the applications of interest rate model
DESCRIPTION
The Applications of Interest Rate Model. Jiakou Wang. in Swap and Bond Market. Presentation in March 2009. Contents. 1. Motivation. 2. Pricing Swap and Bond. 3. EFM Model. 4. Applications of EFM. Investment Banking Business. Why do we need IR model?. - PowerPoint PPT PresentationTRANSCRIPT
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