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Using Quanto Adjustments for Caps under Different Collateral Currencies

Article Quant Q&A · Author: ababoua
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Summary

The question concerns pricing a US three-month Libor cap under euro collateral, where the discounting and forward curves reflect the collateral agreement. It asks whether the volatility quoted for a US-collateralized cap can be reused, given that the pricing measure changes with collateral currency and the Libor rate may covary with the US–euro spread.

The response treats the change of collateral measure like a quanto adjustment. Under an assumed geometric Brownian motion for the forward rate, the measure change adjusts its drift through covariance with the change-of-measure factor, while leaving volatility unchanged. It therefore suggests using the US-collateralized volatility alongside an adjusted quanto forward. This is a modeling convention rather than a general result for every volatility representation: the answer cautions that a Black formula does not prove geometric Brownian dynamics, particularly when a volatility smile matters.

Key ideas

  • A collateral currency change alters the pricing measure used for the forward rate.
  • Under the stated geometric Brownian motion assumption, covariance with the measure change adjusts the forward drift.
  • The response says the volatility remains unchanged under that assumption.
  • It suggests combining an adjusted quanto forward with the non-quanto volatility at the relevant strike.
  • Smile effects and the limits of the model make this a qualified pricing practice.

Tags

OptionsDerivatives pricingVolatilityFixed income
Full text
# Which volatility to use in cap pricing with CSA discounting?


# Which volatility to use in cap pricing with CSA discounting?












I'm currently trying to price a cap on a Libor 3M (US) collateralized in EUR. I understand that my discount curve should be the CSA and the price of a caplet should be using a Black-scholes price: $$Cplt(t,K,T_{k-1},T_{k})=\tau_k \times DF(t,T_{k}) \times Black(K, L_{k}(t),\sigma_{k})$$ where $L_{k}(t)=\mathbb E^{CSA}_t(F_k(T_{k-1}))$ is the expected value of the forward implied from the CSA 3M Libor (US) curve and $DF$ is the discount curve implied from the CSA curve. My question is what is the olatility to use since the bloomberg volatilities are just US collaterlized Cap (discounted with OIS). Can i use the same volatility as the ones in Bloomberg? but in that case, i think i forget the covariance between US Libor 3M and the spread US/EUR.

Thank you in advance.

## Answer by Antoine Conze (score 0, accepted)

https://quant.stackexchange.com/a/34921

It works like a quanto option: you know the forward Libor dynamics under $P^{USD}$ but you want to price under $P^{CSA}$. If you assume a Black & Scholes geometric brownian motion dynamics under $P^{USD}$, then under $P^{CSA}$ the drift is adjusted with the covariance between the forward libor and the change of measure $dP^{CSA} / dP^{USD}$, but the volatility does not change.

Of course pricing with the Black formula is not equivalent to saying that the underlying follows a geometric brownian motion, especially in the presence of a smile, but again the practice for quanto options is to use the adjusted quanto forward and the non quanto volatility at strike $K$, so I would say that pricing with USD collateralized volatility is fine.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.