Why Collateral Choice Option Valuation Lacks a Known Closed Form

Summary

The document describes the collateral choice option as an expectation involving the time integral of the maximum among several spread processes, typically modeled as Gaussian. It notes that this expectation is difficult to calculate and that a cited treatment reports no known closed-form expression. Published work referenced in the discussion instead proposes efficient numerical methods for approximating the option’s value.

The post asks whether research has since produced a closed-form solution or proved that none can exist. It provides no resolution to either question, so it should be read as a statement of the author’s awareness rather than a definitive survey of the literature. The central practical takeaway is that valuation is treated as a numerical problem in the cited work; the document does not explain or compare the approximation techniques.

Key ideas

• Collateral choice option valuation can be expressed as an expectation involving the integrated maximum of multiple spread processes.
• The spreads are commonly assumed to follow Gaussian distributions in the formulation described.
• The cited discussion states that no closed-form expression was known and points to numerical approximation methods.
• The document leaves open whether later research found a closed form or established its nonexistence.

Tags

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


# 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.

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.