Consider a stock forward contract that has a life of $5$ months. The risk$-$free interest rate is $6\%$ p$.$a$.($continuously compounded$).$

$1.$ a$)$ If the spot price is $$108,$ what is the $$fair$$ futures price?

$2.$ b$)$ If the futures contract is selling for $$111\mathrm{.}1,$ what should an arbitrageur do to get a riskless, positive profit? What is this profit?

Consider a company whose assets are estimated to have the current market value of at $$120$ mil. Assem that the value of assets follows a lognormal process with parameters $\backslash$mu $=0\mathrm{.}1$ and s$=0\mathrm{.}4.$ Suppose that the company is financed by common stock and debt in the form of zero$-$coupon bonds. The bonds mature in $10$ years. The face value of debt is $$30$ mil.

Assume that the risk$-$free rate is $5\%$ p$.$a$.$

The company wants to go public, and is interested in finding the fair value of its equity.

To answer the questions below, you may use an online Black$-$Scholes calculator, like the one discussed in class. You must explain what inputs to the Black$-$Scholes formula you used in your calculations.

$1.$ a$)$ What are the current market values of the firm$$s equity and debt? Explain your reasoning.

$2.$ b$)$ How would your answer to part $($a$)$ change, if the firm$$s debt had an additional clause in its indenture: at maturity the bondholders can choose between receiving the face value of $$30$ mil, or they can exchange the entire bond issue for $25\%$ ownership in the firm$$s assets $($the equity holders will have to transfer $25\%$ of their equity to the bondholders, if the latter exercise their right to convert the bonds into stocks$).$ What would be the current market values of the firm$$s equity and debt in this case? Explain your reasoning.

Hint: First, identify the $$threshold$$ future value of the firm$$s assets, at which bondholders will be indifferent between getting the face value of the debt or converting their bonds into equity. Next, identify and price new options, which appear due to debt$$s convertibility.