Problem 8: Consider the following Long Crude Call option: 1. Strike Price K=$70, T=0.5 2. Strike Price K=$75, T=0.5 3. Strike Price K=$80, T=0.5 c. The current spot price of oil is $70. The CCIR is 4%. The volatility of oil (sigma) is 10%. Assume no convenience yield. a. Find the price of each option using B-S. b. Suppose that an oil producer expects to receive 43 barrels of oil six months from now and wants to hedge its position with a call option. Which option would the firm prefer and why? Let's compute the delta of each option: dCt = e-87 N(dit) dSt Where 8 is the assets dividend or convenience yield, in this case assumed to be zero. Which of the options is more sensitive to changes in stock prices? d. If the firm is concerned about the volatility of its assets (for instance, an option, if entered, is recorded on the firm's assets side), which of the option should the firm select? Now let's compute the vega of the option: e. dct e - (27)0.5 =e do2 -8T St * VT Which option is more sensitive to changes in the volatility of the oil? f. Consider an investor that is planning to sell the option before expiration. If the investor forecasts uncertainty in the production of oil in the near future, which option would she prefer? g. Let's now compute the theta: oe 2 dCt dt (21T)0.5 Which option exhibits the lowest theta? h. To illustrate the use of theta, think about executive compensation. Suppose that a manager is given stock options today and that the manager has the chance to sell them before expiration. In which case is the wealth (or pay) of the option the least affected by time? i. Let's compute gamma. Gamma is the change in delta (which is already a change) or: d dAt dS12 dst SO(217)0.5 j. Which option exhibits the smallest/largest I? d Ct 2 = T = e-ST