The Applications of Interest Rate Model in Swap and Bond Market Jiakou Wang Presentation in March 2009.

The Applications of Interest Rate Model

in Swap and Bond Market

Jiakou Wang

Presentation in March 2009

Contents

1. Motivation 1. Motivation

2. Pricing Swap and Bond 2. Pricing Swap and Bond

3. EFM Model 3. EFM Model

4. Applications of EFM 4. Applications of EFM

Investment Banking Business

Trading

Equities FID

IR ExoticsCreditFX

Why do we need IR model?Case 1: Japanese Government Bonds Market on 2/20/2009(Source: Bloomberg)

TERM COUPON MATURITY PRICE/YIELD1-Year 0.7 1/15/2010 100.33 / .322-Year 0.4 2/15/2011 99.96 / .423-Year 1.4 12/20/2011 102.56 / .484-Year 0.8 12/20/2012 100.65 / .625-Year 0.8 12/20/2013 100.25 / .746-Year 1.3 12/20/2014 102.7 / .817-Year 2 3/20/2016 107.72 / .848-Year 1.7 12/20/2016 105.45 / .959-Year 1.5 12/20/2017 102.63 / 1.17

10-Year 1.3 12/20/2018 99.96 / 1.3015-Year 1.8 12/20/2023 100.42 / 1.7620-Year 1.9 12/20/2028 99.5 / 1.9430-Year 2.4 9/20/2038 108.29 / 1.96

Why do we need IR model?

40 year JGB bond is not liquid. Assume there is no quoted price in the market at the present time. If your client is calling you to buy this bond, how much price would you like to offer?

Why do we need IR model?

Case 2: The portfolio of U.S. Treasuries on 2/20/2009 (Bloomberg)

TERM COUPON MATURITY YIELD NOTIONAL 3-Month 0 5/ 21/ 2009 0.28 $100,0006-Month 0 8/ 20/ 2009 0.48 $200,000

12-Month 0 2/ 11/ 2010 0.64 $4,000,0002-Year 0.875 1/ 31/ 2011 0.92 $300,0003-Year 1.375 2/ 15/ 2012 1.29 $50,000,0005-Year 1.75 1/ 31/ 2014 1.82 ($45,000)

10-Year 2.75 2/ 15/ 2019 2.79 $560,00030-Year 3.5 2/ 15/ 2039 3.61 ($1,000,000)

Why do we need IR model?

How much is the risk of this portfolio? What risk does the portfolio have? If your client needs an optimized interest rate risk free portfolio with positive carry, how do you adjust by going long or short the treasuries?

Why do we need IR model?

Case 3: U.S. Interest Rate Swap Market on Feb.20,2009(Federal Reserve)

Term Rate

1Year 1.37%

2Year 1.60%

3Year 1.93%

5Year 2.46%

7Year 2.75%

10Year 2.99%

12Year 3.10%

15Year 3.20%

20Year 3.21%

Why do we need IR model?

Some interest rate swaps are not as liquid as 2 year, 10 year, 20 year swaps etc. Their prices might be richer or cheaper comparing with the liquid swaps. How do you find the trading opportunity?

Why do we need IR model?

In order to answer the above three questions, we need to answer the following specific questions:

What are the principles of the asset pricing? How are bond and swap priced? How to calculate the interest rate risk of the bond

and swap? How is interest rate model linked to the price

and risk valuation? What is interest rate curve? How is it linked to

model, pricing and risk?

Pricing Principles

All financial instruments can be visualized as bundles of cash flows.Arbitrage freeSynthetic replicationMarket interpolation

Pricing Principles

Example : Synthetic market deposit rate

Term Type Rate

1 year deposit 1.5%

2 year deposit 2.1%

3 year deposit 2.9%

4 year deposit 3.4%

What is the present value of the $1 coupon paid one year later?

What is the (1year, 3year) forward loan rate? What is the 2.5 year deposit rate?

Pricing Principles

Calculate the discount factors based on the current market interest rates.

Calculate the forward rate for given discount factors.

Discount all the cash flows to the present time. Forward rate is given by:

Summary:

),0(

),0(),0(),(

tttd

ttdtdtttf

-7

3

Pricing Swap and Bond

Interest rate swap has floating leg and fixed leg.

floatfixswap

n

n

iiiifloat

n

iiifix

PVPVPV

ddLPV

rdPV

11

1

Pricing Swap and Bond

The cash flow of a bond with annual coupon c

0

10

20

n

n

iibond

n

n

iibond

yy

cPV

dcdPV

)1(

1

)1(1

1

Pricing Swap and Bond

We need to interpolate the market interest rates to get the discount factors d(0,t) for all t.

We can use either curve fitting or interest rate model to calculate the discount factors.

Any difference and common features for curve fitting and interest rate model?

Interest Rate Curve Fitting

U.S. Interest Rate Swap curve on Feb.20,2009(Federal Reserve)

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

Feb.20

Curve Pricing and Risk Valuation

For given market rates, possible choices for curve fitting are: piecewise linear, cubic spline etc.

Once curve is set up, we use it to price the swap and bond.

The PV01 (delta) is calculated on each bucket by bumping the interest rate

The Delta PnL is calculated as

ii y

PVPV

01

K

iiidelta yPVPnL

1

01

iy

Curve Pricing and Risk Valuation

Market Rates

Curve Fitting

Pricing Risk valuation

Examples of Short Rate Models

Single factor short rate model Vasicek: CIR:

Multi-factor short rate model

tdWdtrkdr )(

tdWrdtrkdr )(

dtdWdW

dWdtykdy

dWdtxkdx

yxr

yx

xyyy

xxxx

),(

)(

)(

Model Pricing and Risk Valuation

Interest rate model is also an interpolation method to the market.

Interest rate model describes the interest rate dynamics.

The model parameters are obtained by fitting the market data.

Model Pricing and Risk Valuation

A simple one factor short rate model

By solving the model, we get the discount factor

The deposit rate is given by

The swap/bond delta risk is given by

tt dWdtdr

220 6

1

2

1)( ttrty

3220 6

1

2

1

),0(tttr

etd

0/01 rPPV

Model Pricing and Risk Valuation

Example: by calibrating the market rates, we get

Term Type Market (%) Model (%) R/C (bps)

1 month deposit 0.82 1.003 -18.3

6 month deposit 1.20 1.162 3.8

1 year deposit 1.30 1.318 -1.8

2 year deposit 1.66 1.618 4.2

3 year deposit 2.01 1.902 10.8

4 year deposit 2.33 2.170 16.0

5 year deposit 2.57 2.422 14.8

7 year deposit 2.87 2.878 -0.8

10 year deposit 3.22 3.442 -22.2

30 year deposit 3.55 3.533 1.7

.219.0;648.0;00.10 r

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1M 3M 6M 1Y 2Y 3Y 4Y 5y 7Y 10Y 30Y

Model Market

Model Pricing and Risk Valuation

Building the interest rate curve by model

Model Pricing and Risk Valuation

Example: The table gives the cash flow of a mortgage bank. Use the short rate model to price the cash flow and valuate the risk. If the bank wants to issue 15 year bond to hedge the interest rate risk, how much face value of bonds it should issue?

Date Cash Flow01-Apr-09 $1,500,00001-Apr-10 $2,400,00001-Apr-11 $3,300,00001-Apr-12 $4,200,00001-Apr-13 $5,100,00001-Apr-14 $6,000,00001-Apr-15 $6,900,00001-Apr-16 $7,800,00001-Apr-17 $8,700,00001-Apr-18 $9,600,00001-Apr-19 $10,500,00001-Apr-20 $11,400,00001-Apr-21 $12,300,00001-Apr-22 $13,200,00001-Apr-23 $14,100,000

Comparing Curve Fitting and IR Model

Curve Fitting

Short Rate Model

Market interpolation Yes Yes

Market duplication Yes No

Long end extension No Yes

Interest rate dynamics No Yes

Pricing Yes Yes

Risk valuation Yes Yes

Price/Risk are the functions of

Market rates

Model factors

Introduction to EFM

The economic factor model is a three factor short rate model which is based on the observation that the market's perceived level for the short rate may not be the same as the actual short rate trading on the market.

The long rate x The slope y = target rate –x The short rate z is mean reverting to the target

rate.

Introduction to EFM

The three driven equations

dtdWdW

dtdWdW

dtdWdW

dWdtzyxkdz

dWydtkdy

dWdtxkdx

yzzy

xzzx

xyyx

zzz

yyy

xxx

),(

),(

),(

)(

)(

Solving the Model

The price of the zero coupon bond with Maturity t, denoted by P(t), is given by

M(t) and V(t) are the mean and variance

The forward rate f(t,t+dt) is given by

))(2

1)(()(

)()( 0tVtmdssz

eeEtPt

)1)(

)((

1),(

dttP

tP

dtdtttf

).)(()();)(()(00 tt

dssrVartVdssrEtm

Historical Market Calibration

With initial guess , calibrate 6M Libor rate, 2 year,10 year,20 year swap rates to get back to 10 year.

Use the time series of to update

Repeat the process until the converge.

),,,( ,, yzxzxyzyx

),,,( 000 zyx

),,,( ,, yzxzxyzyx ),,( 000 zyx

),,,( ,, yzxzxyzyx

dtdzdxdtdzdz

dtdzdydtdydy

dtdydxdtdxdx

xzzxz

yzzyy

xyyxx

),(;),(

),(;),(

),(;),(

2

2

2

Applications of EFM

Example: Price JPY LIBOR 40 year swap.

0.015 0.35 1.00 0.012 0.016 0.003 -0.85 0.129 -0.342

xk zkyk x zy xy yzxz

Term Type Market(Aug.31,07)

6M Deposit 1.0675%

2Y Swap 1.08523%

10Y Swap 1.81023%

20Y Swap 2.29398%

0.0139

-0.006

0.0112

0.1306

40 yr 2.56%

0x

0y

0z

Applications of EFM

Example: JPY swap butterfly trading Rich/Cheapness (rc) = market rate – model rate

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

1Y 2Y 3Y 4Y 5Y 7Y 10Y 12Y 15Y 20Y 25Y 30Y

Rich/ Cheapness

Applications of EFM

Trade 7 year swap rich/cheapness by going long/short 2 year and 20 year swap to hedge hedging the model factors . ),( yx

3

1

01i

mkii yPVPnL

3

1

3

1

3

1

01

01)(01

iii

i

mdii

i

mdi

mkii

rcPV

yPVyyPV

Applications of EFM

R/C is an mean reverting process.

-20

-15

-10

-5

0

5

10

15

20

Jan-

96

Jul-9

6

Jan-

97

Jul-9

7

Jan-

98

Jul-9

8

Jan-

99

Jul-9

9

Jan-

00

Jul-0

0

Jan-

01

Jul-0

1

Jan-

02

Jul-0

2

Jan-

03

Jul-0

3

Jan-

04

Jul-0

4

Jan-

05

Jul-0

5

Jan-

06

Jul-0

6

Jan-

07

Jul-0

7