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?


1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$ Oct 18, 2022 at 6:43
  • 2
    $\begingroup$ 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. $\endgroup$
    – Kurt G.
    Oct 18, 2022 at 7:44
  • $\begingroup$ 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. $\endgroup$
    – Kevin
    Oct 18, 2022 at 14:06
  • $\begingroup$ 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. $\endgroup$ Oct 18, 2022 at 17:19

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