FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE INTEREST

1. FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE INTEREST RATE DERIVATIVES LECTUREDynamic Hedging and Portfolio Insurance

2. Topics Caplet, Cap, Floorlet, Floor,Collar Swaption Forward Swap Mortgage Backed Securities

3. Caplet, Caps Floorlets, Floors and Collars LECTUREDynamic Hedging and Portfolio Insurance

4. Caplet and Floorlet Interest rate option gives holder the right but not the obligation to receive one interest rate (eg. floating\LIBOR) and pay another (eg. the fixed strike rate K%). Caplet (payoff at maturity) (Excel T15.1): [15.1] Q { max (0,LIBORT - Kc ) days/360 } Floorlet (payoff at maturity)(Excel T15.2) : [15.2] Q max (0,KFL - LIBORT ) days/360 } LECTUREDynamic Hedging and Portfolio Insurance

5. Fig 15.1: Payoff Caplet on 90 - day LIBOR Caplet fixes effective max cost of loan at Kc Strike rate Kc fixed in the contract Expiry \ Valuation of option, (LIBORT - Kc) Cash Payout 0 T=30 t=120 days 90 days

6. INPUT IS FROM EXCEL FILE T15.1 Fig15.2:Planned Borrowing+ Caplet (Call) Loan only Loan plus long call

7. INPUT IS FROM EXCEL FILE T15.2 Figure 15.3 : Loan+ Interest Rate Floorlet (Put) Loan plus interest rate put Return on loan only Note : Payoff profile is like a protective put or long call.

8. Payoffs to (Loan+Cap) and (Deposit+Floors) See Excel files T15.3 and 15.4

9. Collar (Excel T15.5) Comprises a long cap and short floor. It establishes both a floor and a ceiling on a corporate or bank’s (floating rate) borrowing costs. Effective Borrowing Cost with Collar (at T = t – 90) = [LIBORt-90+ max[{0,LIBORt-90 - Kcap} -max {0, KFL–LIBOR(t-90)}]Q (90/360) = Kcap Q(90/360) if LIBORt-90 > Kcap = KFL Q(90/360) if LIBORt-90 < KFL = LIBORt-90 (90/360) if KFL < LIBORt-90 < Kcap Collar involves borrowing cost at eachpayment date of either Kcap = 10% or KFL = 8% or LIBOR if the latter is between KFL = 8% and Kc = 10%. LECTUREDynamic Hedging and Portfolio Insurance

10. Swaption Forward Swap and MBS

11. Swaption OTC option to enter into a swap either as a fixed rate payer and floating rate receiver (ie. payer swaption) or vice versa US corporate may need to borrow $10m over 3 years at a floating rate, beginning in 2 years time. Wishes to swap the floating rate payments for fixed rate Corporate therefore needs a $10m swap, to pay fixed and receive floating beginning in 2 years time and an agreement that swap will last for further 3 years

12. Swaption Suppose the corporate thinks that interest rates will rise over the next 2 years and hence the cost of the fixed rate payments in the swap will be higher than at present. The corporate can hedge by purchasing a 2 year European payer swaption, on a 3 year “pay fixed-receive floating” swap, at say K = 10%. Payoff is the annuity value of Q max{cpT – K, 0} Value of Swaption at T [15.15] Vswpo(T=2) = $10m [cpT – K] [(1+r23)-1 + (1+r24)-2 + (1+r25)-3]

13. Figure 15.4 : Forward Swap A long forward swap is “pay fixed-receive floating” swap that will start in the future but at a swap rate agreed today. It ‘locks in’ a swap rate, agreed today or, can be used to speculate on future swap rates(see below) f25 f24 f23 0 1 2 3 4 5 Swap’s Life Enter into forward swap Expiration of forward swap

14. Pricing a Forward Swap Example Long a 2-year forward contract on a 3-year swap, on a notional principal of Q=$10m. How do we price this swap at time t=0 (see figure 15.4) ?

15. Pricing a Forward Swap • The forward swap rate at t=0 is the fixed coupon rate cpf • that makes the swap have zero expected value at T=2. • [15.16] Q = C e-f23(1) + C e-f24(2) + C e-f25(3) + Q e-f25(3) • fij =forward rates ( known at t=0) • Fixed coupon rate cpf = C/Q, hence • 15.17] At t=0, cpf(2-5) = [ 1 - ]/ AN2-5 • AN2-5 = • ~ annuity value of $1 using the forward rates at t=0. • ~ cpf is the forward swap rate agreed at t=0.

16. Value of Forward Swap at T Value at expiration (T=2) to the fixed rate payer After 2 years, current swap rate is cp2(2-5) Value is 3-yr annuity value of (cp2-cpf) per $1 principal. [15.18] Vfs(at T=2) = (cp2 – cpf) [e-r23(1) + e-r24(2) + e-r25(3)] Cash value at expiry = $(Q.Vfs) , paid to “the long”. Note: the r2j are the actual spot rates (ex-post) known at t=2 which is now ‘the present’ , that is, two years after inception of the forward swap. Speculation: If at t=0 you believe cp2 will exceed cpf then go long a forward swap

17. MBS: Mortgage Pass-Throughs and Strips Mortgages bundled up into portfolio and sold to investors in the form of mortgaged backed securities (MBS). Interest only (IO) strip entitles the investor to receive only the interest payment from the portfolio of mortgages. Principal only(PO) strip, only receives payments of principal [15.22] PVPT = PVIO + PVPO [Table 15.6 here - Excel] LECTUREDynamic Hedging and Portfolio Insurance

18. End of Slides LECTUREDynamic Hedging and Portfolio Insurance