Question

Stock ABC is currently trading at $22.31. The annualized risk-free rate is 6%, compounded monthly, and the stock's annualized dividend yield is 5%. You want to buy a 9-month put option. What is the strike price of the 6-month option that is at-the-money?

Answer 1

Fair Future Price = S_o (1 + i - d)ⁿ

where, S_o = Spot Price of Stock; i = Risk rate; d = Dividend yield; n = Time Period

Requirement: Strike Price of 6-Month Option that is at the money (At the money refers to a point where Strike Price (K) = S_o )

Here,  S_o = $ 22.31; K = $ 22.31; n = 6 Months; d = 5%; r = 5%

Fair Future Price = $ 22.31 * (1 + 0.06 - 0.05)^6/12

Fair Future Price = $ 22.31 * (1.01)^0.5

Fair Future Price = $ 22.4213

Price of 6-Month Put Option = ($ 22.4213 - $ 22.31) / (1.01)^0.5

Price of 6-Month Put Option = 0.1113 / 1.005

Price of 6-Month Put Option = 0.1107 (approx)

Note:

Price of 9 Month Put Option:

Fair Future Price = $ 22.31 * (1 + 0.06 - 0.05)^9/12

Fair Future Price = $ 22.31 * (1.01)^0.75

Fair Future Price = $ 22.477

Price of 9-Month Put Option = ($ 22.477 - $ 22.31) / (1.01)^0.75

Price of 9-Month Put Option = 0.167 / 1.0075

Price of 9-Month Put Option = 0.1658 (approx)