Chaper 4: Continuous-time

interest rate modelsLin Heng-Li

December 5, 2011

The general principles in this development are that

◦ is Markov

◦ Prices depend upon an assessment at time t of

how will vary between t and T

◦ The market is efficient, without transaction costs

and all investors are rational.

4.3 The PDE Approach to Pricing

Suppose that

Where W(t) is a Brownian motion under PThe first two principles ensure that

Thus, under a one-factor model, price changes for all bonds with different maturity dates are perfectly (but none-linearly) correlated.

4.3 The PDE Approach to Pricing

By Itô’s lemma

Where

4.3 The PDE Approach to Pricing

(a.1)

(a.2)

Consider two bonds with different maturity dates T1and T2 (

At time t, suppose that we hold amounts in the -bond and in the -bond

Total wealth

4.3 The PDE Approach to Pricing

(1)

The instantaneous investment gain from t to t+dt is

4.3 The PDE Approach to Pricing

We will vary and in such a way that the portfolio is risk-free.◦ Suppose that, for all t,

then

◦ By (1) and (2)

4.3 The PDE Approach to Pricing

(2)

Hence, the instantaneous investment gain

Since the portfolio is risk-free, by the principle of no arbitrage

and

4.3 The PDE Approach to Pricing

This must be true for all maturities. Thus, for all T>t

is the market price of risk.◦ Cannot depend on the maturity date◦ Can often be negative.

(Since is usually negative, suppose the volatility be positive, we have Thus, must be negative to ensure that expected returns are greater than the risk-free rate.)

4.3 The PDE Approach to Pricing

(b)

From (b), we have And from (a.1) Equate the two expressions,

4.3 The PDE Approach to Pricing

This is a suitable form to apply the Feynman-Kac formula

◦ The boundary condition for this PDE

4.3 The PDE Approach to Pricing

By the Feynman-Kac formula there exists a suitable probability triple with filtration under which

◦ (s) () is a Markov diffusion process with ◦ Under the measure Q, satisfies the SDE

◦ is a standard Brownian motion under Q

4.3 The PDE Approach to Pricing

Suppose that◦ satisfies the Novikov condition

◦ We define

By Girsanov Theorem, there exists an equivalent measure Q under which (for ) is a Brownian motion and with Radon-Nikodym derivative

4.3 The PDE Approach to Pricing

Note that we have

)

4.3 The PDE Approach to Pricing

The Feynman-Kac formula can be applied to interest rate derivative contracts.

Let be the price at time t of a derivative which will have only a payoff to the holder of at time T

4.3 The PDE Approach to Pricing

Suppose that

By Itô’s lemma

From market price of risk

4.3 The PDE Approach to Pricing

From above, we will have

Apply to Feynman-Kac formula

◦ subject to

4.3 The PDE Approach to Pricing

By the Feynman-Kac formula , we have

where

4.3 The PDE Approach to Pricing