Glossary — M10: Valuing a Derivative Using a One-Period Binomial Model

Term	Definition
Binomial model	A discrete-time option pricing framework where the underlying can move to one of exactly two possible values (up or down) in each period
Up factor ( $u$ )	The multiplicative factor by which the stock price increases in the up state: $S^{+} = S_{0} \times u$ where $u > 1$
Down factor ( $d$ )	The multiplicative factor by which the stock price decreases in the down state: $S^{-} = S_{0} \times d$ where $d < 1$
Hedge ratio ( $h$ )	The number of shares of the underlying needed to replicate one option: $h = (c^{+} - c^{-}) / (S^{+} - S^{-})$ ; also called delta
Risk-neutral probability ( $π$ )	The probability of an up move in the risk-neutral framework: $π = (1 + r - d) / (u - d)$ ; not the actual probability of an up move
Risk-neutral pricing	A valuation method that discounts expected payoffs (computed using risk-neutral probabilities) at the risk-free rate: $c_{0} = [π c^{+} + (1 - π) c^{-}] / (1 + r)$
One-period binomial model	The simplest binomial framework with a single time step from today to expiration; the foundation for multi-period models
Replicating portfolio (binomial)	A portfolio of $h$ shares of stock plus borrowing/lending at the risk-free rate that exactly replicates the option payoff in both up and down states
No-arbitrage condition (binomial)	The requirement that $d < (1 + r) < u$ — the risk-free return must lie between the down and up returns to prevent arbitrage
Risk-neutral world	A theoretical construct where all assets earn the risk-free rate; option prices derived in this world are valid in the real world because of no-arbitrage

See Also