What is the risk neutral measure?

Mastering risk neutral probabilities took me a while so I thought it worthwhile to share my experience.

---

The risk neutral probability is the probability measure that applies when asset prices are martingales. The expected future value of an asset is equal to its value today. In

mathematical terms, assuming that the risk-free rate is zero, we have:

Where:

is the filtration (or the set of scenarios that could happen up to time 0)

is the asset price at time t

In a risk neutral world the expected future value of an asset is its value today. This result is a consequence of the no-arbitrage principle. If the expected future price of the asset is different from its current price, the market would purchase or sell the asset until it reaches the equilibrium level. Hence the risk neutral probability is based on the reachability of an asset.

We know that under the historical probability measure the above equation is untrue. The expected value of the asset can be higher or lower than its current value. Assume that the expected future value of a stock is higher than its value today. In other words, one expects the asset price to increase. Given the risk associated with the asset, however, no one “dares” to purchase it at a higher price.

Hence the historical expectation of an asset is not a sufficient indicator of its price. The risk neutral probability adjusts 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 shifts the density to a new center.

Let's assume that we have an asset with;

drift and

volatility

The relationship between a Brownian path in the historical world

and the risk neutral world

is a shift in the drift:

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.

Asset paths in a risk neutral and historical world

This blog post will be continued….