Consider a call option in the Black$-$Scholes model and denote by $C(t)$ the

process of the call option prices.

a$)$ Prove that $C(t)$ is an Ito process. $(5$ marks$)$

b$)$ Assume that

$dC(t)={}_C(t)C(t)dt+{}_C(t)C(t)dW(t)$

for some processes ${}_C(t),{}_C(t).$ Find the expressions for these processes.

marks$)$

c$)$ Derive the equation satisfied by $C(t)$ in the risk$-$neutral world, that is

with respect to $W_Q(t).(15$ marks$)$

d$)$ For $=30\%,S(0)=100,K=100,T=0\mathrm{.}25,r=5\%,=10\%$ find the

values of ${}_C(0),{}_C(0)$ and the market prices of risk $(5$ marks$)$

e$)$ Sketch the graph showing the dependence of ${}_C$ on Kin$\{85,90,$dots,$115\}.$

$(10$ marks$)$