Question $2(20$ points$)$

Consider the European call and put options that have the same underlying stock, the same

expiration date $(T=0\mathrm{.}5)$ and the same strike price $(K=$$$30).$ The current stock price is

$$30.$ The transaction cost to short a share is $$1,$ and the cost to long a share is $$0\mathrm{.}5.$ Suppose

the call option price is $$5$ and the annualized risk$-$free rate is $5\%.$

a$)$ Construct the synthetic put. $(6$ points$)$

b$)$ Find the maximum lower bound and the minimum upper bound of the put option price

such that there are no arbitrage opportunities. $(14$ points$)$