Binominal Changing Volatility Approach toSubprime Loan Problem
YASUAKI WATANABE*Abstract
Recently, subp rime lo an pro blem is a h otly d ebated issu e aro und th e wo rld.
In this paper, binominal option approach is adopted to surmise thebino minal op tion value o f ABX.HE Ind ex 1 b y ch anging its volatility. A
nonrecombining lattice is required to so lve th e option. Supp ose that th ecurrent value of ABX.HE Index is US$α ,its annual volatility isβ percent,risk free rate isγ percent. We can calcu late the value o f a th ree year op tion
to abandon for a price of US$δ . We simulate the change of solution when
the annual volatility changes through time.

Key Words : Subprime Loan Problem, Binominal Option, Volatility
JEL Classification : C-15, C-22, G-10
* Affiliation : Kochi University of Technology, Professor of Finance atManagement Department (Former Stanford University Graduate School ofBusiness, Visiting Scholar)
Mailing Address : 185 Miyanokuchi, Tosayamada-cho, Kami City, Kochi
Prefecture 782-8502, JapanEmail Address : watanabe.yasuaki@kochi-tech.ac.jp
Home Phone No. :+81-887-57-2776
Fax Number : +81-887-57-2811

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HE of ABX.HE Index means Home Equity. This Index is used as a hedge tool ofRMBS(Residential Mortgage Backed Securities) which is related to subprime loan.
11. Introduction
The movement of subprime loan problem will affect not only U.S. economy, butalso world economy. Many economists and analysts predict this precarious situation
will continue for the time being. At first, many people believed that the loss wasrestrictive, because the related financial institutions can manage the losses.
However, when hedge fund companies involved these transactions and bankrupt due
to high leverage, market environments suddenly became worse. Originally, they
invested CDO (Collateralized Debt Obligation) which includes RMBS (Residential
Mortgage Backed Securities). Here, we use the ABX.HE Index which is derived
from CDS (Credit Default Swap) of RMBS. It is noteworthy that highly rated
indexes such as AAA and AA are also decreasing the prices sharply since last July
2007. Thus, we simulate the future scenario of subprime loan problem by applying
into the binominal option approach with changing index’s volatility. The reason isthat the volatility taken up in the binominal option is based on twomethods. One method uses a historical volatility. The other method
uses some cash flow scenarios. These volatilities are constant,however, the actual volatility usually changes through time. Thus, thescenario analysis of volatility is very important for the future
prediction in the option value. Here, we use present value event tree,binominal option analysis decision tree and optimal execution tree.

2. ModelWe adopt the binominal option approach to surmise the binominal option
value of ABX.HE Index by changing its volatility. These indexes
include five credit ratings from AAA, AA, A, BBB, BBB_. Here, we
compare the affect of subprime loan problem in each credit rating.
Suppose that the current value of ABX.HE Index is US$ α, its annual
volatility is β percent, risk free rate is γ percent. We can calculate thevalue of a three year option to abandon for a price of US$ δ. We
simulate the four scenarios in this analysis. The following Table-1shows the details.

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Table 1. Four Scenarios in the Volatility
Conditions 2BestFirst Period
βBetter/WorseβWorse/BetterβWorstβ

Second Periodβ − θ d1()(β − θ )(β + θ )(β + θ )Third Periodβ − θ d2()

d1

β

u1β

(β + θ )u1

u2

Namely, we simulate the change of solution when the annual volatilitychanges from β percent the first year to β − θ d1 Q β ≥ θ d1 percent the(second to) ()(β − θ ) (Q β ≥ θ ) percent or β percent the third year 3. Wed2d2also simulate the reverse case. The annual volatility changes from β
percent the first year to β + θ u1 percent the second to β or β + θ u2()()

percent the third year 4 .
At first step, we calculate a present value tree for the ABX.HE Index in each credit
ratings. While, the annual volatility is β percent and the beginning ABX.HE
Index value 5 is US$ α. Next, we compute the BOA value by first
calculating the optimal execution values for the end nodes of thedecision tree. We use the replicating portfolio approach 6 to computethe value of each previous node with risk free rate γ percent. The
option to abandon of it is US$ δ. In addition, when the volatility per(period decreases from β percent to β − θ d1) in second period and to (β − θ )d2

or increase again to β percent in third period, and increases from β percent to(β + θ ) in second period and to (β + θ ) or decrease again to β percent in thirdu1

u2period, the present value tree for the ABX.HE Index does not recombine in eachcase.

Conditions mean the subprime loan problem’s situations. For example, if the problem
settle down smoothly by the policy of FRB, etc., the conditions become better with lowervolatility. And if it continue two periods successively, the conditions become best withlowest volatility and vice versa.
3 The value of θ is smaller than θ in this case.
d1d2
24

The value of θu1 is smaller than θu2 in this case.

The price is based on at the end of Dec.31 2007.
Replicating portfolio is composed of m shares of the twin securities and B US $ of the risk freebond that present value is US$ 1 per bond.
563

3. Result of Data Analysis
We examine the case of 07-17 BBB_ in the ABX.HE Index. This rating is thehighest credit risk and often referred to the symbol of subprime loan
problem. The present value of BBB_ is α=18.89. Its annual volatility is
β=45.93 percent. The PV event tree is illustrated in Figure-1.
Figure-1[PV Event Tree]
74.92
47.33
29.90
29.90
PV=18.89
18.89
11.93
11.94
7.54
4.76
0123We calculate the BOA value by computing the optimal execution values
for the end nodes of the decision tree. We use the replicating portfolio
approach to get the value of each previous node in Figure 2.
Suppose that the abandonment value of it is δ=5.00.
Figure-2[BOA Decision Tree]

74.92
47.33

29.90

29.90

8BOA =18.93

18.89
12.00

11.94
7.67

01

5.00
32

7

There are two indexes in 2007. The one is 07-1 and the other is 07-2. 07-1 index starts fromJanuary 2007. But, 07-2 index starts from July 2007. Calculation is based on a daily data of theindex.
8 Calculation Fo rmu la : BOA = Max[m * PV + B, δ ](Qm =(BOA − BOA ) , B = (BOA − m * PV ) )(1 + γ )(PV − PV )upupdownup

up

down

4

The result of BOA value (=18.93) is higher than the PV (=18.89).
We can see that the option to abandon for US$5.00 is executed only whenthe value of it has declined continuously for three periods.
While, the volatility per period decreases from β=45.93 percent to(β − θ ) =35.93 percent in period 2 and to (β − θ ) =25.93 percent in period 3, thed1d2

present value tree for it does not recombine 9 .
Figure-3[PV Event Tree]

55.50
42.83
33.05
29.90
27.06
20.88
16.11

PV=18.89
22.15
17.09
13.19
11.93
10.80
8.33
0

912

θd =10 percent and θ d =20 percent in this case.
125

6.43
3

Figure-4[BOA Decision Tree]

55.50
42.83
33.05
29.90
27.06
20.88
16.11

BOA=18.89
22.15
17.09
13.19
11.94
10.80
8.33
0

12

6.43
3

The BOA value decreases with the decrease of volatility. It is same as
the PV and then there is no flexibility. However, the decrease in thevolatility of it also decreases the BOA value from 18.93 to 18.89.

(When the volatility per period decreases from β=45.93 percent to β − θ d1
=35.93 percent in period 2 and increases to β=45.93 percent in period 3, the present
value tree for it also does not recombine 10 .

10θd =10 percent in this case.
16)

Figure-5[PV Event Tree]

67.79
42.83
27.06
29.90
33.05
20.88
13.19

PV=18.89
27.06
17.09
10.80
11.93
13.19
8.33
012

5.26
3

Figure-6[BOA Decision Tree]

67.79
42.83
27.06
29.90
33.05
20.88
13.19

BOA=18.89
27.06
17.09
10.80
11.93
13.19
8.33
0

127

5.26
3

The BOA value decreases with the decrease of volatility. It is same as thePV and then there is no flexibility. The optimal execution of theabandonment option remains the same. However, we notice that thefluctuation of volatility decreases the BOA value from 18.93 to 18.89.
When the volatility per period increases from β=45.93 percent to(β + θ ) =55.93 percent in period 2 and decreases to β=45.93 percent in period 3, theu1

present value tree for it also does not recombine 11Figure-7[PV Event Tree]

82.80
52.31
33.05
29.90
27.06
17.09
10.80

PV=18.89
33.05
20.88
13.19
11.93
10.79
6.82
01112

θu =10 percent in this case.
18

4.31
3

Figure-8[BOA Decision Tree]

82.80
52.31
33.05
29.90
27.06
17.79
10.80

BOA=19.00
33.05
20.88
13.19
12.15
10.80
7.20
01

5.00
32

The BOA value is higher than the PV and then the flexibility is valuable.
The optimal execution of the abandonment option remains the same.
However, we notice that the fluctuation of volatility increases the BOAvalue from 18.93 to 19.00.
Furthermore, the volatility per period increases from β=45.93 percent to(β + θ ) =55.93 percent in period 2 and to (β + θ ) =65.93 percent in period 3, theu1

u2

present value tree for it also does not recombine 12 .

12

θu =10 percent and θu =20 percent in this case.
129

Figure-9[PV Event Tree]

101.13
52.31
27.06
29.90
33.05
17.09
8.84

PV=18.89
40.36
20.88
10.80
11.93
13.19
6.82
012

3.53
3Figure-10[BOA Decision Tree]

101.14
52.31
27.06
29.90
33.05
17.09
8.84

BOA=19.17
40.36
20.88
10.80
12.44
13.19
7.71
01

2

10

5.00
3

The BOA value increases with the increase of volatility. It is higher than
the PV and then the flexibility is valuable. The optimal execution of theabandonment option remains the same. However, we notice that thefluctuation of volatility increases the BOA value from 18.93 to 19.17.
Likewise, we execute the scenario analyses of other credit ratings such as 07-1BBB,07-1A, 07-1AA, 07-1AAA respectively. Judging from the Table-2, we notice there is atendency that higher credit ratings and worse scenarios have a leeway of more
flexibility.
Table-2.

Scenario Analyses in Each Credit Ratings

Best07-1BBB_*
Not Flexible
BOA-PV(=18.89)
007-1BBB**
Flexible
BOA-PV(=19.87)
0.12
07-1A***
Flexible
BOA-PV(=25.9)
0.41
07-1AA****
Flexible
BOA-PV(=45.53)
0.33
07-1AAA***** Not Flexible
BOA-PV(=75.64)
0Better/Worse Worse/Better
Not Flexible
Flexible00.11
Flexible
Flexible
0.32
0.53
FlexibleFlexible
0.69
0.97
Flexible
Flexible
0.83
1.46
Flexible
Flexible
0.24
1.12

WorstFlexible
0.28
Flexible
0.56
Flexible
1.25
Flexible
1.89
Flexible
2.68*Annual Volatility=0.4593, Abandonment=US$5.00**Annual Volatility=0.4669, Abandonment=US$7.50***Annual Volatility=0.5232, Abandonment=US$10.00****Annual Volatility=0.4243, Abandonment=US$20.00*****Annual Volatility=0.2018, Abandonment=US$50.00

Finally, we can confirm the diffusion of bad affection of subprime loanproblem from Table-3 to Table-6.The correlated coefficients in eachquarter shows the difference. Especially, we can find out the significant
difference between second quarter and third quarter. Apparently, we cansee the tendency that higher credit ratings indexes are highly correlatedwith lower credit ratings indexes.

11Table-3. Correlated Coefficients in Each Credit Ratings (Jan.2007 - Mar.2007)
07-1BBB_ 07-1BBB
07-1A
07-1AA
07-1AAA
07-1BBB_1
07-1BBB
0.9298107-1A
0.81720.8237
107-1AA
0.45540.45630.3528107-1AAA
0.50050.39710.4734 0.7075906
1
Table-4. Correlated Coefficients in Each Credit Ratings (Apr.2007 - Jun.2007)
07-1BBB_ 07-1BBB07-1A07-1AA07-1AAA
07-1BBB_107-1BBB
0.8960107-1A
0.66710.7140107-1AA
0.17330.25430.3800107-1AAA
0.17450.16900.16450.2240
1
Table-5. Correlated Coefficients in Each Credit Ratings (Jul.2007 - Sep.2007)
07-1BBB_ 07-1BBB
07-1A07-1AA07-1AAA07-1BBB_1
07-1BBB
0.8318107-1A
0.69430.7476107-1AA
0.68740.73700.8537
107-1AAA0.6026
0.58900.64050.746798
1Table-6. Correlated Coefficients in Each Credit Ratings (Oct.2007 - Dec.2007)
07-1BBB_ 07-1BBB07-1A07-1AA07-1AAA07-1BBB_107-1BBB
0.8910107-1A
0.68620.7383
107-1AA
0.62830.66230.8458107-1AAA0.58940.56940.7419 0.8620666
1

4. Conclusion
When we simulate the change of volatility in ABX.HE Index, we notice in most casesthat BOA value is higher than the PV and the flexibility is valuable irrespective of thechange of volatility. We use a nonrecombining lattice method to solve this kind ofoption problem. In addition, we notice that if the volatility increases in the second
period and/or in the third period, the BOA value increases. If not, the BOA value
decreases. We can confirm the tendency that higher credit ratings indexes
are highly correlated with lower credit ratings indexes during the period
between second quarter and third quarter. Because subprime loan problem
quivered the economy not only U.S. but the rest of the world. Namely,
investor ’s psychological factors spur the vile situations. It is necessarythat U.S. government sets up the public company to purchase all thestructured bonds related to subprime loan to settle down this problem
smoothly.

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REFERENCES
Amram, Martha, and Nalin Kulatilaka. REAL OPTIONS.: Harvard Business SchoolPress 1999
Copeland, Tom, and Vladimir Antikarov. Real Options.
: TEXERE 2001
Howell, Sydney, Andrew Stark, David Newton, Dean Paxson, Mustafa Cavus, JosePereira, and Kanak Patel. Real Option ; Evaluating Corporate Investment
Opportunities in a Dynamic World. : Prentice Hall 2001
Imai, Junichi. Real Options. Chuokeizai 2004
Japan Real Option Association. Real Options and Management Strategy. : Sigma
Base Capital 2006
Mun, Johnathan. MODELING RISK. : WILEY 2006
Mun, Johnathan. . Real Options Analysis: WILEY 2006
Yamamoto, Daisuke. Real Options : Toyokeizai 2001

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