# Marketing-AW617

## Marketing-AW617 Online Services

Question 1 [10 Points].The Option for the Undecided. A call option has a positive payoff if a stock is above a strike, so you would buy a call option if expect stock price increase. A put option has a positive payoff if a stock price is below a strike, so you buy a put option if you expect stock price to decrease. But what do you do if you expect the stock price to move, but you are not sure in which direction? Maybe up, maybe down?

Consider the following option. The stock price right now is S_0=100.00. The maturity of the option is two years, T = 2.0. If the stock price at maturity, S_T is greater or equal to X_1=120.00, then you get paid 95% of the difference between the stock price and X_1, 0.95*(S_T-X_1 ). If the stock price at maturity, S_T is smaller or equal to X_2=90, then you get paid 75% of the difference between X_2 and the stock price, 0.75*(X_2-S_T ). For all other stock prices, the option pays you zero. In sum, the payoff on the option is

H(S_T )={■(0.75*(90-S_T ),&S_T≤90@0.00&90here.

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Question 2 [15 Points].The Option to Exchange one Asset for Another. This option belongs to the class of “Rainbow Derivatives Contracts” (see class notes for numerical methods). There are two stocks, Pfizer (PFE) and Merck (MRK). Stock prices today are: S_(0,PFE)=\$40, S_(0,MRK)=\$40. The correlation between returns on these stocks is ρ=0.75. The standard deviations are: σ_PFE=0.35,σ_MRK=0.25. The risk-free rate is r=2%. Time to maturity of the option is T=1.5 years. The stocks do not pay dividends.

The option gives you the right, but not the obligation, at maturity to give away 1 (one) share of Merck and to receive 1 (one) share of Pfizer.

Set up the spreadsheet to price this option by Monte Carlo simulation. For each simulation (each row) you will have to generate two uniform random variables, then convert each uniform random variable into a normally distributed random variable, then turn these two uncorrelated normal random variables into two correlated random variables (see class notes on Rainbow derivative contracts).

1. Use 5,000 simulations (rows) to determine the price and the standard error of the price of this option.

2. Use 15,000 simulations (rows) to determine the price and the standard deviation of the price of this option.

3. What happens to the price of the option if correlation between the two stocks decreases to become ρ=-0.25? What happens to the price if the correlation increases to ρ=0.975?

Question 3 [5 points].Black-Scholes Option Pricing.Consider a regular call option. Stock price today is S = 50.00. Strike price is X = 60.00. Time to maturity is 18 months. Risk free rate is 4.5%. The call option is selling for \$5.0593. What is the implied volatility (sigma)? [Use Black-Scholes call option pricing formula to find implied volatility. Program it in Excel and use Solver to find the value of sigma).

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