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Risk-neutral Measures

Under the risk-neutral measure Q\mathbb{Q}, every tradeable asset is expected to grow at the risk-free rate:

The martingale condition
EQ[STerT ∣ Ft]=StertE^{\mathbb{Q}}\left[\frac{S_T}{e^{rT}}\,\Big|\,\mathcal{F}_t\right] = \frac{S_t}{e^{rt}}

Under Q the discounted price is a martingale, which is the formal way to say there is no arbitrage.

Discounted prices are martingales, and every derivative is priced as

V0=e−rTEQ[f(ST)]V_0 = e^{-rT}E^{\mathbb{Q}}[f(S_T)]

The name is misleading

Nobody is claiming investors are indifferent to risk. They plainly are not, which is why equities have historically returned more than bonds.

Q\mathbb{Q} is a mathematical device: a reweighting of probabilities under which the algebra of pricing simplifies. Risk preferences do not disappear from the world; they are absorbed into the probability weights.

The fundamental theorem of asset pricing makes this precise: a market is arbitrage-free if and only if such a measure exists, and it is unique if and only if the market is complete.

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