Question 2.

Background:  Metallgesellschaft AG was formerly one of Germany's largest industrial conglomerates based in Frankfurt. It had over 20,000 employees and revenues in excess of 10 billion US dollars. It had over 250 subsidiaries specializing in mining, specialty chemicals (Chemetall), commodity trading, financial services, and engineering (Lurgi). In 1993, the company lost 1.3 billion dollars suffering from flawed long hedge strategy in near term futures contracts that was meant to protect against forward sales commitments. A fall in spot prices forced margin calls for the company and the contracts were closed at a loss. Subsequently, the spot price increased and the company suffered even greater losses covering its customer commitments. It is debated whether the company was speculating after unwinding the long futures hedge since they became essentially exposed or naked against their forward customer commitments.  – from Wikipedia [https://en.wikipedia.org/wiki/Metallgesellschaft]

The following illustrates how important it is to consider tailing factor in taking a correct hedge strategy. 0.5 marks for filling each blank> 

Suppose the current spot price is $3. With 50 percent of the probability, the spot price will increase or decrease by 1 dollar for first year and then remain the same as shown in the graph below. 

Please answer the following questions. 

(a) If the annual discrete compounding risk-free rate is 10%, and the cost of carry offsets the convenience yield exactly, then the future price is equal to the spot price. Please explain. 

(b) Assume hedger takes hedge ratio as h*, i.e, if the risk exposure is a long position of 100 units of spot commodity, to hedge the risk, hedger will short 100h* futures underlying on that commodity. Please answer the questions in the right panel in analogy to the left panel, by filling the blanks in j) –r) below

(c) To make the hedge portfolio risk-free, the value of the portfolio should not vary no matter stock price rises or falls (or, equivalently, the hedge portfolio should not make profit in one state and make loss in the other, where the state means stock price rises or falls). Please based on the above fact, solve the optimal hedge ratio h*. 
(d) To make the hedge portfolio risk-free, the hedge portfolio should not make profit in one state and make loss in the other, where the state means stock price rises or falls. Please use the optimal hedge ratio solved in (c) to verify this fact.