Quant Finance Foundations, Part 4. Previously: How to Read a Yield Curve.
Forwards, futures and swaps make up the overwhelming majority of derivative volume globally, and every one of them can be priced with a yield curve and arithmetic. No stochastic calculus, no simulation, no volatility input.
In other words, derivatives have a reputation for complexity that the most widely traded ones do not deserve.
They share one structural property that makes this possible: their payoff is a linear function of the underlying. This is the natural place to start before options, where linearity breaks and the mathematics gets serious.
Forwards: an agreement about a future price
A forward contract is an agreement made today to buy or sell something at a set price on a set future date. Nothing changes hands now. On the delivery date, the buyer pays the agreed price regardless of where the market has gone.
The pricing question is: what should that agreed price be? The answer comes from an argument that recurs throughout derivatives pricing — if two strategies produce the same outcome, they must cost the same today.
Consider gold at a spot price S, with a one-year rate r. Two ways to own gold in a year:
- Borrow S, buy gold now, hold it, repay S(1+r) in a year.
- Enter a forward and pay the forward price F in a year.
Both leave you holding gold in one year having paid nothing today. So F must equal S(1+r). If F were higher, you would sell forward, buy spot with borrowed money, deliver, and pocket the difference at zero risk. That trade is called cash-and-carry arbitrage, and its availability is what forces the relationship to hold.
Two adjustments cover most real cases. If the asset pays income at rate q (a dividend yield, or the foreign interest rate for an FX forward), that income offsets the financing cost: F = S(1 + r − q). If holding it costs money — warehousing, insurance — that adds to the carry.
Applied to currencies, this is covered interest parity, and it is why an FX forward rate is not anybody’s forecast of the future exchange rate. It is arithmetic on two interest rates. A corporate treasurer hedging import payments is buying certainty, not buying a view.
Futures: forwards, industrialised
A futures contract is economically a forward with the counterparty risk engineered out. Three mechanisms do the work:
- Standardisation. Fixed sizes, fixed delivery dates, fixed quality specifications. Contracts become fungible, which creates liquidity.
- Central clearing. A clearing house interposes itself between buyer and seller, becoming counterparty to both. Neither side needs to assess the other’s credit.
- Daily margining. Gains and losses settle every day in cash. Exposure never accumulates quietly over the contract’s life.
Daily settlement creates a genuine, if small, pricing difference from forwards. Because margin flows are reinvested or funded at prevailing rates, futures prices are slightly sensitive to the correlation between the underlying and interest rates. On short-dated equity or FX contracts the effect is negligible. On long-dated interest rate futures it is real, and the correction is called the convexity adjustment.
Swaps: a portfolio of forwards
An interest rate swap exchanges a stream of fixed payments for a stream of floating payments on the same notional. The notional itself never changes hands — it only sizes the interest calculation.
Why anyone bothers: a Vietnamese corporate borrows floating from its bank because that is what the bank offers, but its budget needs certainty. It enters a swap to pay fixed and receive floating. The floating leg received offsets the floating loan, leaving a synthetic fixed-rate liability. The loan and the swap are separate contracts with different counterparties, but economically the exposure is transformed.
How a swap is valued
Pricing follows directly from the last two posts. Value each leg as a bond, then take the difference.
- Fixed leg: known cash flows. Discount each at the corresponding zero rate and sum. This is exactly the bond calculation from Part 2.
- Floating leg: the future payments are unknown, but the forward rates implied by the curve tell you what the market says they will be. Project each payment using its forward rate, then discount it.
At inception the fixed rate is set so the two legs have equal value and the swap is worth zero to both sides. That break-even fixed rate is the swap rate — the same number quoted in the market and used to bootstrap the long end of the curve. The circularity is deliberate: swap rates build the curve, and the curve prices swaps.
After inception, rates move and the swap acquires value — positive to one party, negative to the other. That mark-to-market is what the bank reports, and what generates counterparty credit exposure requiring CVA.
Forwards, futures and swaps compared
| Forward | Future | Swap | |
|---|---|---|---|
| Traded | Over the counter | On exchange | Over the counter |
| Terms | Customised | Standardised | Customised |
| Counterparty risk | Bilateral | Clearing house | Bilateral or cleared |
| Cash flows | One, at maturity | Daily margin | Periodic, both legs |
| Priced with | Spot and carry | Spot, carry, convexity | Forward rates and discounting |
What linearity buys you, and what it costs
A forward’s payoff is a straight line: every unit the underlying rises, you gain one unit. This makes hedging conceptually simple. Sell a forward on 1,000 tonnes, buy 1,000 tonnes of the underlying exposure, and the hedge holds regardless of where the price goes. No rebalancing needed.
The cost is symmetry. A forward removes downside and upside together. The exporter who hedges at 25,500 VND/USD is protected if the rate falls to 24,000 — and cannot benefit if it rises to 27,000. Clients who want protection and participation need a payoff that bends rather than runs straight.
That bend is optionality, and it is where the mathematics stops being arithmetic.
Try it yourself
Test covered interest parity against a live quote, in this order:
- Pick a currency pair you follow and note the spot rate.
- Find the two relevant interest rates for the same tenor.
- Compute the theoretical forward from covered interest parity.
- Pull the dealer’s quoted forward rate for that tenor.
- Compare the two and account for the residual.
That residual is bid-offer, credit and — in constrained markets — genuine deviation from parity.
Then price a simple three-year fixed-for-floating swap using a curve you bootstrapped in Part 3. Solve for the fixed rate that makes the swap worth zero. If it lands near the market swap rate for that tenor, your curve is behaving.
Frequently asked questions
What is the difference between a forward and a future?
Economically they are close cousins. However, futures are standardised, exchange-traded and margined daily, which removes counterparty risk and creates liquidity.
Is an FX forward rate a forecast?
No. It is arithmetic on two interest rates, known as covered interest parity. Therefore a treasurer hedging with a forward is buying certainty rather than expressing a view.
Why is the notional of a swap never exchanged?
Because it exists only to size the interest calculation. Consequently the credit exposure is limited to the net mark-to-market rather than the full principal.
Why do these instruments need no volatility input?
Their payoffs are linear in the underlying. As a result, replication requires no rebalancing and volatility never enters the price.
Next in this series: what an option really is, and why optionality costs money. From there the series moves to binomial tree option pricing.