The time it takes for a black hole to evaporate through the process of Hawking radiation can be calculated using the following formula, in which M is the mass of the black hole in kilograms and t is the lifetime of the black hole in seconds: t=10240 x 2 GM hc4 = ( h = = 6.63 1034 kg-m G = 6.67 10-11 x S m kgxs Some scientists speculate that in the far distant future, the universe will consist only of gigantic black holes and scattered subatomic particles. The largest black holes that conceivably might form would have a mass of about a trillion (1012) Suns. 1. Use the formula to calculate the lifetime of a black hole with the mass of the Sun (MSun=2.0x1030kg) (MSun=2.0x1030kg). 2. How does your answer compare to the current age of the universe? Assume the current age of the universe is about 10101010 years. 1. Verify, by computing the partial derivatives, that the Black-Scholes formula given in class, C(s, t,T,02, K) = sN(d) - Ke (1) N(d2) solves the partial differential equation: ac ac 102C + TS+ -s = rC. t s 2 82 To do this, calculate each partial individually, and then show the PDE holds. *2. What are the prices of a call option and a put option with the following characteristics? Apply Black-Sholes model. Stock Price 83, Exercise price =80, Risk-free rate=3% per year, compounded continuously. Maturity - 4 months, Standard deviation = 0.23. (15 points) 8