Constant Maturity Swap dates and conventions

Question

Let's note $L(t,T_i,T_{i+1})$ the libor rate observed at $t$, fixing at $T_i$ with delivery at $T_{i+1}$.

The natural delivery date for this rate is $T_{i+1}$, so a vanilla swap with no pay lag would be priced as :

$Swap(t) = \sum_{i=1}^{n} \tau (T_i,T_{i+1}) P(t,T_{i+1}) \mathop{\mathbb{E}} ^{i+1}\left[ (L(T_i,T_i,T_{i+1})-K) \right]$

with $\mathop{\mathbb{E}} ^{i+1}$ the expectation under the forward measure associated with the bond $P(t,T_{i+1}) $

$\tau (T_i,T_{i+1}) = T_{i+1} - T_i$ for simplification

Under the sum we use $P(t,T_{i+1}) $ for dicounting as the payments occurs at $T_{i+1} $.

To price a CMS swap, we have the know expression for the swap rate :

$$s_{m,n}(t)=\frac{ P(t,T_m) - P(t,T_n)}{ A_{m,n}(t) }$$

with $A_{m,n}$ the annuity given by : $$A_{m,n}(t)= \sum_{i=1}^{n} \tau (T_i,T_{i+1}) P(t,T_{i+1})$$

now a CMS swap can be priced as (Simply replacing the libor rate with the constant maturity swap rate) :

$SwapCMS(t) = \sum_{i=1}^{n} \tau (T_i,T_{?}) P(t,T_{?}) \mathop{\mathbb{E}} ^{?}\left[ (s_{m,n}(T_i)-K) \right]$

I put questions marks in the formula because this is where i'm confused. When is the delivery of the swap rate? I mean the natural pay date so that I can choose the the bond for discounting and the forward measure needed? Or what is the discount I need to use and the forward measure in this case and why?

Thank you

Answer 1

To lighten notation, we assume a constant accrual factor $\tau$, a swap rate $S_n(T)$ which fixes at $T$ and pays at $T_p$ (e.g. $T_p-T=\text{3 months}$) and a simple CMS payoff of the form: $$\Phi(S_n(T))=(S_n(T)-K)$$ fixed at time $T=T_m$. We are interested in pricing under a measure for which the underlying risk factor of interest (i.e. the swap rate) is a martingale. Note that: $$S_n(T)=\frac{1-P(T,T_n)}{A_n(T)}$$ The annuity $A_n(T)$ is a portfolio of traded assets (zero-coupon bonds), therefore it is itself a traded asset which can be used as a numéraire. Hence when pricing CMS payoffs we work under the annuity measure instead of a forward measure because it is the measure which makes the swap rate a martingale, and we normally model the swap rate dynamics directly. Hence: $$\text{CMS Swap}(t)=A_n(t)\mathbb{E}^{A_n}\left[\frac{P(T,T_p)}{A_n(T)}(S_n(T)-K)\bigg|\mathscr{F}_t\right]$$ where $P(T,T_p)$ accounts for the delay between the fixing date and the payment date of the swap rate. The problem now is that we have a complex random variable inside the expectation. In fact, note that the CMS value depends on the whole interest rate curve up to $T$ due to the annuity factor $A_n(T)$. In order to price a CMS payoff, the following method is often followed:

1. Define a reasonable (twice-differentiable) mapping function $f(\cdot)$ such that: $$f(S_n(T))\approx\mathbb{E}^{A_n}\left[\frac{P(T,T_p)}{A_n(T)}\bigg| S_n(T)\right]$$ This simplifies the problem because now you can express the annuity factor as a function of the terminal swap rate only.
2. It comes that: $$\text{CMS Swap}(t)\approx A_n(t)\mathbb{E}^{A_n}\left[g(S_n(T))|\mathscr{F}_t\right]$$ where: $$g(S_n(T))=f(S_n(T))(S_n(T)-K)$$ You can then use the replication technique to price your CMS payoff.

I recommend taking a look at Cedervall and Piterbarg's "Full implications for CMS convexity" (2012, Risk magazine) for a detailed discussion on CMS pricing.