$($a$.)$ Briefly comment on the most important differences between the standard Black$-$Scholes model and

the Heston stochastic volatility model. $[5$ marks$]$

$($b$.)($i$.)$ Write a function SVEM which takes as input $M($number of paths$)$ and $N=\frac{T}{h}($where $h$

is the step size$),$ and outputs an $M(N+1)$ matrix in which each row is a simulated path of

the stock price process $\{S_t$:$t0\}$ in the market described above using the Euler$-$Maruyama

scheme. Explain and justify any corrections or adjustments you make to the standard Euler$-$

Maruyama scheme. $[15$ marks$]$

$($ii$.)$ Write a function SVMil which takes as input $M($number of paths$)$ and $N($as above$),$ and

outputs an $M(N+1)$ matrix in which each row is a simulated path of the stock price process

$\{S_t$:$t0\}$ in the market described above using the Milstein scheme. Explain and justify

any corrections or adjustments you make to the standard Milstein scheme. $[10$ marks$]$

$($c$.)($i$.)$ Use both of the functions you have written in part $($b$.)$ to price your conventional European

call $($i$.$e$.$ payoff is $\{$:max$(S_T-K,0))$ for various values of $M$ and $N($small to large$)$ using the

Monte Carlo Method. Tabulate your results neatly and briefly comment on them.$[15$ marks$]$