Final August 12, 2020 Name: Show all your work. Indicate clearly the methods you use because you will be graded on the correctness of your methods as well as on the accuracy of your final answers. All...

In a word document, complete the final exam. Show all your work. Indicate clearly the methods you use because you will
be graded on the correctness of your methods as well as on the accuracy of your final answers. Provide all relevant solutions performed in excel, as well


Final August 12, 2020 Name: Show all your work. Indicate clearly the methods you use because you will be graded on the correctness of your methods as well as on the accuracy of your final answers. All Financial Data for this exam are posted on Blackboard. 1 Problem 1 (20 points) Assume an asset price St follows the geometric Brownian motion, dSt = µStdt + σStdWt, S0 = s > 0 where µ and σ are constants and r is the risk-free rate. 1. Using the Ito’s Lemma find the stochastic differential equation satisfied by the process Xt = S n t , where n is a constant. 2. Compute E[Xt] and Var[Xt]. 2 Problem 2 (20 points) Consider a position X that follows the uniform random variable over the interval (-10, 100). 1. For some given h compute the V aRh(X 3) 2. Compute the CV aRh(X 3) and the AV ARh(X 3). 3 Problem 3 (20 points) We want to price options using the binomial lattice. The current stock price is 105 and the strike price is 100. Assume that the stock up-trend rate is u = 1.1 with probability p = 0.4 and the down-trend rate is d = 0.9 with probability 1 − p = 0.6. The annual risk-free rate is r = 0.01. Assume that the length of a period is one month. 1. Construct a binomial lattice that gives the price of a 5-month European call option. 4 2. Construct a binomial lattice that gives the price of a 5-month American put option. 5 Problem 4 (40 points) Consider Apple Inc. as the underlying asset, use its daily adjusted closing prices from August 12, 2019 to August 11, 2020 as historical data. 1. Estimate the daily standard deviation of the returns of this stock. 2. Deduce the yearly standard deviation. Consider the yearly standard deviation as the volatility of the stock and use the rate r = 0.0125 as annual risk-free rate. Assume you want to build a portfolio of options containing one call option with strike K1 = 420, and one put option with strike K2 = 460. 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 11, 2020 as the initial stock price. 3. Compute the option prices C1, P2 on that date. 6 4. Compute the Delta (∆) of this portfolio. 5. Compute the Gamma (Γ) of this portfolio . 7 6. 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 find n. 8