Question

Consider an increase in the volatility of the stock in the previous problem. Suppose that if the stock increases in price, it will increase to \(130, and that if it falls, it will fall to \)70. Show that the value of the call option is higher than the value derived using the original assumptions.

Short answer

Answer

$18.18

Step-by-step solution

Calculation of values at expiration

Two possible stock prices are: S+ = $130 and S– = $70.

Two possible call values are: Cu = $30 and Cd = $0 (as exercise price = $100)

Calculation of hedge ratio (H)

H = (Cu – Cd)/(uS0 – dS0)

= (30 – 0)/(130 – 70)

= 0.5

Now the cost of the riskless portfolio is: ($S_0-2C_0$) = 100 – $2C_0$

End-of-year value is $70 (given)

Calculation of present value

Present value of $70 with a one-year interest rate of 10%: $70/1.1 = $63.64

The value of the hedged position equal to the pre sent value of the certain payoff:

The value of the call option (Co):

Current price - 2$C_0$= Present value

$100 – 2$C_0$= $63.54

$C_0$= $18.18

Note that the value of the call is greater than the value of the call in the lower volatility scenario.