Finite Difference Methods:

https://en.wikipedia.org/wiki/Finite_difference_method

This project calculates the Black-Scholes Closed Form, Explicit FDM , Implicit FDM, Crank-Nicholson FDM, and Implicit SOR, * and Crank-Nicholson SOR to price the European and American call and put options. *

• The initial values are defined in the main function for the risk free rate, stock, sigma (volatility), dividend, Strike K, Time T, timestep, and step size in space. These values can be modified accordingly by the user by changing the top of the main function.

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Compile & Run Instructions / Screen Output:

To compile & run: 1.) Run make command in same directory as project 2.) Type ./FDM As follows:

1.) $ make g++ FDM_main.cpp ImplicitFDM.cpp ExplicitFDM.cpp CN_FDM.cpp
ImplicitSORFDM.cpp CN_SORFDM.cpp FDM_utils.cpp
BlackScholesFormula.cpp -o FDM

2.) after first step it will compile to an FDM.exe file which can be executed like this $ ./FDM

Installation / Troubleshooting Tips: Make sure g++ and make are on the os path variable.

For example, if you have a MingGW installation, then put this installation's bin folder on the path. Then make sure the make is available.

For example: MingGW installation: C:\DevTools\MinGW make - mingw32-make.exe g++ - g++

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Author Name: David Turner

Summary and Analysis

The Black-Scholes equation is to be solved numerically and compared to the closed form equations. For the given input parameters, i.e. • Asset price S = 20 • Strike price K = 20 • Risk free rate r = 0.03 • Dividend yield q = 0.04 • Volatility σ = 0.8 • Maturity T = 1 The closed form equation gives values of $5.91 and $6.10 respectively for European call and put options. To solve numerically, a grid must be defined over space and time. The space and time dimensions can be transformed from S, t of the standard Black-Scholes to x, τ where x = log(S/K) and τ = 0.5 σ2

(T-t). The space and time steps can then be chosen as dx = 0.05 and d τ = 0.00125. Using these values, the stability factor λ = d τ/ (dx)2 is equal to 0.5, and hence would be stable for the explicit finite difference scheme.

The equation is solved numerically using several different methods including Explicit FDM (Thomas), Implicit

FDM (Thomas), Crank-Nicholson (Thomas), Implicit FDM (SOR), Implicit FDM (Projected SOR), Crank- Nicholson (SOR), and Crank-Nicholson (Projected SOR).

The results are summarized in the following table on the next page:

Table 1. Summary of different numerical simulation methods of solving the Black-Scholes equation

The Implicit FDM and Crank-Nicolson FDM both require the solution of a matrix equation to solve the backward Euler step. Since the matrix is tridiagonal, this matrix equation can be solved more efficiently than for standard matrices. For this exercise, the matrix equation is solved using Thomas algorithm and Successive Over Relaxation (SOR), or Projected SOR for the case of American options. The results obtained for the different option types are consistent with each other, even though there is some error from the closed form result. The error could possibly come from the discretization step, since the step size is in terms of x = log(S/K) rather than S. While the grid is uniform in x, the step size in S becomes larger as the ratio S/K increases.

Sample Output (should look something like this):

PS C:\Projects\finiteDifferenceMethodsOptionPricer> mingw32-make g++ FDM_main.cpp ImplicitFDM.cpp ExplicitFDM.cpp CN_FDM.cpp
ImplicitSORFDM.cpp CN_SORFDM.cpp FDM_utils.cpp
BlackScholesFormula.cpp -o FDM PS C:\Projects\finiteDifferenceMethodsOptionPricer> .\FDM.exe

Table 1: Summary of values calculated by different numeric methods

|-----------------------------------|--------------|-------------|------|

METHOD	OPTION TYPE	OPTION VALUE	ERROR
Close form Black Scholes	European call	5.907000	N.A.
Close form Black Scholes	European put	6.100122	N.A.
-----------------------------------	--------------	-------------	------
Explicit FDM	European call	6.220736	5.31%
Explicit FDM	European put	6.421977	5.28%
Explicit FDM	American call	6.220736	N.A.
Explicit FDM	American put	6.421977	N.A.
-----------------------------------	--------------	-------------	------
Implicit FDM	European call	6.190003	4.79%
Implicit FDM	European put	6.388773	4.73%
Implicit FDM	American call	6.156959	N.A.
Implicit FDM	American put	6.352955	N.A.
-----------------------------------	--------------	-------------	------
Crank-Nicholson FDM	European call	6.182996	4.67%
Crank-Nicholson FDM	European put	6.380880	4.60%
Crank-Nicholson FDM	American call	6.182325	N.A.
Crank-Nicholson FDM	American put	6.380138	N.A.
-----------------------------------	--------------	-------------	------
Implicit (SOR) FDM	European call	6.190003	4.79%
Implicit (SOR) FDM	European put	6.388753	4.73%
Implicit (Projected SOR) FDM	American call	6.156958	N.A.
Implicit (Projected SOR) FDM	American put	6.352934	N.A.
-----------------------------------	--------------	-------------	------
Crank-Nicholson (SOR) FDM	European call	6.179574	4.61%
Crank-Nicholson (SOR) FDM	European put	6.377203	4.54%
Crank-Nicholson (Projected SOR) FDM	American call	6.164172	N.A.
Crank-Nicholson (Projected SOR) FDM	American put	6.360504	N.A.
-----------------------------------	--------------	-------------	------