Hedging the Asset Swap
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Hedging the Asset Swapof the JGB Floating Rate Notes
Jiakou Wang
Presentation at SooChow University March 2009
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Contents
1. Introduction
2. Pricing the ASW
3. Hedging the ASW
4. Conclusion
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Asset Swap An asset swap enables an investor to buy a bond
and then hedge out the interest rate risk by swapping the coupon payments to floating.
Bond Investor
Interest rate risk Credit risk
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Asset Swap An asset swap enables an investor to buy a bond
and then hedge out the interest rate risk by swapping the coupon payments to floating.
Bond Seller
Investor
ASW Seller
Libor + s
c p
c
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JGB Floating Rate Notes The cash flow structure
FRN coupon = Max(Reference rate – K,0) Reference rate = recent issued 10 year bond
yield on the coupon reset date Participants bid on the level of K
2 3 4 5 6
|15| Year
|6| M
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The JGB FRN Asset SwapThe FRN asset swap deal between Lehman
and the client
ClientLehman
JGB FRN floating coupon
3M LIBOR+spread
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The JGB FRN Asset SwapQuestions for Lehman
How to price the FRN asset swap?
What are the risks of the FRN asset swap?
What are the proper hedging instruments?
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Asset Pricing Key Points Recall the pricing formula for any traded asset
and the numeraire TXTN
)(00T
T
NXENX
Under the risk neutral measure with the money market account as the numeraire, the pricing formula is written as
)(),0(0 TXETdX
The interest rate curve and volatility surface are the most important concepts for the interest rate asset pricing in practice.
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Asset Pricing Key Points
0.0%0.2%0.4%0.6%0.8%1.0%1.2%1.4%1.6%1.8%2.0%
JGB Feb.25 2009 JGB Mar.2 2009
An example of interest rate curve (Bloomberg)
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Asset Pricing Key Points An example of Yen swaption ATM Volatility Surface
(in %) on Sept. 1,2008.
0
5
10
1520
25
3035
40
45
1Y2Y4Y5Y10Y15Y20Y
1Y
6Y
20Y
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Asset Pricing Key PointsWhat are the functions of Interest Rate Model ?
Interest Rate Model describes the interest rate curve dynamics as a stochastic process I(t).
Today’s interest rate curve and the volatility surface are fitted to get the model parameters. It is called Market Calibration.
If we know the interest rate curve dynamics, we know the asset payoff dynamics. Furthermore, we can calculate .
Interest rate discount curve gives the discount factor
)( TXE
).,0( Td
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Pricing FRN Asset Swap Denote the FRN coupon payment dates by
Denote the discount factor by
Denote the 10 year JGB yield covering the time interval by
),...,( 2,1 nttt
),0( itd
)( itF)10,( 11 ii tt
n
iiii tKtFMaxEtdPV
1
))0,)(((),0(
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Pricing FRN ASW by SABR Model The SABR model is a two factor volatility model
used widely to price interest rate derivatives.
.)0(;)0(21
2
1
fFdtdWdW
dWddWFdF
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Calibrating the SABR Model
0
0
Fitting the interest rate curve and the volatility surface
f
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Calibrating the SABR Model
f ,
Target 1
Target 2
Target 3
Build the bond yield curve on today’s market to calculate the forward yield
Fitting the ATM volatility trace (backbone) to get
Fitting the swaption volatility surface to get
,
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Building the JGB CMT Curve
The forward yield can be calculated as
),0(),0(),0(),,0(
tttdttdtdtttf
0.5%0.7%0.9%1.1%1.3%1.5%1.7%1.9%2.1%2.3%2.5%
JGB CMT curve on Sept.1,2008
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Fitting the Swaption Market Singular perturbation techniques are used to
obtain the European option price. The swaption implied volatility is given by
)1
21log()(
,/log)(
...]2432
)(41
)(24)1([1
...))(
(.../log
24)1(1)(
),(
2
2/)1(
22
2/)1(1
22
22
2/)1(
zzzzx
KffKz
tfKfK
M
Mzxz
KffKfK
ex
Blk
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Fitting the Swaption Market The implied volatility can be approximated by
1
22221 .../log)32()1(
121/log)1(
211),(
f
fKfKf
fKBlk
Managing Smile Risk, Patrick S. Hagan, Deep Kumar etc.
The ATM implied volatility has an approximated relation with the exponent :
fffATM log)1(log),(log
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Fitting the Swaption Market
5%
10%
15%
20%
25%
30%
.;0 1 fATM
Fitting to the backbone of the volatility smiles
The interest rate is normal
,
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Fitting the Swaption Market
5%
10%
15%
20%
25%
30%
.;1 1 fATM
Fitting to the backbone of the volatility smiles
The interest rate is log normal
,
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Fitting the Swaption Market Recall the implied volatility can be approximated
by
1
22221 .../log)32()1(
121/log)1(
211),(
f
fKfKf
fKBlk
Skew term:
Smile term:
fK /log)1(21
fK /log)32()1(21 2222
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Fitting the Swaption Market
5%
7%
9%
11%
13%
15%
17%
19%
Fitting to the swaption implied volatility curve ,
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Fitting the Swaption Market Alpha on Sept. 1 2008
Alpha 1Y 5Y 10Y 15Y 20Y 30Y1Y 0.26 0.26 0.32 0.34 0.36 0.362Y 0.24 0.27 0.33 0.35 0.37 0.374Y 0.26 0.29 0.35 0.36 0.38 0.385Y 0.26 0.29 0.36 0.38 0.39 0.397Y 0.29 0.32 0.38 0.39 0.40 0.4010Y 0.33 0.36 0.41 0.41 0.41 0.4115Y 0.37 0.39 0.41 0.41 0.41 0.4120Y 0.41 0.41 0.41 0.41 0.41 0.4130Y 0.41 0.41 0.41 0.41 0.41 0.41
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Fitting the Swaption Market Correlation on Sept. 1 2008
Rho% 1Y 5Y 10Y 15Y 20Y 30Y1Y 53 63 47 40.5 34 342Y 65.5 63 47 40.5 34 344Y 76 62.5 46.5 41 35.5 35.55Y 77 62 46 41 36 367Y 72.2 58.8 24 38 34 3410Y 65 54 36 33.5 31 3115Y 55.5 47.5 33.5 31 28.5 28.520Y 46 41 31 28.5 26 2630Y 46 41 31 28.5 26 26
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Fitting the Swaption Market Vol of vol on Sept. 1 2008Vol of v 1Y 5Y 10Y 15Y 20Y 30Y
1Y 33 33 33 31.5 30 302Y 27.5 27.5 26 25.5 25 254Y 19 19 18.5 18 17.5 17.55Y 16 16 16 15.5 15 157Y 14.4 14.4 14.4 14.1 13.8 13.810Y 12 12 12 12 12 1215Y 10 10 9.5 9.5 9.5 9.520Y 8 8 7 7 7 730Y 8 8 7 7 7 7
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Pricing FRN Asset Swap
Calculate the caplet
Calculate implied volatility
Fitting volatility curve
Build JGB curve
n
iiii tKtFEtdPV
1
))0,)((max(),0(
)0,)(( KtFMaxE i
iiii ,,,
),,,;,(),( iiiiiiii fKfK
if
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The Risks of FRN Asset Swap
1
Interest Rate risk1.Delta2.Gamma
3
Other Risks1.Theta2.Other risks depending on the model
2
Volatility Risk1.Vega2.Nova3.Vol of vol
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The Risks of FRN Asset SwapDelta: The first order derivative of the
price with respect to the interest rate;Gamma: The second order derivative of
the price with respect to the interest rate;Theta: The first order derivative of the
price with respect to the time;Vega: The first order derivative of the
price with respect to ATM volatilitySensitivity of the volatility of the volatilitySensitivity of the correlation
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An example: Synthetic JGB FRN Assume an synthetic JGB FRN starting to accrue
interests on Sept. 1, 2008 with coupon payment every 6 month.
Face value 100 yen. The expiration date is Sept. 1, 2023. The first coupon payment is on March 1, 2009. The coupon will be reset every 6 month. Assume strike K= 0.65. Assume the asset swap is based on this synthetic
JGB Floating Rate Notes.
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IR Risk of FRN Asset Swap The Delta risk(cents/bp) by bumping the interest
rate curve on Sept. 1, 2008
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Solution: Hedge the Delta risk by going long or short general JGB bonds such that the hedged portfolio is Delta neutral.
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The Volatility Risk of FRN ASW The Vega risk(cents/bp) by bumping the volatility surface
Solution: Hedge the Vega risk by going long or short swaption such that the hedged portfolio is Vega neutral.
0
0.5
1
1.5
2
2.5
1Y 5Y 10Y 15Y 20Y 25Y 30Y1Y
5Y
15Y
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Hedging strategy and conclusion Use SABR model to price and calculate the risk of
the JGB FRN asset swap. Hedge the Delta risk by going long or short
general JGB bonds such that the hedged portfolio is Delta neutral. Rebalance the portfolio when time is progressing.
Hedge the Vega risk by going long or short swaption such that the hedged portfolio is Vega neutral. Rebalance the portfolio when time is progressing.
A historical simulation is done for the past 5 years which shows a good hedging result.
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