Derivative Pricing

1. Derivative Pricing • Black-Scholes Model • Pricing exotic options in the Black-Scholes world • Beyond the Black-Scholes world • Interest rate derivatives • Credit risk

2. Interest Rate Derivatives Products whose payoffs depend in some way on interest rates.

3. Underlying Interest rates Basic products Zero-coupon bonds Coupon-bearing bonds Other products Callable bonds Bond options Swap, swaptions …… Underlying Stocks Basic products Vanilla call/put options Exotic options Barrier options Asian options Lookback options …… Interest Rate Derivatives vs Stock Options

4. Why Pricing Interest Rate Derivatives is Much More Difficult to Value Than Stock Options? • The behavior of an interest rate is more complicated than that of a stock price • Interest rates are used for discounting as well as for defining the payoff For some cases (HJM models): • The whole term structure of interest rates must be considered; not a single variable • Volatilities of different points on the term structure are different

5. Outline • Short rate model • Model calibration: yield curve fitting • HJM model

6. Zero-Coupon Bond • A contract paying a known fixed amount, the principal, at some given date in the future, the maturity date T. • An example: maturity: T=10 years principle: $100

7. Coupon-Bearing Bond • Besides the principal, it pays smaller quantities, the coupons, at intervals up to and including the maturity date. • An example: Maturity: 3 years Principal: $100 Coupons: 2% per year

8. Bond Pricing • Zero-coupon bonds • At maturity, Z(T)=1 • Pricing Problem: Z(t)=? for t<T • If the interest rate is constant, then

9. Continued • Suppose r=r(t), a known deterministic function. Then

10. Short Rate • r(t) short rate or spot rate • Interest rate from a money-market account • short term • not predictable

11. Short Rate Model • dr=u(r,t)dt+(r,t)dW • Z=Z(r,t;T) • Z(r,T;T)=1 • Z(r,t;T)=? for t<T

12. Short Rate Model (Continued)

13. Remarks • Risk-Neutral Process of Short Rate dr=(u(r,t)-(r,t)(r,t))dt+(r,t)dW • The pricing equation holds for any interest rate derivatives whose values V=V(r,t)

14. Interest rate HIGH interest rate has negative trend Reversion Level LOW interest rate has positive trend Tractable Models • Rules about choosing u(r,t)-(r,t)(r,t) and (r,t) • analytic solutions for zero-coupon bonds. • positive interest rates • mean reversion

15. Named Models • Vasicek • Cox, Ingersoll & Ross • Ho & Lee • Hull & White

16. Vasicek Model dr=(  - r) dt+cdW • The first mean reversion model • Shortage: the spot rate might be negative • Zero-coupon bond’s value

17. Cox,Ingersoll & Ross Model • Mean reversion model with positive spot rate • Explicit solution is available for zero-coupon bonds

18. Ho Lee Model • The first no-arbitrage model

19. Extending Vasicek Model:Hull White Model dr(t)=( (t) - r) dt+cdW • A no-arbitrage model

20. Yield Curve Fitting • Ho-Lee Model • Hull-White Model

21. Tractable Models • Rules about choosing u(r,t)-(r,t)w(r,t) and w(r,t) • analytic solutions for zero-coupon bonds. • positive interest rates • mean reversion • Equilibrium Models: • Vasicek • Cox, Ingersoll & Ross • No-arbitrage models • Ho & Lee • Hull & White

22. General Form

23. Empirical Study about Volatility of Short Rate

24. Other Models • Black, Derman & Toy (BDT) • Black & Karasinski

25. Coupon-Bearing Bonds

26. Callable Bonds • An example: zero-coupon callable bond

27. Bond Options

28. HJM Model

29. Disadvantage of the Spot Rate Models • They do not give the user complete freedom in choosing the volatility.

30. HJM Model • Heath, Jarrow & Morton (1992) • To model the forward rate

31. The Forward Rate

32. The Instantaneous Forward Rate

33. Discretely Compounded Rates

34. Assumptions of HJM Model

35. The Evolution of the Forward Rate

36. A Risk-Neutral World

37. HJM Model

38. The Non-Markov Nature of HJM

39. Continued • The PDE approach cannot be used to implement the HJM model • Contrast with the pricing of an Asian option. • In general, the binomial tree method is not applicable, too.

40. Monte-Carlo Simulation Assume that we have chosen a model for the forward rate volatility v(t,T) for all T. Today is t*, and the forward rate curve is F(t*;T). • Simulate a realized evolution of the risk-neutral forward rate for the necessary length of time. 2. Using this forward rate path calculate the value of all the cash flows that would have occurred. 3. Using the realized path for the spot interest rate r(t) calculate the present value of these cash flows. Note that we discount at the continuously compounded risk-free rate. • Return to Step 1 to perform another realization, and continue until we have a sufficiently large number of realizations to calculate the expected present value as accurately as required.

41. Disadvantages • The simulation may be very slow. • It is not easy to deal with American style options

42. Links with the Spot Rate Models • Ho-Lee Model • Vasicek Model

43. Multi-factor Models • HJM model • Spot rate model

44. BGM Model • It is hard to calibrate the HJM model • BGM is a LIBOR Model. • Martingale theory and advanced SDE knowledge are involved.