Why the Risk-Neutral Measure Implies Risk-Free Expected Returns

Summary

The document asks why an equivalent martingale measure is called risk-neutral, given that martingale pricing makes the conditional expectation of a discounted asset price equal to its current value. The key clarification is that martingales apply to appropriately discounted prices under the measure; this does not mean every asset’s undiscounted expected price simply remains unchanged when interest rates are positive.

Under the risk-neutral measure, each spot asset has an expected return equal to the risk-free rate, regardless of its risk. This is the pricing convention associated with a risk-neutral agent. In contrast, a risk-averse investor may require a positive expected excess return for an asset whose payoff covaries positively with shocks to consumption. The short answer explains the economic interpretation, but does not derive the pricing formula or discuss the assumptions needed for an equivalent martingale measure to exist.

Key ideas

• Under an equivalent martingale measure, discounted asset prices have the martingale property.
• The expected return on each spot asset under the risk-neutral measure equals the risk-free rate.
• Risk-neutral pricing does not mean that every undiscounted asset price is expected to remain constant.
• A risk-averse investor may demand excess return for exposure that covaries with adverse consumption shocks.
• The explanation gives the intuition but does not derive the valuation formula or its assumptions.

Tags

# Why is a martingale a risk-neutral measure


# Why is a martingale a risk-neutral measure












We have the risk-free valuation formula $$ \pi^X_i = B_T^{-1}B_iE_{P^*}[X|F_i]$$ Where $P^*$ is an equivalent martingale measure.

Why is this martingale measure considered risk-neutral? All I know is that an expected price with a martingale prob measure just predicts the last known value again. $E_{P^*}[X|F_i] = X_i$

How does this make it risk-neutral?

## Answer by LocalVolatility (score 1, accepted)

https://quant.stackexchange.com/a/32020

The expected return of every spot asset under $\mathbb{P}^*$ is the same and equal to the risk-free interest rate, irrespective of the risk. Only a risk-neutral agent would price securities this way. A risk-averse agent would demand a positive expected excess return for an asset that covaries positively with shocks to her consumption.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.