Pricing Options and Bonds with the Method of Lines
In these notes we discuss some of the numerical complications which have to be
resolved when the partial differential equations modeling option and bond prices
are to be solved numerically. These complications arise from
discontinuities in the solution or in its derivatives, from early
exercise boundaries or other nonlinearities, from vanishing diffusion
coefficients, and from the high dimensionality of some of the more
advanced applications. The complications are inherent in the problem
formulation and must be overcome by whatever numerical method is chosen
to solve the model equations.
The thrust of these notes is a demonstration that the method of lines
time discretization of the partial differential equation coupled with
a Riccati transformation method (sort of an LU decomposition method)
provides an effective numerical method for the pricing problem in finance.
The method of lines (MOL) discretization is introduced in Chapter 1.
Chapter 2 contains an exposition of the Riccati transformation method
(also known as invariant imbedding method) for the solution of the
boundary value problems obtained from the MOL discretization.
Chapter 3 deals with European options and contains an expansive discussion
of the following model problems, their numerical complications, and
their resolution with the MOL technique:
3.1 A plain European call
3.2 A binary cash or nothing European call
3.3 A binary call with low volatility
3.4 The Black Scholes Barenblatt equation for a CEV process
with barriers.
Chapter 4 concentrates on American options and treats the following
model problems:
4.1 An American put
4.2 A put on an asset with a fixed dividend
4.3 An American strangle
4.4 The Black Scholes Barenblatt formulation for American
options with jump diffusion.
Subsequent chapters (yet to be written) will deal with the bond equation,
with other exotic options and with multi-factor models.
