The fundamental theorem of asset pricing
FTAP (Harrison & Pliska, 1981): A market is arbitrage-free if and only if there exists an equivalent martingale measure $\mathbb{Q}$ (risk-neutral measure) such that discounted asset prices $S_t/B_t$ are $\mathbb{Q}$-martingales.
Change of measure: the Radon-Nikodym derivative
Moving from $\mathbb{P}$ to $\mathbb{Q}$ via the Girsanov theorem: if $W_t^\mathbb{P}$ is a $\mathbb{P}$-Brownian motion, then:
\[W_t^\mathbb{Q} = W_t^\mathbb{P} + \int_0^t \theta_s\,ds\]is a $\mathbb{Q}$-Brownian motion, where $\theta_t = (\mu - r)/\sigma$ is the market price of risk.
Under $\mathbb{Q}$, all assets grow at the risk-free rate $r$:
\[dS_t = r S_t\,dt + \sigma S_t\,dW_t^\mathbb{Q}\]The pricing formula
Any derivative with payoff $H_T$ at time $T$ is priced at:
\[V_t = e^{-r(T-t)}\mathbb{E}^\mathbb{Q}[H_T \mid \mathcal{F}_t]\]Black-Scholes as a special case: for a European call $H_T = (S_T - K)^+$, this integral evaluates to the Black-Scholes formula.
Incomplete markets
When the market has more risk sources than traded assets (e.g. stochastic volatility), the risk-neutral measure is not unique — there is a range of arbitrage-free prices. Additional assumptions or market instruments are needed to pin down a unique price.