Collateral Rates, Numeraires, and Martingale Pricing Questions
Summary
The document raises questions about applying risk-neutral and numeraire-based pricing when securities are collateralized. It describes the standard setup in which a fully collateralized payoff in the same currency is discounted using the collateral rate, and notes familiar derivative-pricing methods that use a forward bond or swap annuity as numeraire. The author asks how the usual martingale result for a tradable asset divided by a numeraire should be understood when collateral payments occur over time.
A specific concern is that a collateralized zero-coupon bond may not behave like a risky zero-coupon bond, while a swap’s floating leg divided by a collateralized annuity is sometimes treated as a martingale under the associated measure. The text also asks whether the collateral rate must match for the asset and numeraire, and how different collateral currencies affect the argument. It provides no answer, derivation, or pricing example, so these are unresolved conceptual questions rather than established conclusions.
Key ideas
- The document frames collateralized pricing in terms of discounting payoffs at the collateral rate.
- Forward-bond and swap-annuity numeraires are presented as familiar tools for derivative pricing.
- The author questions how intermediate collateral payments affect the tradable-asset martingale argument.
- The text leaves unresolved whether an asset and numeraire require matching collateral rates or currencies.
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Full text
# Pricing with collateral # Pricing with collateral 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 ? Many thanks
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