The basic setup for all questions begins the same. Frederica Johnson has been offered the position as CIO of a large public technology company. As part of the deal, Frederica has been offered employee stock options in the company. The details of the option offer is as follows:

• The package is for 250,000 options over the company’s stock
• The strike price of each options has been set at $120 – that is the current stock price of the company – meaning the options have been priced at-the-money
• The options are European and mature in 5 years
• For the purpose of this questions, the annualized volatility of the company’s stock has been estimated at 30%
• The company does not pay a dividend and is not expected to over the next five years
• An appropriate annualized risk-free rate is 1.0% (professor’s current favorite)
• Question 1: Using the Black-Scholes model, what is each option worth today, and what are they worth in total?
• Question 2: Use a binomial tree approach to value the same options – please use “years” as the basic unit of time, meaning your tree should contain 5 steps. a) What are the ”up” and “down” factors for constructing the tree (values u and d in my examples)?
• b) Construct the price tree for the underlying stock using the opening stock price given, and the up and down factors calculated in a).
• c) At the terminal nodes of the tree, what is the option worth?
• d) What are the risk-neutral probabilities p and 1 – p?
• e) Using the terminal node values, the risk-neutral probabilities, and the risk-free rate, construct the equivalent price tree for the option.
• f) What is the value of the option using this approach?

• Question 3: Using the C-R-R equation, calculate the value of the option.

  Question 4: In questions 1, 2 and 3, the value of the option should be approximately the same (actually the answer from Q2 and Q3 should be identical). In this question, we add a “twist”. As part of the negotiation, the company offers Frederica the following “sweetener” – if the stock price is at $200 or above on her 3^rd or 4^th anniversary with the firm, the number of options will be increased to 300,000. The new 50,000 options are identical to the originals, same strike price, but of course, now with less time to expiration.[1] This only happens ONCE, so if the stock is above $200 on third and fourth anniversary, she only gets the “bump” once on the third anniversary. AND furthermore, if the stock subsequently falls below $200, she does not return the new options.

  How does this change the value of the options? Hint, the value of the option at the terminal nodes is now PATH DEPENDENT.