I have been confused about many things concerning the princing of securities with collateral.
We can prove that today's price of a security( fully collateralized and within the same currency) is the expectation under the risk neutral measure of the discounted(with the collateral rate) payoff. This first application of this theorem is the introduction of the zero coupon bond collateralized . Now, for a matter of simplicity , we would like to work with this zero coupon bond as a numéraire to price classic derivatives such as cap/floor and swaptions. We all remember how to proceed to price those derivatives, change to the T forward measure, the forward libor is a martingale under the T forward measure , assuming a simple log normal process, we have Black's formula. We apply the same thing for the swaption working with the annuity as a numeraire. My confusion is that we often use the rule that says that for all tradable asset and a given numeraire, their ratio is a martingale under the measure related to the numeraire. Let apply this rule to the Zero coupon bond collateralized,... it does not work because it would mean that the zero coupon collateralized and the risky zero coupon have the same price. The result was kind of expected because it is (i think), "all tradable asset without any intermediate payments....." Under the collateralization , there are intermediate payments .
However, we can see that for the pricing of swaption with collateral , the swap rate can be written as a ratio between a floating leg with collateralization( which is tradable) and the annuity with collateralization . In some papers I have read, they quickly conclude that because the floating leg is a tradable asset, its ratio with the annuity collateralized is a martingale under the measure related to this annuity. Why??? We saw a counter example with the zero coupon bond collateralized.
Shall we say : For all collateralized tradable asset, its ratio with a numeraire collateralized at the same collateral rate as the tradable asset is a martingale under the measure related to this numeraire? What about when the tradable asset is collateralized with one currency, and the numeraire with another one ?