1. Jan-1999 T.Bjork, Arbitrage Theory in Continuous TimeForeign Currency, Bank of Israel Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Financial Models 15

2. Zvi WienerFinModels - 15 slide 2 Bonds and Interest Rates Zero coupon bond = pure discount bond T-bond, denote its price by p(t,T). principal = face value, coupon bond - equidistant payments as a % of the face value, fixed and floating coupons.

3. Zvi WienerFinModels - 15 slide 3 Assumptions F There exists a frictionless market for T- bonds for every T > 0 F p(t, t) =1 for every t F for every t the price p(t, T) is differentiable with respect to T.

4. Zvi WienerFinModels - 15 slide 4 Interest Rates Let t < S < T, what is IR for [S, T]? F at time t sell one S-bond, get p(t, S) F buy p(t, S)/p(t,T) units of T-bond F cashflow at t is 0 F cashflow at S is -$1 F cashflow at T is p(t, S)/p(t,T) the forward rate can be calculated...

5. Zvi WienerFinModels - 15 slide 5 The simple forward rate LIBOR - L is the solution of: The continuously compounded forward rate R is the solution of:

6. Zvi WienerFinModels - 15 slide 6 Definition 15.2 The simple forward rate for [S,T] contracted at t (LIBOR forward rate) is The simple spot rate for [S,T] LIBOR spot rate is

7. Zvi WienerFinModels - 15 slide 7 Definition 15.2 The continuously compounded forward rate for [S,T] contracted at t is The continuously compounded spot rate for [S,T] is

8. Zvi WienerFinModels - 15 slide 8 Definition 15.2 The instantaneous forward rate with maturity T contracted at t is The instantaneous short rate at time t is

9. Zvi WienerFinModels - 15 slide 9 Definition 15.3 The money market account process is Note that here t means some time moment in the future. This means

10. Zvi WienerFinModels - 15 slide 10 Lemma 15.4 For t  s  T we have And in particular

11. Zvi WienerFinModels - 15 slide 11 Models of Bond Market F Specify the dynamic of short rate F Specify the dynamic of bond prices F Specify the dynamic of forward rates

12. Zvi WienerFinModels - 15 slide 12 Important Relations Short rate dynamics dr(t)= a(t)dt + b(t)dW(t)(15.1) Bond Price dynamics(15.2) dp(t,T)=p(t,T)m(t,T)dt+p(t,T)v(t,T)dW(t) Forward rate dynamics df(t,T)=  (t,T)dt +  (t,T)dW(t) (15.3) W is vector valued

13. Zvi WienerFinModels - 15 slide 13 Proposition 15.5 We do NOT assume that there is no arbitrage! If p(t,T) satisfies (15.2), then for the forward rate dynamics

14. Zvi WienerFinModels - 15 slide 14 Proposition 15.5 We do NOT assume that there is no arbitrage! If f(t,T) satisfies (15.3), then the short rate dynamics

15. Zvi WienerFinModels - 15 slide 15 Proposition 15.5 If f(t,T) satisfies (15.3), then the bond price dynamics

16. Zvi WienerFinModels - 15 slide 16 Proof of Proposition 15.5

17. Zvi WienerFinModels - 15 slide 17 Fixed Coupon Bonds

18. Zvi WienerFinModels - 15 slide 18 Floating Rate Bonds L(T i-1,T i ) is known at T i-1 but the coupon is delivered at time T i. Assume that K =1 and payment dates are equally spaced.

19. Zvi WienerFinModels - 15 slide 19 This coupon will be paid at T i. The value of -1 at time t is -p(t, T i ). The value of the first term is p(t, T i-1 ).

20. Zvi WienerFinModels - 15 slide 20 Forward Swap Settled in Arrears K - principal, R - swap rate, rates are set at dates T 0, T 1, … T n-1 and paid at dates T 1, … T n. T 0 T 1 T n-1 T n

21. Zvi WienerFinModels - 15 slide 21 Forward Swap Settled in Arrears If you swap a fixed rate for a floating rate (LIBOR), then at time T i, you will receive where c i is a coupon of a floater. And at T i you will pay the amount Net cashflow

22. Zvi WienerFinModels - 15 slide 22 Forward Swap Settled in Arrears At t < T 0 the value of this payment is The total value of the swap at time t is then

23. Zvi WienerFinModels - 15 slide 23 Proposition 15.7 At time t=0, the swap rate is given by

24. Zvi WienerFinModels - 15 slide 24 Zero Coupon Yield The continuously compounded zero coupon yield y(t,T) is given by For a fixed t the function y(t,T) is called the zero coupon yield curve.

25. Zvi WienerFinModels - 15 slide 25 The Yield to Maturity The yield to maturity of a fixed coupon bond y is given by

26. Zvi WienerFinModels - 15 slide 26 Macaulay Duration Definition of duration, assuming t=0.

27. Zvi WienerFinModels - 15 slide 27 Macaulay Duration What is the duration of a zero coupon bond? A weighted sum of times to maturities of each coupon.

28. Zvi WienerFinModels - 15 slide 28 Meaning of Duration r $

29. Zvi WienerFinModels - 15 slide 29 Proposition 15.12 TS of IR With a term structure of IR (note y i ), the duration can be expressed as:

30. Zvi WienerFinModels - 15 slide 30 Convexity r $

31. Zvi WienerFinModels - 15 slide 31 FRA Forward Rate Agreement A contract entered at t=0, where the parties (a lender and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T]. Assuming continuous compounding we have at time S:-K at time T: Ke R*(T-S) Calculate the FRA rate R* which makes PV=0 hint: it is equal to forward rate

32. Zvi WienerFinModels - 15 slide 32 Exercise 15.7 Consider a consol bond, i.e. a bond which will forever pay one unit of cash at t=1,2,… Suppose that the market yield is y - flat. Calculate the price of consol. Find its duration. Find an analytical formula for duration. Compute the convexity of the consol.