I NEED MATLAB CODE for the following question:

Assume that an underlying follows the Heston model dS

$($

t

$)$

$=$

$0$

$.$

$1$

$5$

S

$($

t

$)$

dt

$+$

v

$($

t

$)$

S

$($

t

$)$

dW

$$

$($

t

$)$

$,$

$$S

$($

$0$

$)$

$=$

$1$

$2$

$,$

$$dv

$($

t

$)$

$=$

$1$

$.$

$2$

$-$

$($

$0$

$.$

$0$

$1$

$-$

v

$($

t

$)$

$)$

dt

$+$

$0$

$.$

$3$

$/$

v

$($

t

$)$

dW

$$

$($

t

$)$

$,$

$($

$0$

$)$

$=$

$0$

$.$

$0$

$1$

$,$

$$where E

$($

W

$$

$($

t

$)$

W

$$

$($

t

$)$

$)$

$=$

$-$

$0$

$.$

$3$

$2$

$.$

Determine the fair price of

a

$)$

$$a European call option,

b

$)$

$$a digital call option,

based on the above asset and a strike price of

$1$

$0$

$,$

$$time to expiration equal

$1$

$$and riskless interest rate equal

$2$

$.$

$5$

$\%$

$$using

$1$

$.$

$$Standard Monte Carlo with control variates,

$2$

$.$

$$Multilevel Monte Carlo based on the "Lamperti backward" Euler

$-$

Maruyama method,

$3$

$.$

$$Multilevel Monte Carlo based on the "Lamperti backward" Milstein method with neglected Levy areas with and without antithetic treatment,

explaining the Heston model and how you simulate it

$,$

$$as well as the

$($

Multilevel

$)$

$$Monte Carlo methods,

and for the digital call using and explaining both

a

$.$

$$the standard payoff function, and

b

$.$

$$a smoothed version of the payoff function.

Document simulation results, a sufficient accuracy, the code and how efficient your simulation is

$($

including showing that variances, simulation numbers and computational effort behave as expected when increasing the accuray

$)$

$,$

$$and how the different

$($

multilevel

$)$

$$Monte Carlo methods compare to each other.

Discuss also whether you efficiently can use Control Variates in combination with the Multilevel Monte Carlo methods, and document by corresponding simulations.

using the monte carlo various method one is standard monte carlo with control variates $,$second is Multilevel Monte Carlo based on the "lamperti backward"Euler$-$Maruyama Method and $3$rd is Multilevel monte carlo based on the "Lamperti backwarc"Milstein method with neglected levy areas with anf without antithetic treatment