Risk, Return, and the Payoff Taxonomy: Linear vs. Non-Linear Instruments

Forwards, futures, and swaps are linear: their payoff at delivery is a constant times the change in the underlying price. Their value changes by the same dollar amount for a dollar move up as for a dollar move down. Replication with a single static position prices them exactly, with no model needed — the cost-of-carry framework from the previous lesson is sufficient.

Options are non-linear: the buyer's payoff is $\max(S_T - K, 0)$ for a call, which is convex in $S_T$. A call on a rising stock earns increasingly more than a falling stock loses. This asymmetry means a single forward position cannot replicate an option, and a model of how $S_T$ is distributed becomes necessary. The split between the two families — static replication for the linear world, dynamic hedging and distributional assumptions for the non-linear world — organises the rest of this unit and all derivative pricing theory that follows.

• define and distinguish linear (delta-one) payoffs from non-linear (convex) payoffs with examples from each family;
• describe the payoff diagrams for a long call, short call, long put, and short put;
• explain why the convexity of an option payoff prevents static replication from pricing it;
• quantify simple and logarithmic returns, and explain why log-returns are preferred in continuous-time models;
• define volatility as the annualised standard deviation of log-returns and explain its role as the key non-observable in option pricing;
• explain the relationship between futures prices and expected future spot prices, and define normal backwardation and contango;
• identify the correct pricing framework for a given instrument: cost-of-carry vs model-based.

1. The linear family: delta-one instruments
2. Options: non-linear payoffs and the convexity premium
3. Returns and volatility
4. Futures prices and the expected future spot price
5. The two-tier pricing hierarchy