Valuation Commentary
How to
Validate an Interest Rate Model?
Part I: Pricing Swaps & Bonds
by Alex Levin
Mortgage practitioners are more liberal and grateful consumers of term structure modeling analytics than typical Street derivative traders. They pay much less attention to such things as the assumed rate distribution (leave that to statisticians - they've got to earn their bread somehow), pricing standard rate derivatives struck away from at-the-money (ATM) or exotics (who cares?). Symptomatically, MBS complexity that traces its roots to the behavioral uncertainty often masks the need for rigor.
Yet, most good models can and should be tested using a few simple steps. Below, I outline several validation exercises for readers and describe how AD&Co.'s financial engineering ensures positive results. In upcoming Pipeline issues, I will discuss the testing of volatility calibration and options.
Test 1: Valuation of fixed-rate swaps and option-free bonds
It may seem trivial, but let us follow Ronald Reagan's favorite Russian saying "Trust, but check!" Pay attention to swap and bond maturities used as inputs (AD&Co. offers up to 14 entry points), and price the bond or swap with some other maturity. For example, a user may input rates for the 5-yr swap and the 10-yr swap (among others), but opt to skip the 7-yr point. What 7-yr rate is calculated by the system? Our OAS spreadsheet easily enables a user to turn a pass-through MBS entry into a bullet bond. And our library is equipped with even more tools to run various tests with non-MBS instruments.
We would first set volatility
to zero and find that the 7-yr bond or swap has an internally interpolated
rate of, say, 4.5%. Let us introduce volatility into the system and rerun
the analysis. The rate on an option-free instrument should not change, should
it? Increase volatility further and see if the system holds and reports the
same rate or the same value for the bond or swap.
How did AD&Co. pass this test? We used three financial engineering methods:
· Spline interpolation
· Arbitrage Removal
· Closed-form analytics, or pseudo-analytics
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?.)
Figure 2 exhibits the convergence pace for various maturities as well as the difference between Monte-Carlo or quasi-Monte-Carlo (click here to read our description of quasi-Monte-Carlo.)
Figure 2. Convergence of Monte-Carlo and quasi-Monte-Carlo for bullet bonds
As seen in
Figure 2, quasi-Monte-Carlo, which features shock pre-processing, is about
twice as accurate as the regular Monte-Carlo (antithetic reflection is included
in each model), for the same number of paths. Pricing errors grow almost linearly
with maturity beyond 10-yr. 
To be continued…