AppliedMultiple choiceFree solution

You are convinced a stock will rise with probability 0.90.9 rather than the 0.50.5 implied by a binomial tree. What does that do to the option price you should quote?

  1. ARaises it, in proportion to your belief
  2. BNothing – the price is set by the hedge
  3. CLowers it, because the option is more likely to be exercised
  4. DRaises the implied volatility you should quote
Show the answer and worked solution

Answer: B – Nothing – the price is set by the hedge

Nothing. The replicating portfolio reproduces the option payoff in both states, so its cost is the price whatever the real probabilities are, and those probabilities cancel out of the algebra entirely. If you quoted higher because you were bullish, a counterparty could sell you the option, buy the hedge and lock in the difference regardless of what the stock did. The right response to a genuine 0.90.9 belief is to buy the option at the quoted price, or buy the stock: a directional view is expressed through the position, never through the volatility you quote.

Worked solution

  1. Formula
    V=ΔS0+B,Δ=fu−fdS0(u−d)V = \Delta S_0 + B, \qquad \Delta = \frac{f_u - f_d}{S_0(u-d)}
  2. Substitute
    real probability p appears nowhere\text{real probability } p \text{ appears nowhere}
  3. Solve
    the hedge matches the payoff in both states\text{the hedge matches the payoff in both states}
  4. so its cost is the price, whatever p is\text{so its cost is the price, whatever } p \text{ is}
  5. Answer
    The price does not move\text{The price does not move}

Sanity check. If it did, a counterparty could sell you the option, buy the hedge and lock in the difference regardless of the outcome.

  • A. If it did, someone could hedge against you for a certain profit.
  • B. Correct. The real probability cancels out of the replication. If you genuinely believe 0.9, buy the option at the quoted price rather than re-pricing it.
  • C. Wrong direction, and the premise is wrong: the real probability does not enter at all.
  • D. A directional view is not a volatility view; the tree width is unchanged.

Takeaway: The real probability of an up-move never enters an option price.

Answer it in practice – your answer is marked and recorded.

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