Question

Consider a 1-year one period European call option where X = 26. The stock price is currently $24 and at the end of one year it will be either $30 or $18. The risk-free interest rate is 5%.

a. What position in the stock is necessary to hedge a short position in one call option? (5 points)

b. Assume C is equal to $2.86, what is the possible values of the portfolio you created in part (a) above at expiration (hint, find Vu and Vd)? (5 points)

Answer 1

a) Let long position in X units of Stock be taken along with one short position

Value of option in upmove= max(St-K,0)= max(30-26,0)=$4

Value of option in downmove= max(St-K,0)= max(18-26,0)=0

So, value of portfolio in case of upmove = value of portfolio in case of downmove

=> X*30-4=X*18-0

=>X=4/12=1/3=0.3333

A long position in 1/3 stock is necessary to hedge a short position in one call option.

b) If C= $2.86

Value of portfolio at expiration (in case of upmove) = X*30-4 = 1/3*30-4 = $6

Value of portfolio at expiration (in case of downmove) = X*18 = 1/3*18 = $6

So, the value of portfolio at expiration will be $6 irrespective of the value of call option today