In a two$-$step tree for a stock, suppose that over each period $($one month$)$ the stock price

is either to rise by $8\%$ or fall by $5\%.$ The stock does not pay dividends.

Assume that the monthly effective risk$-$free interest rate is $2\%.$

Consider an Asian option, which is an exotic option whose payoffs depend on the average

of certain prices of the stock over time. Early exercise of the option is not allowed.

For instance, the payoff of an Asian call option at the expiration date is:

max$(\frac{S_0+S_1+S_2}{3}-x,0)$

where $S_0,S_1$ and $S_2$ are the price of the underlying asset today, the prices after $1$ and $2$

months, respectively. Its payoff is otherwise defined in the same way as a standard call

option. Assume that the strike price for the Asian call option is $x=48$ and the stock

price today is $S_0=50.$

$($a$)$ Using the risk neutral valuation approach discussed in class, compute the price of

the Asian call option. Explain your steps.

$($b$)$ Using the replication approach discussed in class, compute the price of the Asian

call option. Explain your steps.

$($c$)$ In general, do Asian call and put options satisfy the put$-$call parity for standard

European options with the same strike and the same expiration date? Explain your

intuition without calculations.