In this section a comparison between the methodologies is presented. In particular, we look at the effects of mean reversion and local volatility on the drift and the spread in the short rates. We present numerical results for the term structures, volatility, and mean reversion in Exhibit 3.3. The exhibit also includes the bond information for use later.
Original Term Structure with No Mean Reversion
We first consider the original term structure with no mean reversion for the HL and HW models. In Exhibit 3.4 we present the binomial tree for the HL model and the trinomial for the HW model using the HW trino- mial methodology. We use a 10% volatility throughout the trees. We see that the spread in the short rates increases over time in the models as expected. We also see that the HL model can give negative short rates.
In Exhibit 3.5 we present the binomial tree for the KWF model, the trinomial for the BK model using the HW trinomial methodology, and the BDT binomial model. The KWF and BK models use the 10% volatility throughout the tree and no mean reversion. Note the volatile nature of the BDT model. This is due to the time varying volatility structure and the way mean reversion is incorporated into the BDT model through this
j 6
3φτ ---
< 0.816 ---φτ
≈
jk 3– 6 3φτ
--- 0.184
---φτ
≈
>
A Review of No Arbitrage Interest Rate Models 61
decreasing volatility structure. Note that all the short rates are positive and that the spread in the rates is significantly less than in Exhibit 3.4.
Exhibit 3.6 presents the trinomial lattices for the HW and BK mod- els using the information in Exhibit 3.3 and a mean reversion of 5%.
The volatility is 10%. Notice the pruning that takes place within the lat- tice when we have mean reversion. This produces lattices that are signif- icantly different than those shown in Exhibits 3.4 and 3.5. This is a peculiarity of the Hull and White methodology. The pruning is a result of incorporating mean reversion into the model and ensuring that the distributional characteristics of the SDE’s are retained.
Comparison of the Models Using Common Risk and Value Metrics
Here we contrast the effective duration, effective convexity, and the option- adjusted spread (OAS) for 10-year callable and putable bonds each with a one-year delay on the embedded option. The information in Exhibit 3.3 is used for the analysis. We computed the effective duration for the original term structures shown in Exhibit 3.3 using a yield change of 25 basis points. The original term structure is then shifted up and down in a parallel manner by ±250 basis points and by ±500 basis points, respectively. In other words, we computed the effective duration at five different term structure levels using a yield change of 25 basis points.
EXHIBIT 3.3 Input Information
Original TS Volatility Mean Reversion
6.20% 10.00% 5%
6.16% 10.00%
6.15% 9.00%
6.09% 9.00%
6.02% 8.00%
6.02% 8.00%
6.01% 7.00%
6.01% 7.00%
6.00% 7.00%
6.01% 7.00%
Bond Information for ED, EC, and OAS
Call Price (Regular Callable) $102.50 Put Price (Regular Putable) $95.00 Annual Coupon ($ per $100) $6.00 Time Option Starts (years from now) 1
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EXHIBIT 3.4The HL Binomial and HW Trinomial Trees for the Original Term Structure with No Mean Reversion a. The Ho-Lee Interest Rate Lattice 136.31% 118.42% 101.50%116.31% 85.20% 98.42% 69.85% 81.50% 96.31% 54.99% 65.20% 78.42% 41.49% 49.85% 61.50% 76.31% 28.93% 34.99% 45.20% 58.42% 17.05% 21.49% 29.85% 41.50% 56.31% 6.20% 8.93% 14.99% 25.20% 38.42% −2.95% 1.49% 9.85% 21.50% 36.31% −11.07% −5.01% 5.20% 18.42% −18.51%−10.15% 1.50% 16.31% −25.01%−14.80% −1.58% −30.15%−18.50% −3.69% −34.80%−21.58% −38.50%−23.69% −41.58% −43.69% Time in Years0.01.02.03.04.05.06.07.08.09.0
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EXHIBIT 3.4 (Continued) b. The Hull-White Trinomial Interest Rate Lattice Using the HW Method with No Mean Reversion 203.31% 172.58% 185.99% 142.30% 155.26% 168.67% 131.70% 124.98% 137.94% 151.35% 107.78%114.38% 107.66% 120.62% 134.03% 84.92% 90.46% 97.06% 90.34% 103.30% 116.71% 63.71% 67.60% 73.14% 79.74% 73.02% 85.98% 99.39% 43.65% 46.38% 50.28% 55.82% 62.42% 55.70% 68.66% 82.07% 24.39% 26.33% 29.06% 32.96% 38.50% 45.10% 38.38% 51.34% 64.75% 6.20% 7.07% 9.01% 11.74% 15.64% 21.18% 27.78% 21.06% 34.02% 47.43% −10.25% −8.31% −5.58% −1.68% 3.86% 10.46% 3.74% 16.70% 30.11% −25.63%−22.90%−19.00%−13.46% −6.86% −13.58% −0.62% 12.79% −40.22%−36.32%−30.78%−24.18% −30.90% −17.94% −4.53% −53.64%−48.10%−41.50% −48.22% −35.26% −21.85% −65.42%−58.83% −65.54% −52.58% −39.18% −76.15% −82.86% −69.90% −56.50% −100.18% −87.22%−73.82% −104.54% −91.14% −108.46% Time in Years0.01.02.03.04.05.06.07.08.09.0
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EXHIBIT 3.5The BDT and KWF Binomial and the BK Trinomial Trees for the Original Term Structure with No Mean Reversion a. The Kalotay, Williams, and Fabozzi Interest Rate Lattice 14.72% 12.92% 11.87%12.05% 10.65%10.58% 9.76% 9.72% 9.87% 8.43% 8.72% 8.66% 7.89%7.99% 7.96% 8.08% 7.44%6.90% 7.14% 7.09% 6.73%6.46%6.54% 6.52% 6.61% 6.20%6.09%5.65% 5.84% 5.81% 5.51%5.29%5.36% 5.34% 5.41% 4.98%4.62% 4.78% 4.75% 4.33%4.39% 4.37% 4.43% 3.79% 3.92% 3.89% 3.59% 3.58% 3.63% 3.21% 3.19% 2.93% 2.97% 2.61% 2.43% Time in Years0.01.02.03.04.05.06.07.08.09.0
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EXHIBIT 3.5 (Continued) b. The Black-Karasinski Trinomial Interest Rate Lattice Using the HW Method with No Mean Reversion 28.45% 23.21%23.92% 19.82%19.52%20.12% 16.52%16.67%16.41%16.92% 14.08%13.89%14.02%13.80%14.23% 11.31%11.84%11.68%11.79%11.61%11.97% 9.82% 9.51% 9.96% 9.83% 9.92% 9.76%10.06% 8.60%8.26% 8.00% 8.37% 8.26% 8.34% 8.21% 8.46% 7.25%7.23%6.95% 6.73% 7.04% 6.95% 7.01% 6.90% 7.12% 6.20%6.09%6.08%5.75% 5.66% 5.92% 5.84% 5.90% 5.81% 5.98% 5.12%5.11%4.91% 4.76% 4.98% 4.91% 4.96% 4.88% 5.03% 4.30%4.13% 4.00% 4.19% 4.13% 4.17% 4.11% 4.23% 3.47% 3.37% 3.52% 3.48% 3.51% 3.45% 3.56% 2.83% 2.96% 2.92% 2.95% 2.90% 2.99% 2.49% 2.46% 2.48% 2.44% 2.52% 2.07% 2.09% 2.05% 2.12% 1.75% 1.73% 1.78% 1.45% 1.50% 1.26% Time in Years0.01.02.03.04.05.06.07.08.09.0
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EXHIBIT 3.5 (Continued) c. The Black, Derman, and Toy Interest Rate Model 11.39% 10.29% 6.47% 9.89% 9.52% 8.93% 7.36%6.34% 8.59% 8.12%8.10% 7.76% 7.24%6.79%6.21% 7.46% 7.44%6.78%6.90% 6.74% 6.73%6.30%6.26%6.08% 6.47% 6.20%6.09%5.66%5.88% 5.85% 5.51%5.49%5.77%5.95% 5.62% 4.98%4.73%5.00% 5.09% 4.78%5.32%5.83% 4.88% 3.95%4.26% 4.42% 4.91%5.71% 4.24% 3.63% 3.84% 5.59% 3.68% 3.33% 3.20% Time in Years123456789
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EXHIBIT 3.6Trinomial Model a. The Hull-White Trinomial Interest Rate Lattice Using the HW Method with Mean Reversion of 5% 83.50%87.60%91.92%96.84%101.89%107.24% 63.14%66.18%70.28%74.60%79.52%84.57%89.91% 43.51%45.82%48.86%52.96%57.28%62.20%67.25%72.59% 24.39%26.18%28.50%31.54%35.64%39.96%44.88%49.93%55.27% 6.20% 7.07% 8.86%11.17%14.22%18.32%22.64%27.56%32.61%37.95% −10.25%−8.46%−6.15%−3.10%1.00%5.32%10.24%15.29%20.63% −25.78%−23.47%−20.42%−16.32%−12.00%−7.09%−2.03%3.31% −40.79%−37.75%−33.64%−29.32%−24.41%−19.35%−14.01% −55.07%−50.96%−46.64%−41.73%−36.67%−31.33% Time in Years123456789 b. The Black-Karasinski Trinomial Interest Rate Lattice Using the HW Method with Mean Reversion of 5% 11.34%11.87%11.73%11.84%11.67%12.03% 9.83%9.53% 9.99%9.86%9.96% 9.81%10.12% 8.60%8.27%8.02% 8.40%8.29%8.38% 8.25% 8.51% 7.25%7.26%6.95%6.74% 7.06%6.98%7.04% 6.94% 7.16% 6.20%6.09%6.08%5.85%5.67% 5.94%5.87%5.92% 5.84% 6.02% 5.12%5.11%4.92%4.77% 4.99%4.93%4.98%4.91% 5.06% 4.30%4.14%4.01% 4.20%4.15%4.19%4.13% 4.26% 3.48%3.37%3.53%3.49%3.52% 3.47%3.58% 2.84%2.97%2.93%2.96%2.92%3.01% Time in Years123456789
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Exhibit 3.7 presents the effective duration and convexity results for the two securities for each model. The results are interesting. It is clear that the normal models do not agree with the lognormal models. Specif- ically, the normal models do not match the characteristics of the price yield relationship at extreme interest rate levels.20 Furthermore, each model gives slightly different results. This is an important finding and must be appreciated by any user of these models.
Exhibit 3.8 presents the OAS results. We used a market price that is 3% below the model price for the OAS computation. They are consis- tent with the results in Exhibit 3.7. Note that the normal models pro- duce OAS values larger than any of the lognormal models. This is due to the distributional differences and the property of allowing very low and negative interest rates. Clearly, normal models are not desirable when evaluating securities with embedded options.21