The value of a six-month European call option on the stock can be calculated using both no-arbitrage arguments and risk-neutral valuation arguments. Both approaches give us the approximate value of $8.15 for the call option.
The value of a European call option can be determined using both no-arbitrage arguments and risk-neutral valuation arguments.
No-arbitrage arguments:
By applying the principle of no-arbitrage, we can create a risk-free portfolio that replicates the payoffs of the call option. Let's assume we buy 1 option and short-sell x shares of the stock. The value of this portfolio at the end of six months can be expressed as:
$60 - x(Price of stock in 6 months) = $48. Solving this equation, we find x = 1/6.
So, the value of the portfolio today is:
$60 - (1/6)($50) = $51.67. Since this is the initial cost of the portfolio, it must be equal to the value of the call option today.
Risk-neutral valuation arguments:
Under the risk-neutral valuation approach, we assume that the expected return on the stock is the risk-free rate of interest, which is given as 12% per annum. Using this assumption, we can calculate the risk-neutral probability of the stock price reaching $60 or $42.
Let p be the probability of the stock price being $60. The probability of the stock price being $42 is then (1 - p). The expected value of the stock price at the end of six months is:
(p x $60) + ((1 - p) x $42)
Setting this equal to the risk-free rate of interest, we have:
(p x $60) + ((1 - p) x $42) = exp(0.12 x 0.5)
Solving this equation, we find p = 0.815.
The value of the call option can now be calculated using the risk-neutral probabilities:
Value of call option = (0.815 x $10) + (0.185 x $0) ≈ $8.15
Both the no-arbitrage arguments and risk-neutral valuation arguments give us the same value for the call option, which is approximately $8.15.