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 FE Problem Set #12 Team LatteJul 16, 2006 Problem #1 The forward is trading at 100 and the volatility of the ATM forward is 20%. Using an analytical (closed form) method estimate the value of a one year log contract on this forward (assume flat vol)? If a trader wants to replicate this log contract to hedge a variance swap then estimate the prices of the set of vanilla call and put options that will hedge this log contract and compare the value of the sum of calls and puts to the analytical value of the log contract that you calculate in (a); (Assume the strike price of calls and puts in the increment of 2 from the ATM value of 100). Design a replicating (hedging) strategy using these calls and puts to hedge the one year variance swap on the above volatility? ( Hint : The value of a log contract is given by the closed form approximation formula as . Take increments of 2 from the ATM value of 100 and take a range of strike prices for calls and puts; then calculate the delta hedge of the calls and puts by taking the gradient of the log contract price and the strike price and then using a flat vol of 20% calculate the Black-Scholes price of these options and the proceeds from hedge. Then estimate the portfolio value by summing up the proceeds from all the calls and puts. This value should closely approximate the analytic value of the log forward calculated using the above formula .) Problem #2 A call option and a put option have identical characteristics (same strike, maturity, risk free rate and volatility). Without using any closed form formula (such as Black-Scholes equation, etc.) show that the gamma of both the call and the put are exactly equal. Now use a Black-Scholes calculator and assume certain parameter values and show the same result. ( Hint : Gamma is the second mathematical derivative of the option with respect to the underlying and put-call parity states that call minus a put is equal to the forward .) Problem #3 An analyst is trying to estimate the value of a property company in an emerging market in Asia the valuation would go as a key input to the M & A division of the investment bank. Despite the numerous constraints in parameter estimation and insufficiency of data it is imperative that the valuation of the company be done as accurately as possible. The company has a land bank on its balance sheet with an estimated market value of \$500 million. This land bank is expected to grow at the rate of 15% per annum for the next five years. The company also has fixed assets (other than land) to the tune of \$150 million and it is expected to grow at the rate of 15% as well. The depreciation is on a straight line basis and is 10%. The net working capital as of today is \$45 million and it is estimated to grow at 15% per annum for the next five years. As of today the company has goodwill of \$100 million on its balance sheet. There is \$30 million debt on the balance sheet on which the company pays 9% interest per annum. There are 100 million common shares outstanding with a par value of \$1.00.The present revenue is \$200 million and is expected to grow at 15% p.a. with cost of goods sold being 75% of the sales and the selling and administrative expenses (including depreciation) being 15% of the sales. The company is not a listed company but there are a few other comparable companies that have their stocks listed on the stock exchange and the closet one has a stock with a beta of 0.75 and historical volatility of 18%. The stock index of the country has returned an average of 12% per annum over the last five years and the risk free rate is 5%. The tax rate is 30%. Can an option valuation framework be applied to value this company? And if so, how much will be the value of the equity of the company? If an option pricing framework is not suitable for valuing this company then what other valuation techniques can you suggest? Use a Discounted Cash Flow (DCF) analysis and estimate both the enterprise value of this company and the value of the equity of the company. ( Hint: DCF model uses free cash flow available to the shareholders to value the company. Free cash flows are estimated from NOPLAT, which is net operating profit less adjusted taxes. NOPLAT adjusted for the movement (change) in working capital, goodwill and the capital expenditures is the free cash flows available to the shareholders. Discount this by WACC to get the enterprise value .) Problem #4 You have entered into a long position in a cap and a short position in a floor with the same strike rate of , where is an interest rate. If floating cash flows are all based on LIBOR show that your position is equivalent to a paying a fixed rat of in a vanilla fixed-for-floating swap. Company ABC enters into a 10-year interest rate swap with Bank XYZ whereby the company's receives 7% and pays LIBOR. However, the Company has the right to cancel this swap at the end of 6 years. Will the fair value of this cancelable swap rate be greater than 7% or less than 7%? Why? A hedge fund managers reports in its memorandum to a fund of funds investor that the average return of its fund over the last five years has been 4.40%; when asked for details the fund managers states that over the last five years the following returns were generated (measure using annual compounding): 12%, 9%, 14%, -24%, 11%. The fund of fund manager points out that there is an inconsistency in the above measurement of return. Comment. Any comments and queries can be sent through our web-based form. 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