Why Numeraire-Discounted Asset Prices Are Martingales

Summary

The document asks why an asset price divided by a reference asset, or numeraire, is a martingale under the probability measure associated with that numeraire. It gives the example of a forward price divided by a zero-coupon bond price under the bond measure.

No derivation or answer is provided, so the central result is posed rather than explained. The question points to the change-of-numeraire framework in asset pricing, where choosing a numeraire defines a measure under which appropriately normalized asset prices have the martingale property. The document does not state assumptions, such as the relevant pricing setup or tradability conditions, and supplies no evidence or caveats beyond the example.

Key ideas

• The question concerns the martingale property of asset prices normalized by a chosen numeraire.
• It uses a forward price divided by a zero-coupon bond price as an example.
• The document poses the result but does not provide a derivation or answer.

Tags

# 68724


# Why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire?












why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire?

For example, if I have the price of a forward price $f_t$ and a zero-coupon bond price $B_t$, then $\frac{f_t}{B_t}$ would be martingale under the measure $Q^B$.

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