# Pricing callable range accruals on spreads

What is an efficient method of pricing callable range accruals on rate spreads? As an example:

A cancellable 30 year swap which pays 6M Libor every 6M multiplied by the number of days the spread of 10-year and 2-year CMS rate is above 0, in exchange for a fixed or floating coupon.

Using LMM for this is dog slow, and 1 factor models are not enough because of both Libor and swap rates involved.

-
Can I ask why 1 factor models are not enough if both LIBOR and swap rates are involved? Thanks. – Jack Jan 6 '14 at 18:26

I know this may sound extreme, but in my experience for these kind of payoffs you do need a LMM and possibly with at least 8 factors. There is no shortcut unfortunately, even a 3 factor gaussian model, which you can use to price faster using trees, will not capture the possible dynamics of the curve implicit in an 8 factor LMM. Just my modest opinion obviously.

-

I've priced similar animals with a naive N-factor method, adding a convexity adjustment for the swap rates. But I'm not sure this is very orthodox...

-
How did you do handle the callability? – quant_dev Mar 29 '11 at 19:56
Simulating all the early call dates in each Montecarlo trajectory, and taking as the trajectory's price the lowest of those (since I assumed the issuer would call the note in that case). – finitud Mar 30 '11 at 14:16
Then I think you're overpricing the cancellation option, as you permit the caller to "look ahead in the future". – quant_dev Mar 30 '11 at 14:34

It is possible to price CRAs using the LMM using a Brownian bridge technique. You simulate to each coupon date and then infer the expectation of the coupon given the values of the rates at the start and end of the accrual period.

http://ssrn.com/abstract=1461285

Interpolation Schemes in the Displaced-Diffusion LIBOR Market Model and the Efficient Pricing and Greeks for Callable Range Accruals

Christopher Beveridge

University of Melbourne - Centre for Actuarial Studies

Mark S. Joshi

University of Melbourne - Centre for Actuarial Studies

August 25, 2009

Abstract:
We introduce a new arbitrage-free interpolation scheme for the displaced-diffusion LIBOR market model. Using this new extension, and the Piterbarg interpolation scheme, we study the simulation of range accrual coupons when valuing callable range accruals in the displaced-diffusion LIBOR market model. We introduce a number of new improvements that lead to significant efficiency improvements, and explain how to apply the adjoint-improved pathwise method to calculate deltas and vegas under the new improvements, which was not previously possible for callable range accruals. One new improvement is based on using a Brownian-bridge-type approach to simulating the range accrual coupons. We consider a variety of examples, including when the reference rate is a LIBOR rate, when it is a spread between swap rates, and when the multiplier for the range accrual coupon is stochastic.

-