Explain replication and valuation under risk neutrality
The idea
Risk-neutral valuation does not claim investors are indifferent to risk. It is a computational device: the risk-neutral probabilities are the weights under which the underlying's expected return equals the risk-free rate, and using them lets the expected payoff be discounted at that rate instead of at an unknown risk-adjusted rate. It gives the same answer as replication because both rest on the same no-arbitrage condition. The device generalises far beyond the binomial tree and underlies essentially all derivative pricing.
Formulas
Risk-neutral probability
π = [(1 + r) − d] ÷ (u − d)
- π
- the risk-neutral probability of an up move
- r
- the risk-free rate for the period
- u
- the up factor, the multiple the underlying rises by
- d
- the down factor
Not anybody's forecast. It is the probability that would make the underlying grow at the risk-free rate, and it is what the replicating portfolio implies — the option can be hedged, so what investors believe about the odds drops out of the price.
A practice question
What do risk-neutral probabilities represent?
- The market's true forecast of future pricesWhy not: They are a computational device. The real probabilities cancel out of the argument and are never needed.
- Weights under which the underlying's expected return equals the risk-free rate — correct
- The historical frequency of up movesWhy not: History plays no part; the weights are derived from today's prices and the risk-free rate.
They are derived from the no-arbitrage condition, not estimated from data. Their role is to make discounting at the risk-free rate valid.
A harder one
Why do replication and risk-neutral valuation give the same option price?
- Because both assume investors are indifferent to riskWhy not: Neither assumes it. Risk aversion is already embedded in the observed price of the underlying.
- Because both ignore the time value of moneyWhy not: Both handle it explicitly, through borrowing in one case and discounting in the other.
- Because both are consequences of the same no-arbitrage condition — correct
Replication builds the payoff from traded instruments; risk-neutral valuation is the algebraic restatement of that construction. Neither is an assumption about preferences.
The hardest tier
A candidate objects that risk-neutral valuation is invalid because investors are plainly risk averse. What is the correct response?
- Risk aversion is already in the underlying — correct
- Risk aversion is irrelevant because options are hedgedWhy not: Hedging is how replication works, but the answer to the objection is where the risk aversion has already been priced in.
- The objection is correct and the model applies only to risk-free assetsWhy not: The method is used to price risky assets specifically, and it is standard throughout derivative pricing.
The risk-neutral probabilities are derived from the observed price of the underlying. The technique values the derivative relative to that underlying, so whatever risk premium the market demands is already in the price and need not be estimated again.
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