Question

A stock is trading at $80 per share and will either increase or decrease by 15% over the next period. Assume a risk free rate of 6% and exercise price on the call of $80. 1. Find the value of the stock with an up move and with a down move. 2.b. Find the value at expiration of the call with an up and with a down move 3.c. Find the expected value of the call today. 4.d. Find the hedge ratio 5.e. If the price of call is currently $7.92 what riskless return can be earned using a riskless hedge.

Answer 1

1

Value of stock with upmove = $80*1.15 = $92

Value of stock with downmove = $80*0.85 = $68

2. Value at expiration of the call with Strike of $80

In case of upmove= ma(92-80,0) = 12

In case of downmove = max(68-80,0) =0

3. Consider a portfolio of long Δ shares and short 1 call option, the portfolio is riskless if at expiration , its value remain the same irrespective of change in stock price

In case of upmove , value of portfolio = 92*Δ - 12

In case of downmove , value of portfolio = 68*Δ

So, for portfolio to be riskless

92*Δ - 12 = 68*Δ

=> Δ = 0.5

and value of portfolio after the period

= 68*Δ = 68*0.5 = $34

So, present value of the riskless portfolio = 34/1.06 =32.07547

which is equal to long 0.5 shares and short 1 call option (worth C)

So, 0.5*80-C = 32.0757

=> C   = $7.92458

4. The Hedge ratio is the same as Δ  

So, hedge ratio = 0.5