inverse floater valuation parameters

8/18/2019 Inverse Floater Valuation Parameters

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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

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Interest Amount:

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.

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


Apr06 – Oct06 25.50% - 4.8 * 3m USD LIBOR IN ARREARS


Apr07 –Oct07 32.50% - 4.8 * 3m USD LIBOR IN ARREARS

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 inrespect 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.

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US0003M Index

0

0.5

1

1.5

2

2.5

2002/10/1 2003/4/1 2003/10/1 2004/4/1 2004/10/1

Source: Bloomberg 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.

3Libor 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).

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Table 4.2 LIBOR on 2004/10/04

LIBOR on Oct. 04, 2004

3m LIBOR 2.03125%

6m LIBOR 2.21%

9m LIBOR 2.365%

1y LIBOR 2.50375%

Table 4.3 Swap Rate on 2004/10/04

Swap Rate on Oct. 04, 2004

2y swap rate 2.808%

3y swap rate 3.368%

4y swap rate 3.659%

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

0, 0,

111 1.25,1.5,...,3,3.25 0.25

(1 ) (1 )

j j

j i ji i j

S S j

y y

τ

τ

τ τ τ

τ τ

−

=

++ = = =

+ +∑ .4 (4.1)

S j 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. τ is the time fraction of year. Take for instance the

case when j=1.25. The bootstrapping equation will be as follows.

1.25 1.25 1.25 1.25 1.25

0.25 0.5 0.75 1 1.25

0,0.25 0,0.5 0,0.75 0,1 0,1.25

0.25 0.25 0.25 0.25 1 0.251

(1 0.25 ) (1 0.25 ) (1 0.25 ) (1 0.25 ) (1 0.25 )

S S S S S

y y y y y

++ + + +

+ + + + + =

.

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.

4 See Appendix for the detail derivation of the equation.

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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

1 2.5038 2 2.8142 3 3.3937

3.25 3.5046

0 0.5 1 1.5 2 2.5 3 3.50.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

Maturity

Y T M

USD LIBOR 3-Month Zero Curve

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.

0, 0,

0,0.25

(1 ) (1 ) (1 (0, , )) i 0.5, , 3, 3.25 0.25

(0,0,0.25) .

i i

i i y y L i i

and L y

τ

τ τ τ τ τ τ −

−+ = + + − = … =

= (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; τ , again, denotes the time fraction of

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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, 1] 2.9203 [1.75, 2] 3.4037 [2.75, 3] 4.9743

[3, 3.25] 4.8376

III. Calibration of Instantaneous Volatility and Instantaneous Correlation Coefficient

To implement the simulation, we have two more parameters to estimate, the

instantaneous correlation coefficient ρij between forward LIBOR ( )10, ,i i L T T − and

forward LIBOR and the instantaneous volatility( 10, , j j L T T − ) ,1iσ of forward

LIBOR ( )10, ,i i L T T − .

For the instantaneous correlation coefficient ρ, we assume that all the

instantaneous correlation coefficients equal one, ρ = 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 ,1iσ , we use the relationship (3.47) to extract

the caplet volatilities.

( ) ( )( )

( ) ( )( )1

1 1

1

1 1

1

0, 0, , , ,

0, 0, , , , .

j

i

j

i i i i i T cap

i

j

i i i i i T caplet

i

P T Black L T T K T v

P T Black L T T K T v

τ

τ −

− − −=

− − −=

=

∑

∑

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.

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

Maturity (year) 1 2 3 4 5

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,

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )2 2

0 1

1 0 1 0,1 0 2 1 2 0,2

1 0 1 0,1 2 1 2 0,2

0, 0, , , , 0 , 0, , , ,

0, 0, , , , 0, 0, , , , .

T cap T cap

T caplet T caplet

P T Black L T T S v P T Black L T T S v

P T Black L T T S v P T Black L T T S v

− −

− −

+

= +

The only unknown variable is , 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.

1T caplet v −

Table 4.7 Implied Caplet Volatility on 2004/10/04


Caplet Vol. (%) 26.28 31.40 26.46 26.21 22.72

1 1.5 2 2.5 3 3.5 4 4.5 50.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.3

0.31

0.32

Maturity

V o l a t i l i t y

Implied Caplet Volatility

Fig 4.3 Implied Caplet Volatility on 2004/10/04

We then assume that the instantaneous volatility structure follows equation

(3.42),

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( ) ( ) ( )11 1( ) ; , , , : ,ib T t

i i it T t a b c d a T t d eσ ψ −− −

− −= − = − + +⎡ ⎤⎣ ⎦ c

and run the Fminsearch (multidimensional unconstrained nonlinear minimization)

of Matlab with the target function

( ) ( )( )1 11

22

1 10

,i

i

i

T b T t

i T caplet iT v a T t d e 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 . Hence the forward LIBOR follow three kinds

of dynamics.

3Q

For the forward LIBOR with maturity less than 3 years, they evolve according to

equation (3.37)

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( ) ( )

( ) ( )1

0 1

0 1 0 1 0 0 ,

1 0 1

0 1 0 0

( , , )( , , ) ( , , )

1 ( , , )

( , , ) ,n

n j j

i i i i i j i j

j i j j

T

i i i i

L T T T dL T T T L T T T T T dt

L T T T

L T T T T dW T

τ σ σ

τ

σ +

++ +

= + +

+

= −+

+

∑ ρ

where ,0 0T = 0.25,0.5, ,2.75iT = … , and 1 3nT + = .

For the forward LIBOR L(0,2.75,3), it evolves as a martingale according to equation

(3.38) ( ) ( )10 1 0 1 0( , , ) ( , , ) nT

n n n n n ndL T T T L T T T T dW T σ ++ += 0 .

For the forward LIBOR L(0,3,3.25), it evolves according to equation (3.39)

( ) ( )

( ) ( )1

0 1

0 1 0 1 0 0 ,

1 0 1

0 1 0 0

( , , )( , , ) ( , , )

1 ( , , )

( , , ) .n

k j j

i i i i i j i j

j i j j

T

i i i i

L T T T dL T T T L T T T T T dt

L T T T

L T T T T dW T

τ σ σ

τ

σ +

++ +

= + +

+

=

++

∑ ρ

( )0i T σ 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 0.25τ = for simplicity. Due to

the assumption of perfect instantaneous correlation ρ = 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 0.25t τ ∆ = = , and the spot LIBOR ( )0,0,0.25 L ceases in this time step, so

that it is set to be zero. Repeatedly proceeding, we complete the forward LIBOR

simulation for one time.

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( )

( )

( )

( )( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )( )

( )

00,0,0.25

0.25,0.25,0.50,0.25,0.5

0.25,0.5,0.750,0.5,0.75

0.25,0.75,10,0.75,10.25,1,1.250,1,1.25

0.25,1.25,1.50,1.25,1.5

0.20,1.5,1.75

0,1.75,2

0,2,2.25

0,2.25,2.5

0,2.5,2.75

0,2.75,3

L

L L

L L

L L L L

L L

L L

L

L

L

L

L

( )

( )

( )

( )

( )

( )

( )

( )( )

( )

( )

( )

( )

( )

( )

0

0

0.5,0.5,0.75

0.5,0.75,10.5,1,1.25

0.5,1.25,1.5

5,1.5,1.75 0.5,1.5,1.75

0.25,1.75,2 0.5,1.75,2

0.25,2,2.25 0.5,2,2.25

0.25,2.25,2.5 0.5,2.25,2.5

0.25,2.5,2.75 0.5,2.5,2.75

0.25,2.75,3 0.

L

L L

L

L

L L

L L

L L

L L

L L ( ) ( )

0 0

0 0

0 0

0 00 0

0 0

0 0

0 0

0 0

0 0

0 0

5,2.75,3 2.75,2.75,3 0 L

( )0,3,3.25 L ( )0.25,3,3.25 L ( ) ( ) ( )0.5,3,3.25 2.75,3,3.25 3,3,3.25 L L L

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 3,3,3.25 ,0i L , 1, ,i 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

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redemption date plus the nominal principal.

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

( ) ( )( )

1

1

,1, ,

1 , ,

nk

j j

k j i j k k

cashflow i T E T i M j n

L T T T

+

= +

= =+

∑ … = (4.3)

where ( )1, k cashflow i T + denotes the cash flow at 1k T + for the simulation path i. The

simulated spot LIBOR at for path i, jT ( )1, ,i j j j L T T T + and its square

( )2

1, ,i j j j L T T T + are chosen as the dependent variables. For example, when

the independent variable is

1 3nT + =

( )( )( )

1 max 32.50% - 4.8 3,3,3.25 ,0

1 2.75,2.75,3

i

i

L

L

+

+ and the

dependent variables are ( )2.75,2.75,3i L and .( )2

2.75,2.75,3i L

Running the regression ( ) ( ) ( )2

1 1 2 1, , , , j j j j j i j j j E T L T T T L T T T α β β + + ε = + + + ,

then we obtain the estimate of the exercise values at , jT

( ) j j E T . The holding values

for each simulation path at , , is one plus interest payment at (nominal

principal plus interest, calculated in percentage). Finally comparing

jT ( ) j j H T jT

( ) j j E T with

, we make the exercise strategy that call the inverse floater if

and make change to the cash flow structure. If the contract is called

at , all the cash flows after are set to be zero; otherwise the cash flow structure

remains unchanged.

( ) j j H T

( ) ( ) j j j j E T H T ≥

jT jT

Recursively for , 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

1, 2, ,0 j n n= − − …

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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 PlcSpecified

Denomination:

USD 10,000 per Note

Issue Price: 100.00% (subject to market conditions)

Issue Date: 27 August 2004

Maturity Date: 27 August 2008

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Interest Amount:

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

semiannually.

Period Coupon Rate

Year 1 5.50%

Year 2 Previous Coupon + 3.00% - USD 6 Month LIBOR set in Arrears



Daycount: 30/360

Interest Payment

Dates:

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

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 . 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

4Q

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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.


LIBOR on Aug. 27,2004

6m LIBOR 1.99%

1y LIBOR 2.3%


Swap Rate on Aug. 27, 2004

2y swap rate 2.83%

3y swap rate 3.241%

4y swap rate 3.566%

5y swap rate 3.838%


Term (year) Yield (%) Term (year) Yield (%)

0.5 1.99 2.5 3.0641 2.3 3 3.2624

1.5 2.5845 3.5 3.4404

2 2.8395 4 3.6021

4.5 3.7513

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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



Cap Vol. (%) 31.84 34.88 33 30.745 28.76



Caplet Vol. (%) 31.84 33.43 27.619 23.579 20.733

1 1.5 2 2.5 3 3.5 4 4.5 50.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

Maturity (year)

V o l a t i l i t y ( % )


Fig 4.4 Implied Caplet Volatility on 2004/08/27

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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

Denomination:

USD 10,000 per Note

Issue Price: 100.00%

Issue Date: 15 July 2004

Maturity Date: 15 July 2009

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estimated daily 3-month yield rates. We then simulate the forward LIBOR under the

forward measure .5Q

The time step is set to bet ∆ 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.

( )( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

0,0,90 0 0

0,1,91 1,1,91 0

0,90,180 1,90,180 90,90,180

0,180,270 1,180,270 90,

0,1530,1620 1,1530,1620

0,1620,1710 1,1620,1710

0,1710,1800 1,1710,1800

0,1800,1890 1,1800,1890

L

L L

L L L

L L L

L L

L L

L L

L L

( )

( )

( )

( )

( )

( )

( ) ( )

0 0

0 0

0 0

180,270 0 0

90,1530,1620 0 0

90,1620,1710 0 0

90,1710,1800 1710,1710,1800 0

90,1800,1890 1710,1800,1890 1800,1800,1890

L

L

L L

L L L

We only have to record the forward LIBOR on the 90 th, 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

61


19/20

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.


LIBOR on July 15, 2004

3m LIBOR 1.62%6m LIBOR 1.88%

9m LIBOR 2.09%

1y LIBOR 2.31%


Swap Rate on July 15, 2004

2y swap rate 3.021%3y swap rate 3.504%

4y swap rate 3.872%

5y swap rate 4.153%

6y swap rate 4.609%


Term (year) Yield (%) Term (year) Yield (%)

0.25 1.62 2.75 3.42160.5 1.88 3 3.5331

0.75 2.0913 3.25 3.6397

1 2.31 3.5 3.7405

1.25 2.5216 3.75 3.8341

1.5 2.7103 4 3.9191

1.75 2.8809 4.25 3.9952

2 3.0352 4.5 4.0668

2.25 3.1752 4.75 4.13952.5 3.3031 5 4.2193

5.25 4.3121


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, 1] 2.31 [3.25, 3.5] 3.7405

62


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[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, 2.25] 3.1752 [4.5, 4.75] 4.1395

[2.25, 2.5] 3.3031 [4.75, 5] 4.2193

[5, 5.25] 4.3121



Cap Vol. (%) 25.47 32.66 30.89 28.53 27.07



Caplet Vol. (%) 25.47 32.838 25.478 21.035 20.376

1 1.5 2 2.5 3 3.5 4 4.5 50.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

Maturity (year)

V o l a t i l i t y ( % )


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

63

inverse floater valuation parameters

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