Valuing a Derivative Using a One-Period Binomial Model

How a one-period binomial model values a European option: let the underlying move up or down, combine the option with a hedge ratio of the underlying to build a riskless portfolio, discount at the risk-free rate, and see why the same answer comes from risk-neutral probabilities that ignore real-world probabilities and investor risk preferences.

1. 1. Why options need a price model: the one-period binomial setupBecause an option's payoff is one-sided, valuing it requires a model of where the underlying can go; the binomial model assumes it can only move up to one price or down to another.Free11 min
2. 2. The hedge ratio and the riskless portfolioCombine an option with just the right number of units of the underlying, the hedge ratio, and the portfolio is worth the same whether the price goes up or down.Video · 6 min12 min
3. 3. From the riskless hedge to the option priceA riskless portfolio must earn the risk-free rate, so discounting its certain end value tells you what it costs today, and the option's price is whatever makes that cost hold.13 min
4. 4. Risk neutrality and risk-neutral probabilitiesAn option's no-arbitrage value equals its expected payoff, computed with risk-neutral probabilities rather than real ones, discounted at the risk-free rate.Video · 5 min12 min
5. 5. What does and does not move a binomial option valueIn the binomial model, an option's value responds to the size of the up and down moves and to the risk-free rate, but not to the real probability of a move or to how investors feel about risk.11 min
6. 6. Integrated cases: an underpriced put and a tree with a negative rateA binomial case hands you a data panel and chains the questions: read the tree, find the hedge ratio and π, price the option, then test the quote against the model or price it with a risk-free rate that may be negative.14 min