Final answer:
The price of a European call option can be calculated using the binomial option pricing model, considering the stock's current price, strike price, interest rate, time to expiration, up and down factors, and the absence of dividends. The risk-neutral probability is used to find the expected payoff, which is discounted to present value to determine the call option price. Arbitrage opportunities are identified by comparing this price to observed market prices.
Step-by-step explanation:
Calculate the Price of a European Call Option
To calculate the price of a European call option using the given parameters—where S = $100 is the current price of the stock, K = $90 is the strike price, r = 8% is the risk-free interest rate, T = 0.5 is the time to expiration in years, and δ = 0 is the dividend yield—you would use the binomial option pricing model as it is suitable for a discrete period of time, represented by n = 1, and the up and down factors, u = 1.3 and d = 0.8, respectively.
To find the call option price, you calculate the possible prices at expiry, then find the payoff of the call option at these prices, discount the expected payoff to the present value, and adjust for the probability of reaching these payoffs.
If the stock price goes up (uS), the call value is max(uS - K, 0). For a down move (dS), the value is max(dS - K, 0). In this case, uS = $130, and dS = $80. So the up-move call value is $40, and the down-move call value is $0 (since the stock price is below the strike price).
Using the risk-neutral probability p = (erT - d) / (u - d), we can calculate p and use it to find the expected payoff. Here,
p = (e0.08*0.5 - 0.8) / (1.3 - 0.8) which is approximately equal to 0.6847.
The expected payoff is (p*uS_call_price + (1-p)*dS_call_price), and the present value of this expected payoff is the call option price. The present value is e-rT * (expected payoff).
After calculating the present value, with the parameters given, we arrive at the price of the European call option, which should then be compared with the observed market prices to identify potential arbitrage opportunities.