CMS Swaps With A Smile

Unlike vanilla interest-rate swap prices, constant maturity swap prices depend on volatility. This Learning Curve reviews the key points in CMS swap pricing and highlights the impact that the interest rate volatility smile can have in pricing.

 

CMS Swaps And Their Valuation

In vanilla interest-rate swaps the LIBOR floating leg has the characteristics of a so called "par leg". In other words, the floating payments at each payment date represent the fair accrual of the principal of the swap over the accrual period. This means the floating leg of a vanilla interest-rate swap, with its implied notional exchanges included, prices to par and that when pricing vanilla swaps the valuation of the floating leg can be ignored.

In CMS swaps or yield curve swaps, the payments of the floating leg are based upon a floating rate other than the par rate for the period. For instance, one can swap a set of annual fixed cashflows for a set of semi-annual floating cashflows based on semi-annual resets of the 10-year swap rate. Clearly the amount of a floating coupon based on the 10-year swap rate and accrued over the six-month period following the rate reset is not the fair accrual amount. Such floating leg will not value to par. Therefore, when pricing CMS swaps we are required to value the floating leg.

 

Convexity Adjustment

The valuation of CMS swaps requires estimation of the value of each floating cashflow and this is done through calculating the expectation at each reset time. T of the applicable floating rate.

Strictly speaking, the desired expectation is the reset time T forward measure expectation. For the purposes of this tutorial, the theoretical framework of interest rate probability measures is omitted in favour of simpler wording.

For the purpose of investigating the value of the expectation at time T of the swap rate Y :

the swap rate is approximated by the par yield of the corresponding bond. There is little question on how to calculate the forward yield Y₀ from the forward bond price and it is tempting to take

Why is this not correct?

Remembering that bond prices and bond yields enjoy a non-linear relationship and the forward yield is defined as the yield corresponding to the forward bond price, it is easy to see that the expected yield and the forward yield are close, but not the same. The difference between the two is called the convexity adjustment (CA) such that:

One of the most commonly used expressions to calculate convexity adjustment is given by:

where B(Y) is the non-linear function relating the bond price and its yield, Y₀ is the initial forward yield, T is the CMS reset time and is the at-the-money swap rate volatility. B' and B" are used to represent the first and second derivatives of the price function, and therefore correspond to the so-called bond modified duration and bond convexity, respectively.

This calculation of convexity adjustment has been reviewed in the literature. Note that it relies on a simple Taylor expansion of the bond price function around Y₀ and two main additional assumptions. Firstly, the yield is assumed to have a lognormal distribution with mean not far from the initial forward yield itself.

The actual approximation is:

Secondly, for purposes of calculation of this distribution's second moment, it is assumed that the ATM volatility of the swap rate under the annuity measure is approximately the same as the ATM volatility under the forward measure.

This is an important consideration for further studies of convexity adjustment problems, but the full discussion of forward versus annuity measures is beyond the scope of this tutorial.

Some examples of forward swap rates and convexity adjustments are given in the table below.

 

Convexity Adjustment In The Presence Of Smile

Implicit in the evaluation of the difference between the expectation of the yield and the forward yield was an assumption on its probability distribution. The inadequacy of the usual lognormal assumption manifests itself in the options market through an observed dependence of the implied volatility with strike level. This dependence is known as either skew or smile. Much like the usual approach for pricing vanilla options with a smile, it is possible to account for this departure from lognormality in the calculation of the convexity adjustment by considering alternative probability distributions.

In the case of CMS swap pricing, risk managers are faced with the challenge of capturing the essence of the smile effect whilst keeping their solution simple and computationally non-intensive. It would not always be feasible to rely on an exotic model to price a book of CMS swaps

One way to represent the smile is to consider the Mixed Lognormal Distribution. By taking m lognormal processes with corresponding volatilities,, the probability density of a Mixed Lognormal process is a weighted average of the lognormal densities:

where,  are positive constants such that.

It is clear that under such a definition, the Mixed Distribution density function is a valid probability density function. Some of the noteworthy properties of Mixed Lognormals are:

* first and second moments are linear combinations of first and second moments of the original lognormal distributions;

* swaption prices are weighted average prices of swaptions with respective volatilities ;

* its probability density function presents fat tails and therefore lends itself well to the representation of smiles, such as seen for the European interest-rate markets at the moment.

The problem of convexity adjustment is particularly simply handled in the framework of the Mixed Lognormals, provided the approximations used in the lognormal case are deemed acceptable. The convexity adjustment is shown to be a linear combination of convexity adjustments calculated by using the different equivalent volatilities and combined weights:

Using swaption data, the parameters can be chosen in order to fit the volatility smile. The figures show the fit and adjustment when three lognormals are mixed.

Figures 2 (a) and (b) - The implied volatility smile for the 10-year tenor with expiries at 2 and 10 years. Data is for GBP in May 2001.

Figure 3 ­ Convexity Adjustment with smile for GBP on May 24, 2001

Conclusion

When pricing CMS swaps, the convexity adjustment is used in the calculation of floating cashflows. A brief review of a simple formula often used in estimating the adjustment is offered and despite some of the approximations that it entails, it is easy to apply, requiring only today's yield curve and swaption implied volatilities. Perhaps for its simplicity, this formula has retained popularity amongst market practitioners.

A simple way to take the volatility smile into consideration is used to illustrate the impact of smile in convexity. The technique employs the Mixed Lognormal distribution and lends itself to markets displaying a volatility smile.

    This week's Learning Curve was written by Leda Braga, head of valuation services, and Katia Babbar, quantitative analyst at Cygnifi in London.