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. Interest Rate Derivatives vs Stock Options
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 ……

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 HJM 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 If the interest rate is constant, then
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. 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
Interest
rate
HIGH interest rate has negative trend
LOW interest rate has positive trend
Reversion
Level

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 Equilibrium Models: No-arbitrage 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.