Question $4$

Use the binomial tree model to price a European call option on a Singapore dollar. The option expires in $6$ months, and its strike price is $55$ Hong Kong dollars. Suppose the current exchange rate of Singapore dollar is $5\mathrm{.}6$ Hong Kong dollars. The exchange rate could either

rise to $5\mathrm{.}8$ in the up state, or drop to $5\mathrm{.}3$ in the down state. The annual risk$-$free rate in Hong Kong is $\backslash (2\backslash \%\backslash ),$ the annual risk$-$free rate in Singapore is $\backslash (1\backslash \%\backslash ).$

a$)$ Find the current price of the call option in Hong Kong dollar.

b$)$ Suppose $\backslash (1\backslash \%\backslash )$ is the annual risk$-$free lending rate in Singapore, and the annual risk$-$free borrowing rate is $\backslash (1\mathrm{.}5\backslash \%\backslash )$ in Singapore.

Find the minimum upper bound and the maximum lower bound for the call premium in Hong Kong dollar such that there is no arbitrage opportunity.