Question

2. Suppose So 4, sl (H) 8, si (T)-2 and the risk-free interest rate is r 0, Som eone is willing to buy or sell European Call options with strike price k = 10 for the price Vó 2. Explain why there exists an arbitrage opportunity; ie construct a portfolio which starts with nothing, has a positive chance of earning money and zero probability of losing money.

Answer 1

Let's create the following portfolio

1. Sell the call option. Proceed from sale of call option = V₀ = 2
2. Buy 0.5 number of stock at current price of S₀ = 4
3. Initial investment required = 2 - 0.5 x 4 = 0
4. Hence, there is no initial investment required.

Value of this portfolio in up state = Value of 0.5 number of stock + payoff from sold call option = 0.5 x S₁(H) - max (S₁(H) - K, 0) = 0.5 x 8 - max (8 - 10, 0) = 4

Value of this portfolio in down state = Value of 0.5 number of stock + payoff from sold call option = 0.5 x S₁(T) - max (S₁(T) - K, 0) = 0.5 x 2 - max (2 - 10, 0) = 1

The portfolio is making money in either state.

Thus we have created a portfolio by selling the call option and buying 0.5 number of stock that requires nothing but has a full chance of earning money and no chance of losing money.