Every instrument encountered so far in this unit — forwards, futures, swaps — has a linear payoff: a one-dollar move in the underlying generates exactly one dollar of gain or loss (scaled by the notional). An option breaks this symmetry. The buyer of a call option gains when the underlying rises but loses only the premium when it falls — the payoff is convex. This asymmetry has a price, the option premium, and computing that price correctly is the central problem in derivatives mathematics.
This topic introduces options from two directions simultaneously: the institutional (how options trade, what their mechanical terms mean, what strategies they enable) and the mathematical (what constraints no-arbitrage places on option prices even without a model, and how the binomial tree provides the first complete pricing framework).
Three ideas will recur throughout this topic and the rest of the unit:
- Replication. Any derivative can be priced by finding a portfolio of basic assets that replicates its payoff. The option's price is the cost of the replicating portfolio.
- Risk-neutral pricing. The unique probabilities under which all assets earn the risk-free rate give option prices directly by discounting expected payoffs — no utility function required.
- The continuum limit. As the binomial tree becomes finer, it converges to the Black-Scholes formula — the connection between discrete and continuous time option pricing.
In this lesson
- Prerequisites
- Contents
- The big picture
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