It is claimed that the credit default swap (CDS) spread should approximate the risky par bond yield or coupon rate spread from the riskless bond on the same entity. This comes about when we assume discount factor $B(t)=e^{-rt}$ with constant riskless interest rate $r$ together with infinitesimal coupon period. However, this is not true in general and it is dubious under what condition this holds even approximately.
Let us examine this claim mathematically. In general, for par bond coupon rate $c$ and CDS spread $s$ $$c-s=\frac{\int_0^T P(t)\mathrm d(-B(t))}{\sum_{i=1}^n \delta_iB_iP_i} \ge 0,$$ where $P(t)$ is the survival probability and $B(t)$ the discount factor at time $t$, $t_i$ is the $i$'th coupon date and $P_i=P(t_i)$ and $B_i=B(t_i)$. It is true $B(t)\searrow 0 \Longrightarrow c-s\searrow 0$, for any given $P$. However, once can device decreasing $P$ and $B$ such that $c-s$ is unbounded from above in the set of admissible $P$ and $B$ (decreasing positive function on $[0,\infty)$ taking value $1$ at $t=0$) for any given $\{\delta_i>0\}_{i=1}^n$. Consider very small $P_i$.
The riskless par bond coupon rate of the same coupon schedule is $$c_0=\frac{\int_0^T \mathrm d(-B(t))}{\sum_{i=1}^n \delta_iB_i}.$$ So $$c-s-c_0=\frac{\int_0^T P(t)\mathrm d(-B(t))}{\sum_{i=1}^n \delta_iB_iP_i}-\frac{\int_0^T \mathrm d(-B(t))}{\sum_{i=1}^n \delta_iB_i}$$ We already know from the previous paragraph that the above expression is unbounded from above. To explore the range of the above expression, consider the following case. $$ P(t) = \begin{cases} 1, & t=0 \\ P_1, & t\in (0,t_1] \\ 0, & t\in (0,\infty) \end{cases}, \quad B(t) = \begin{cases} 1, & t=0 \\ B_1, & t\in (0,t_1] \\ 0, & t\in (0,\infty) \end{cases}. $$ then $$c-s-c_0 = \frac{1}{\delta_1B_1}-1-\frac{1}{\delta_1B_1}=-1.$$ So $c-s-c_0$ ranges at least from $-1$ to positive infinity.
Therefore, all we can say is that in the very low interest rate regime like recently, $c$ is not much higher than $s$. Then what additional conditions are imposed or specific models are assumed to justify the folklore statement that $c-s\approx c_0$? Can someone provide a mathematical derivation or supply some references to that effect? Many authors have cited Darrell Duffie's paper as the source of the claim. However, I do not see a mathematical derivation or justification there --- the aforementioned link is a draft version not the published one and perhaps therein lies the rub.
One of the papers making such claim is this. There Assumption 4. is the most pertinent, acknowledging the assumption of constant interest rate, presumably implying $B(t)=e^{-rt}$ with $r$ a positive constant, just as I have stated in the first paragraph. How good an approximation is this? This and this are two other papers amongst many making the claim. None of these authours are "dubious" folks.
Can anyone elucidate?