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I've read the OpenGamma paper https://quant.opengamma.io/CDS-Options-OpenGamma.pdf on CDS Options, and noticed a small discrepancy. So I wanted to double-check my understanding.

In Section 6.4 the option payoff is defined as

OptionPayoff

with the default-adjusted forward portfolio swap price given by

PortfolioSwapPrice

And the exercise price given by

ExercisePrice

The portfolio swap price $V$ includes the index factor $f(T_e)$ as of option expiry $T_e$, as one would expect. Now, what I am wondering, is the exercise price $G$ not missing the index factor $f(T_{Inception})$ as of trade inception $T_{Inception}$? That is, formula (70) should be:

CorrectedExercisePrice

For example, say, we buy a swaption on Version 2 (index factor 0.99 at trade time), and by expiry 2 more names have defaulted (so we are on Version 4 with factor 0.97). Then the exercise price we pay (at expiry) should use the factor as of when we "entered into" the forward CDS (so 0.99 as of trade inception), and not the current index factor. Whereas the live portfolio swap obviously uses the latest factor 0.97.

Does this make sense?

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    $\begingroup$ You may also like this paper: arxiv.org/abs/1201.0111v3 section 3. Unless I'm confused, an exercise is on the portfolio consisting of version 4 of the index (having factor .97) and the recoveries of the 2 names that defaulted since the swaption was written on version of the index (having factor .99)? $\endgroup$
    – Dimitri Vulis
    Commented Oct 14, 2021 at 8:22
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    $\begingroup$ You are right, when you exercise, you enter into version 4 of the index + you receive the 2 protection payments. However, you exercise price you pay should be based on the idx factor of the original version, see e.g. i.sstatic.net/DFWBk.png $\endgroup$
    – Phil-ZXX
    Commented Oct 14, 2021 at 8:50

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