 seeded paths, so a setting repeats. Profit is measured at expiry for the writer after hedging. No trading costs.
A swap exchanges payments. In an interest-rate swap, one side pays a fixed rate on an agreed notional amount and receives a floating rate (tied to a benchmark that resets); the other does the reverse. No notional changes hands, only the net difference. A company with a floating-rate loan that wants a predictable cost pays fixed and receives floating, and the floating receipts offset its loan interest. Swaps became public in 1981, when IBM and the World Bank agreed a currency swap.
The fixed rate that makes a new swap worth nothing is the par swap rate. It comes straight from the discount factors of the yield curve: s = (1 − Dn) ÷ (D1 + … + Dn), the reasoning being that the fixed leg is a bond, and a floating leg is worth its face value on a reset date. After rates move, the swap's value is the difference between the new par rate and the old fixed rate, times the sum of the discount factors and the notional. In the tool, a 5-year swap on 10 million is fixed at the par rate on today's curve. Then everything shifts. The fixed payer gains when rates rise: it is locked in at the old, lower rate.
Notional 10 million, annual payments, continuously compounded zero rates rising in a straight line over years 1 to 5. Invented curve; the swap is between invented parties. Counterparty credit risk, which real markets price separately, is ignored.
A hedge is a position that offsets another exposure; the test is whether the combined position is less risky than the business alone. Mexico's Finance Ministry announced in January 2020 that it had hedged the government's 2020 oil exports at an average of 49 dollars a barrel using put options, at a reported cost of around one billion dollars; when oil prices collapsed that spring, press reports put the payout at more than two billion (official figures were not published). A clean case of an option as insurance. Others went the other way.
| Case | What happened | The lesson |
|---|---|---|
| Barings, 1995 | A Singapore trader took large unhedged Nikkei 225 futures positions and hid the losses (£827 million). The bank went insolvent on 26 February 1995 and was sold to ING for £1. | Controls: one person ran both the trading and the back office. |
| Metallgesellschaft, 1993 | A German group hedged long-term fixed-price oil sales with short-dated futures. When prices fell, margin calls forced it to close out; the loss was about 1.3 billion dollars. Economists still dispute whether the hedge itself was sound. | A hedge can be right in value and still fail in cash. |
| LTCM, 1998 | A highly leveraged hedge fund, with Scholes and Merton among its partners, lost about 4.6 billion dollars in under four months; a private rescue of about 3.6 billion, organised by the New York Fed, was agreed on 23 September 1998. | Models assume correlations hold; leverage leaves no room when they do not. |
| AIG, 2008 | A unit sold credit default swaps insuring hundreds of billions of dollars of securities. Downgrades demanded collateral AIG lacked; the New York Fed announced an 85 billion dollar facility on 16 September 2008. | Insurance sold without reserves is a leveraged bet. |
| Oil, 20 April 2020 | The expiring May WTI futures contract fell to about −37 dollars a barrel. Holders who could not take physical delivery or find storage had to pay to be rid of it. | A futures price converges on the physical thing, even when that is a tank. |
Warren Buffett called derivatives "financial weapons of mass destruction" in Berkshire Hathaway's 2002 annual letter; defenders reply that moving risk to those who want it is how airlines, farmers and governments sleep. The scale is large. The Bank for International Settlements reported about 845 trillion dollars of notional amount outstanding in over-the-counter derivatives at the end of June 2025, but the gross market value, what it would cost to replace the contracts, was about 22 trillion dollars: notional is the size of the yardstick, not the size of the risk. Finally, not every tradition accepts the tools. Islamic finance, which forbids gharar (excessive uncertainty) and maysir (speculation), generally rules out conventional options and futures, and Shariah scholars differ by school and country on which hedging structures are acceptable.
A derivative's price is the cost of copying it. Forwards cost what carrying the asset costs; options cost what a hedger would spend replicating them, so volatility, not opinion, is the price. Hedging does not remove risk; it trades one risk for another and often demands cash early. The people who use these tools well know exactly which risk they have swapped.
Try this: find something in your life with the shape of an option (a refundable fare, a warranty, a fixed-rate loan you can repay early, a free trial). Say what the premium is, what the strike is, and what volatility you would be paying for. This is for learning, not advice.
Sources. Pricing maths is standard textbook finance (Hull; Shreve). Black and Scholes, Journal of Political Economy 81(3) (1973); Merton, Bell Journal of Economics and Management Science (1973); Cox, Ross and Rubinstein, Journal of Financial Economics (1979); Bachelier (1900). Nobel 1997: Merton and Scholes. CBOE opened 26 April 1973. Aristotle, Politics I.11. Dōjima: licensed 1697, futures-like trading about 1710. Swaps: IBM and the World Bank, 1981. Barings: insolvent 26 February 1995, £827 million, sold to ING for £1. LTCM: rescue 23 September 1998. AIG: facility announced 16 September 2008. Metallgesellschaft: Culp and Miller (1995) against Mello and Parsons (1995). Oil: May 2020 WTI contract negative on 20 April 2020. Mexico: Finance Ministry statement, January 2020; cost and payout from press reports, not official data. BIS over-the-counter derivatives statistics, end of June 2025 (notional about 845 trillion, gross market value about 22 trillion dollars, summed over the BIS risk categories). Buffett, Berkshire Hathaway letter for 2002. Numbers computed in node: forward 309.14; margin calls of 550 and 400; call 9.149, put 7.661, delta 0.570, gamma 0.0185, vega 0.278, theta −0.0268; tree 12.16 against Black–Scholes 10.45 (200 steps give 10.44); hedge spread 8.20 falling to 0.67; swap par rate 4.04%, then 5.07%, worth about 0.45 million. The hedging figures come from 300 seeded paths.