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Hedging the Asset Swap

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Hedging the Asset Swap

Jiakou Wang

of the JGB Floating Rate Notes

Presentation at SooChow University March 2009

1. Introduction

2. Pricing the ASW

3. Hedging the ASW

4. Conclusion

Bond Investor

Interest rate risk

Credit risk

- 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

ASW Seller

Investor

- 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.

c

p

Libor + s

c

- 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

Lehman

Client

- The FRN asset swap deal between Lehman and the client

JGB FRN floating coupon

3M LIBOR+spread

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?

- 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.

An example of interest rate curve (Bloomberg)

- An example of Yen swaption ATM Volatility Surface (in %) on Sept. 1,2008.

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

- Denote the FRN coupon payment dates by
- Denote the discount factor by
- Denote the 10 year JGB yield covering the time interval by

- The SABR model is a two factor volatility model used widely to price interest rate derivatives.

Fitting the interest rate curve and the volatility surface

Target 3

Fitting the swaption volatility surface to get

Target 2

Target 1

Fitting the ATM volatility trace (backbone) to get

Build the bond yield curve on today’s market to calculate the forward yield

- The forward yield can be calculated as

- Singular perturbation techniques are used to obtain the European option price. The swaption implied volatility is given by

- The implied volatility can be approximated by

- The ATM implied volatility has an approximated relation with the exponent :

Managing Smile Risk, Patrick S. Hagan, Deep Kumar etc.

- Fitting to the backbone of the volatility smiles

- The interest rate is normal

- Fitting to the backbone of the volatility smiles

- The interest rate is log normal

- Recall the implied volatility can be approximated by

- Skew term:
- Smile term:

- Fitting to the swaption implied volatility curve

- Alpha on Sept. 1 2008

- Correlation on Sept. 1 2008

- Vol of vol on Sept. 1 2008

Calculate the caplet

Calculate implied volatility

Fitting volatility curve

Build JGB curve

1

2

3

Interest Rate risk

Delta

Gamma

- Volatility Risk
- Vega
- Nova
- Vol of vol

Other Risks

1.Theta

2.Other risks depending on the model

- 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

- 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.

- 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 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.

- 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.

Thank You !