Please be aware these notes are no longer maintained

They are now part of the book, The Time-Discrete Method of Lines for Options and Bonds: A PDE Approach, published by World Scientific.

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.

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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