Suppose that the 12-month spot rate is 3%,

Suppose that the 12-month spot rate is 3%,

Subject: Business    / Finance    

Question
Question 1: Suppose that the 12-month spot rate is 3%, the 15-month spot rate is 3.5%, and the 24-month spot rate is 4%. All rates are annual rates with annual compounding.

    [10 marks] Determine the no-arbitrage forward price for a bond with annual coupon rate 10%, face value $100, and maturity two years. The maturity of the forward contract is 15 months.
    [15 marks] Assume that a MADBANK is willing to trade (buy/sell) the forward from part (a) with you at price $120. Assume also that you can trade riskless zero-coupon bonds with face value $100 with any maturity and that you can trade the 2-year coupon bond from part (a). Set up an abritrage trade. Give details: What assets you buy/sell at what quatities? What is the arbitrage profit per unit of trade at the time you are entering into it (i.e, at t = 0)?

Question 2: A wine wholesaler needs to buy 100,000 gallons of Cheap Chardonnay for delivery on October 14th 2012. A producer offers to deliver the wine on October 14th 2012 for $500,000 paid today, i.e., on April 14th 2011. The wholesaler can also buy Cheap Chardonnay futures contracts maturing on October 14th 2012. The current futures price is $50,500 for each 10,000 gallon futures contract. The wholesaler is determined to lock in the cost of the 100,000 gallons needed on October 14th 2012.

    [5 marks] The wholesaler considers the futures contract, but worries that the con- tract will not lock in her cost, because futures prices may fluctuate widely between now and October. Is her concern justified? Why or why not?
    [10 marks] Do you recommend that the wholesaler pay the producer now or take a long position in Cheap Chardonnay futures? Be precise, show based on what should the wholesaler make her decision.

Question 3: A stock is currently selling at $50. Every 6 months the stock price goes eitherupbyafactorofu=1.3ordownbyafactorofd=0.7. Theprobabilityofanup move is 0.7. (Be careful: this is not the risk-neutral probability, this is the probability of an up move when investors are risk averse.) The riskless rate is 5% semiannual APR. The stock pays no dividends in the next year.

    [5 marks] Compute the price of a forward contract on the stock with a maturity of 1 year.
    [5 marks] Compute the price of a European call option on the stock, that has a maturity of 1 year and a strike price of $55.
    [5 marks] Suppose that you are risk averse and you buy the stock today. Compute your expected return on the stock over the next year. Express the expected return as a semiannual APR.

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d. [5 marks] Suppose that you are risk averse and you buy the call option from part (b) today. Compute your expected return on the call option over the next year. Express the expected return as a semiannual APR. How does the expected return on the call compare to the expected return on the stock? Explain why.

Question 4: The price today of SAUDER.com stock is $40. After 6 months, it will either increase by 25% or decrease by 20%. Over the next 6 month period the stock price will again either increase by 25% or decrease by 20%. The stock does not pay any dividends over the next year. The 6 month interest rate is 10% (for both the 6 month periods). In answering the following questions, assume that there are no arbitrage opportunities.

    [5 marks] Calculate the price today of a 1 year European put option on this stock which has a strike price of $40.
    [6 marks] Calculate the price today of a 1 year American put option on this stock which has a strike price of $40.
    [9 marks] An Asian option is an option whose payoff depends on the average value of the stock price over the life of the option. (Early exercise of the option is not allowed.) One example is a fixed strike Asian call option, whose payoff at maturity is max[A ? X, 0], where A is the average value of the stock price and X is the strike price. Calculate the value today of a 1-year fixed strike Asian call option on this stock with a strike price of X = $35 (include today’s stock price when you calculate the average price over the year).

Question 5: A butterfly spread is a combination of option positions that involves three strike prices. To create a butterfly spread, a trader purchases an option with a low strike price and an option with a high strike price, and sells two options with an intermediate strike price. For this problem, assume that the intermediate strike price is halfway between the low and the high strike prices and that the options are European. Denote the intermediate strike price by X, the low strike price by X ? a, and the high strike price by X + a, a > 0.

    [5 marks] Graph the payoff diagram at maturity of the butterfly spread in which the underlying options are call options.
    [5 marks] Graph the payoff diagram at maturity of the butterfly spread in which the underlying options are put options.
    [10 marks] Using put-call parity, show whether the initial investment (i.e., the amount of money you need to pay at t = 0) to create the butterfly spread is higher/lower/the same when you use puts instead of calls.

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