# 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?

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.
• So if I understand your answer correctly: $E_{P^*}[X|F_i] = X_i$ tells us the expected value will be the same. Only risk-neutral agents will be drawn to this kind of expectation which makes us land in the risk-neutral world. Now because we are in this risk-neutral world, the value grows with the risk-free interest rate. And to know the current value, we discount it? – xrdty Jan 21 '17 at 23:57
• Just to be precise - not the prices are martingales under $\mathbb{P}^*$ but the discounted prices. $\mathbb{P}^*$ is an equivalent measure that simplifies the pricing problem as we can price as if risk-preferences didn't matter - thus the name. There are quite a few related question here - see e.g. quant.stackexchange.com/questions/103. – LocalVolatility Jan 22 '17 at 0:15