It took me a while to master the risk neutral probability and it is worth sharing my experience. The risk neutral probability is the probability where the asset is a martingale; the future value of the asset is equal to its today's value.
In a risk neutral world the future value of an asset is its today's value. This is a consequence of the non arbitrage principle; if the future value of the asset would have been different, the market would have sold or purchased until the asset price reaches its equilibrium level. Hence the risk neutral probability is based on the reach-ability of an asset.
We know that in a historical measure the above equation is untrue. The expected value of the asset can be higher or lower than its current value. Let's assume the case that the expected future value of the stock is higher, one would expected the asset price to rise. But because of the risk related to the asset, no one "dare" to purchase it at a higher price.
Hence the historical expectation of an asset is not a sufficient indicator of the asset price. The risk neutral probability re-positions the expected return of the asset. So the risk neutral probability is nothing more than a linear shift of the historical density; it does not change the variance or correlation but only shift the density into a new center.
Brownian paths in risk neutral and historical world. The green lines are the risk neutral paths , the blue lines are the historical paths. The green process do not seem to be a martingale, but is a martingale in a risk neutral world.
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