Hedging the Asset Swapof the JGB Floating Rate Notes

Jiakou Wang

Presentation at SooChow University March 2009

Contents

1. Introduction

2. Pricing the ASW

3. Hedging the ASW

4. Conclusion

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

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

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

The JGB FRN Asset SwapThe FRN asset swap deal between Lehman

and the client

ClientLehman

JGB FRN floating coupon

3M LIBOR+spread

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?

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.

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)

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

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

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(

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

Calibrating the SABR Model

0

0

Fitting the interest rate curve and the volatility surface

f

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

,

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

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

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

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

,

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

,

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

Fitting the Swaption Market

5%

7%

9%

11%

13%

15%

17%

19%

Fitting to the swaption implied volatility curve ,

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

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

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

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

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

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

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.

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.

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

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.