ТОП просматриваемых книг сайта:
Numerical Methods in Computational Finance. Daniel J. Duffy
Читать онлайн.Название Numerical Methods in Computational Finance
Год выпуска 0
isbn 9781119719724
Автор произведения Daniel J. Duffy
Жанр Ценные бумаги, инвестиции
Издательство John Wiley & Sons Limited
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:
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
(3.7)
We can see that the series is beginning to look like
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:
In the cases
(3.9)
3.3.2 Riccati ODE
The Riccati ODE is a non-linear ODE of the form:
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
Duffie and Kan proved that P(t, T) is exponential-affine if and only if the drift