$(25$ marks$)($Stochastic Volatility Model$)$ The stochastic volatility model for pricing op$-$

tions is given by

$\frac{dS}{S}=rdt+vd{Z}^{(1)}$

$dv=(\frac{?}{\text{b}\text{a}\text{r}}(v)-v)dt+d{Z}^{(2)}$

$E(d{Z}^{(1)}d{Z}^{(2)})=dt$

where $$ is the speed of reversion of the voltility $v$ to its long$-$term mean $\frac{?}{\text{b}\text{a}\text{r}}(v).$

$($a$).(5$ marks$)$ Show that $dx=(r-\frac{v^2}{2})dt+vd{Z}^{(1)}$ given $x=ln(S).$

$($b$.(20$ marks$)$ Write a Matlab code to compute the value of a European call with strike

$K$ and expiry $T$ using the Monte Carlo method whose path can be generated as follows

$x_{i+1}=x_i+rt-\frac{v_i^2}{2}t+v_i\sqrt[\phantom{2}]{t}{}_i^{(1)}$

$v_{i+1}=v_i+((\frac{?}{\text{b}\text{a}\text{r}}(v))-v_i)t+\sqrt[\phantom{2}]{t}{}_i^{(2)}$

where $x=ln(S),{}_i^{(1)}$ and ${}_i^{(2)}$ are standard normals with correlation $.$ If at any time

$v_i<mn\; id="EditorElement\_481780">0</mn>$ then simply set $v_i$:$=|v_i|$ and continue simulation. Using data in Table $1,$ with

$10000$ simulations in Monte Carlo method, and $t=\frac{1}{100},$ compute the initial value of

an at$-$the$-$money call $($i$.$e$.,K=S_0)$ with the expiry $T=1.$

Table $1$: Data for the stochastic volatility model