1.       Background

Let us consider a European Put option.

Recall that the buyer of a put option has the right to sell stock at time T for the exercise price K (the terminology“strike price K”is also used with the same meaning of“exercise price K”). Assume the stock price S paying a continuous dividend yield q follows the process below:

where r is the risk-free interest rate and W is a Wiener process under the risk-neutral probability measure.

We know that the price of the option, V (S, t), on the above stock satisfies the Black-Scholes equation below:

The payoff of the option at time T is

V(S, T) = max{K – S, 0}

With a change of time variable τ = T − t the equation satisfied by the option becomes

and the payoff of the option becomes the initial condition

  (1)

Let the minimum and maximum stock prices be Smin  = 0 and  Smax  much bigger than the strike price (we will use Smax = 4 K) . The boundary condition is (European put option)

            (2)

The project aims to investigate the properties of the numerical solution of the BS equation as follows:

(a)  Reduce the BS equation to the heat equation

(b) Solve the HE using a Finite Difference Method (FDM) scheme of your choice; namely the

explicit, the implicit, or the Crank-Nicholson scheme

(c)  Use (a) and (b) to solve the BS equation numerically.

2.  Project Tasks (max total of 30 marks)

You  are   asked  to   implement  the  following   programming  tasks   (1-5)   and  to  write   PDF documentation (task 6).

1.    (4 marks) Consider the following domain: x interval [a,b],    t  interval [0,T]. Generate the grid in the x, t coordinates as follows:

•    Space interval divided into N equal parts (N+1 boundary points). The index values should be consistent with MATLAB array index convention running from 1 to N+1.

•   Time interval divided into M equal parts (M+1 boundary points). The index values should be consistent with MATLAB array index convention running from 1 to M+1.

2.    (4  marks)  Using the change  of variables  described  in the  lectures  (going from the  unknown function V(S,t) to the unknown function v(x, τ) and then to the unknown function u(x, τ)), convert the BS equation to the heat equation

Use the values Smin = 0.001,   Smax = 4 K . Write the MATLAB code to generate the x interval and the τ interval (from the values of Smin ,   Smax and T calculate the range of the x variable and the τ variable). Note that the change of variables is

and that the above time variable is different  by a factor from the time variable  used in the Background section (where we used the change of time variable τ = T – t ).

Calculate the corresponding grid.

3.    (4 marks)    Translate the initial and boundary conditions (1) and (2) in the correct initial and boundary conditions for the new unknown function u(x, τ).

4.    (4 marks) Use a scheme of your choice (Implicit, Explicit, or Crank-Nicholson) to solve the heat equation for u(x, τ). In the case of the explicit finite difference scheme, demonstrate that the scheme is conditionally stable and find the condition under which the scheme would be stable.

5.    (8 marks) Work your way back from u(x, τ) to V(S,t). to solve the Black-Sholes equation.

MATLAB code to solve the Black-Scholes equation in the conditions described in this project, using a direct derivation of a finite difference scheme (with non-constant coefficients) is given as file “ImplicitFDMBSEurPricePut.m” . You can use this script to check the validity of your results by comparing the curve V(S, t = 0) calculated by your code with the same curve (represented by the value of the variable “Vold” at the end of the execution of “ImplicitFDMBSEurPricePut.m”).

6.    (6 marks) The code should be accompanied by detailed documentation, split into:

•    MATLAB  comments  explaining  the  structure  of  the  code,  available  functions,  main variables and any other information to help to understand the code;