Introduction to Derivatives: Forwards, Futures, Options, and Swaps

What Are Derivatives (and Why Should You Care)?

A derivative is a financial contract whose value depends on — is derived from — the price of something else. That something else is called the underlying: a stock, a bond, an interest rate, an exchange rate, a commodity, even the weather.

The global derivatives market is vast. By notional value, it is measured in the hundreds of trillions — dwarfing the equity and bond markets combined. And it is where some of the most mathematically sophisticated work in finance happens.

If you want to work in quant finance, derivatives are very likely going to be part of your world. This post covers the four main types and the intuition behind them.

---

Forwards and Futures

A forward contract is an agreement to buy or sell an asset at a fixed price on a future date. No money changes hands today (unlike buying the asset outright).

Example: You agree today to buy 1,000 barrels of oil at $75/barrel in 3 months. In 3 months, if oil is at $80, you have saved $5,000. If it is at $70, you have lost $5,000 compared to the market.

Futures are standardised forwards traded on exchanges. The key differences:

The Forward Price

The forward price is determined by no-arbitrage. If the spot price is ( S_0 ), the risk-free rate is ( r ), and maturity is ( T ):

[ F_0 = S_0 \cdot e^{rT} ]

This is the cost-of-carry model. If the forward price were higher, you could buy the asset, sell the forward, and earn a risk-free profit. If lower, the reverse. Arbitrageurs ensure the price stays close to this formula.

This is your first taste of risk-neutral pricing — the foundational principle of derivatives valuation.

---

Options

An option gives the holder the right, but not the obligation, to buy or sell an asset at a specified price (the strike) by a certain date (the expiry).

• Call option: right to buy
• Put option: right to sell

The asymmetry — you can choose not to exercise — is what makes options interesting. It also makes them harder to price than forwards.

Payoff Diagrams

At expiry:

Call payoff: ( \max(S_T - K, 0) ) — you profit if the stock ends above the strike

Put payoff: ( \max(K - S_T, 0) ) — you profit if the stock ends below the strike

These hockey-stick-shaped payoffs are the signature of options. They are piecewise functions — and if mathematical notation is fresh in your mind, you will recognise the ( \max ) function immediately.

Intrinsic Value vs Time Value

An option's price has two components:

• Intrinsic value: the payoff if you exercised right now
• Time value: the extra premium for the chance of further favourable moves

Time value decays as expiry approaches (this is theta decay). A deep out-of-the-money option with a week to expiry has almost no time value — it needs a miracle.

Put-Call Parity

A beautiful no-arbitrage relationship links puts and calls:

[ C - P = S_0 - K e^{-rT} ]

If this does not hold, arbitrageurs can earn risk-free profits. It is one of the cleanest examples of how no-arbitrage pricing works.

---

Swaps

A swap is an agreement to exchange cash flows. The most common type — an interest rate swap — exchanges fixed-rate payments for floating-rate payments.

Example: Company A pays a fixed 3% and receives SONIA (the floating rate). If SONIA averages 4% over the swap's life, Company A profits. If SONIA averages 2%, it loses.

Interest rate swaps are the single largest derivative market by notional value. Banks use them to manage interest rate risk, and corporations use them to convert between fixed and floating debt.

---

Why Derivatives Exist

Derivatives serve three fundamental purposes:

1. Hedging

A wheat farmer can sell futures to lock in a price, removing the risk of a price drop before harvest. An airline can buy oil futures to manage fuel costs. A bank with a fixed-rate mortgage book can use swaps to manage interest rate risk.

2. Speculation

If you believe a stock will rise, buying call options gives you leveraged exposure with limited downside (you can only lose the premium). More bang for your buck — and more potential bang for your loss.

3. Arbitrage

If a derivative is mispriced relative to its underlying, traders can construct risk-free profits. This activity — and the threat of it — keeps markets efficient and ensures derivatives prices stay closely linked to their underlyings.

---

Pricing Derivatives: The Big Ideas

Pricing forwards is relatively straightforward (cost of carry). Pricing options is harder because of the asymmetric payoff. Two major approaches:

The Binomial Model

Start with a simple world: the stock can only go up or down each period. Build a tree, calculate payoffs at expiry, and work backward using risk-neutral probabilities. The binomial model is the first formal pricing framework, and it is surprisingly powerful.

Black-Scholes

In the limit of infinitely many small time steps, the binomial model converges to the Black-Scholes formula — the most famous equation in finance. It uses calculus, probability, and Brownian motion to produce a closed-form solution for European option prices.

---

The Greeks

Once you can price an option, you need to understand how the price changes when market conditions move. The Greeks — delta, gamma, theta, vega, rho — measure these sensitivities. They are partial derivatives (the maths kind) of the option price with respect to its inputs.

Traders use the Greeks every day to manage risk. A delta-neutral portfolio is insensitive to small stock price moves. Adding gamma hedging protects against larger moves. The Greeks transform abstract calculus into practical risk management.

---

Getting Started with Derivatives

Derivatives are where maths, technology, and finance meet most intensely. The pricing models use stochastic calculus, the risk management uses linear algebra, and the implementation requires solid programming.

Quantt takes you from the basics through to pricing models and the Greeks, with Python implementations at every step. The finance, maths, and technology streams are designed to converge — because that is how derivatives work in practice.

Hull's Options, Futures, and Other Derivatives is the standard textbook if you want a comprehensive reference. And Investopedia's derivatives section is good for quick lookups.