Explain how a European option's value can be viewed as its expected payoff at expiry discounted to today
The idea
A European option is worth its expected payoff at expiry, taken with risk-neutral probabilities and discounted at the risk-free rate. In a two-period tree, the end nodes have probabilities π², 2π(1 − π) and (1 − π)², with the middle node counted twice because two paths reach it, and the weighted sum is discounted over both periods. Real-world probabilities and a risky discount rate would also work in principle, but the correct risky rate is unknown; the risk-neutral route sidesteps that. Investors' attitudes to risk affect the option only through the underlying's current price. The trap is using an analyst's forecast probability of an up move, which gives a value that is not arbitrage free.
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