# Formula to price a CDS Index Option?

Say I wanted to buy an option on the CDX US HY Index (specifics here are irrelevant, but the point is that I'm looking for an option on a CDS Index).

What would be the pricing formula given inputs of spread vol, strike, time to maturity, etc?

If anyone could help, that would be great.

I have had a look through Dominic O'Kane's book (2008) but haven't found an explicit reference. I'm looking to try and build this out in excel, so if anyone is able to show/refer to something done there that would be even more useful.

A good reference for the valuation of a CDS index option is the paper by Massimo Morini and Damiano Brigo, where they discussed the Bloomberg CDS index option valuation, which is based on Black's formula given the forward par index spread vol, strike, and time to maturity.

The issue with this formula is that the numeraire, the Index Defaultable Present Value per Basis Point, or the forward PV01, can be zero with a positive probability, and, as a consequence, the swap measure is not equivalent to the risk-neutral probability measure.

Morini and Brigo addressed the issues above, using a sub-filtration method, where they defined the so called no-Armageddon pricing measure that is equivalent to the risk-neutral probability measure, and consequently, they obtained an arbitrage-free option valuation.

You will see a discussion of how to do this in O'Kane's book in section 11.7. It shows that you can adjust the forward CDS index spread in such a way that you can then use Black's swaption pricing formula set out in section 11.3.1.

You will need to calculate a few extra terms to make these adjustments which are all set out. These will need you to extract the index survival probability curve from index spreads using a standard CDS valuation model.

You then have to adjust the forward rate that you enter. To do so you must calculate the value of the front end protection (PV of a protection leg to the forward start date) and then amortise this over the life of the index swap by dividing by the forward period index risky PV01.

You also have to adjust the strike of the CDS index option. Once again this is explained in the text.

Table 11.2 gives some example values and shows that this model gives option prices which are very close to the full model described before this section.

You want to price the HY options given spread vols and other inputs. First of all, you need to understand HY indices are quoted in prices rather than spreads. With spread vols, you essentially assumed the spread dynamics, but your pay off is related to the price (or upfront payment). So you need to first derive the price dynamics from the spread dynamics using Ito's lemma. Then you should be able to calculate expected payoff under the right measure.

I believe Bloomberg and some other firms are currently using this approach.

However, if you are able to get price vols, things become easier, because you have price dynamics automatically and can simply plug in to the Blacks Formula.

There are a couple of other complications you need to consider in credit index option pricing.

1. front end protection;
2. possible zero numeraire.

A more interesting problem is to price IG options. IG indices are quoted in spreads. And you don't have an easy way to price them since you need to run a spread to upfront conversion on the exercise date to make the exercise decision. Please check out this paper on SSRN, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2616370 which provides an alternative approach to solve the problem.