# Put option valutionWatch

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IBM's share price is 100. In 3 months it can increase to 110 or fall to 95. From 110, it can increase to 120 or fall to 100. From 95, it can increase to 100 or fall to 90. What is the value of a 6 month put option on IBM with a strike price of 95?

Risk free interest rate is 4% per quarter

In 3 months the option is either worthless or worth pd in the down state. From
the down state its payoff is then either 0 if IBM’s price is 100 or \$5 if \$90.
Lending 50/1.04 and selling short 0.5 IBM shares pays off either 50 – 0.5*100
= 0 or 50 – 0.5*90 = 5 and replicates the put in the down state. Its cost is
(50/1.04) – (0.5*95) = 0.58. At date zero we need to replicate an option that is
worth either zero (if IBM goes to 110) or 0.58 (if IBM goes to 95). Short
0.58/15 = 0.038462 shares and lend 110x0.038462/1.04 = \$4.068. This costs
4.068 – 3.846 = \$0.222. So the put is worth \$0.222.

I don't see where the answer is getting the lending 50 from or the 50/50 probability... can someone explain please?
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