Spline interpolation smoothly extends the benchmark set to 22 points. Arbitrage removal adjusts the internal calibrating function of time so that the backward induction on the AD&Co. probability lattice values all 22 bonds at par. Using exact (for the Hull-White case) or approximate analytics (Black-Karasinski or Squared Gaussian models) for a so-called "convexity adjustment" limits errors to near non-existing levels for swaps or bonds maturing between the benchmarks. This is a much-preferred way of calibrating the lattice over fudging it for all 360 rates, directly. Below is a sample test that I performed with the AD&Co. library, using market as of 01/30/2004. The benchmark set includes 180-mo, 240-mo, 300-mo, and 360-mo maturities (shown in bold) among others; testing points were selected at 210-mo, 270-mo, and 330-mo (for shorter maturities, it is hopeless to see any error at all.) I ran the term structure models with a zero mean reversion and a constant, market-comparable volatility.

Figure 1. Coupon rate reconstruction in AD&Co system

Valuation	180	210	240	270	300	330	360
Static	4.990	5.1149	5.193	5.2396	5.264	5.2773	5.286
HW, 100 bp vol		5.1148		5.2396		5.2772
BK, 20% vol		5.1148		5.2397		5.2773
SqG, 0.25 vol		5.1147		5.2397		5.2772

Forward rates, critical to valuation, are sensitive to differences in spot rates. For example, a 10-yr receive-fixed swap starting forward in 5 years is a long position in a 15-yr bond coupled with a short position in a 5-yr bond with the same coupon. Therefore, having made sure each option-free bond is priced model-independent, we nailed all forward rates too.

Test 2: Testing random sampling for option-free instruments

Rates in Figure 1 were calculated by pricing bonds backwards on the AD&Co. lattice. Alternatively, rates or values could be computed using our Monte-Carlo or quasi-Monte-Carlo methodology available for both OAS spreadsheet users and software developers. Even option-free instruments will be priced with some error because every path yields a different present value. In such a case, results in Figure 1 should be interpreted as the asymptotic rates achieved with a very large number of paths (Does your Monte-Carlo converge to the same price or rate with change of volatility or mean reversion?.)

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