Only looking for answer to #9. #8 At t=0, the price of a certain stock is S(0)=$

Question

Only looking for answer to #9.

#8 At t=0, the price of a certain stock is S(0)=$50. At t=1, the price is either S(1)=$80 or S(1)=$30. A certain option contract is worth $10 if the stock price is $80, and is worth $0 if the stock price is $30. Assuming no arbitrage opportunities, and continuously compounded interest of 5%, what is the price of the option at time t=0?

#9 Suppose we are in the situation of problem 8, but a certain bank thinks that the option should be worth $5 for some reason and they are willing to sell you options at $5.1 and buy options from you at $4.9. Choose a portfolio of x shares of the stock and y options that you buy or sell at time 0 that will guarantee you a net return of $10^6 dollars at time 1.

Explanation / Answer

Answer to 1st part :-

Lets take the rate as r=5 %, which is compounded continously, so in Year 1 the interest will be Square of 1.05 which will be 1.1025(R).

Given U(Upward movement)= 80/50 = 1.60 and D(Downward movement) = 30/50 = 0.60.

P(Probablity of movement) = R - D/ U- D, i.e. 1.1025 - 0.60/ 1.60-0.60 = 0.5025.

So Probablity of upward movement is 50.25 % and Probablity of downward movement is 49.75 %.

Now using the binomial theorom we can calculate the Price of call at t=0 as follows:-

0.5025 * 10 (+) 0.4975*0/1.05 = 4.7857.

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