Question

Suppose that a stock price is currently 54 dollars, and it is known that five months from now, the price will be either 21 percent higher or 21 percent lower. Find the value of a European put option on the stock that expires five months from now, and has a strike price of 52 dollars. Assume that no arbitrage opportunities exist, and a risk-free interest rate of 5 percent.

Answer 1

Price of stock will be either 21 percent higher or 21 percent lower (only 2 possibilities).

So, we will use binomial pricing model to price the option.

Price of stock 5 months from now = 54 x (1+21%) or 54 x (1-21%)

Price of stock 5 months from now = $65.34 or $42.66

Value of put option if prices goes up ($65.34) = 0

Value of put option if prices goes up ($42.66) = 54-42.66 = $11.34

For calculating risk neutral probabilites

let Probability of stock price going up = p

Therefore probabilty of stock price going down = (1-p)

(1+21%) p + (1 - 21%) (1-p) = (1+5%x5/12)

1.21p -0.79p = 1.02083 - 0.79

Therefore p =54.96%

Probability of stock price going up = 54.96%

Probability of stock price going down = (1-54.96%) = 45.04%

Value of option after 5 months = 0 * 54.96 + 11.34 * 45.04% = $5.107

Value of put option now = $ 5.11 / (1+Riskfree rate) = 5.11 / (1+5%*5/12) = $ 5.00

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