# Has a closed-form formula for the collateral choice option been found?

The collateral choice option problem has been formulated in e.g. Fujii and Takahashi (2011), Piterbarg (2012) or Antonov and Piterbarg (2013), as the computation of an expectation of the following form: $$E^\mathcal{Q}\left(e^{-\int_0^T\max_is_i(u) \text{ d}u}\right)$$ where $$s_i$$ represents a spread between 2 rates, normally assumed to have a Gaussian distribution.

This expectation is known to be hard to calculate. Piterbarg (2012) writes:

There appears to be no closed-form expression for an option like this.

Different efficient numerical techniques have been put forward to approximate the collateral choice option, see for example Antonov and Piterbarg (2013).

However, it does seem like no progress has been made on the derivation of a closed-form solution. At least I am not aware of any published paper that has found an expression.

Does anybody know whether such a solution has been found since then? Alternatively, has it been proved that no closed-form solution exists?

References

Masaaki Fujii and Akihiko Takahashi. “Choice of collateral currency”, Risk, 2011.

Vladimir Piterbarg. “Cooking with Collateral”, Risk, 2012.

Alexander Antonov and Vladimir Piterbarg. “Collateral Choice Option Valuation”, SSRN, 2013.