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

dSt

St

$=$ rdt $+\backslash$sigma dZt

using a finite difference method as discussed in class. Use forward$/$backward$/$central differencing 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 $\backslash$tau $=$ T $/25,$ where $\backslash$tau is the time to maturity, i$.$e$.,\backslash$tau $=$ T $$ t$,$ and

the grid

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

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

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

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

$1\mathrm{.}12*$K:$0\mathrm{.}02*$K:$1\mathrm{.}2*$K$,...$

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

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

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

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

$1$

Table $1$: Data for Put Example

$\backslash$sigma $\mathrm{.}4$

r $\mathrm{.}02$

Time to expiry $($T$)1\mathrm{.}0$ years

Strike Price $$100$

Initial asset price S

$0$ $$100$

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$((\backslash$tau $)$

$2$

$,($S$)$

$2$

$)$ ; $$S $=$ max

i

$($Si$+1$ Si$)(1)$

Let

h $=$ C$1$S

h $=$ C$2\backslash$tau

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$)=$ Vexact $+$ A $$ h

$2$

V $($h$/2)=$ Vexact $+$ A $($h$/2)2$

V $($h$/4)=$ Vexact $+$ A $($h$/4)2$

$(2)$

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

V $($h$)$ V $($h$/2)$

V $($h$/2)$ V $($h$/4)\text{'}4(3)$

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 convergence 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