3.1 Options, Futures & Markets

Basic Prerequisites: None. This is the entry point to Pillar 3 - start here if you have never priced a derivative. Everything else in the pillar assumes the vocabulary built on this page.

1. Intuition & Practical Objective

A derivative is a contract whose value is derived from something else - a stock, an index, a currency, a commodity, an interest rate. That is the whole idea. The stock itself pays you cash flows; a derivative pays you cash only because the stock moved, and the contract says exactly how. The practical objective of this folder is to give you the product menu and the conventions - what each instrument is, what it pays, where it trades, and what the no-arbitrage relationships between its prices must be - before a single model is written.

The reason this is a separate topic and not a preface is that the four instruments below are not variations on one idea; they are four different shapes of the same contract. A forward is symmetric (both sides can win or lose, costless to enter). An option is asymmetric (one side has a right, the other an obligation, and the right is paid for up front). A swap is a portfolio of forwards. A futures contract is a forward that is settled every day. Get the shapes wrong and no amount of Black–Scholes will save the answer.

The one-sentence essence. “A derivative is a contract that pays a specified function of an underlying price; its fair value is fixed by the cost of replicating that function with traded assets, and the first place that constraint shows itself is in the payoffs, the bounds, and put–call parity - all of which hold without assuming any model.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). All statements below are transcribed from the verified corpus (Hull 11th ed. Ch 1–12, Haug §1, Shreve Vol I §1/§5) and every number in the check column was re-executed in §3.

Notation: $S$ spot, $S_T$ terminal spot, $K$ (or $X$) strike, $F$ futures/forward price, $T$ time to expiry (years), $r$ risk-free rate, $q$ dividend yield, $r_f$ foreign rate, $b$ cost-of-carry, $\sigma$ vol, $N(\cdot )$ standard normal CDF.

2.1 The derivative product menu (Hull’s taxonomy)

The four option positions (Hull Ch 10.2 - the whole payoff algebra in one table):

2.2 The no-arbitrage skeleton (model-free)

2.3 Conventions that must be carried into every calculation

• Cost-of-carry dictionary (one master formula, many models): $b=r$ stock · $b=r-q$ index/yield $q$ · $b=0$ futures (Black-76) · $b=r-r_f$ currency. Wrong $b$ = wrong price (Haug §1).
• Contract multiplier. Equity/index options: $100\times$ premium per contract; index options are $100\times$ the index (Hull Ch 10.3).
• Moneyness. In-the-money / at-the-money / out-of-the-money by $S$ vs $K$; option value $=$ intrinsic value $+$ time value (Hull Ch 10.4).
• European vs American. European: exercise only at $T$. American: any time $\le T$ - a free option on the option, hence $\ge$ the European value (Hull Ch 10.1).
• Exchange-traded vs OTC. Exchange = standardised, cleared, margined; OTC = bespoke, bilateral/CCP, ISDA + CSA. Dec-2019 scale: OTC notional $\approx$ $558.5T vs exchange ≈$96.5T (Hull Ch 1).

3. Computational Implementation - the payoff engine and the parity checks

Standard library only (math.erf gives the exact normal CDF). This block produces the payoff table, the Haug-verified parity number, and the Black-76 anchor used throughout the folder.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s full failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Confusing the instrument shapes - treating a short forward like a short option (the obligations differ), or an American for a European (the early-exercise premium is real: $0.4000$ on the Shreve tree).
2. Applying an option formula to a forward, or vice versa - the cost-of-carry dictionary ($b=r$ vs $b=0$ vs $b=r-r_f$) is the first thing to check; the wrong $b$ misprices by exactly the missing factor.
3. Ignoring the basis and the friction of real hedges - a textbook hedge assumes the underlying and the hedging instrument are the same asset; in practice $b_2\ne 0$ and hedge effectiveness is only ${\rho }^2$ (Hull Ch 3).

5. References

• Hull, John C.: Options, Futures, and Other Derivatives (11th ed., 2022)
• Shreve, Steven E.: Stochastic Calculus for Finance I
• Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed., 2006)
• Hull, J. & White, A.: background on OIS discounting and post-2008 clearing (via Hull

6. Connected Graph Bridges

• Foundational base: Probability & Measure Theory · Stochastic Calculus & Itô
• Forward topic-page (next step): No-Arbitrage Foundations & Binomial Trees
• Sibling topic-folders (in-pillar): Black–Scholes–Merton · No-Arbitrage & the Binomial Model · Volatility Surfaces & Smiles · Exotic & Path-Dependent Options · Numerical Methods · Counterparty Risk & XVA · Interest Rate & Term Structure
• Sub-pages (in-folder): 01 What Is a Derivative · 02 Options Mechanics & Payoffs · 03 Markets & Products · 04 No-Arbitrage & Bounds · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05