a$).$ Develop Matlab code for pricing European options under a Black Scholes model

$\frac{d{S}_t}{S_t}=rdt+d{Z}_t$

using a finite difference method as discussed in class. Use forward$/$backward$/$central differ$-$

encing as appropriate to ensure a positive coefficient discretization. The code should be able

to use fully implicit, and Crank$-$Nicolson method.

You will have to solve a tridiagonal linear system. Use lu Matlab function to solve linear

systems. Avoid unnecessary LU factorization computation.

b$).$ Solve for a European put option, using the data given in the Table $1,($with fully implicit,

and Crank$-$Nicolson method $)$ and compare with the exact solution using blsprice in Matlab.

Use constant timestep sizes.

c$).$ Show a convergence table, with a series of grids. Show the option value at $t=0,S=100.$

Begin with a timestep of $=\frac{T}{25},$ where $$ is the time to maturity, i$.$e$.,=T-t,$ and

the grid

$S=[0$:$0\mathrm{.}1**K$:$0\mathrm{.}4**K,$dots

$0\mathrm{.}45**K$:$0\mathrm{.}05**K$:$0\mathrm{.}8**K,$dots

$0\mathrm{.}82**K$:$0\mathrm{.}02**K$:$0\mathrm{.}9**K,$dots

$0\mathrm{.}91**K$:$0\mathrm{.}01**K$:$1\mathrm{.}1**K,$dots

$1.12**K$:$0\mathrm{.}02**K$:$1\mathrm{.}2**K,$dots

$1\mathrm{.}25**K$:$\mathrm{.}05**K$:$1\mathrm{.}6**K,$dots

$1\mathrm{.}7**K$:$0\mathrm{.}1**K$:$2**K,$dots

$2\mathrm{.}2**K,2\mathrm{.}4**K,2\mathrm{.}8**K,$dots

$3\mathrm{.}6**K,5**K,7\mathrm{.}5**K,10**K]$;

Table $1$: Data for Put Example

where $K$ is the strike, and we are interested in the solution near $S=K.$

Note: to carry out a convergence study, you should solve the pricing problem on a sequence of

grids. Each grid has twice as many intervals as the previous grid $($new nodes inserted halfway

between the coarse grid nodes$)$ and the timestep size is halved.

Assume that

Error $=O(({)}^2,(S{)}^2)$;$S=ma{x}_i(S_{i+1}-S_i)$

Let

$h=C_1*S$

$h=C_2*$

Suppose we label each computation in the above sequence by a set of $h$ values. Then the

solution on each grid $($at a given point$)$ has the form

$V(h)=V_{\text{e}\text{x}\text{a}\text{c}\text{t}}+A*h^2$

$V(\frac{h}{2})=V_{\text{e}\text{x}\text{a}\text{c}\text{t}}+A*(\frac{h}{2}{)}^2$

$V(\frac{h}{4})=V_{\text{e}\text{x}\text{a}\text{c}\text{t}}+A*(\frac{h}{4}{)}^2$

where we have assumed that the mesh size and timestep are small enough that the coefficient

$A$ in equation $(2)$ is approximately constant. Now, equation $(2)$ implies that

$\frac{V(h)-V(\frac{h}{2})}{V(\frac{h}{2})-V(\frac{h}{4})}4$

Check the theory by examining the rate of convergence of your pricer.

Carry out the above tests using fully implicit, and Crank Nicolson method. Show a conver$-$

gence table for each test.

d$).$ Show plots of the option value for the range $S=[50,150],$ for your solution on the finest

grid for CN method.

Submit your matlab code.