A petrochemical firm will purchase 100,000 barrels of oil in 4 months and is concerned about the uncertainty in the price it will have to pay. The firm can use either forward contract or options on oil to hedge its risk. Assume the strike price of the option is the same as the forward price. Please specify both hedging strategies and explain. Please show firm's total payoff in 4 months in a payoff table and graph the payoff for both hedging strategies (use notations such as S0, ST, K, F0, etc.). Please list the difference of using option to hedge vs. using forward contract. (7 points)

In this assignment, you estimate the weighted average cost of capital (WACC) for Microsoft, Campbell Soup and Merck. Assume their before-tax costs of debt are (in the same order): 2.2%, 3%, and 2.5%.

Compute the market value of equity using the average of the last years's closing prices for the sample stocks downloaded from finance.yahoo.com multiplied by the number of share outstanding which you can find in finance.yahoo.com "Key Statistics" section. Compute the Net Debt using the latest balance sheet numbers you find on the SEC EDGAR website Compute the equity weight E% and debt weight D% . Use the 5-year period monthly stock return data you collected and calculated and linear regression to estimate the betas of these three stocks. Calculate the WACC for these three firms.

3.5.3 EXERCISE: GAMMA OF AN OPTION AND ITS RELATION TO OPTION PRICE AND DELTA. Gamma, the second derivative of the option price with respect to the stock price S, can be interpreted in the following ways: as a curvature of the graph of V (S) and as a measure of how much the delta changes 51 with a change of the stock price. Explain these interpretations using Figure 3.12 and make similar graphs for a put option

3.5.4 MAXIMUM OF GAMMA. Find the stock price S, for which the option gamma attains its maximum and compare the answer with previous graphs. Alternatively, for a given stock price S, which exercise price of an option leads to the highest gamma? 3.5.5 EXERCISE: MAXIMAL GAMMA AND THE ILLUSIONS OF RISK. The title of this exercise is taken from the section title in the paper Know Your Weapon: Part 1 by E. G. Haug, published in the Wilmott Magazine. The author writes: One day in the trading room of a former employer of mine, one of the BSD traders suddenly got worried over his gamma. He had a long dated deep-out-of-the money call. The stock price had been falling, and the further the out-of-themoney the option went the lower the gamma he expected. As with many option traders he believed the gamma was largest approximately at-the-money-forward. Looking at his Bloomberg screen, however, the further out of the money the call went the higher his gamma got. Another BSD was coming over, and they both tried to come up with an explanation for this. Was there something wrong with Bloomberg?2 This can indeed occur, as it can be seen by plotting 3-dimensional graph of gamma as a function of stock price S and time remaining to expiration , see Figure 3.13. The OTM options really have high gammas, when there is a long time remaining to maturity.

One fine day in the dealing room my risk manager asked me to get into his office. He asked me why I had a big outright position in some stock index futures - I was supposed to do "arbitrage trading". That was strange as I believed I was delta neutral: long call options hedged with short index futures. I knew the options I had were far out-of-themoney and that their DdeltaDvol was very high. So I immediately asked what volatility the risk management used to calculate their delta. As expected, the volatility in the risk-management-system was considerable below the market and again was leading to a very low delta for the options. This example is just to illustrate how a feeling of your DdeltaDvol can be useful. If you have a high DdeltaDvol the volatility you use to compute your deltas becomes very important. 4 Explain the assertions made in the quote above: Why do those options have high DdeltaDvol5? Why is the choice of volatility important in this situation and why does using too low volatility lead to too low deltas?

3.5.7 EXERCISE: DELTA FOR SIMPLE "DERIVATIVES". Recall that V (S, t) = S and V (S, t) = Ker(T t) are solutions to the Black-Scholes PDE with q = 0. What is their delta? Give an interpretation of this result. Recall the solution to the general Black-Scholes equation (general q 0) with the terminal condition V (S, t) = S. What is its delta? Explain the effect of dividends. 3.5.8 EXERCISE: GREEKS FOR A CASH-OR-NOTHING OPTION. Recall the binary options from the lectures and for simplicity assume q = 0. Consider cash-or-nothing binary option with payoff V (S, T ) = 0 for S E, 1 for S > E. We know that its price is V (S, t) = e q N(d2), where d2 is the expression (3.4) from the Black-Scholes formula, and it is shown in Figure 3.14 as a function of the stock price. Compute its delta and plot its dependence on the stock price S for different times remaining to expiration. See Figure 3.15 for a sample result. By relating it to hedging such an option, explain, why such a behaviour could have been expected. Explain the following quote on a cash-or-nothing option: What starts off as a placid instrument turns into an unmanageable monster over the last few hours of its life as the at-the-money delta becomes so high that the option becomes unhedgeable.6 Based on the graph of delta, sketch the graph of gamma. How does the option price depend on volatility? Plot the dependence of vega on the stock price for different times remaining to expiration and explain its behaviour. See Figure 3.16 for a sample result. How does the option price depend on time? Plot the dependence of theta on the stock price for different times remaining to expiration and explain its behaviour. See Figure 3.17 for a sample result.