All data collection should be done from the following website: https://?nance.yahoo.com/

Problem 1
Assume an asset price St follows the geometric Brownian motion, dSt = µStdt + σStdWt, where µ and σ are constants and r is the risk-free rate.

1. Using the Ito's Lemma ?nd the stochastic di?erential equation satis?ed by the process Xt = Stn, where n is a constant.

2. Compute E[Xt] and Var[Xt].

3. Using the Ito's Lemma ?nd the stochastic di?erential equation satis?ed by the process Yt = Stert.

Problem 2
Consider these two stocks: AT&T Inc. (T) and Verizon Communications Inc. (VZ). Use the daily adjusted closing prices from March 1, 2016 to August 8, 2016 as historical data.

1. Estimate the mean rate of return and the standard deviation of each of these assets. Moreover estimate their correlation coe?cient ρ, and their covariance.

2. Using their correlation coe?cient ρ, ?nd the weight of each of these assets that will give an e?cient portfolio with minimum variance. Deduce the return of that portfolio.

Problem 3
We want to price options using the binomial lattice. The current stock price is 54 and the strike price is 50. Assume that the stock up-trend rate is u = 1.2 with probability p = 0.4 and the down-trend rate is d = 0.8 with probability 1-p = 0.6. The annual risk-free rate is r = 0.005. Assume that the length of a period is one month.

1. Construct a binomial lattice that show the evolution of the stock price during the 5 months.

2. Construct a binomial lattice that gives the price of a 5-month European call option.

3. Construct a binomial lattice that gives the price of a 5-month American put option.

Problem 4
Consider Apple Inc. as the underlying asset, use its daily adjusted closing prices from August 10, 2015 to August 8, 2016 as historical data. Estimate the daily standard deviation of the returns of this stock. Deduce the yearly standard deviation. Consider the yearly standard deviation as the volatility of the stock and use the rate r = 0.005 as annual risk-free rate. Assume you want to build a portfolio of options containing one call option with strike K1 = 100, and one put option with strike K2 = 110. Let C1(t,x) denotes the call option pricing function. Let P2(t,x) denotes the put option pricing function. Let the maturity T = 12 months. Using the adjusted closing price of August 10, 2016 as the initial stock price.

1. Compute the option prices C1, P2 on that date.

2. Compute the Delta (?) of this portfolio.

3. Compute the Gamma (Γ) of this portfolio.

4. Assume we want to build a new portfolio with 3 call options with strike K1 and n put options with strike K2. Is there a value of n that will make this new portfolio delta neutral? If yes ?nd n.

5. Assume that on August 16, 2016 the stock price will close at $111. Using the ?, Γ, Θ, estimate the value of the call option on that date.