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)?
