Derivative Pricing
• Black-Scholes Model
• Pricing exotic options in the Black-Scholes world• Beyond the Black-Scholes world• Interest rate derivatives
• Credit risk
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– ……
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
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
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
Bond Pricing
• Zero-coupon bonds– At maturity, Z(T)=1 – Pricing Problem: Z(t)=? for t<T
• If the interest rate is constant, then
Short Rate
• r(t) short rate or spot rate
• Interest rate from a money-market account– short term– not predictable
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)
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
Interestrate
HIGH interest rate has negative trend
LOW interest rate has positive trend
ReversionLevel
Vasicek Model
dr=( - r) dt+cdW
• The first mean reversion model
• Shortage: the spot rate might be negative
• Zero-coupon bond’s value
Cox,Ingersoll & Ross Model
• Mean reversion model with positive spot rate
• Explicit solution is available for zero-coupon bonds
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
Disadvantage of the Spot Rate Models
• They do not give the user complete freedom in choosing the volatility.
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.
Monte-Carlo SimulationAssume 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).
1. 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.
4. 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.
Disadvantages
• The simulation may be very slow.
• It is not easy to deal with American style options