3. Combine of Pricing Tree with State Variables of Snowball Notes
3.2 Numerical Approach to Snowball Notes
3.2.1 Construct States Variables of Snowball Price and Coupon
In this segment, there are three stages to construct the state variables of coupon rates in each node (i, j). To start with, in order to evaluate coupons from equation (3.1),
we compute negative summation of number of spacing between interest rate without freeze at zero article, ,
1
−
∑
, in the Hull-White preliminary tree. Next, consider the freeze at zero coupon article, the negative coupons should be eliminated and adjust the states of coupon variables with previous zero coupons. Lastly, for articles of redemptive and freeze on zero coupon, we introduced back-ward tracking and linear interpolation method to price snowball notes.First stage: building the maximum and minimum of negative summation of number of spacing rate in the Hull-White preliminary tree without freeze at zero article
Sum (i, j) is denoted as the negative summation of number of spacing rate which final rate is the rate at node (i, j) without the article of freeze at zero in the Hull-White preliminary tree:
and equation (3.5) satisfies the following formulas:
1
where c is one case of coupons at node (i, j) without the article of freeze at zero and node (i, j) is the child of node (i, j’).
Therefore, focus on the maximum and minimum of negative summation of number of spacing rate at each node, the all possible negative summation of number of spacing rate at node could be known. Nevertheless, determinate maximum and minimum of Sum (i, j) at node (i, j) is considered the previous nodes by different types
of branches.
Node C is the child of these nodes, so Sum (3, 1) calculated by equation (3.6) is following:
Follow equation (3.6), we can get the Max(Sum(i, j)) and Min(Sum(i, j)) at each node from figure 3.3:
Period of interest rate (i)
Spacing between inverse interest rate (-j)
(-10,-10) (-10,-14)
Period of interest rate (i)
(-10,-10) (-10,-14)
Period of interest rate (i)
Spacing between inverse interest rate (-j)
Figure 3.3 An Example of the Maximum and Minimum Summation of Each Node. The unit of x-axis is period time and unit of y-axis is the spacing between interest rate on the preliminary tree; ( M, m ) represents that M is the maximum and m is the minimum summation number of spacing rate.
When the maximum and minimum summation of each node is determinated, the
number of possible path in each node is also known. For instance, ( M, m ) at node (3, 2) where at time 3 tΔ and inverse rate − Δ is (-3,-5) which means that the node (3, 2 R 2) has three kinds of negative summation of number of spacing rate. It is following:
(3, 2) { 3, 4, 5}
Sum = − − −
Consequently, as the summations of node, Sum (i, j), and adjustments of Hull-White tree αi calculated, the coupons without the article of freeze at zero in each node could be given and the next stage will solve the problem of non-negative coupons.
Second stage: Check that the coupons at each node are non-negatives
Continuously, the problem of non-negative coupons would be solved by the following two steps. There are two situations we must adjust the states of coupons.
One phenomenon is that the previous coupon is positive but present coupon becomes negative. Therefore, the first step is to eliminate the states with negative coupons for the article of freeze on zero coupons. The other phenomenon is how to find the maximum and minimum of Sum (i, j) at present nodes when the precious coupons are reset to zero coupons. The second step would be introduced to solve this circumstance.
The First Step: Eliminate the states with negative coupons.
We define the lower bond of integers of each period such that the maximum and minimum summations which are larger than those integers would enable positive coupons of each node. The mathematical description is
1
. . ( α ) 0
=
∃ ∈i
∑
i k− k + Δ ≥ik
k Z s t S k R
Then solution of the lower bond of integers in each period is
1( ) 1( )
α , α
= − ⎡ = − ⎤
⎢ ⎥
≥ =
Δ ⎢⎢ Δ ⎥⎥
∑
ik k k∑
ik k ki i
S S
k k
R R (3.7) where ki is to eliminate the states with negative coupons in i-th period. If the elements
in Sum (i, j) less than or equal to that integer in the period, it means that resetting coupon rate to zero is in node (i, j).
Take the example from figure 3.3 with the condition for non-negative coupons, if the node has a situation of resetting coupon rate to zero, there is a symbol of 0* in (M, m) of that node. Supposing the node is (0*,0*), it means that the maximum summation is less the lower bond. That is to say, there is only one case of coupon rate in that node and the coupon rate is zero.
Period of interest rate (i) -1
Period of interest rate (i) -1
Spacing between inverse interest rate (-j)
Period of interest rate (i)
1 2 3
Spacing between inverse interest rate (-j)
1 2 3
Spacing between inverse interest rate (-j)
-1
Spacing between inverse interest rate (-j)
Figure 3.4 Adjust the Maximum and Minimum Summation of Each Node in the First Step of Second Stage.
Assume the estimation of k2 = −2 from equation (3.7), the maximum and minimum summations of node (2, 2) is (-3,-3) where touch the lower bond of integer, so as (-1,-2) at node (2, 1).
Thus, (-3,-3) changes to N(0*,0*) where 0* is a symbol of a reset coupon rate in that node.
are M-m+1 kinds of coupon rates in (M, m) and M-m+2 kinds of coupon rates in (M, m, 0*) which can be seen in figure 3.4 and figure 3.6.
The Second Step: Adjust the sates with previous zero coupons
We define the reset integers of each period,δi,j, such that change the present coupon value to the form of maximum and minimum summations at this time where the previous coupon is reset to zero. The mathematical description is
, , , coupon is zero. Then solution of the reset integers of each node is
1 the maximum and minimum of negative summation of number of spacing rate in the Hull-White preliminary tree in i=3 period. After that, change present coupon which the previous coupon is reset to zero to the form of maximum and minimum of summation in figure 3.5 by the equation (3.8).
Spacing between inverse interest rate (-j)
Period of interest rate (i)
-1
Period of interest rate (i)
-1
Period of interest rate (i)
-1
Figure 3.5 Adjust the Maximum and Minimum Summation of Each Node in the Second Step of Second Stage.
From equation (3.7), assume the estimations ofδ = − , 3,3 7 δ3,2 = − ,6 δ = − . The maximum 3,1 5 and minimum summations of node (3, 3) is calculated by type-(A) in first stage, so it has only reset
coupon in the node and change summation into (-7,-7). About node (3, 2) calculated by type-(B), the
maximum summation is from the maximum one of node (2, 1) and the minimum summation is from
the minimum one of node (2, 2) which has zero coupon rate, in this case, its minimum at node (3, 2) is
δ . Hence we can analogize the all node in third period. 3,2
After the first and second stages are completed, the determination of Sum (i, j) is built in figure 3.6 and the formula for coupons at node (i, j) is following:
,
Period of interest rate (i)
Period of interest rate (i)
-1
Figure 3.6 Maximum and Minimum Summation of Each Node in Forward-Tracking Method.
The maturity is i=5, the example is the same from figure 3.3 to figure 3.5. Assume the value of principal and the cost of redeeming snowball notes are $1. The numbers of coupon rates is -2-(-5) +2=5
according to (-2, -5, 0*) at node (4, 2) and the set of coupon rates at node (4, 2) is following:
Furthermore, the next stage is to evaluate the Snowball bonds from formula (3.2) and early redemptive time is considered in equation (3.4).
Third stage: Pricing the value by backward-tracking method
In this segment, the third stage is divided from two parts. One is how to discount the bond value with redemptive article; the other one is to use linear interpolation method when the discounted node has the situation of resetting zero coupons.
The First Step: Price the bond value of snowball notes
Suppose coupons at node (i, j) are built in figure 3.6, the equation (3.2) and (3.4)
for pricing snowball value can be rewritten as:
C face value if i t maturity
f R t t coupon fixed at node (i, j) and its summation situation is a times spacing between interest rate in Hull-White tree; B i j Sum a( , , ( )) is the snowball value at node (i, j) with redemptive article and B i( +1,j Sum ak, ( k))is the snowball value at node (i+1, jk) which is the child of node (i, j) ; C is the coupon paid at node(i, j) and fixed at node(i-1, j*) which is the parent of node(i, j) with Sum(i-1,j*)-j equal to a.
Choice of the following probability P and discounted nodes of up, median and k down are according to the styles of branch in figure 2.3 and its following:
( ) ( )
However, there is a problem when discounted bond value with its summation situation 0* in pricing snowball process. Thus, we will give examples to price snowball note value by equation (3.9) and linear interpolation method in next step.
The Second Step: Interpolation of coupons in discounted process
In corroding to simplify algorithm, we take the integer value to approach the reset coupon in second stage, thus we use linear interpolation method to find actual the discounted coupons. Given two examples from figure 3.6 which one must use interpolation method in figure 3.7(b) and the other one is without this character in figure 3.7(a).
-4 -5
Figure 3.7 Two Examples for Discounted Process.
One case is the discounted process ofD(3,1,Sum( 4))− and the other case is the discounted process of D(3, 2,Sum(0*)) which the coupon rate is reset to zero at node (3, 2). Moreover, assume the lower bond of integers is k4 = −5 by equation 3.7 and the reset integersδ < − ,4,3 5 δ4,2< − ,5 δ < − by equation 3.8. 4,1 5
On one hand, in the case of D(3,1,Sum( 4))− at node (3, 1), its tree is a type-(a) in figure 2.3. Hence its according discounted nodes are from node (4, 2), node (4, 1), and node (4, 0), and corresponding summations are
( ) ( 4) 4 2 6 4
the solution is rewritten by
( ) (0*)
and the solution of probabilities is type-(a) solution in Hull-White tree.
(a) (b)
On the other hand, in the case of D(3, 2,Sum(0*))at node (3, 2), its tree is also type-(a) in figure 2.3. Hence the children of node (3, 2) are node (4, 3), node (4, 2), and node (4, 1). Because of its zero coupons in node (3, 2), the coupons of these discounted nodes are following the equation (3.1):
, ,
χi j =max(Si−(αi+ fi jΔR) ,0) (3.10) where χi,j is actual coupon at node (i, j) with the previous coupon reset to zero.
Hence, we must check the actual coupons in these nodes for formula (3.10).
Assume at node (4, 3) and node (4, 2), the actual coupons are also reset to zero with previous coupon reset to zero; nevertheless, the actual coupon at node (4, 1) is positive. We must use interpolation method to find the actual discounted value in node (4, 1) because its actual coupon is positive. It is following
4,1, ( 5)
The solution of actual discounted value is
4,1, ( 5)
Actual bond value at node(4,1)
Actual reset coupon at node(4,1)*[ (4,1, ( 5)) (4,1, (0*))]
(4,1, (0*)) explain this method is in figure 3.8.
Pu
Coupon Rate at Sum(-5), node=(4,1)
Coupon Rate at Sum(-5), node=(4,1)
Coupon Rate at Sum(-5), node=(4,1)
Coupon Rate at Sum(-5), node=(4,1)
Coupon Rate at Sum(-5), node=(4,1) Actual coupon rate
0 .0 0 0 3 6 6 4 0.00125
0
Coupon Rate at Sum(-5), node=(4,1) Actual coupon rate
0 .0 0 0 3 6 6 4
0
Coupon Rate at Sum(-5), node=(4,1) Actual coupon rate
0
Coupon Rate at Sum(-5), node=(4,1) Actual coupon rate
0
Coupon Rate at Sum(-5), node=(4,1) Actual coupon rate
0 .0 0 0 3 6 6 4
Node(4, 2)
Node(4, 1)
Figure 3.8 The Example of Interpolation Process from Figure 3.7 (b).
Assume actual coupon rate is 0.0003664, the coupon rate at Sum (-5) is 0.00125, and as we known 0* means that coupon rate is 0. Therefore, the actual discounted value from node (4, 1) could be
calculated by equation (3.11).From equation (3.9), the discounted value of D(3, 2,Sum(0*)) is following:
*
3 4 3
* 2,
(3, 2, (0*))
( (4,3, (0*)) (4, 2, (0*)) *Actual bond value(4,1)) * 1
1 ( 2 * ) * ( )
(3, 2, (0*)) {min( (3, 2, (0*)), 1) ; C { : (2, *) 2 0*, =1,2}}
u m d
j
D Sum
P B Sum P B Sum P
R t t
B Sum D Sum C C Sum j j
= + + α
+ + Δ −
= + ∀ ∈ − =
and the solution of probabilities is type-(a) solution in Hull-White tree.
We can use this linear interpolation method to find the actual bond value at the reset node (4, 1), and so on. After the procedure of third stage, the determination of variables in backward-tracking is built and snowball value at all nodes are known.
3.2.2 Creating Proper Recursive Steps for Pricing Snowball Based on Hull-White