Risk, return, and ross recovery

The risk return relation is a staple of modern finance. When risk is measured by volatility, it is well known that option prices convey risk. One of the more influential ideas in the last twenty years is that the conditional volatility of an asset price can also be inferred from the market prices of options written on that asset. Under a Markovian restriction, it follows that risk-neutral transition probabilities can also be determined from option prices. Recently, Ross has shown that real-world transition probabilities of a Markovian state variable can be recovered from its risk-neutral transition probabilities along with a restriction on preferences. In this article, we show how to recover real-world transition probabilities in a bounded diffusion context in a preference-free manner. Our approach is instead based on restricting the form and dynamics of the numeraire portfolio.