# 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?

The fundamental theory says only that the ratio of asset prices A/B under the measure associated with B, is a martingale. The short rate r is not an asset.

• 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
• An asset is something you can buy and sell for a price that is nonnegative. Examples are bonds, stocks, foreign currency, commodities. BTW: the martingale requirement applies only when the asset does not pay dividends. When it does see this answer. Oct 18, 2022 at 7:44
• As @KurtG says it’s really about tradable assets. Interest rates, variances and some commodities are not directly tradable and thus they don’t need to be martingales after discounting. Oct 18, 2022 at 14:06
• 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