LO 13.6: Describe the process o f constructing a simple and recombining tree for a

LO 13.6: Describe the process o f constructing a simple and recombining tree for a short-term rate under the Vasicek M odel with mean reversion.
The Vasicek model assumes a mean-reverting process for short-term interest rates. The underlying assumption is that the economy has an equilibrium level based on economic fundamentals such as long-run monetary supply, technological innovations, and similar factors. Therefore, if the short-term rate is above the long-run equilibrium value, the drift adjustment will be negative to bring the current rate closer to its mean-reverting level. Similarly, short-term rates below the long-run equilibrium will have a positive drift adjustment. Mean reversion is a reasonable assumption but clearly breaks down in periods of extremely high inflation (i.e., hyperinflation) or similar structural breaks.
The formal Vasicek model is as follows:
dr = k(0 – r)dt + crdw
where: k = a parameter that measures the speed of reversion adjustment 0 = long-run value of the short-term rate assuming risk neutrality r = current interest rate level
In this model, k measures the speed of the mean reversion adjustment; a high k will produce quicker (larger) adjustments than smaller values of k. A larger differential between the long- run and current rates will produce a larger adjustment in the current period.
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Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
Similar to the previous discussion, the drift term, A, is a combination of the expected rate change and a risk premium. The risk neutrality assumption of the long-run value of the short-term rate allows 6 to be approximated as:
a X 9 rj H k
where:
= the long-run true rate of interest
Lets consider a numerical example with a reversion adjustment parameter of 0.03, annual standard deviation of 130 basis points, a true long-term interest rate of 6%, a current interest rate of 6.2%, and annual drift of 0.36%. The long-run value of the short-term rate assuming risk neutrality is approximately:
A 9 ~ 6% H———= 18%
0.36% 0.03
It follows that the forecasted change in the short-term rate for the next period is:
0.03(18% – 6.2%)(1/12) = 0.0293%
The volatility for the monthly interval is computed as 1.5% x y/l /12 = 0.43% (43 basis points per month).
The next step is to populate the interest rate tree. Note that this tree will not recombine in the second period because the adjustment in time 2 after a downward movement in interest rates will be larger than the adjustment in time 2 following an upward movement in interest rates (since the lower node rate is further from the long-run value). This can be illustrated directly in the following calculations. Starting with rQ = 6.2%, the interest rate tree over the first period is:
Figure 5: First Period Upper and Lower Node Calculations
(0.03)(18% – 6.200%)
6.200% + —– —————- – +
12
,
= 6.663%
1.5% Vl2
6.200% + ——————— – – -4=r = 5.796%
(0.03)(18%- 6.200%)
12
1.5% yjl2
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Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
If the interest rate evolves upward in the first period, we would turn to the upper node in the second period. The interest rate process can move up to 7.124% or down to 6.238%.
Figure 6: Second Period Upper Node Calculations
6.663% +(Q-Q3)(18%- 6-663%)+ ^ = 7.124%
12
yjil
6.663%
(0.03)(18%-6.6630/0) _ +5% = 6^258%
12
Vl2
If the interest evolves downward in the first period, we would turn to the lower node in the second period. The interest rate process can move up to 6.260% or down to 3.394%.
Figure 7: Second Period Lower Node Calculations
5.796%+ (0-03)(18%-5.796%) L5% = ^
12
Vl2
5.796% + i8 % – 5 7 9 6 % ) _ + 5 % = ^
12
yjl2
Finally, we complete the 2-period interest rate tree with mean reversion. The most interesting observation is that the model is not recombining. The up-down path leads to a 6.258% rate while the down-up path leads to a 6.260% rate. In addition, the down-up path rate is larger than the up-down path rate because the mean reversion adjustment has to be larger for the down path, as the initial interest rate was lower (5.796% versus 6.663%).
Figure 8: 2-Period Interest Rate Tree with Mean Reversion
6.200%
6.663%
5.796%
At this point, the Vasicek model has generated a 2-period non-recombining tree of short- term interest rates. It is possible to modify the methodology so that a recombining tree is the end result. There are several ways to do this, but we will outline one straightforward method. The first step is to take an average of the two middle nodes (6.258% + 6.260%) 12 = 6.259%. Next, we remove the assumption of 50% up and 50% down movements by generically replacing them with {p, 1 p) and (q, 1 q) as shown in Figure 9.
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Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
Figure 9: Recombining the Interest Rate Tree
0.5
6.200%
The final step for recombining the tree is to solve for p and q and ^ and p and q are the respective probabilities of up movements in the trees in the second period after the up and down movements in the first period. yjm an(j yddare the respective interest rates from successive (up, up and down, down) movements in the tree.
We can solve for the unknown values using a system of equations. First, we know that the average of p x ruu and (1 – p) x 6.259% must equal:
6.663% + 0.03(18% – 6.663%)(1/12) = 6.691%
Second, we can use the definition of standard deviation to equate:
Vp(ruu 6.691%)2 + (1 p)(6.259% 6.691%)2 = 1.50% x
We would then repeat the process for the bottom portion of the tree, solving for q and If the tree extends into a third period, the entire process repeats iteratively.