Practice Question $2$: Valuation with a Binomial Lattice Approach

The current market price of a stock is $45\mathrm{.}5.$ Over the next two quarters, this stock is projected

to either increase by $7\%$ or decrease by $6\%$ each period, as per a binomial lattice model. The

risk$-$free interest rate is $6\mathrm{.}1\%$ per annum, with continuous compounding.

$($a$)$ Calculate the price of a $6-$month European call option on this non$-$dividend$-$paying stock

with an exercise price of $46\mathrm{.}5.$

$($b$)$ Calculate the price of a $6-$month European put option on the same non$-$dividend$-$paying

stock with an exercise price of $46\mathrm{.}5.$

$($c$)$ Ensure that the calculated prices from $($a$)$ and $($b$)$ are consistent with the put$-$call parity

relationship.

$($d$)$ In the given market model, an investment bank has introduced an Asian call option that

accounts for the full history of the stock's value. The payoff for this option is:

from $n=0$ to $N,$

where $N$ is the total number of periods in the market model. Determine the initial price of the

Asian call option with an exercise price of $46\mathrm{.}5$ at time $t=0.$

Hint: Apply a similar pricing strategy as used for the European call and put options, but with

more precision. The value of the option at the final interval's midpoint $($i$.$e$.,$ S$\_2($ud$)$ or S$\_2($du$))$

will differ based on the stock's price path. Careful calculation is required to capture this

distinction.