# Risk neutral valuation formula

I am totally new to Finance and Arbitrage theory and I have started reading Björk (2018) Arbitrage theory in continuous time. Can anyone please explain to me what is the risk-neutral valuation formula in an intuitive manner? Moreover is there any introductory textbook on "Arbitrage theory"?

• In the simplest terms (perhaps too simple) the intuition of the RNVF is that the value is equal to the Expectation (a linear operator on the different possible outcomes), but this Expectation must be taken under a specific Probability Measure, the so called Risk Neutral Measure $Q$, which is perhaps rather surprising.And this result is a consequence of the no-arbitrage assumption. – noob2 Nov 7 '18 at 21:45
• "No arbitrage" is a basic principle of Economics which says that in well-functioning markets it should not be possible for someone to make infinite profits without any risk. It is too simple and intuitive a principle to deserve a whole introductory textbook. – noob2 Nov 7 '18 at 22:34

## 1 Answer

The absolute reference for starters which does not dwelve too much into mathematical details but enough to be accurate is Hull so i suggest you have a look at this book first.

Options, Futures, and Other Derivatives

That being said to give a simple illustration: sometimes people try to value derivative assets (like options eg) by looking at the historical distribution of the underlying asset. This is called working with the historical measure.

While this can be informative by itself this can lead to so-called arbitrage opportunities because using the prices derived from the historical measure you would be able to construct a portfolio of these instruments which would be free of any market risk but would offer you a higher return than the risk-free rate.

For instance if you imagine a market where you were the single market maker for options on a stock S which has say a pretty large drift (say 20% per year) and if you were using the historical measure to price options on S you would very quickly get out-of-business because you would most likely be bidding the otm calls way too high and offering the otm puts way too cheap which would make your pricing of implied forward completely out-of-line with reality (and make you out of business :) ).

By contrast, the risk-neutral measure is another probability measure which has the property that when assets are priced using it then there can be no arbitrage opportunity. This leads to a consistent view of both options and the underlying spot price.