# Index CDS Option (Spread Quoted) - Black's Formula

I have looked at the question and answers here and I have read Chapter 11 of Dominic O'Kane's book Modelling Single-name and Multi-name Credit Derivatives. The book is very clear and has some in-depth practical examples but I am still having trouble with the steps to get to Black's formula for spread quoted index CDS options and was hoping that someone could help.

Question 1

The payoff at expiry $$t_E$$ for the holder of payer swaption on an index CDS with maturity $$T$$, strike spread $$K$$, fixed running coupon $$C(T)$$ and $$M$$ underlying reference entities each with weight $$1/M$$ is given in formula (11.2) as:

$$V(t_E)=\Big[\frac{1}{M} \sum_{m=1}^{M} \mathbb{1}_{\tau_m \leq t_E} (1 - R_m) + \left(S_I(t_E; t_E, T) - C(T) \right) RPV01(t_E; t_E, T, S_I(t_E; t_E, T)) - \left(K - C(T) \right) RPV01(t_E; t_E, T, K) \Big]^+$$

where $$S_I(t_E; t_E,T)$$ is the market index spread at expiry, $$R_m$$ is the recovery rate on the $$m$$-th reference entity and $$RPV01(t_E; t_E, T, x)$$ is the risky annuity at $$t_E$$ using the spread $$x$$ on the index entered into at $$t_E$$.

Should the formula include the index factor in the middle term? In other words, should the formula be as follows:

$$V(t_E)=\Big[\frac{1}{M} \sum_{m=1}^{M} \mathbb{1}_{\tau_m \leq t_E} (1 - R_m) + \frac{1}{M} \sum_{m=1}^{M} \mathbb{1}_{\tau_m > t_E} \left(S_I(t_E; t_E, T) - C(T) \right) RPV01(t_E; t_E, T, S_I(t_E; t_E, T)) - \left(K - C(T) \right) RPV01(t_E; t_E, T, K) \Big]^+$$

The Example immediately following formula (11.2) suggests to me that it should be included here.

Question 2

My second question is regarding how to get from this payoff (without the amendment in Question 1) at $$t_E$$ to Black's formula (with the reasonable assumption that almost surely not all reference entities default before a short dated option expiry) giving the value at time $$0$$:

$$V(0) = RPV01(0; t_E, T, S_I(0; t_E, T)) \bigl[ \Phi(d_1)F'(0) - \Phi(d_2)K' \bigr]$$

where

$$F'(0) = S_I(0; t_E, T) + \frac{FEP(0; t_E)}{RPV01(0; t_E, T, S_I(0; t_E, T))} = S_I(0; t_E, T) + \frac{D(t_E) \frac{1}{M} \sum_{m=1}^{M} (1 - Q_m(t_E)) (1 - R_m)}{RPV01(0; t_E, T, S_I(0; t_E, T))}$$

$$K' = C(T) + \left( K - C(T) \right) \frac{RPV01(0; t_E, T, K)}{RPV01(0; t_E, T, S_I(0; t_E, T)) Q_I(t_E)}$$

$$d_1 = \frac{1}{\sigma \sqrt{t_E}} \left[ \log \frac{F'(0)}{K'} + \frac{\sigma^2 t_E}{2} \right]$$

$$d_2 = \frac{1}{\sigma \sqrt{t_E}} \left[ \log \frac{F'(0)}{K'} - \frac{\sigma^2 t_E}{2} \right]$$

If we assume that almost surely not all reference entities default before $$t_E$$, we can use $$RPV01(t; t_E, T, S_I(t; t_E, T))$$ on $$[0,t_E]$$ as numeraire with associated measure $$\mathbb{P}$$ equivalent to the risk neutral bank account measure $$\mathbb{Q}$$ (I think we can in any case). This in turn means that we can write the value of the option as:

$$V(0) = RPV01(0; t_E, T, S_I(0; t_E, T) E^{\mathbb{P}} \Big[ \bigl[F'(t_E) - K^*(t_E)\bigr]^+ \Big]$$

where

$$K^*(t_E) = C(T) + \left( K - C(T) \right) \frac{RPV01(t_E; t_E, T, K)}{RPV01(t_E; t_E, T, S_I(t_E; t_E, T))}$$

If we assume that the front end protection (FEP) adjusted forward spread has the $$\mathbb{P}$$ dynamics:

$$dF'(t) = \sigma F'(t) dW^{\mathbb{P}}(t)$$

then we are close to getting the Black formula result above. However, the $$K^*(t_E)$$ term has a dependence on $$S_I(t_E; t_E, T)$$ in the denominator of the second term which in turn has a dependence on $$F'(t_E)$$. I am stuck here on how to get to the Black type result above. Is there just a simplification that we replace the stochastic quantity $$K^*(t_E)$$ with the deterministic term $$K'$$ (with the constraint that put call parity is satisfied by the resulting value)?

• While Dom's 2008 book is excellent, there have been a few important papers after that on index options. I suggest you look at Morini, Brigo doi.org/10.1111/j.1467-9965.2010.00444.x (2010) and (a good summary of the current state) Martin: A CDS Option Miscellany arxiv.org/abs/1201.0111v3 (2019) Commented May 11, 2021 at 13:19
• @DimitriVulis thanks. Section 3 of Martin's paper is good and essentially answers my first question i.e. I think the index factor should be in the middle term of the payoff equation. However, it doesn't really answer 2. Morini and Brigo paper is useful also but doesn't deal with the differing $RPV01$s, i.e. $S_I$ vs. $K$, in the payoff. Markit (by providing FEP adjusted spread volatilities) and ICE (paper from 2018) still use the methodology and formulae in Section 11.7 from Dominic O'Kane's book so I was hoping to understand question 2 better i.e. the assumptions to get to $K'$. Commented May 11, 2021 at 14:27