# Strictly increasing asset price under a risk-neutral probability measure?

I am reading a paper on option pricing under jump processes in continuous time. There is a section labeled examples where the authors work under a risk neutral probability measure and derive option pricing formulas. This is done for asset price processes that are strictly increasing or strictly decreasing.

My question is: Is it not impossible for the asset price to be strictly increasing/decreasing under a risk-neutral measure?

• If the underlying asset is not a tradable asset then its risk-neutral drift does not have to equal the risk-free rate. If the underlying asset is tradable then you can still have a strictly increasing price process for the asset if the risk free rate $r > 0$. Let's say the risk free is 5% per annum. You can easily imagine a process, starting at 100, that can only go up but still have an expectation 105 after 1 year. Eg in a binomial model, asset starts at 100, in 1 year it can either have value 103 or 107. All you need is find the risk-neutral probability $q$ so that $q*107 + (1-q)*103 =105$. – ilovevolatility Oct 8 '19 at 6:54