A stock currently trades at $100, and its price can increase or decrease by 10%

Question

A stock currently trades at $100, and its price can increase or decrease by 10% over the following year. The risk-free rate is 3% per year. (a) We have call and put options written on this stock with strike prices of $105. What are the prices of these options? (b) How many shares would you need to buy to replicate the call option; that is, what is of the call option? (c) Suppose now that the stock price today is $101 instead of $100. What is the new price of the call option? (d) Suppose now that the stock price is again $100, but that instead of increasing or decreasing by 10% over the following year, the stock price will increase or decrease by 30%. What are the new prices of the call and put options?

Explanation / Answer

Stock price = $ 100

u = 10%

d = 10%

so the stock price can move to either $ 110 (case 1) or $90 (case 2)

now in case of call option, strike price $ 105 for case 1 would give us valuation of call option as $ 5 (which is 110-105)

or in case 2, valuation would be 0, since in this case it wont be exercised

now we know 110x-5=90x

thus x=0.25

So the present value of price of the call option = 90 * 0.25 * e^(-.03*1) = $ 21.83, and similarly can be found out for put option.

Now if the stock price is 101,

then case 1 would give us = $ 111.1

case 2 would give us = $ 90.9

then valuation of call in case 1 : $ 6.1

case 2 : 0

or 111.1y - 6.1 = 90.9y thus y = 0.30

so option price = 90.9 * 0.3*e^(-.03) = $ 26.46

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