Q$1)$

The price of a stock is governed by the stochastic differential equation:

$\frac{dS(t)}{S(t)}=0\mathrm{.}10dt+0\mathrm{.}02dZ(t)$

where $\{Z(t),t0\}$ is a standard Brownian motion. Consider the geometric average

$G=[S(a)S(2a)S(3a)S(4a){]}^{\frac{1}{4}}$

Calculate a such that the volatility of $lnG$ is $0\mathrm{.}35.$

Q$2)$

A non$-$dividend paying stock has a current price of R$100.$ In any unit of time the price of the stock is expected to increase by $10\%$ or decrease by $5\%.$ The continuously compounded risk$-$free interest rate is $4\%$ per unit of time.

A European call option is written with a strike price of R$103$ and is exercisable after two units of time, at $t=2.$

Establish, using a binomial tree, the replicating portfolio for the option at the start and end of the first unit of time, i$.$e$.$ at $t=0,1.$ Hence, calculate the value of the option at $t=0.$

Q$3)(15$ marks$)$