A stock at 100 goes to 125 or 80 in one period with zero rates. Price the 100-strike call, give the replicating portfolio, and explain why the true probability of the up move plays no part.

A stock at 100 goes to 125 or 80 in one period with zero rates. Price the 100-strike call, give the replicating portfolio, and explain why the true probability of the up move plays no part.

Approach: Solve for the share and cash holding that reproduces both payoffs, price the option as the cost of that portfolio, then read the risk-neutral probability off the same equations.

100/9. The hedge ratio is Delta = (25 - 0)/(125 - 80) = 5/9 shares, and to match the down state the replicating portfolio borrows (5/9)*80 = 400/9, so the call costs (5/9)*100 - 400/9 = 100/9 = 11.11. The same answer comes from the risk-neutral probability p = (1 - 0.8)/(1.25 - 0.8) = 4/9, giving (4/9)*25 = 100/9. The true probability of the up move never enters because the payoff is exactly reproducible with stock and cash, so no-arbitrage fixes the price whatever anyone believes about direction. A trader who thinks the up move has a 90% chance still cannot quote the call above 11.11 without being arbitraged, and the correct expression of that view is to buy the stock.

Follow-up: The stock can also finish at 100, three states now. What happens to the replication and what is the widest arbitrage-free range for the call?

Key concepts: replicating portfolio, risk-neutral probability, hedge ratio, no-arbitrage.