Hedging the Asset Swap of 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 SellerInvestorASW Seller Libor + s cp 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

The JGB FRN Asset Swap  The FRN asset swap deal between Lehman and the client ClientLehman JGB FRN floating coupon 3M LIBOR+spread

The JGB FRN Asset Swap  Questions 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  Under the risk neutral measure with the money market account as the numeraire, the pricing formula is written as  The interest rate curve and volatility surface are the most important concepts for the interest rate asset pricing in practice.

Asset Pricing Key Points  An example of interest rate curve (Bloomberg)

Asset Pricing Key Points  An example of Yen swaption ATM Volatility Surface (in %) on Sept. 1,2008.

Asset Pricing Key Points What 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

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

Pricing FRN ASW by SABR Model  The SABR model is a two factor volatility model used widely to price interest rate derivatives.

Calibrating the SABR Model Fitting the interest rate curve and the volatility surface

Calibrating the SABR Model 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

Fitting the Swaption Market  Singular perturbation techniques are used to obtain the European option price. The swaption implied volatility is given by

Fitting the Swaption Market  The implied volatility can be approximated by Managing Smile Risk, Patrick S. Hagan, Deep Kumar etc.  The ATM implied volatility has an approximated relation with the exponent :

Fitting the Swaption Market  Fitting to the backbone of the volatility smiles  The interest rate is normal

Fitting the Swaption Market  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  Skew term:  Smile term:

Fitting the Swaption Market  Fitting to the swaption implied volatility curve

Fitting the Swaption Market  Alpha on Sept Alpha1Y5Y 10Y15Y20Y30Y 1Y Y Y Y Y Y Y Y Y0.41

Fitting the Swaption Market  Correlation on Sept Rho%1Y5Y 10Y15Y20Y30Y 1Y Y Y Y Y Y Y Y Y

Fitting the Swaption Market  Vol of vol on Sept Vol of v1Y5Y 10Y15Y20Y30Y 1Y Y Y Y Y Y12 15Y Y Y887777

Pricing FRN Asset Swap Calculate the caplet Calculate implied volatility Fitting volatility curve Build JGB curve

The Risks of FRN Asset Swap 1 Interest Rate risk 1.Delta 2.Gamma 3 Other Risks 1.Theta 2.Other risks depending on the model 2 Volatility Risk 1.Vega 2.Nova 3.Vol of vol

The Risks of FRN Asset Swap  Delta: 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 volatility  Sensitivity of the volatility of the volatility  Sensitivity 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,  The first coupon payment is on March 1,  The coupon will be reset every 6 month.  Assume strike K=  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  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.

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.