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As a new graduate in RBC Capital Market's rotational program, I have been assigned to the derivatives desk in the front office. My boss has

As a new graduate in RBC Capital Market's rotational program, I have been assigned to the derivatives desk in the front office. My boss has asked me to determine the fair value of a call option and a put option on AAPL with an expiration date of February 18 and an exercise price of $165 using a two-step binomial tree. It is currently January 25 and AAPL is closed at $159.78. According to my research on Yahoo Finance, the current price of these options on January 25 is $4.00 for the call option and $9.55 for the put option. As these options are American style, I will need to consider the possibility of early exercise at each node before the terminal nodes of the tree. I will also assume that there are no dividends until the maturity date. In order to compute the "up" and "down" factors, I need to determine AAPL's stock volatility. I will use 3-year volatility from FactSet, as I believe it is the most relevant measure for the February 18 expiration. On January 25, it is 33.06%. I will use a three-month T-bill rate of 0.17% per year, which is continuously compounded, but will adjust it to 0.42% following the FOMC announcement on January 26. I am aware that the put-call-parity equation is only applicable to European options on the same underlying with the same strike price and maturity, but I will still evaluate the equation for these options

What is the fair value for these call and put options? Is there any possibility that either call or put will be exercised early?

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