(a) (5 points) Assume that the price S of a stock follows geometric Brownian motion. Its current value is $50, with yearly expected return 12% and yearly volatility 30%. What is the probability that the stock price will be larger than $80 in two years? (b) (5 points) As in (a) a European put is written on the stock with expiration date 6 months from now and exercise price $60. What is the probability that this put will not be exercised? (c) (5 points) Suppose a European call option is currently priced at c = $110. You want to estimate the volatility, o, by trial and error assuming o is the value which makes Black-Scholes formula for the call price c equal to $110. You start with an initial guess of o = 0.30 that gives c = $115. What should be your next guess for o? Explain. (d) (5 points) Consider the binomial option pricing model for a European put with exercise price $104, current stock price $100. Let u = 1.25 and d = 0.8 for a 40-period binomial tree. Find the maximum number of up-movements so that the put will be "in the money (meaning the strike price is higher than the underlying stock price) at expiration. (a) (5 points) Assume that the price S of a stock follows geometric Brownian motion. Its current value is $50, with yearly expected return 12% and yearly volatility 30%. What is the probability that the stock price will be larger than $80 in two years? (b) (5 points) As in (a) a European put is written on the stock with expiration date 6 months from now and exercise price $60. What is the probability that this put will not be exercised? (c) (5 points) Suppose a European call option is currently priced at c = $110. You want to estimate the volatility, o, by trial and error assuming o is the value which makes Black-Scholes formula for the call price c equal to $110. You start with an initial guess of o = 0.30 that gives c = $115. What should be your next guess for o? Explain. (d) (5 points) Consider the binomial option pricing model for a European put with exercise price $104, current stock price $100. Let u = 1.25 and d = 0.8 for a 40-period binomial tree. Find the maximum number of up-movements so that the put will be "in the money (meaning the strike price is higher than the underlying stock price) at expiration