d1 and d2
Definition · Level 11 · Pricing toolkit
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The two numbers fed into N() in Black–Scholes: d1 = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T) and d2 = d1 − σ√T.
Example
S = K = 100, T = 1, r = 5%, σ = 20% → d1 = 0.35, d2 = 0.15.
Where Tradecraft teaches it
Level 11 · Pricing toolkit, in the lesson “Black–Scholes & risk-neutral pricing”: Price an option by building a hedge: the formula, N(d1) and N(d2), why real-world odds do not matter, and the 0.4 rule for quick estimates.
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.
- N(d1)The Black–Scholes call delta (times e^(−qT) with dividends): the number of shares that hedge one call, and the weight on the stock in the price…
- N(d2)The risk-neutral probability that a call finishes in the money; discounted at r, it is the price of a digital call paying 1.
- Risk-neutral pricingValue = expected payoff discounted at the risk-free rate, computed as if every asset drifted at r.
- Arbitrage boundsPrice limits any option must respect, or someone locks in a riskless profit: call ≤ S, European put ≤ K·e^(−rT), call ≥ max(0, S − K·e^(−rT))…
- Backward induction (pricing tree)Solving a problem from the last step back to the first.