Risk-neutral pricing
Definition · Level 10 · Derivatives pricing
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Value = expected payoff discounted at the risk-free rate, computed as if every asset drifted at r. Justified by hedging, so the real drift and investors’ risk appetite drop out.
Example
Analysts’ 90% odds of a rally don’t change the option price.
Where Tradecraft teaches it
Level 10 · Derivatives pricing, in the lesson “Black–Scholes, risk-neutral pricing & the binomial tree”: What BS assumes, what N(d1) and N(d2) mean, the 0.4 rule, √T scaling, the smile and a one-step tree.
Related terms
- 0.4 ruleAt-the-money-forward call or put ≈ 0.4 × S × σ × √T (small dividends, modest σ√T), because 1/√(2π) ≈ 0.4.
- Binomial risk-neutral probabilityUp-move weight q = (1 + r − d)/(u − d) that makes the stock earn the risk-free rate in a one-step tree; option = discounted q-weighted payoff.
- N(d1)Black–Scholes call delta (times e^(−qT) with dividends); also the in-the-money probability measured with the stock as numeraire.
- N(d2)Risk-neutral probability that a call finishes in the money; discounted at r, it is the price of a digital call paying 1.
- Replicating portfolioPosition in the underlying plus borrowing or lending that reproduces an option’s payoff; its cost must equal the option’s price or there is an…
- Arbitrage boundsPrice limits any option must respect or someone locks in riskless profit: call ≤ S, European put ≤ K·e^(−rT), call ≥ max(0, S − K·e^(−rT)) without…