We wish to emphasize that we DO NOT present new financial models or resolve practical problems of finance. The data are synthetic and the calibration of models is ignored. Instead, we consider in great detail a number of standard, and some perhaps not quite so common, applications and discuss where mathematical and numerical problems arise. Of particular interest here are nonlinear problems arising in pricing bonds and options related to early exercise, transaction costs, and implied volatility.

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.

We favor this approach for one-dimensional problems because it applies directly to the differential equations of finance with variable coefficients, requires no change of variables, computes simultaneously price, delta and gamma, explicitly tracks early exercise boundaries without iteration or interpolation, and allows easy mesh refinement to resolve discontinuities in the solution near barriers and non-smooth pay-offs. Some of these features are retained when inherently nonlinear equations like the Black Scholes Barenblatt equation, when the equation for jump diffusion, and when multi-dimensional and multi-factor problems can be solved iteratively as a sequence of one dimensional problems.

The method is introduced primarily through its application to a number of pricing problems of various complexity. Throughout, our intent has been to give a complete mathematical description of the equations to be solved, of the initial and boundary conditions which are imposed, and of the parameters chosen for the computation. Many numerical values and graphs of the solutions obtained with the method of lines are presented which may serve as benchmarks for readers engaged in developing their own numerical algorithms and codes.

All computations were carried out on a desktop computer and relied on
a few basic homegrown Fortran subroutines for the Riccati transformation
(which may eventually be appended to these notes).
No external program libraries were used.

**The Time-Discrete Method of Lines -- A PDE Approach**

** Table of Contents**

**Chapter 1:** Comments on the Pricing Equations in Finance

1.1 Solutions and their properties

Example 1.1 Positivity of option prices and the Black Scholes formulas

Example 1.2 The early exercise boundary for plain American puts and calls

Example 1.3 Exercise boundaries for options with jump diffusion

Example 1.4 The early exercise premium for an American put

Example 1.5 The early exercise premium for an American call

Example 1.6 Strike price convexity

Example 1.7 Put-call parity

Example 1.8 Put-call symmetry for a CEV and Heston model

Example 1.9 Equations with an uncertain parameter

1.2 Boundary conditions for the pricing equations

1.2.1 The Fichera function for degenerate equations

Example 1.11 Boundary conditions for the heat equation

Example 1.12 Boundary condition for the CEV Black Scholes equation at S = 0

Example 1.13 Boundary condition for a discount bond at r = 0

Example 1.14 Boundary conditions for the Black Scholes equation on two assets

Example 1.15 Boundary condition for the Black Scholes equation
with stochastic volatility v at at S = 0 and v = 0

Example 1.16 Boundary conditions for an Asian option

1.2.2 The boundary condition at "infinity"

Example 1.17 CEV puts and calls

Example 1.18 Puts and calls with stochstic volatility

Example 1.19 The European max option

Example 1.20 An Asian average price call

1.2.3 The Venttsel boundary condition on "far but finite" boundaries

Example 1.21 A defaultable bond

Example 1.22 The Black Scholes equation with stochastic volatility

1.2.4 Free boundaries

**Chapter 2:** The Method of Lines (MOL) for the Diffusion Equation

2.1 The method of lines with continuous time (the vertical MOL)

2.2 The method of lines with continuous x (the horizontal MOL)

Appendix 2.2 Stability of the time-discrete three level scheme for the heat equation

2.3 The method of lines with continuous x for multi-dimensional problems

Appendix 2.3 Convergence of the line Gauss Seidel iteration for a model problem

2.4 Free boundaries and the MOL in two dimensions

**Chapter 3:** The Riccati Transformation Method for Linear Two Point Boundary Value
Problems

3.1 The Riccati transformation on a fixed interval

3.2 The Riccati transformation for a free boundary problem

3.3 The numerical solution of the sweep equations

Example 3.1 A real option for interest rate sensitive investments

Appendix 3.3 Connection between the Riccati transformation, Gaussian
elimination and the Brennan-Schwartz method

**Chapter 4:** European Options

Example 4.1 A plain European call

Example 4.2 A binary cash or nothing European call

Example 4.3 A binary call with low volatility

Example 4.4 The Black Scholes Barenblatt equation for a CEV process

**Chapter 5:** American Puts and Calls

Example 5.1 An American put

Example 5.2 An American put with sub-optimal early exercise

Example 5.3 A put on an asset with a fixed dividend

Example 5.4 An American lookback call

Example 5.5 An American strangle for power options

Example 5.6 Jump diffusion with uncertain volatility

**Chapter 6:** Bonds and Options for One-Factor Interest Rate Models

Example 6.1 The Ho-Lee model

Example 6.2 A one-factor CEV model

Example 6.3 An implied volatility for a call on a discount bond

Example 6.4 An American put on a discount bond

**Chapter 7:** Two-Dimensional Diffusion Problems in Finance

7.1 Front tracking in Cartesian coordinates

Example 7.1 An American call on an asset with stochastic volatility

Example 7.2 A European put on a combination of two assets

Example 7.3 A perpetual American put -- MOL with overrelaxation

Example 7.4 An American call, its deltas and a vega

Example 7.5 American spread and exchange options

Example 7.6 An American call option on the maximum of two assets

7.2 American puts and calls in polar coordinates

Example 7.7 The basket call in polar coordinates

Example 7.8 A call on the minimum of two assets

Example 7.9 A put on the minimum of two assets

Example 7.10 A perpetual put on the minimum of two assets with uncertain correlation

Example 7.11 Implied correlation for a put on the sum of two assets

7.3 A three-dimensional problem

Example 7.12 An American call with Heston volatility and a stochastic interest rate

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Chapters 0-6 | Sat Jun 7 14:31:19 EDT 2014 |