Q1: Consider a binomial tree problem for an American option. On December 8^th, 2022, the stock price of APPLE was $174.07.

1. Use one year of historical data to estimate appropriate values for u and d for the use in a two-step binomial option pricing model with a time period of 1 day. The historical stock prices are given in the data.xlsx file. (Tips: u=et;d=1u. The volatility () is calculated as the standard deviation of past daily returns.)

2. Estimate the price of an American call option on APPLE stock with a strike price of $134 and a maturity of December 12^th, 2022 (the duration is two trading days). Assume that the continuously risk-free rate is 1.76% per annum.

1 Here are the price history of calls and puts written on APPLE stocks with the strike price = 131, and the expiry date on Dec 9,2022