Question

The price of ULL stock is currently $110. The stock will either increase by a factor of u=1.2 or decrease by a factor of d=0.6 in the following year. The annual risk-free rate is 3% and the stock does not pay dividends.

Consider a European call option, C₁, written on ULL stock with a strike price of $66 that matures in one year.

1. Suppose that an investor wants to create a portfolio today that generates the same payoffs in both states of the world in 1 year, using ULL stocks and the call option C₁. Calculate the number of ULL stocks that an investor should purchase for each call option C₁ that he sells.
   
2. What is the price of the call option C₁ today?

Answer 1

S is 110 (spot price )

Su = 1.2×110 = 132

Sd = 0.6×110 = 66

Strike price is 66

We use risk less Hegde approach to value the option here

We will create a portfolio by buying the stock and selling call option which shall have same cash flows for any expiration price

When price is 110 cash flow will be

110(h) - c

When price is 132 call option value will be 132-66 = 66

Portfolio = 132(h)-66

When price is 66 call value will be o

Portfolio = 66(h) -0

As we have discussed above Portfolio value shall be equal at all expiration prices

132(h) -66 = 66h

66(h)=66h

H = 1

So hedge ratio is 1

Cash flow from portfolio = 66(1) =6

We know present portfolio = 110(h)-c

This shall be equal to present value of cash flow from portfolio

Risk free rate is 3% and we know hedge ratio is 1

110(1) -c = 66/e^0.03

C = 45.95

So value of call at strike price 66 is 45.95