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

Products whose payoffs depend in some way on interest rates.

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

Outline

• Short rate model– Model calibration: yield curve fitting

• HJM model

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

Continued

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

Short Rate

• r(t) short rate or spot rate

• Interest rate from a money-market account– short term– not predictable

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

Short Rate Model (Continued)

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

Named Models

• Vasicek

• Cox, Ingersoll & Ross

• Ho & Lee

• Hull & White

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

Ho Lee Model

• The first no-arbitrage model

Extending Vasicek Model:Hull White Model

dr(t)=( (t) - r) dt+cdW

• A no-arbitrage model

Yield Curve Fitting

• Ho-Lee Model

• Hull-White Model

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

General Form

Empirical Study about Volatility of Short Rate

Other Models

• Black, Derman & Toy (BDT)

• Black & Karasinski

Coupon-Bearing Bonds

Callable Bonds

• An example: zero-coupon callable bond

Bond Options

HJM Model

Disadvantage of the Spot Rate Models

• They do not give the user complete freedom in choosing the volatility.

HJM Model

• Heath, Jarrow & Morton (1992)

• To model the forward rate

The Forward Rate

The Instantaneous Forward Rate

Discretely Compounded Rates

Assumptions of HJM Model

The Evolution of the Forward Rate

A Risk-Neutral World

HJM Model

The Non-Markov Nature of HJM

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

Links with the Spot Rate Models

• Ho-Lee Model

• Vasicek Model

Multi-factor Models

• HJM model

• Spot rate model

BGM Model

• It is hard to calibrate the HJM model

• BGM is a LIBOR Model.

• Martingale theory and advanced SDE knowledge are involved.