European Call Price

Q: In the BSM call formula, N(d_2) represents:

The risk-neutral probability that S_T > K at expiry. N(d_2) is the risk-neutral probability that the call finishes in the money (S_T > K). It multiplies the discounted strike K e^-rT to give the expected cost of exercise under the risk-neutral measure. N(d_1), not N(d_2), is the option delta. Since d_1 > d_2, we always have N(d_1) > N(d_2). The delta exceeds the exercise probability because the delta also accounts for the fact that, conditional on exercise, the stock tends to be worth more than the strike.

The Black-Scholes-Merton (BSM) European call price is $C = S_0 N(d_1) - K e^{-rT} N(d_2)$ , where $N(\cdot)$ is the standard normal CDF, $r$ is the continuously compounded risk-free rate, and $T$ is time to expiry in years. The formula prices a European call under the assumptions of geometric Brownian motion, constant volatility, no dividends, and frictionless markets. It was derived by Black and Scholes (1973) and extended by Merton (1973), and remains the benchmark against which all subsequent single-asset option models are measured.

Try it yourself

Black-Scholes explorer

Move the inputs and watch the call and put reprice. The smooth curve is the Black-Scholes value; the dashed kink is intrinsic value max(S−K, 0). The gap between them is time value (which can go slightly negative for a European put when rates are high).

Curve Black-ScholesDashed IntrinsicTime value A$6.63

Call valueA$6.63

Put valueA$4.65

d₁0.2121

d₂0.0707

N(d₁)0.584

N(d₂)0.528

Put-call parity: C − P = A$1.98 = S − Ke^−rT = A$1.98

Spot S100

Strike K100

Volatility σ20%

Maturity T0.50 yr

Rate r4.0%

Why it matters

Think of the call as a leveraged bet. You might end up owning the stock if the market rises (good), but only if you pay the strike (bad). $S_0 N(d_1)$ is the present value of receiving the stock conditional on exercise. $N(d_1)$ is the option delta, so this term answers 'how much of one share am I effectively long right now?'. $K e^{-rT} N(d_2)$ is the present value of the strike payment, scaled by $N(d_2)$ , the risk-neutral probability that $S_T > K$ at expiry. For an at-the-money ASX SPI 200 call with three months to expiry, $N(d_2)$ will be close to 0.5, meaning the market treats the outcome as roughly a coin-flip, and the call price reflects that uncertainty.

Before you read on — recall

In the BSM call formula, $N(d_2)$ represents:

Formulas

BSM call

C = S_0 N(d_1) - K e^{-rT} N(d_2)

d₁

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

d₂

d_2 = d_1 - \sigma\sqrt{T}

Worked examples

Scenario

Spot A$50, strike A$50, risk-free rate $r = 5\%$ p.a. (continuously compounded), volatility $\sigma = 30\%$ , time to expiry $T = 1$ year, no dividends. These parameters are representative of a mid-cap equity option on the ASX.

Solution

First compute $d_1 = [\ln(50/50) + (0.05 + 0.045) \times 1] / (0.30 \times 1) = 0.095/0.30 = 0.3167$ and $d_2 = 0.3167 - 0.30 = 0.0167$ . From standard normal tables, $N(0.3167) = 0.6243$ and $N(0.0167) = 0.5067$ . The call price is $C = 50 \times 0.6243 - 50 \, e^{-0.05} \times 0.5067 = 31.21 - 24.10 = 7.11$ , i.e. A$7.11. The no-arbitrage lower bound is $\max(50 - 50 e^{-0.05}, 0) = 2.44$ , and $C = 7.11$ well exceeds this, reflecting substantial time value.

Common mistakes

✗Students often equate the BSM call price with intrinsic value. The BSM price always exceeds intrinsic value $\max(S_0 - K, 0)$ for $T > 0$ and $\sigma > 0$ , because the option carries time value even when it is at the money or out of the money.
✗ $N(d_2)$ and $N(d_1)$ are often confused as the same probability. $N(d_1)$ is the option delta (sensitivity of $C$ to $S_0$ ), while $N(d_2)$ is the risk-neutral probability of expiring in the money. Because $d_1 > d_2$ always (for $\sigma > 0, T > 0$ ), $N(d_1) > N(d_2)$ . Confusing them misstates both the hedge ratio and the exercise probability.
✗The BSM formula does not apply to American calls. It prices European calls only. For a non-dividend-paying stock an American call has the same price as its European counterpart (early exercise is never optimal), but this equivalence breaks down once dividends are introduced.

Revision bullets

• $C = S_0 N(d_1) - K e^{-rT} N(d_2)$
•Delta $N(d_1) =$ option sensitivity to $S_0$
•Exercise probability $N(d_2) =$ risk-neutral prob. $S_T > K$
• $d_1 - d_2 = \sigma \sqrt{T}$ (always positive)
•Lower bound $C \geq \max(S_0 - K e^{-rT}, 0)$
•Time value $> 0$ for $T > 0$ and $\sigma > 0$
•European call only, with no dividends and constant $\sigma$

Quick check

In the BSM call formula, $N(d_2)$ represents:

Connected topics

d1 & d2 BSM Put

More in Black-Scholes-Merton

The Greeks BSM Intro Normal Dist.Z-table BSM + Divs Impl. Vol

In learning paths

Pricing Models Path

Sources

Black & Scholes (1973), JPE 81(3)
Fischer Black and Myron Scholes, 'The Pricing of Options and Corporate Liabilities,' Journal of Political Economy, Vol. 81, No. 3 (May–Jun. 1973), pp. 637–654, The University of Chicago Press.
Original derivation of the closed-form European call and put pricing formula using the no-arbitrage replicating portfolio argument.
Merton (1973), Bell J. Econ. 4(1)
Robert C. Merton, 'Theory of Rational Option Pricing,' Bell Journal of Economics and Management Science, Vol. 4, No. 1 (Spring 1973), pp. 141–183, The RAND Corporation.
Extends Black-Scholes using continuous-time stochastic calculus; establishes the risk-neutral valuation framework and the interpretation of N(d₂) as an exercise probability.
Hull (2022), §15.1–15.4
John C. Hull, Options, Futures, and Other Derivatives, 11th ed., Pearson, 2022. ISBN 978-0-13-693997-9.
Anchor textbook treatment of the BSM call formula, d₁/d₂ derivation, and the risk-neutral probability interpretation of N(d₂).
ASX, Understanding Options Trading (2022)
Australian Securities Exchange (ASX), 'Understanding Options Trading,' ASX Investor Resources, 2022. Available at asx.com.au.
Provides Australian market context for exchange-traded options (ETOs) on ASX-listed equities and the S&P/ASX 200 Index, illustrating where BSM pricing is applied in practice.

How to cite this page

Dr. Phil's Quant Lab. (2026). European Call Price. Derivatives Atlas. https://phucnguyenvan.com/concept/bsm-call

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