1. The Applications of Interest Rate Model in Swap and Bond Market Jiakou Wang Presentation in March 2009

2. Contents 1. Motivation 2. Pricing Swap and Bond 3. EFM Model 4. Applications of EFM

3. Investment Banking Business Trading EquitiesFID IRFXCreditExotics

4. Why do we need IR model? Case 1: Japanese Government Bonds Market on 2/20/2009 (Source: Bloomberg)

5. 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?

6. Why do we need IR model? Case 2: The portfolio of U.S. Treasuries on 2/20/2009 (Bloomberg)

7. 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?

8. Why do we need IR model? Case 3: U.S. Interest Rate Swap Market on Feb.20,2009 (Federal Reserve) TermRate 1Year1.37% 2Year1.60% 3Year1.93% 5Year2.46% 7Year2.75% 10Year2.99% 12Year3.10% 15Year3.20% 20Year3.21%

9. 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?

10. 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?

11. Pricing Principles All financial instruments can be visualized as bundles of cash flows.  Arbitrage free  Synthetic replication  Market interpolation

12. Pricing Principles  Example : Synthetic market deposit rate TermTypeRate 1 yeardeposit1.5% 2 yeardeposit2.1% 3 yeardeposit2.9% 4 yeardeposit3.4%  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?

13. 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:

14. Pricing Swap and Bond  Interest rate swap has floating leg and fixed leg.

15. Pricing Swap and Bond  The cash flow of a bond with annual coupon c

16. 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?

17. Interest Rate Curve Fitting U.S. Interest Rate Swap curve on Feb.20,2009 (Federal Reserve)

18. 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

19. Curve Pricing and Risk Valuation Market Rates Curve Fitting Pricing Risk valuation

20. Examples of Short Rate Models  Single factor short rate model Vasicek: CIR:  Multi-factor short rate model

21. 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.

22. 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

23. Model Pricing and Risk Valuation  Example: by calibrating the market rates, we get TermTypeMarket (%)Model (%)R/C (bps) 1 monthdeposit0.821.003-18.3 6 monthdeposit1.201.1623.8 1 yeardeposit1.301.318-1.8 2 yeardeposit1.661.6184.2 3 yeardeposit2.011.90210.8 4 yeardeposit2.332.17016.0 5 yeardeposit2.572.42214.8 7 yeardeposit2.872.878-0.8 10 yeardeposit3.223.442-22.2 30 yeardeposit3.553.5331.7

24. Model Pricing and Risk Valuation  Building the interest rate curve by model

25. 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?

26. Comparing Curve Fitting and IR Model Curve Fitting Short Rate Model Market interpolationYes Market duplicationYesNo Long end extensionNoYes Interest rate dynamicsNoYes PricingYes Risk valuationYes Price/Risk are the functions of Market rates Model factors

27. 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.

28. Introduction to EFM  The three driven equations

29. 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

30. 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.

31. Applications of EFM  Example: Price JPY LIBOR 40 year swap. 0.0150.351.000.0120.0160.003-0.850.129-0.342 TermTypeMarket(Aug. 31,07) 6MDeposit1.0675% 2YSwap1.08523% 10YSwap1.81023% 20YSwap2.29398% 0.0139 -0.006 0.0112 0.1306 40 yr2.56%

32. Applications of EFM  Example: JPY swap butterfly trading  Rich/Cheapness (rc) = market rate – model rate

33. 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.

34. Applications of EFM  R/C is an mean reverting process.