BDT Model Applications © SimCorp Financial Training A/S .

BDT Model Applications

© SimCorp Financial Training A/S

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AFIX5-900 ©SimCorp Financial Training A/S 2 af 54

Applications

1. Pricing bonds

2. American calls on bonds

3. Step up callables

4. Danish mortgage backed bonds

5. Caps (Floors)

6. Forward starting swap

7. Swaptions

8. CTD


Pricing Bonds3 year 10% Bullet

Binominal tree Bond Prices

10

11

9

12.25

10.50

9.00

97.97

101.13

98.00

99.55

100.92

99.59

100

100

100

100

11.11

102

55.999897.97

Bond prices are found by discounting one period at a time, backwards, begin-ning at maturity.


Pricing Bond Options2 year American call on 3 year 10% bullet, strike 99

10

11

9

12.25

10.50

9.00

97.97

101.13

98.00

99.55

100.92

99.59

100

100

100

100

0.59

0.00

0.00

0.55

1.92

Binominal tree Bond Prices American Call

1.08

0.25

2.13*1.13

* The option is exercised immediately

Using the BDT model the price of the American call option can be found to be 1.08.

• Value of Callable Bond is: NonCall-CallOption = 99.59 – 1.08 = 98.51


Pricing step-up Callables3Y step-up Callable Bond, strike at 100

Step-up Plan0-1Y 9%1-2Y 10%2-3Y 11%

10

11

9

12.25

10.50

9.00

98.80

101.96

98.89

100.45

101.83

99.44

100

100

100

100

0.00

0.00

0.45

1.83

Binominal tree Bond Prices American Call

0.98

0.20

1.961.05

• Value of Callable Note is 99.43 - 0.98 = 98.45


Pricing Mortgage BondsDanish Mortgage-Backed Securities

Bond Pool of underlying Loans

• Callable Bond Model Prepayment Risk of Call Option

• Debtors are not homogenous: Several Call options

• Other Features:

– Cost of Prepaying

– Premium required

– Prepayment behaviour (first, optimal)

– Prepayment Model

– Tax

– DK Cash flows

• Path Dependency?

Debtor Model

P P = P CALLNONDK

1.0 W , W P P = P ,ii

CALLNONDK


Pricing Danish Mortgage-Backed Securities

Zero Yields

Volatility

Short rate model, e.g. BDT

DebtorModel

Short Rates

Price MBS

Price/Risk/Return

RentabilityCalculations


Caps/FloorsProduct description

Long term options based on a money market rate at future dates (often 3M or 6M LIBOR). Caps ensure a maximum funding rate compared to floors which ensure a minimum deposit rate. A purchased collar is a combination of a long cap and a short floor.

Time (months)

Strike

Libor

Libor

Compensation from purchased cap

3 6 9 12 15 18

21 24 ….

Strike

Libor

Libor

Compensation from purchased floor

Time (months)3 6 9 12 15 18

21 24 ….


Pricing an Interest Rate Cap3Y Cap on 1Y rate, strike 10%

3Y Cap (1Y) = 1Y Call IRG (1Y) + 2Y Call IRG (1Y)

10

11

9

12.25

10.50

9

Binomial tree 1Y Call IRG 2Y Call IRG

0.41

0.00

90.011.11011

1.1

090.0 2/1

0.60

1.11

0.21

0.00

00.21225.125.2

45.01050.150.0

11.1

45.000.2 2/111.1

Value 3Y Cap = 0.41 + 0.60 = 1.01 Tree is in Bond yields, strike is Money Market Rate (here is no

difference) Also beware of Day Counts


Pricing a Forward Starting Swap

1Y forward, annual in 2Y, 10%, receive floating

Term structure from Binomial tree

t r (%) Forward (%)1 10.00 10.002 9.99 9.993 10.17 10.54

Fixed legt CF PV0 0 01 0 02 10 8.263 110 82.24

90.51

Floating leg

t CF PV0 0 01 0 02 9.99 8.263 110.54 82.65

90.91

Diff = FloatPV - FixPV = 0.40


Can we construct a forward starting swap using swaptions?

A Payer swaption : Gives the buyer the right to pay fixed

A Receiver swaption : Gives the buyer the right to receivefixed

Concurrent : Underlying swap has fixed expiry date

Non-concurrent : Underlying swap has fixed maturity,i.e. 5Y or 10Y

Swaption - product description

A swaption is a right not a duty to buy/sell a forward starting IRS.


Synthetic Construction of Fwd Starting Swap

• Buy 1 Payers swaption (Strike = 10%)• Sell 1 Receivers swaption (Strike = 10%)• The swaptions are European Style• The swaptions expire 1 year from today• The swaptions have a 2-year swap as underlying instrument

Swap-rate > 10%

BoughtPayers swaption

Swap-rate < 10%

Not exercisedPay 10% and receive floating

Net Profile = forward starting swap

SoldReceiver Swaption

Pay 10% and receive floating

Pay 10% and receive floating

Not exercisedCounterparty asks you to pay 10% and receive floating


Pricing a European Swaption1Y swaption into 2Y swap, strike 10%

99.9711.1

105.1110

1225.1110

21

10

13.10109.1

09.1110

105.1110

21

10

9

10

11

9

12.25

10.50 11

12.25

10.50

9

10.50

9

Binomial tree 2Y swap in 1Y UP-state 2Y swap in 1Y DOWN-state

FixPV = FixPV =

FloatPV = 100Float – Fix = 2.01Strike if PAY fix

FloatPV = 100Float – Fix = -1.13Strike if RECEIVE fix


Pricing a European Swaption

0.9136

2.01

0.00

0.5136

0.00

1.13

Payer Swaption (Pay Fix) Receiver Swaption (Receive Fix)

Swaption Put-Call parity

Payer Swaption - Receiver Swaption = Forward Start Swap0.91 0.51 0.40 (Rounding error)


Pricing CTD Futures

Deliverable Maturity Coupon Conversion Factor

#1 3Y 12% 1.0347

#2 3Y 8% 0.9653

Future with delivery of 2Y bond (at contract maturity) with Notional Coupon of 10% in 1Y

Conversion factors calculated as price at maturity with yield = 10%

1.0347 = 1.1

1.12 +

1.10.12 = CF 21 0.9653 =

1.1

1.08 +

1.10.08 = CF 22


Pricing CTD FuturesCost-of-Carry

Find forward prices of Bonds:

#1:

#2:

Futures prices are:

#1:

#2:

CTD is Bond #2

F = 99.554 Cheapest to deliver

104.57 = ) y + 1 (

C = Pt

t

t

t1

94.63 = ) y + 1 (

C = Pt

t

t

t2

99.57 = 1.0347103.025

99.554 = 0.965396.098

103.025 = C ) y + 1 ( P = f 1111

96.098 = C ) y + 1 ( P = f 2122

Adjust for Conversion Factors


Pricing CTD FuturesThe BDT Model

101.41

104.64

99.78

101.36

102.75

104.57

100

100

100

100

94.57

97.62

96.21

97.74

99.08

94.63

100

100

100

100

11

9

12.25

10.50

9

10

Binomial tree Bond # 1 Bond # 2


Pricing CTD FuturesThe BDT Model

Level 1 - Up State

Futures Prices

Level 1 - Down State

Futures Prices

01.980347.1

41.101 1# 127.101

0347.164.104

1#

97.979653.0

57.94 2# 134.101

9653.062.97

2#

#2 is CTD #1 is CTD

The Futures price is then the expected Forward price =

Delivery option value is 0.4 bp.

55.992

97.97127.101


CTD Futures PricingDeterminants of Delivery Option Value

Futures Contract Maturity ()

Bond Maturity ()

Bond Coupon Differential ()

Volatility of Zero Yields ()

Zero Coupon Yield Curve Shape ()

• Number of Deliverables ()


Processes

1 9 7 2 B l a c k - S c h o l e s

1 9 7 6 B l a c k

1 9 7 7 V a s i c e k

1 9 8 0 H o & L e e

1 9 8 5 C o x , I n g e r s o l l & R o s s

1 9 9 0 B l a c k , D e r m a n & T o y

1 9 9 1 B l a c k & K a r a s i n s k i

1 9 9 3 E x t e n t e d V a s i c e k( H u l l - W h i t e )

dZFdtFdF σμ

dZdtrdr σβα

dZdtdr t σα

dZdtlnrdlnr t'tt σθ σ

dZdtlnrdlnr tt σφθ

dZdtrdr t σβα

dZrdtrdr σβα

dZSdtSdS σμ


The BDT Model

TSOI: 3->5%, TSOV: 25->11%

0%

5%

10%

15%

20%

25%

30%

35%

40%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Mean Reversion


The BDT ModelMean fleeing

TSOI: 3->5%, TSOV: 10% flat

0,0%

0,5%

1,0%

1,5%

2,0%

2,5%

3,0%

3,5%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Low

er

limit

0%

50%

100%

150%

200%

250%

300%

350%

Upper

limit


The BDT Model

TSOI: 3->5%

0%

5%

10%

15%

20%

25%

30%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

TSOV Local volatility

Evolution of local volatility


The BDT ModelEvolution of forward volatility

TSOI: 3->5%, TSOV: 10% flat

0%

2%

4%

6%

8%

10%

12%

14%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


The BDT Model

TSOI: 3->5%; TSOV: 25->11%;0-30Y

0%

5%

10%

15%

20%

25%

0% 5% 10% 15% 20% 25% 30% 35%

Log normality of short rates after 15 years


The BDT Model

Advantages

• Consistency• Term structure of volatility• American options• Mean reverting rates• Log-normal (i.e. postive) rates• Benchmark term structure

model?

Problems

• One-factor model• Link between mean reversion and

forward volatility• No analytical prices (numerical

solution)• Constant steps• Hull/White have provided an

improved model, but this is more• Complicated• Products that depend on more

factors, e.g.:– Quanto’s (diff. swaps)– Spread options– Convertibles

Summary


The first method

The data The treeTIME TSOI TSOV

1 10,00% 20,00% 10,00%2 11,00% 19,00% 9,79% 14,32%3 12,00% 18,00% 9,76% 13,77% 19,42%4 12,50% 17,00% 8,72% 11,83% 16,06% 21,79%5 13,00% 16,00% 8,65% 11,34% 14,86% 19,48% 25,52%

Fair value of a non callable Or using the treet CF discount 97,471 12,00 0,9091 101,47 88,862 12,00 0,8116 103,87 94,95 84,453 12,00 0,7118 104,71 99,30 92,74 84,974 12,00 0,6243 103,08 100,59 97,51 93,74 89,235 112,00 0,5428 100,00 100,00 100,00 100,00 100,00

97,48

Fair value of the option 0,85 1,81 0,05 3,87 0,12 0,00 4,71 0,26 0,00 0,00 3,08 0,59 0,00 0,00 0,00

The calculations

BDT Model Applications © SimCorp Financial Training A/S .

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