1. Zvi WienerContTimeFin - 10 slide 1 Financial Engineering Interest Rate Models Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049 following Hull and White Hull and White

2. Zvi WienerContTimeFin - 10 slide 2 Definitions Hull and White P(t,T) zero-coupon bond price maturing at time T as seen at time t. r - instantaneous short rate v(t,T) volatility of the bond price R(t,T) rate for maturity T as seen at time t F(t,T) instantaneous forward rate as seen at time t for maturity T.

3. Zvi WienerContTimeFin - 10 slide 3 Modeling Bond Prices Hull and White Suppose there is only one factor. The process followed by a zero-coupon bond price in a risk-neutral world takes the form: dP(t,T) = rP(t,T)dt + v(t,T)P(t,T)dZ the bond price volatility must satisfy v(t,t)=0, for all t.

4. Zvi WienerContTimeFin - 10 slide 4 Modeling Bond Prices Hull and White

5. Zvi WienerContTimeFin - 10 slide 5 Forward Rates and Bond Prices $1 today grows by time T 1 to This amount invested at forward rate F(t,T 1,T 2 ) for an additional time T 2 -T 1 will grow to By no arbitrage this must be equal

6. Zvi WienerContTimeFin - 10 slide 6 Forward Rates and Bond Prices

7. Zvi WienerContTimeFin - 10 slide 7 Forward Rates and Bond Prices

8. Zvi WienerContTimeFin - 10 slide 8 Forward Rates and Bond Prices Hull and White

9. Zvi WienerContTimeFin - 10 slide 9 Modeling Forward Rates Hull and White What is the dynamic of F(t,T)?

10. Zvi WienerContTimeFin - 10 slide 10 Modeling Forward Rates Hull and White Applying Ito’s lemma we obtain for F(t,T):

11. Zvi WienerContTimeFin - 10 slide 11 HJM Result Hull and White Suppose We know that for some v Hence

12. Zvi WienerContTimeFin - 10 slide 12 Two Factor HJM Result Hull and White

13. Zvi WienerContTimeFin - 10 slide 13 Volatility Structure The HJM result shows that once we have identified the forward rate volatilites we have defined the drifts of the forward rates as well. We have therefore fully defined the model! Hull and White

14. Zvi WienerContTimeFin - 10 slide 14 Short Rate Non-Markov type of dynamic - path dependence. Hull and White In a one factor model this process is

15. Zvi WienerContTimeFin - 10 slide 15 Ho and Lee Model Rates are normally distributed. All rates have the same variability. The model has an analytic solution. Hull and White

16. Zvi WienerContTimeFin - 10 slide 16 Ho and Lee Model Where F(t,T) is the instantaneous forward rate as seen at time t for maturity T. Hull and White

17. Zvi WienerContTimeFin - 10 slide 17 Bond Prices under Ho and Lee Where Hull and White

18. Zvi WienerContTimeFin - 10 slide 18 Option Prices under Ho and Lee A discount bond matures at s, a call option matures at T Hull and White

19. Zvi WienerContTimeFin - 10 slide 19 Lognormal Ho and Lee Hull and White Is like Black-Derman-Toy without mean reversion. Short rate is lognormally distributed. No analytic tractability.

20. Zvi WienerContTimeFin - 10 slide 20 Black-Derman-Toy Hull and White Black-Karasinski

21. Zvi WienerContTimeFin - 10 slide 21 Hull and White Similar to Ho and Lee but with mean reversion or an extension of Vasicek. All rates are normal, but long rates are less variable than short rates. Is analytic tractability.

22. Zvi WienerContTimeFin - 10 slide 22 Hull and White

23. Zvi WienerContTimeFin - 10 slide 23 Bond Prices in Hull and White Where Hull and White

24. Zvi WienerContTimeFin - 10 slide 24 Option Prices in Hull and White A discount bond matures at s, a call option matures at T Hull and White

25. Zvi WienerContTimeFin - 10 slide 25 Generalized Hull and White f(r) follows the same process as r in the HW model. When f(r) is log(r) the model is similar to Black-Karasinski model. Analytic solution is only when f(r)=r. Hull and White

26. Zvi WienerContTimeFin - 10 slide 26 Options on coupon bearing bond In a one-factor model an option on a bond can be expressed as a sum of options on the discount bonds that comprise the coupon bearing bond. Let T be the bond’s maturity, s - option’s maturity. Suppose C=  P i - bond’s price. Hull and White

27. Zvi WienerContTimeFin - 10 slide 27 Options on coupon bearing bond The first step is to find the critical r at time T for which C=X, where X is the strike price. Suppose this is r*. The correct strike price for each P i is the value it has at time T when r=r*. Hull and White P i (r) is monotonic in r!

28. Zvi WienerContTimeFin - 10 slide 28 Example Suppose that in HW model a=0.1,  =0.015. We wish to value a 3-month European option on a 15-month bond where there is a 12% semiannual coupon. Strike price is =100, bond principal =100. Assume that the yield curve is linear y(t) = 0.09 + 0.02 t Hull and White

29. Zvi WienerContTimeFin - 10 slide 29 Example In this case Hull and White Also

30. Zvi WienerContTimeFin - 10 slide 30 Example Thus Hull and White Substituting into the equation for logA(t,T)

31. Zvi WienerContTimeFin - 10 slide 31 Example The bond price equals the strike price of 100 after 0.25 year when Hull and White This can be solved, the solution is r = 0.0943. The option is a sum of two options on discount bonds. The first one is on a bond paying 6 at time 0.75 and strike 6x0.9926e -0.4877*0.0943 =5.688.

32. Zvi WienerContTimeFin - 10 slide 32 Example Hull and White This can be solved, the solution is r = 0.0943. The option is a sum of two options on discount bonds. The first one is on a bond paying 6 at time 0.75 and strike 6x0.9926e -0.4877*0.0943 =5.688. The second is on a bond paying 106 at time 1.25 and strike 106x0.9733e -0.9516*0.0943 =94.315.

33. Zvi WienerContTimeFin - 10 slide 33 Example The first option is worth 0.01 The second option is worth 0.41 The value of the put option on the bond is 0.01+0.41=0.42 Hull and White

34. Zvi WienerContTimeFin - 10 slide 34 Interest Rates in Two Currencies Model each currency separately (by building a corresponding binomial tree). Combine them into a three-dimensional tree. Include correlations by changing probabilities. Hull and White

35. Zvi WienerContTimeFin - 10 slide 35 Two Factor HW Model Where x = f(r) and the correlation between dz 1 and dz 2 is . Hull and White

36. Zvi WienerContTimeFin - 10 slide 36 Discount Bond Prices When f(r) = r, discount bond prices are Hull and White Where A(t,T), B(t,T), C(t,T) are given in HW paper in Journal of Derivatives, Winter 1994.

37. Zvi WienerContTimeFin - 10 slide 37 General HJM Model Hull and White In addition of being functions of t, T and m, the s i can depend on past and present term structures. But we must have:

38. Zvi WienerContTimeFin - 10 slide 38 General HJM Model Hull and White Once volatilities for all instantaneous forward rates have been specified, their drifts can be calculated and the term structure has been defined.

39. Zvi WienerContTimeFin - 10 slide 39 One-factor HJM Hull and White The model is not Markov in r. The behavior of r between times t and t+  t depends on the whole history of the term structure prior to time t.

40. Zvi WienerContTimeFin - 10 slide 40 Specific Cases of One-factor HJM Hull and White s(t,T) is constant: Ho and Lee s(t,T) =  e -a(T-t) : Hull and White

41. Zvi WienerContTimeFin - 10 slide 41 Cheyette Model Hull and White s(t,T) =  (r)e -a(T-t) : Cheyette where There are two state variables r and Q.

42. Zvi WienerContTimeFin - 10 slide 42 Cheyette Model Hull and White Discount bond prices in the Cheyette model are

43. Zvi WienerContTimeFin - 10 slide 43 Simualtions Hull and White P(i,j) price at time i  t of discount bond maturing at time j  t. F(i,j) price at time i  t of a forward contract lasting between j  t and (j+1)  t. v(i,j) volatility of P(i,j) s(i,j) standard deviation of F(i,j) m(i,j) drift of F(i,j)  random sample from N(0,1).

44. Zvi WienerContTimeFin - 10 slide 44 Modeling Bond Prices with One- Factor Hull and White

45. Zvi WienerContTimeFin - 10 slide 45 Modeling Bond Prices with Two-Factors Hull and White

46. Zvi WienerContTimeFin - 10 slide 46 Modeling Forward Rates with One-Factor Hull and White The Heath-Jarrow-Morton result shows that

47. Zvi WienerContTimeFin - 10 slide 47 Euler Scheme Hull and White The order of convergence is 0.5

48. Zvi WienerContTimeFin - 10 slide 48 Milstein Scheme Hull and White The order of convergence is 1

49. Zvi WienerContTimeFin - 10 slide 49 Trees Hull and White up down

50. Zvi WienerContTimeFin - 10 slide 50 Trees Hull and White   is big  is small

51. Zvi WienerContTimeFin - 10 slide 51 Trees Hull and White

52. Zvi WienerContTimeFin - 10 slide 52 Other Topics F The first two factors – duration – twist F Hedging F Monotonicity in one-factor F Multi currency TS models

53. Zvi WienerContTimeFin - 10 slide 53 Other Topics F Credit spread F Model Risk F Path Dependent Securities F Binomial Trees with Barriers