Question 1: [Airline Dynamic Pricing]
You are the sales manager of an Airline. There is a flight that will take off in six days and there are still two available seats. Therefore, there are five days for you to sell them. Assume that in each day there will be a potential customer browsing your airline's booking website. The customer's valuation is random and is described by a uniform distribution. That is, if you price the ticket at p ∈ [0, 1], a customer has 1 - p chance to purchase the ticket. Unsold tickets generate no profit and overbooking is not allowed. Please find the optimal first-day price of a ticket.
Question 2: [Option Pricing]
There are three assets in the economy, stock, money, and call option. The stock price tomorrow can be either 5 or 15 depending on circumstances. The money price tomorrow is 1 independent of the economy. The call option has a strike price of 10 (if you own one unit of the call option, you are entitled to purchase a share of stock tomorrow at the strike price).
Suppose today's prices for a share of stock is 12, for money is 1, and for an unit of the call option is 4.
Question: Is there an arbitrage (A portfolio that has a negative price today and nonnegative value in all circumstances tomorrow)? If an arbitrage exists, please find one. If no arbitrage exists, explain why