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Real-World vs Risk-Neutral Expectation

Pricing under the risk-neutral measure
P0=EQ[XT(1+r)T]P_0 = E^{\mathbb{Q}}\left[\frac{X_T}{(1+r)^T}\right]

A discounted expectation, but taken under Q rather than under any probability anyone believes.

Derivative prices are expectations under the risk-neutral measure Q\mathbb{Q}, discounted at the risk-free rate. Under Q\mathbb{Q}, every tradeable asset grows at rr in expectation.

Nobody is risk neutral

The name misleads. Investors demand a premium for risk, which is why equities have out-returned bonds.

Q\mathbb{Q} is a reweighting of probabilities that absorbs risk preferences into the weights, so they need not appear separately in the discount rate. It exists because derivatives can be replicated, and replication means the price cannot depend on anyone's view of the underlying's expected return.

Key takeaway

Risk-neutral pricing is a change of measure, not an assumption about attitudes. Preferences are still present; they have been moved from the discount rate into the probabilities.

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