3
$\begingroup$

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

$\endgroup$
  • $\begingroup$ In a standard CMS swap in the US, which isn’t very common nowadays , the payment date of the cms leg is 3months after the observation date of the cms rate. $\endgroup$ – dm63 Jun 20 at 9:51
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.