Why Only Discounted Tradable Asset Prices Must Be Martingales

Summary

The discussion distinguishes tradable asset prices from quantities that influence them. Under the pricing framework described, an asset’s price divided by the chosen numeraire is a martingale under the corresponding pricing measure. A stock price discounted by a money market account fits this rule; a short rate does not, because it is not itself an asset price.

The question contrasts that principle with a Vasicek short-rate model used to price a bond option. The accepted explanation resolves the apparent conflict by noting that the martingale condition applies to tradable asset prices, such as the bond, rather than automatically to every state variable in a model. The thread offers a conceptual distinction, not a derivation of the Vasicek model or a discussion of measure changes, market completeness, or practical estimation.

Key ideas

• The martingale pricing condition applies to tradable asset prices expressed relative to a numeraire.
• A short interest rate is a model state variable, not itself an asset price.
• A bond price can satisfy the discounted martingale condition even when the short rate divided by the cash account does not.
• The distinction clarifies why pricing rules for assets do not automatically extend to all model processes.

Tags

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


# 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?

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/73518

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.

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