Q$-5.$ XYZ Inc. is considering investing $$100$ million to create a Spanish version of its popular

television channel to serve the growing Mexican market. Financial analysis of the cash

flows from this investment suggests that the present value of the cash flows from this

investment to XYZ will be only $$80$ million. Thus, by itself, the new channel has a

negative NPV of $$20$ million. However, if the market in Mexico turns out to be more

lucrative, than currently anticipated, XYZ could expand its reach to all of Latin America

with an additional investment of $$150$ million any time over the next ten years. Although

the current expectation is that the cash flows from having a $xYZ$ channel in Latin

America will have a present value of only $$100$ million, there is considerable uncertainty

about both the potential for such a channel and the shape of the market itself, leading to

significant variance in this estimate. The standard deviation in the estimate of the project

value is estimated from the annualized standard deviation of firm value of publicly traded

entertainment firms in the Latin American markets, which is approximately $30$ percent.

The risk$-$free rate is $4\%$ per annum. First discuss the setting of an option to expand and

determine whether the applicable option will be a put option or a call option. Then

compute the value of the option to expand using the Black$-$Scholes Option Pricing Model.