Let's first check that 0.633 is the call option price. The risk-neutral probability $p^*$ of an up-tick in the stock is computed by assuming the stock earns a risk-free rate of return:
$$20 = e^{-0.03}(22p^* + 18(1-p^*)) \qquad \Rightarrow \qquad p^* = 0.65227.$$
The price of the option is the risk-neutral expected payoff, discounted at the risk-free rate. In this case, the option has payoff 1 in an up-tick and payoff 0 in a down-tick, so
$$c = e^{-0.03}p^* = 0.633.$$
The delta-hedge ratio needed for a replicating portfolio is the ratio of the change in the option price to the change in the stock price:
$$\Delta = \frac{1 - 0}{22-18} = 0.25.$$
If the option is trading at $C < 0.633$, then you should be able to capture the $0.633 - C$ spread by employing the buy-cheap sell-expensive strategy:
- Buy the option at $C$
- Sell $\Delta = 0.25$ shares of the stock
If $C = 0.62$, then selling $\Delta$ shares of the stock and buying the option gives you an initial cash position of
$$0.25*20 - 0.62 = 4.38,$$
which grows to $4.38e^{0.03} = 4.5133$ at expiry. We now consider two cases:
- $S_T = 22$: the option has payoff 1, and the short position has payoff $-0.25*22 = -5.50$, so the total payoff is $1 - 5.50 + 4.5133 = 0.0133$.
- $S_T = 18$: the option has payoff 0, and the short position has payoff $-0.25*18 = -4.50$, so the total payoff is $0 - 4.50 + 4.5133 = 0.0133$.
If the option is trading at $C > 0.633$, then again employ the buy-cheat sell-expensive strategy. The main point is that you can completely replicate the payoff of the option using just a portfolio of stock and cash; by the law of one price, the cash required to set up the replicating portfolio is the price of the option.
