Callable range accrual swap
A callable range accrual swap has two legs: a regular floating leg and a structured coupon leg. The structured coupon rate of the j-th period () is given by
(40a)
where
(40b)
where R is the fixed rate, and are the accrual range of the j-th period, is the LIBOR rate, is the range accrual index term, is the total number of the business days in the j-th period.
We choose a real 10 years maturity trade. The floating leg has a quarterly payment frequency and the structured coupon leg has a semi-annually payment frequency with varying accrual ranges. It starts with the first call opportunity being in 3 years from inception, and then every year until the last possibility being 9 years from inception. The range accrual index term is 6 months.
The lattice implementation procedure for a callable capped floater swap or a callable range accrual swap is quite similar to the one for a callable bond except the valuation for the underlying instrument.
The convergence diagrams of pricing calculations are shown in Exhibits 5 and 6. Each curve in the diagrams represents the convergence behavior for a given grid space as nodes are increased. All of the lattice results are well converged. If the grid space is smaller, the algorithm has better convergence accuracy but a slower convergence rate, and vice verse.
We benchmarked our model under different drift approximation methods with several standard market approaches, e.g., the regression-based in the full LMM and the HJM trinomial tree. The model comparisons for the accuracy and speed are shown in Exhibits 7 and 8. With regards to accuracy, as expected, the FD performs very badly. AAFR and GAFR do a little better but errors go in different directions. The same conclusions can be drawn for AADT and GADT. Both CEFR and CEDT are the best. In terms of CPU times, FD, AAFR, AADT, GAFR and GADT are the same. But CEFR and CEDT are slower, especially in the callable range accrual swap case.
  1. CONCLUSION
In this paper, we proposed a lattice model in the LMM to price interest rate products. Conclusions can be drawn, supported by the previous sections. First, the model is quite stable. The fast convergence behavior requires fewer discretization nodes. Second, this model has almost equivalent accuracy to the current pricing models in the market. Third, the implementation of the model is relatively easy. The calibration is very simple and straightforward. Finally, the performance of the model is probably the best among all known approaches at the time of writing.
We use the following techniques in our model: shifted forward measure, drift approximation, probability distribution structure exploitation, long jump, numerical integration, incomplete information handling, and calibration. Combining these techniques, the model achieves sufficient accuracy in relatively few time steps and discrete nodes, which makes it a very efficient method.
For ease of illustration, we present the lattice model based on the Trapezoidal Rule integration. A better but slightly more complicated solution is to spline the payoff functions. The cubic spline of the option payoffs can achieve higher accuracy, especially for Greeks calculations, and higher speed. Although cubic spline takes some time, the lattice will require much fewer nodes (23 ~ 28 nodes are good enough) and can perform a much faster integration. In general, the spline method can provide a speedup factor around 3 ~ 5 times.
We have implemented the lattice model to price a variety of interest rate exotics. The algorithm can always achieve a fast convergence rate. The accuracy, however, is a bit trickier, depending on many factors: drift approximation approaches, numerical integration schemes, volatility selections, and calibration, etc. Some work, such as calibration, is more of an art than a science.