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Explain replication and valuation under risk neutrality

CFA® Level I · Derivatives and Risk Management · Valuing a Derivative Using a One-Period Binomial Model · DE.9.b

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

From the question bank, with the reason each wrong option is wrong.

What do risk-neutral probabilities represent?

  1. The market's true forecast of the chance of each price moveWhy not: They are a computational device. The real probabilities cancel out of the argument and are never needed.
  2. Weights that make the expected return equal the risk-free rate - correct
  3. The historical frequency of up moves in past price dataWhy not: History plays no part; the weights are derived from today's prices and the risk-free rate.

Risk-neutral probabilities are not forecasts. They are weights worked out so that the expected price, discounted at the risk-free rate, equals today's price. Formula: p = (1 + r − Down factor) / (Up factor − Down factor). Example: price 100, going to 120 or 90, zero rate. Solve p × 120 + (1 − p) × 90 = 100, which gives p = 1/3, and 1/3 × 120 + 2/3 × 90 = 100. Investors may think an up move is more or less likely, but that does not matter. The weights only make discounting at the risk-free rate valid.

A harder one

Why do replication and risk-neutral valuation give the same option price?

  1. Because both assume investors are indifferent to riskWhy not: Neither assumes it. Risk aversion is already embedded in the observed price of the underlying.
  2. Because both ignore the time value of moneyWhy not: Both handle it explicitly, through borrowing in one case and discounting in the other.
  3. Because both are consequences of the same no-arbitrage condition - correct

Replication builds a portfolio of the underlying and borrowing that pays exactly what the option pays, then uses its cost as the option's price. Risk-neutral valuation takes the two option payoffs, weights them with risk-neutral probabilities and discounts at the risk-free rate. Option value = [p × Up payoff + (1 − p) × Down payoff] / (1 + r). It is the same arithmetic written another way. Both come from the no-arbitrage condition, so they must agree. Neither assumes anything about investors' attitude to risk.

The hardest tier

A candidate objects that risk-neutral valuation is invalid because investors are plainly risk averse. What is the correct response?

  1. Risk aversion is already in the underlying - correct
  2. 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.
  3. 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 objection sounds fair but misses a point. The risk-neutral probabilities are worked out from the observed price of the underlying. Whatever extra return investors demand for bearing risk is already built into that price. The method values the option relative to the underlying, so the risk premium (the extra return for taking risk) does not need to be estimated again. Rule: Option value = [p × Up payoff + (1 − p) × Down payoff] / (1 + r), where p comes from today's prices. It prices risky assets, not only risk-free ones.

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