Numerical Methods in Computational Finance. Daniel J. Duffy

can be proved that the solution of (3.1) is also the solution of (3.5) and vice versa. We see then that Picard iteration is based on (3.5) and that we wish to have the iterates converging to a solution of (3.5).

We note that the Picard method is used primarily to prove the existence of solutions and it is not a numerical method as such.

      3.2.1 An Example

      We take a simple autonomous non-linear scalar ODE to show how to calculate Picard iterates:

(3.6)

      whose solution is given by:

      We now compute the Picard iterates (3.4) for this ODE in order to determine the values of t for which the ODE has a solution. For convenience, let us take . Some simple integration shows that:

      (3.7)

We can see that the series is beginning to look like . We know that this series is convergent for . A nice exercise is to compute the Picard iterates in the most general case (that is, ) and to determine under which circumstances the ODE (3.6) has a solution. In this case we have represented the solution of an ODE as a series, and we then analysed this series for which there are many convergence results, such as the root test and the ratio test.

3.3 OTHER MODEL EXAMPLES

      We take some model ODEs for motivation.

      3.3.1 Bernoulli ODE

      The Bernoulli ODE is named after Jacob Bernoulli. It is special in the sense that it is a non-linear equation having an exact solution:

(3.8)

In the cases and and when P and Q are constants, this equation is linear, and an exact solution can then be found. We reduce Equation (3.8) to a linear one by the substitution to produce the following linear ODE:

      (3.9)

      3.3.2 Riccati ODE

      The Riccati ODE is a non-linear ODE of the form:

(3.10)

This ODE has many applications, for example to interest-rate models (Duffie and Kan (1996)). In some cases a closed-form solution to Equation (3.10) is possible, but in this book our focus is on approximating it using the finite difference method.

      We now discuss the relationship between the Riccati equation and the pricing of a zero-coupon bond P(t, T), which is a contract that offers one dollar at maturity T. By definition, an affine term structure model assumes that P(t, T) has the form:

      Let us assume that the short-term interest rate is described by the following stochastic differential equation (SDE):

      where is a standard Brownian motion under the risk-neutral equivalent measure and and are given functions.

      Duffie and Kan proved that P(t, T) is exponential-affine if and only if the drift and volatility Скачать книгу