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1.5 Exercise 1.6 Xi's are independent, exponentially distributed random variables with a mean value of 1/a, a > 0, i = 1,2. ..., n. Calculate P(X 42.31 (E) The range is empty. 12. Consider a European put option on a stock index without dividends, with 6 months to expiration and a strike price of 1,000. Suppose that the effective six-month interest rate is 2%, and that the put costs 74.20 today. Calculate the price that the index must be in 6 months so that being long in the put would produce the same profit as being short in the put. (A) 922.83 (B) 924.32 (C) 1,000.00 (D) 1,075.68 (E) 1,077.17 IFM-01-18 Page 8 of 105 13. A trader shorts one share of a stock index for 50 and buys a 60-strike European call option on that stock that expires in 2 years for 10. Assume the annual effective risk-free interest rate is 3%. The stock index increases to 75 after 2 years. Calculate the profit on your combined position, and determine an alternative name for this combined position. Profit Name (A) -22.64 Floor (B) -17.56 Floor (C) -22.64 Cap (D) -17.56 Cap (E) -22.64 "Written" Covered Call16. The current price of a non-dividend paying stock is 40 and the continuously compounded risk-free interest rate is 8%. The following table shows call and put option premiums for three-month European of various exercise prices: Exercise Price Call Premium Put Premium 35 6.13 0.44 40 2.78 1.99 45 0.97 5.08 A trader interested in speculating on volatility in the stock price is considering two investment strategies. The first is a 40-strike straddle. The second is a strangle consisting of a 35-strike put and a 45-strike call. Determine the range of stock prices in 3 months for which the strangle outperforms the straddle. (A) The strangle never outperforms the straddle. (B) 33.56 K. Determine an algebraic expression for the investor's profit at expiration.Evaporating Black Holes 52 electron-positron pairs In 1973, Stephen Hawking deduced that rotating black hole can evaporate and lose mass, thanks to the quantum mechanical properties of 'empty' space. electron-positron pairs Pairs of electrons and anti- electrons are constantly appearing and disappearing in space. If this happens near the event horizon, one particle escapes, while the other carries 'negative mass' into the black electron-positron pairs hole. This causes the black hole to lose mass. The formula for the evaporation time of a black hole with a mass of M in kilograms is given by 1 = - 102567 G'M 3 he The formula for the temperature of a black hole with a mass of M in kilograms is given by T = = 167 GMk where *= 3.141, and Boltzmann Constant k = 1.3806503 x 102 m kg s* K'. Planck's Constant h = 6.628x10 Joules sec, and the Newtonian Constant of Gravity G = 6.67x10' Newton Meter /kg Problem 1 - By substituting the values for the physical constants into the two formulae above, show that t(seconds) = 8.39 x 10"" M' and T (Kelvins) = 1.23x102 / M where M is the mass of the black hole in kilograms. Problem 2 - The universe has an age of 13.7 billion years. What is the mass of a black hole that, by now, should have completely evaporated if it had formed at the Big Bang? (Assume 3.1 x 10' seconds = 1 year)
