$d_1$ and $d_2$

Q: The relationship between d_1 and d_2 is

d_2 = d_1 - σ√T. By definition, d_2 = d_1 - σ√T. The shift σ√T is the total return volatility over the option's life. As T → 0 the two converge and the call collapses to max(S_0 - K, 0).

Q: N(d_1) in the BSM call formula is most directly interpreted as

The call's delta, shares of stock per option in the hedge. N(d_1) is the call's delta, the partial derivative ∂ C / ∂ S_0. The risk-neutral probability of exercise is N(d_2), the other piece of the formula. Mixing the two is the textbook trap.

$d_1$ and $d_2$ are the two inputs that feed into $N(\cdot)$ inside the BSM call formula. They are defined by $d_1 = [\ln(S_0/K) + (r + \sigma^2/2)T] / (\sigma\sqrt{T})$ and $d_2 = d_1 - \sigma\sqrt{T}$ . $N(d_1)$ is the call's delta (shares to hold per option), and $N(d_2)$ is the risk-neutral probability the call finishes in the money (Hull, 2022, §15.6).

Try it yourself

d₁ & d₂ on the normal curve

d₁ and d₂ are the two points fed into N(·) in the Black-Scholes formula. Move the inputs and watch them slide along the standard normal curve. The blue area up to d₂ is N(d₂), the risk-neutral probability the call ends in the money. Add the gold strip from d₂ to d₁ and the whole shaded area becomes N(d₁), the call's delta — so N(d₁) = blue + gold.

N(d₂) = P(ITM), blue 0.528N(d₁) = delta, blue+gold 0.584

d₁0.2121

d₂0.0707

N(d₁)0.584

N(d₂)0.528

d₂ = d₁ − σ√T, so d₂ always sits to the left of d₁ by the total volatility σ√T.

Spot S100

Strike K100

Volatility σ20%

Maturity T0.50 yr

Rate r4.0%

Why it matters

Think of $d_2$ as a standardised moneyness. It measures how far the log forward $\ln(S_0 e^{rT}/K)$ sits above zero, divided by the total return volatility $\sigma\sqrt{T}$ . When $d_2$ is large and positive, the option is deep in the money and almost certain to be exercised. $d_1$ is the same quantity shifted up by $\sigma\sqrt{T}$ , which adjusts for the fact that the conditional expected payoff of the stock at exercise is higher than its forward price.

Before you read on — recall

The relationship between $d_1$ and $d_2$ is

Formulas

$d_1$

d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}

For a stock paying continuous dividend yield

q

, replace

r

with

r - q

$d_2$

d_2 = d_1 - \sigma\sqrt{T}

The

\sigma^2/2

term in

d_1

is the Itô correction that pulls the median of the lognormal below its mean.

Worked examples

Scenario

At-the-money one-year call, $S_0 = K =$ A$50, $r = 5\%$ , $\sigma = 30\%$ , $T = 1$ .

Solution

$\ln(S_0/K) = 0$ , so $d_1 = [0 + (0.05 + 0.045)(1)] / (0.30 \sqrt{1}) = 0.095 / 0.30 = 0.3167$ . Then $d_2 = 0.3167 - 0.30 = 0.0167$ . From a z-table $N(d_1) \approx 0.6242$ and $N(d_2) \approx 0.5067$ . Delta is 0.6242 shares per option.

Scenario

Deep out-of-the-money call, $S_0 =$ A$40, $K =$ A$60, other inputs as above.

Solution

$\ln(40/60) = -0.4055$ . $d_1 = (-0.4055 + 0.095) / 0.30 = -1.035$ . $d_2 = -1.335$ . $N(d_1) \approx 0.150$ and $N(d_2) \approx 0.091$ . The risk-neutral probability of exercise is only about 9%, and the call's delta is roughly 0.15.

Common mistakes

✗ $d_1$ is the same as $N(d_1)$ . $d_1$ is the input to the standard normal CDF. $N(d_1)$ is the output between 0 and 1. Mixing the two is the single most common BSM arithmetic error.
✗ $N(d_2)$ is the real-world probability of exercise. It is the risk-neutral probability. Real-world probability uses the actual stock drift $\mu$ in the numerator instead of $r$ and is generally higher.

Revision bullets

• $d_1 = [\ln(S_0/K) + (r + \sigma^2/2)T] / (\sigma\sqrt{T})$
• $d_2 = d_1 - \sigma\sqrt{T}$
• $N(d_1)$ is the call delta
• $N(d_2)$ is risk-neutral probability of exercise
•Replace $r$ with $r - q$ for dividend yield $q$

Quick check

The relationship between $d_1$ and $d_2$ is

$N(d_1)$ in the BSM call formula is most directly interpreted as

Connected topics

BSM Intro Normal Dist.Z-table BSM Call BSM Put

More in Black-Scholes-Merton

The Greeks BSM + Divs Impl. Vol

In learning paths

Pricing Models Path

Sources

Hull (2022), §15.6
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Derives $d_1$ and $d_2$ from the lognormal expectation and gives the interpretation as delta and risk-neutral probability.
Black & Scholes (1973), JPE
Black, Fischer, and Myron Scholes. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81, no. 3 (1973): 637-654.
Original paper where $d_1$ and $d_2$ first appear inside the closed-form European call price.
Merton (1973), Bell J. Econ.
Merton, Robert C. "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4, no. 1 (Spring 1973): 141-183.
Generalises $d_1$ and $d_2$ to include a continuous dividend yield, replacing $r$ with $r - q$ .

How to cite this page

Dr. Phil's Quant Lab. (2026). $d_1$ and $d_2$. Derivatives Atlas. https://phucnguyenvan.com/concept/d1-d2