Question

PLZ FAST!

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

Ans a)

We have to calculate Option Delta first

Delta = Spread of Call Option /  Spread of Share Price

Spread of Call Option = Pay off in Up move from call - Pay off in down move from call  

= ( 30 - 26) - 0  

= 4

Pay off in down move is zero due to security price is below the option price

Spread of Share Price = Pay off in Up move from share -  Pay off in down move from share

= 30 - 18 = 12

Delta = 4 / 12 = 1 / 3 = 0. 333

So the Portfolio Should have long 0.333 Share to Short 01 Call. (Ans)

Portfolio Value in Up move

Vu = Pay off from Call + Pay off from Share * 0.333 = - ( 30 -26) + 0.333 * ( 30 - 24 ) = -- 4 + 2 = - 2

Vd = Pay off from Call + Pay off from Share * 0.333 = 0 + 0.333( 18 -24) = - 2

Now Pay off from the Portfolio will be = (Present Value of Call Premium - Interest Rate ) + Portfolio Value

Amount Required for Short Sell = 0.333 * Share Price = 0.333* 24 = 8

Interest for that Amount = Amount * Risk free rate = 8 * 0.05 = 0.40

Present Value of Call Premium = Call Premium * ( 1+ Risk free Rate) = 2.86 * 1.05 = 3

[ Premium received at the time of selling ]

Pay off from the Portfolio will be = ( 3 - 0.40) - 2 = 0.60

At expiration, risk-neutral profit will be = 0.60