4 Case Study

In this section, we will show the valuation of callable inverse floater issued by UBS
AG, callable cumulative inverse floater issued by Royal Bank of Scotland Plc, and
callable daily range accrual note issued by UBS AG respectively. The least-squares
Monte Carlo simulation of lognormal forward LIBOR is applied to all these three
cases.

4.1 Callable Inverse Floater Note

4.1.1 Term Sheet
Part of the term sheet of our first case, Callable Inverse Floater Note, is shown in
Table 4.1.
Table 4.1 Term Sheet of Callable Inverse Floater Note

3 YEAR USD CALLABLE INVERSE FLOATER NOTE

Description:

3 year USD denominated Callable Inverse Floater Notes

(the Notes) with Interest linked to the USD 3 month
LIBOR. The Notes are callable quarterly by the Issuer on
each Interest Payment Date commencing on or after 3
months from the Issue Date.

Issuer:

UBS AG

Specified Denomination: USD 10,000 per Note

Issue Price:

100.00%

Issue Date:

04 October 2004

Maturity Date:

04 October 2007

44

Interest Amount:
Period

Coupon Rate

Oct04 Apr05

15.00% - 4.8 * 3m USD LIBOR IN ARREARS

Apr05 Oct05

18.50% - 4.8 * 3m USD LIBOR IN ARREARS

Oct05 Apr06

22.00% - 4.8 * 3m USD LIBOR IN ARREARS

Apr06 Oct06

25.50% - 4.8 * 3m USD LIBOR IN ARREARS

Oct06 Apr07

29.00% - 4.8 * 3m USD LIBOR IN ARREARS

Apr07 Oct07

32.50% - 4.8 * 3m USD LIBOR IN ARREARS

Interest is subject to minimum of 0% and Capped at 12%

per annum, For the purpose of Coupon calculation Libor
fixings will be floored at zero.
Daycount:

30/360

Interest Payment Dates:

04 October, 04 January, 04 April and 04 July each year

commencing on 04 January 2005, adjusted as per the
Business Day Convention.

Early Redemption
Option (Call):

The Issuer may redeem the Notes, in whole but not in

part, on each Interest Payment Date commencing on 04
January 05. The note holder will be entitled to any
Interest payments due on the Early Redemption Date.

Early Redemption Date:

If the Notes are called, the Interest Payment Date in

respect of which the Early Redemption Option is
exercised.

Optional Redemption
Amount:

100% of the Aggregate Nominal Amount

Source: UBS Investment Bank

The trend of the USD 3-month LIBOR for two years is illustrated in Figure 4.1.
Though it has had an upward movement recently, the USD 3-month LIBOR still
remains a low level. The investors still might have good chance to gain profits from
the inverse floater.

45

US0003M Index
2.5
2
1.5
1
0.5
0
2002/10/1

2003/4/1

Source: Bloomberg

2003/10/1

2004/4/1

2004/10/1

Fig. 4.1 USD 3-Month LIBOR

4.1.2 Valuation
I. Construction of Yield Curve
To value the contract, first we have to construct a yield curve on the issue date,
corresponding to the maturity of the contract. In this case of callable inverse floater,
we, at least, construct a 3.25-year (due to the term in arrears)3 quarterly yield curve
on Oct. 04, 2004.
The LIBOR with maturity not longer than one year (3 months, 6 months, 9
months, and one year) is accessible from the market. They are simply the quarterly
yield rates with terms within one year. The LIBOR is shown in Table 4.2.

Libor in Arrears: Libor set in arrears and paid in arrears. At the maturity of the contract, year 3, the
available LIBOR in the market will be L(3, 3, 3.25).
46

Table 4.2 LIBOR on 2004/10/04

Table 4.3 Swap Rate on 2004/10/04

LIBOR on Oct. 04, 2004

Swap Rate on Oct. 04, 2004

3m LIBOR

2.03125%

2y swap rate

2.808%

6m LIBOR

2.21%

3y swap rate

3.368%

9m LIBOR

2.365%

4y swap rate

3.659%

1y LIBOR

2.50375%

For the yield rates with terms longer than one year, we have to get the market data of
swap rate, displayed in Table 4.3. Then we use the Cubic Spline function of Matlab
software to estimate the quarter-united swap rate, i.e. swap rate for 1.25, 1.5, 1.75
years, and so on.
Next, we apply the bootstrapping method to swap rates to obtain the quarterly
yield rate with terms longer than one year. The equation of bootstrapping method is as
j

Sj
i =

1+ S j
1
+
=1
i
(1 + y0,i ) (1 + y0, j ) j

j = 1.25,1.5,..., 3,3.25 = 0.25 .4

(4.1)

Sj is the swap rate for the interest rate swap terminating at time j; y0,i is the yield to
maturity for time i at time zero. is the time fraction of year. Take for instance the
case when j=1.25. The bootstrapping equation will be as follows.
0.25S1.25
0.25S1.25
0.25S1.25
0.25S1.25
1+ 0.25S1.25
+
+
+
+
= 1.
0.25
0.5
0.75
1
(1+ 0.25y0,0.25)
(1+ 0.25y0,0.5)
(1+ 0.25y0,0.75)
(1+ 0.25y0,1) (1+ 0.25y0,1.25)1.25

In this equation, the only unknown variable is the yield to maturity for time 1.25 at
time zero, y0,1.25. We can solve for y0,1.25 with simple algebra manipulation. Proceeding
recursively, we then bootstrap the entire yield to maturity we need. The yields to
maturity are shown in Table 4.4, and the yield curve illustrated in Figure 4.2.

See Appendix for the detail derivation of the equation.

47

Table 4.4 3-Month Yield Rate on 2004/10/04

Term (year)

Yield (%)

Term (year)

Yield (%)

Term (year)

Yield (%)

0.25

2.0313

1.25

2.6020

2.25

2.9391

0.5

2.2100

1.5

2.6681

2.5

3.0915

0.75

2.3650

1.75

2.7301

2.75

3.2503

2.5038

2.8142

3.3937

3.25

3.5046

USD LIBOR 3-Month Zero Curve

0.038
0.036
0.034

YTM

0.032
0.03
0.028
0.026
0.024
0.022
0.02

0.5

1.5
2
Maturity

2.5

3.5

Fig. 4.2 USD LIBOR 3-Month Zero Curve

II. Extraction of the Initial Forward LIBOR

After construction of yield curve of LIBOR, we calculate the initial forward LIBOR
implied by the yield to maturity.
The initial forward LIBOR are extracted through the following no-arbitrage
relationship.

(1 + y0,i )i = (1 + y0,i )i (1 + L(0, i , i )) i = 0.5, , 3, 3.25 = 0.25

and L(0,0,0.25) = y0,0.25 .

(4.2)

L(0, i, j) is the forward rate prevailing between time i and time j known at time zero;
y0,j is the yield to maturity for time j at time zero; , again, denotes the time fraction of
48

year.
Iteratively, we obtain the whole sets of forward LIBOR, which will be used for
simulation. The result of initial forward LIBOR is shown in Table 4.5.
Table 4.5 Initial 3-Month Forward LIBOR L(0, i, j)

Time (year)

Fwd. LIBOR (%)

Time (year) Fwd. LIBOR (%)

Time (year)

Fwd. LIBOR (%)

[0,

0.25]

2.0313

[1,

1.25]

2.9954

[2,

2.25]

3.9391

[0.25, 0.5]

2.3888

[1.25, 1.5]

2.9985

[2.25, 2.5]

4.4662

[0.5, 0.75]

2.6752

[1.5, 1.75]

3.1024

[2.5, 2.75]

4.8416

[0.75,

2.9203

[1.75,

3.4037

[2.75,

3]

4.9743

3.25]

4.8376

1]

2]

[3,

III. Calibration of Instantaneous Volatility and Instantaneous Correlation Coefficient

To implement the simulation, we have two more parameters to estimate, the
instantaneous correlation coefficient ij between forward LIBOR L ( 0, Ti 1 , Ti ) and
forward LIBOR L ( 0, T j 1 , T j ) and the instantaneous volatility i ,1 of forward
LIBOR L ( 0, Ti 1 , Ti ) .
For the instantaneous correlation coefficient , we assume that all the
instantaneous correlation coefficients equal one, = 1, for the simplicity of
implementation. It is reasonable to make this assumption, according the historical
instantaneous correlation coefficients between forward LIBOR.
For the instantaneous volatilities i ,1 , we use the relationship (3.47) to extract
the caplet volatilities.

P ( 0, T ) Black ( L ( 0, T
j

i =1

i 1

, Ti ) , K , Ti 1 vT j cap

= i P ( 0, Ti ) Black L ( 0, Ti 1 , Ti ) , K , Ti 1 vTi 1 caplet .

i =1

First we obtain the swap rates (for the strike price K) and the market quotes of caps on
the issue date of the contract, 2004/10/04, as shown in Table 4.6.
49

Table 4.6 Market Quotes of Cap on 2004/10/04

Maturity (year)

Cap Vol. (%)

26.28

30.91

31.22

30.16

28.82

Applying the relationship (3.47), we extract the caplet volatilities implied by the
market quotes of caps. Take for example when j=2,

P ( 0, T1 ) Black L ( 0, T0 , T1 ) , S0,1 , vT2 cap + P ( 00 , T2 ) Black L ( 0, T1 , T2 ) , S0,2 , vT2 cap

= P ( 0, T1 ) Black L ( 0, T0 , T1 ) , S0,1 , vT0 caplet + P ( 0, T2 ) Black L ( 0, T1, T2 ) , S0,2 , vT1 caplet .

The only unknown variable is vT1 caplet , which can be solved by simple algebra
manipulation. Repeatedly, we then extract all the caplet volatilities we need. The
result of extraction is shown in Table 4.7 and illustrated in Figure 4.3.
Table 4.7 Implied Caplet Volatility on 2004/10/04

Maturity (year)

Caplet Vol. (%)

26.28

31.40

26.46

26.21

22.72

Implied Caplet Volatility

0.32
0.31
0.3

Volatility

0.29
0.28
0.27
0.26
0.25
0.24
0.23
0.22

1.5

2.5

3
Maturity

3.5

4.5

Fig 4.3 Implied Caplet Volatility on 2004/10/04

We then assume that the instantaneous volatility structure follows equation

(3.42),

50

 i (t ) = (Ti 1 t; a, b, c, d ) := a (Ti 1 t ) + d e b(T

i 1 t

+ c,

and run the Fminsearch (multidimensional unconstrained nonlinear minimization)

of Matlab with the target function

Ti 1vT2i 1 caplet =

Ti 1

( a (T

i 1 t ) + d
e

b(Ti 1 t )

+ c dt ,

to estimate the parameters. We get a = -0.040558, b = -0.028386, c = 0.35325, d =

-0.017911. Next, these four parameters are used to recover the whole set of
instantaneous volatilities. The result is in Table 4.8.
Table 4.8 Instantaneous Volatility of 3-Month Forward LIBOR

Fwd. LIBOR

Instant. Vol. (%)

Fwd. LIBOR

Instant. Vol. (%)

Fwd. LIBOR

Instant. Vol. (%)

L(0,0,0.25)

32.50

L(0,1,1.25)

28.22

L(0,2,2.25)

23.69

L(0,0.25,0.5)

31.45

L(0,1.25,1.5)

27.11

L(0,2.25,2.5)

22.52

L(0,0.5,0.75)

30.39

L(0,1.5,1.75)

25.98

L(0,2.5,2.75)

21.33

L(0,0.75,1)

29.31

L(0,1.75,2)

24.84

L(0,2.75,3)

20.13

L(0,3,3.25)

18.91

IV. Simulation of Forward LIBOR

As mentioned in Section 3.2, to ensure consistent comparisons for all exercise values
to holding values, it is necessary to simulate the forward LIBOR under a single
forward measure.
We use the zero coupon bond whose maturity is the same as that of our callable
inverse floater, which is three years, as the numeraire. The forward LIBOR is
simulated under forward measure Q 3 . Hence the forward LIBOR follow three kinds
of dynamics.
For the forward LIBOR with maturity less than 3 years, they evolve according to
equation (3.37)

51

L(T0 , T j , T j +1 )

1 + L(T , T , T

dL(T0 , Ti , Ti +1 ) = L(T0 , Ti , Ti +1 ) i (T0 )

j =i +1

+ L(T0 , Ti , Ti +1 ) i (T0 ) dWi

Tn +1

j +1 )

j (T0 ) i , j dt

(T0 ) ,

where T0 = 0 , Ti = 0.25,0.5, , 2.75 , and Tn +1 = 3 .

For the forward LIBOR L(0,2.75,3), it evolves as a martingale according to equation
(3.38) dL(T0 , Tn , Tn +1 ) = L(T0 , Tn , Tn +1 ) n (T0 ) dWnTn +1 (T0 ) .
For the forward LIBOR L(0,3,3.25), it evolves according to equation (3.39)

dL(T0 , Ti , Ti +1 ) = L(T0 , Ti , Ti +1 ) i ( T0 )

L(T0 , T j , T j +1 )

j =i +1

1 + L(T , T , T
j

j +1

j (T0 ) i , j dt

+ L(T0 , Ti , Ti +1 ) i (T0 ) dWi Tn +1 ( T0 ) .

i (T0 ) are the instantaneous volatilities we recover from the market quotes of caps.
Discretizing these dynamics, then we can start simulate the forward LIBOR at each
time step. The time step is set to be equal to the tenor = 0.25 for simplicity. Due to
the assumption of perfect instantaneous correlation = 1, one random number is
enough for each time step of simulation. Otherwise, we have to generate as many
random numbers as the initial forward LIBOR and perform Cholesky decomposition
for each time step of simulation.
The simulation process is as the following matrix. The first column vector is the
initial forward LIBOR; the second column vector is the simulation for the first time
step t = = 0.25 , and the spot LIBOR L ( 0, 0, 0.25 ) ceases in this time step, so
that it is set to be zero. Repeatedly proceeding, we complete the forward LIBOR
simulation for one time.

52

0
L ( 0,0,0.25) 0
L ( 0,0.25,0.5) L ( 0.25,0.25,0.5) 0
L ( 0,0.5,0.75) L ( 0.25,0.5,0.75) L ( 0.5,0.5,0.75)
L ( 0,0.75,1)

L ( 0.25,0.75,1)

L ( 0.5,0.75,1)

L ( 0,1,1.25) L ( 0.25,1,1.25) L ( 0.5,1,1.25)

L ( 0,1.25,1.5) L ( 0.25,1.25,1.5) L ( 0.5,1.25,1.5)
L ( 0,1.5,1.75) L ( 0.25,1.5,1.75) L ( 0.5,1.5,1.75)

L ( 0,1.75, 2 ) L ( 0.25,1.75, 2 ) L ( 0.5,1.75, 2 )

L ( 0, 2, 2.25) L ( 0.25, 2, 2.25) L ( 0.5, 2, 2.25)
L ( 0, 2.25, 2.5) L ( 0.25, 2.25, 2.5) L ( 0.5, 2.25, 2.5)
L ( 0, 2.5, 2.75) L ( 0.25, 2.5, 2.75) L ( 0.5, 2.5, 2.75)
L ( 0, 2.75,3) L ( 0.25, 2.75,3) L ( 0.5, 2.75,3)
L ( 0,3,3.25) L ( 0.25,3,3.25) L ( 0.5,3,3.25)

0
0

0
0

0
0

0
0

0
0

0
0

0
0
L ( 2.75, 2.75, 3) 0

L ( 2.75, 3,3.25) L ( 3,3, 3.25)

This is the illustration of forward LIBOR simulation for one time. It is necessary to
simulate as many times as possible. In addition, we have to record all the forward
LIBOR of every simulation in order to calculate the exercise values.
V. Calculation of Cash Flows
After simulating for M times, we calculate the interest for each payment date
according to the payoff condition in the term sheet. For example, the interest payment
at the maturity of the contract, time 3, is

max ( 32.50% - 4.8 Li ( 3,3,3.25) ,0 ) , i = 1, , M .

Then we adjust the interest payment structure to fulfill the constraint that the
maximum interest payment is 12% per annum.
VI. Dealing with the Callable Feature
Constructing the cash flow structure without the callable feature, we calculate the
exercise values for each time step. It need not to consider which path is in the money,
since the note holder will be entitled to any interest payments due on the early

53

redemption date plus the nominal principal.

Cash flows are first discounted back to T j , j = n as the independent variables,
n

E j (T j ) =
k= j

cashflow ( i, Tk +1 )

1 + Li (T j , Tk , Tk +1 )

i = 1, , M

j=n

(4.3)

where cashflow ( i , Tk +1 ) denotes the cash flow at Tk +1 for the simulation path i. The
simulated spot LIBOR at

Tj

for path i,

Li ( T j , T j , T j +1 )

and its square

Li (T j , T j , T j +1 ) are chosen as the dependent variables. For example, when Tn +1 = 3

2

the

independent

variable

is

1 + max ( 32.50% - 4.8 Li ( 3,3, 3.25) ,0 )

1 + Li ( 2.75, 2.75,3)

and

the

dependent variables are Li ( 2.75, 2.75, 3) and Li ( 2.75, 2.75,3) .

2

Running the regression E j (T j ) = + 1L (T j , T j , T j +1 ) + 2 Li (T j , T j , T j +1 ) + ,

2

then we obtain the estimate of the exercise values at T j , E j (T j ) . The holding values
for each simulation path at T j , H j (T j ) , is one plus interest payment at T j (nominal
principal plus interest, calculated in percentage). Finally comparing E j (T j ) with

H j (T j ) , we make the exercise strategy that call the inverse floater if

E j (T j ) H j (T j ) and make change to the cash flow structure. If the contract is called
at T j , all the cash flows after T j are set to be zero; otherwise the cash flow structure
remains unchanged.
Recursively for

j = n 1, n 2,,0 , we make the cash flow structure

incorporated with the callable feature. Discounting all the cash flows back to time
zero and averaging the M paths results, we have the inverse floater price. When M =
10000, we might have the reasonably theoretical price of 98.87% with variance
54

3.41%.
The valuation procedures may be summarized as follows.
I. Constructing yield curve through market quotes of LIBOR and swap rates.
II. Extracting the initial forward LIBOR implied by the market yield curve.
III. Calibrating the model to the market.
IV. Implementing the Least-squares Monte Carlo simulation of forward LIBOR.
Step IV varies due to the different characteristics of each contract. What makes this
procedure various is the calculation of cash flows. In addition, it is essential to
simulate the forward LIBOR under a single forward measure to ensure the consistent
comparison between the exercise values and the holding values.

4.2 Callable Cumulative Inverse Floater

4.2.1 Term Sheet
Part of the term sheet of Callable Cumulative Inverse Floater is shown in Table 4.9.
Table 4.9 Term Sheet of Callable Cumulative Inverse Floater

4YNC6M USD RBS CUMULATIVE CALLABLE INVERSE FLOATER

Description:

4yNC6m year USD-denominated Cumulative Callable Inverse Floater (the

Notes) with Interest linked to the USD 6 Month LIBOR set in Arrears. The
Notes are callable by the Issuer after 6 months and semiannually thereafter.

Issuer:

Royal Bank of Scotland Plc

Specified

USD 10,000 per Note

Denomination:
Issue Price:

100.00% (subject to market conditions)

Issue Date:

27 August 2004

Maturity Date:

27 August 2008

55

Interest Amount:
Period

Coupon Rate

Year 1

5.50%

Year 2

Previous Coupon + 3.00% - USD 6 Month LIBOR set in Arrears

Year 3

Previous Coupon + 5.00% - USD 6 Month LIBOR set in Arrears

Year 4

Previous Coupon + 7.00% - USD 6 Month LIBOR set in Arrears

The Coupon Rate is subject to a minimum of 0.00% and is reset

semiannually.
Daycount:

30/360

Interest Payment

27 February and 27 August each year commencing on 27 February 2005,

Dates:

adjusted as per the Business Day Convention.

Early Redemption

The Issuer may redeem the Notes, in whole but not in part, on each Interest

Option (Call):

Payment Date commencing on 27 February 2005. The note holder will be

entitled to any Interest payments due on the Early Redemption Date.

Early Redemption

If the Notes are called, the Interest Payment Date in respect of which the

Date:

Early Redemption Option is exercised.

Optional

100% of the Aggregate Nominal Amount

Redemption
Amount:
Source: UBS Investment Bank

4.2.2 Valuation
The valuation procedure is very similar to that addressed in Section 4.1.2.
First, we have to construct a 4.5-year semiannual yield curve of LIBOR (Table
4.12). Second, the initial forward LIBOR are extracted from these yield rates (Table
4.13). Using market quotes of caps (Table 4.14) to calculate the volatilities of caplets
(Table 4.15, Figure 4.4), then we are able to recover the instantaneous volatility
structure of forward LIBOR (a = -0.083555, b = -0.035927, c = 0.39781, d =
-0.0032122, Table 4.16).
After all these prerequisites are completed, we start to simulate the forward
LIBOR under forward measure Q 4 . It is necessary to keep tracks of all the simulated
forward LIBOR. The next step is to calculate the cash flows according to the interest

56

payment condition in the term sheet. Discounting the cash flows, running the
regression of the discounted cash flows on the simulated spot LIBOR to estimate of
the exercise values, making the call strategy, changing the cash flow structure, again
discounting the cash flow structure back to time zero, we eventually obtain the
reasonably theoretical price of callable cumulative inverse floater. When M = 10000,
we might have the price of 102.43% with variance 0.24%.
The simulation result shows that this callable cumulative inverse floater is issued
at premium. Observing the simulated forward LIBOR and the cash flow structure, we
find that the interest rate payments are relatively high, due to the low level of
simulated forward LIBOR. Besides, the term of interest payment does not impose any
constraints on the possible highest payment, which might be the key factor causing
the issuance at premium.
Table 4.10 LIBOR on 2004/08/27

Table 4.11 Swap Rate on 2004/08/27

LIBOR on Aug. 27,2004

Swap Rate on Aug. 27, 2004

6m LIBOR

1.99%

2y swap rate

2.83%

1y LIBOR

2.3%

3y swap rate

3.241%

4y swap rate

3.566%

5y swap rate

3.838%

Table 4.12 6-Month Yield Rate on 2004/08/27

Term (year)

Yield (%)

Term (year)

Yield (%)

0.5

1.99

2.5

3.064

2.3

3.2624

1.5

2.5845

3.5

3.4404

2.8395

3.6021

4.5

3.7513

57

Table 4.13 Initial 6-Month Forward LIBOR L(0, i, j)

Time (year)

Fwd. LIBOR (%)

Time (year)

Fwd. LIBOR (%)

[0, 0.5]

1.99

[2, 2.5]

3.9645

[0.5, 1]

2.6105

[2.5, 3]

4.257

[1, 1.5]

3.1546

[3, 3.5]

4.5118

[1.5, 2]

3.6065

[3.5, 4]

4.7381

[4, 4,5]

4.9488

Table 4.14 Market Quotes of Cap on 2004/08/27

Maturity (year)

Cap Vol. (%)

31.84

34.88

33

30.745

28.76

Table 4.15 Implied Caplet Volatility on 2004/08/27

Maturity (year)

Caplet Vol. (%)

31.84

33.43

27.619

23.579

20.733

Implied Caplet Volatility

0.34
0.32

Volatility (%)

0.3
0.28
0.26
0.24
0.22
0.2

1.5

2.5
3
3.5
Maturity (year)

4.5

Fig 4.4 Implied Caplet Volatility on 2004/08/27

58

Table 4.16 Instantaneous Volatility of 6-Month Forward LIBOR

Fwd. LIBOR

Instant. Vol. (%)

Fwd. LIBOR

Instant. Vol. (%)

L(0,0.5,1)

35.20

L(0,2,2.5)

16.58

L(0,1,1.5)

30.79

L(0,2.5,3)

11.50

L(0,1.5,2)

26.21

L(0,3,3.5)

6.25

L(0,2,02.5)

21.48

L(0,3.5,4)

00.82

L(0,4,4.5)

00.13

4.3 Callable Daily Range Accrual Note

4.3.1 Term Sheet
Part of the term sheet of Callable Daily Range Accrual Note is shown in Table 4.17.
Table 4.17 Term Sheet of Callable Daily Range Accrual Note

5 YR NC 3 MTH USD CALLABLE RANGE ACCRUAL NOTE

Description:

5 year USD denominated Callable Daily Range Accrual Notes

(the Notes) with interest payments linked to the USD 3 month
LIBOR. The Notes are callable by the Issuer quarterly on any
Interest Payment Date falling on or after the day that is 3 months
after the Issue Date.

Issuer:

UBS AG

Specified

USD 10,000 per Note

Denomination:
Issue Price:

100.00%

Issue Date:

15 July 2004

Maturity Date:

15 July 2009

59

Interest Amount:

{ Coupon Rate 1 x (n/N) + Coupon Rate 2 x (N-n)/N } x

Nominal Amount x Day Count Fraction
n = the number of days in the Interest Period that 3 month USD
LIBOR is within the following ranges at Fixing.
N = the number of calendar days in the Interest Period
Period

Range

CouponRate1

CouponRate2

Year 1

0 4.00%

(Floating Rate + 2.00%)

1.25%

Year 2

0 5.00%

(Floating Rate + 2.00%)

1.25%

Year 3

0 6.00%

(Floating Rate + 2.00%)

1.25%

Year 4

0 7.00%

(Floating Rate + 2.00%)

1.25%

Year 5

0 7.00%

(Floating Rate + 2.00%)

1.25%

Daycount:

30/360

Interest Payment

15 October, 15 January, 15 April and 15 July each year

Dates:

commencing on 15 October 2004, adjusted as per the Business

Day Convention.

Early Redemption

The Issuer may redeem the Notes at the Optional Redemption

Option (Call):

Amount, in whole but not in part, on any Interest Payment Date

falling on or after 15 October 2004. Note holders will be entitled
to any Interest payments due on the Early Redemption Date.

Early Redemption

If the Notes are called, the Interest Payment Date in respect of

Date:

which the Early Redemption Option is exercised.

Optional

100% of the Aggregate Nominal Amount

Redemption
Amount:
Source: UBS Investment Bank

4.3.2 Valuation
As mentioned in Section 4.1.2, we start our valuation procedures from building the
daily 3-month yield curve (Table 4.20), extracting the initial daily 3-month forward
LIBOR (Table 4.21), to calibrating the model to the market (Table 4.23, Figure 4.5, a
= -0.086139, b = -0.037914, c = 0.22377, d = 0.14876.)
The daily 3-month yield rates are cubic-splined out of the quarterly 3-month
yield rates. The initial daily 3-month forward LIBOR are extracted from these

60

estimated daily 3-month yield rates. We then simulate the forward LIBOR under the
forward measure Q 5 .
The time step t is set to be 1 360 because of the term of daycount. We have
to know about the simulated forward LIBOR of every day to calculate the interest
payment.

When simulating, we use an indicator variable to count the days when the

simulated spot LIBOR are in the range, and this indicator variable could be used to
calculate the interest payment. However, it needs not to record all the daily forward
LIBOR for estimating the exercise values. What we need is the forward LIBOR on
the interest payment date. For example, the forward LIBOR structure of one time
simulation could be as follows.

L( 0,0,90)

L( 0,1,91)

L(1,1,91)

L( 0,90,180)

L(1,90,180)

L( 90,90,180)

L( 0,180,270) L(1,180,270)

L( 90,180,270)

L( 0,1530,1620) L(1,1530,1620)

L( 90,1530,1620)

L( 0,1620,1710) L(1,1620,1710)

L( 90,1620,1710)

L( 0,1710,1800) L(1,1710,1800)

L( 90,1710,1800)

L(1710,1710,1800)

L( 0,1800,1890) L(1,1800,1890)

L( 90,1800,1890)

L(1710,1800,1890)

L(1800,1800,1890)

We only have to record the forward LIBOR on the 90th, 180th, 270th day, and so on,
with the time interval of 90 days until the maturity of the contract.
Repeatedly estimating the exercise value and comparing it with the holding value
when the contract is callable, we could obtain the reasonably theoretical price. When

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M = 10000, we might have the price of 100.72% with variance 0.003%.

The valuation result indicates that the contract should be issued at premium.
Examining the cash flow structure, we find that the contract is redeemed at the first or
second interest payment date. The simulated discount rates (the spot LIBOR and the
simulated spot LIBOR prevailing between 3 months and 6 month later) are much
lower than the interest rate payment, resulting in the issuance at premium.
Table 4.18 LIBOR on 2004/07/15

Table 4.19 Swap Rate on 2004/08/27

LIBOR on July 15, 2004

Swap Rate on July 15, 2004

3m LIBOR

1.62%

2y swap rate

3.021%

6m LIBOR

1.88%

3y swap rate

3.504%

9m LIBOR

2.09%

4y swap rate

3.872%

1y LIBOR

2.31%

5y swap rate

4.153%

6y swap rate

4.609%

Table 4.20 3-Month Yield Rate on 2004/07/15

Term (year)

Yield (%)

Term (year)

Yield (%)

0.25

1.62

2.75

3.4216

0.5

1.88

3.5331

0.75

2.0913

3.25

3.6397

2.31

3.5

3.7405

1.25

2.5216

3.75

3.8341

1.5

2.7103

3.9191

1.75

2.8809

4.25

3.9952

3.0352

4.5

4.0668

2.25

3.1752

4.75

4.1395

2.5

3.3031

4.2193

5.25

4.3121

Table 4.21 Initial 3-Month Forward LIBOR L(0, i, j)

Time (year)

Fwd. LIBOR (%)

Time (year)

Fwd. LIBOR (%)

[0,

0.25]

1.62

[2.5, 2.75]

3.4216

[0.25, 0.5]

1.88

[2.75,

3]

3.5331

[0.5, 0.75]

2.0913

[3,

3.25]

3.6397

[0.75,

2.31

[3.25, 3.5]

3.7405

1]

62

[1,

1.25]

2.5216

[3.5, 3.75]

3.8341

[1.25, 1.5]

2.7103

[3.75,

4]

3.9191

[1.5, 1.75]

2.8809

[4,

4.25]

3.9952

[1.75,

2]

3.0352

[4.25, 4.5]

4.0668

2.25]

3.1752

[4.5, 4.75]

4.1395

[2.25, 2.5]

3.3031

[4.75,

5]

4.2193

5.25]

4.3121

[2,

[5,

Table 4.22 Market Quotes of Cap on 2004/07/15

Maturity (year)

Cap Vol. (%)

25.47

32.66

30.89

28.53

27.07

Table 4.23 Implied Caplet Volatility on 2004/07/15

Maturity (year)

Caplet Vol. (%)

25.47

32.838

25.478

21.035

20.376

Implied Caplet Volatility

0.34
0.32

Volatility (%)

0.3
0.28
0.26
0.24
0.22
0.2

1.5

2.5
3
3.5
Maturity (year)

4.5

Fig. 4.5 Implied Caplet Volatility on 2004/07/15

63