# Does every process need to be a martingale under martingale measure?

From the pricing theory, processes need to be martingales when divided by the numeraire asset.

A classical example is a stock option: Consider a money market $$B$$ being the numeraire asset. When we price a stock option with a payoff $$h(S(T))$$, then the money-market discounted stock price process $$S/B$$ has to be a martingale under the martingale measure associated with $$B$$.

But now consider a bond option where the bond's price is driven by and risk-free rate $$r$$ subject to a Vasicek process (under risk-neutral measure). The payoff of the bond option is $$h(r(T))$$. If we consider the dynamics of $$r$$ under the risk-neutral measure, $$dr(t)=k(\theta - r(t))dt + \sigma dW^Q(t)$$, then $$r/B$$ will clearly not be a martingale under $$Q$$.

My question is: How come that the discounted risk-free rate $$r/B$$ doesn't need to be a martingale under $$Q$$ if the stock had to?

I do understand that the discounted bond price in Vasicek model is a martingale under $$Q$$ but why the same doesn't apply to the risk-free rate in the bond option case?

• Yes, that's something I understand. But how is 'asset' defined? Is it a positive price process? It feels to easy to say $r$ is not an asset. I'm looking for the arguments behind. Oct 18, 2022 at 6:43
• Thanks, I accept the solution and I even expected that the 'asset' is the keyword that was needed. Pity that books don't have these kind of counterexamples to explain which processes under $Q$ actually don't need to be martingales and why. Oct 18, 2022 at 17:19