Risk-Neutral Valuation

Pricing derivatives by changing measure — and why it works.

Under the real-world ("physical") probability measure, asset prices have drift related to investors' expected returns. The risk-neutral measure ( $\mathbb{Q}$ ) is a different probability measure under which all assets earn the risk-free rate.

The fundamental theorem of asset pricing: in the absence of arbitrage, there exists a risk-neutral measure such that every traded asset's discounted price is a $\mathbb{Q}$ -martingale. Derivative prices are then expectations under $\mathbb{Q}$ :

V_0 = e^{-rT} E^{\mathbb{Q}}[\text{payoff}_T]

The trick is that we don't need to know investors' actual risk preferences to price derivatives — those preferences are baked into the underlying's price already. Risk-neutral valuation lets us price by replication: simulate paths under $\mathbb{Q}$ , compute expected discounted payoff, that's your fair price.

Practical implication: implied volatility is calibrated to the prices of liquid options, then used to price illiquid options consistently — without ever asking what real-world drift the underlying actually has.