CMS Rate Convexity Adjustment
A method is presented to calculating a particular multiplicative factor, which appears in a formula for a CMS rate convexity adjustment.
Last updated
A method is presented to calculating a particular multiplicative factor, which appears in a formula for a CMS rate convexity adjustment.
Last updated
CMS Rate Convexity Adjustment
A method is presented to calculating a particular multiplicative factor, which appears in a formula for a CMS rate convexity adjustment. A CMS rate convexity adjustment provides a correction term to the forward CMS rate to match the mean value of the CMS rate under the forward probability measure.
We model the probability density of a CMS rate, under the forward swap measure, by a certain weighted sum of three log-normal densities. The defining parameter values for the log-normal densities above are determined by matching, in a least squares sense, the market price for various European style swaptions.
The model returns the ratio of the variance of the CMS rate, under the distributional assumptions above, to the variance of the CMS rate, instead assuming that it is log-normally distributed under the forward swap measure.
Consider a forward starting, fixed-for-floating interest rate swap. Assume that the swap’s respective floating and fixed legs have common reset points, , for . Furthermore, assume that the floating leg pays at time , for , a Libor rate, , which sets at for the accrual period, . Then
is the swap rate at time (here is the price at of a zero coupon bond, which matures at ).
We define a probability density for the swap rate, , as a linear combination of three respective log-normal densities. In particular, We assume that has probability density of the form
where
are unknown parameter values. The parameter values (1a-c) are determined by matching the model price for various European style swaptions, specified by respective
strike levels (in, at or out-of-the money),
diffusion,
and tenor,
Consider the fixed-for-floating rate swap defined above in Section 2.0. We seek to determine
where
Then
We now assume that
where
We calculate a ratio,
where
The convexity adjustment formula is then given by
We consider a fixed-for-floating interest rate swap specified by
forward start, 10 years,
maturity, 5 years,
The swap rate for the above is given by
References:
where denotes the density for a log-normal random variable, , such that is normally distributed with mean, , and standard deviation, . Furthermore, for ,
,
,
,
,
, where is a forward swap rate,
.
We assume that the probability density parameters,, depend on the CMS rate forward start time, , as follows. In particular, for
, (0a)
, (0b)
, (0c)
, (1a)
, and (1b)
, (1c)
against their corresponding market price. Additionally, we determine the volatility parameter, , by matching the price of an at-the-money European style payer swaption.
, for , is the numeraire process for the corresponding forward swap measure,
denotes expectation with respect to the forward swap measure above.
.
(2)
where is deterministic function. Then
is a forward swap rate,
.
,
where has probability density defined in Section 2.2. Here
is a constant volatility parameter,
is a standard Brownian motion.
reset, , for ,
floating leg pays the Libor rate, , at , for ,
fixed rate settlement points, , for .
.
We seek to calculate the convexity adjustment, , where
.
