Delta

Formula

d1 = [ln(S/K) + (r − q + σ²/2) T] / (σ √T)

Δ_call = e^{−qT} N(d1)
Δ_put  = e^{−qT} (N(d1) − 1) = −e^{−qT} N(−d1)

Δ_call − Δ_put = e^{−qT}

q = 0 is the no-dividend textbook case: Δ_call = N(d1), Δ_put = N(d1) − 1.
The on-site calculator uses the same expressions, including q.

How to read the number

• A call delta of 0.50 is not automatically at-the-money once rates or dividends are nonzero. It is the model slope at those inputs. N(d1) equals a half when d1 is zero, which is not the same as S = K.
• Delta of a position is the sum of signed deltas times the multiplier. The function returns the per-unit derivative, not a share count for a book.
• The slope changes as spot moves. That second derivative is gamma. The slope also depends on the volatility you typed in, which is where implied volatility enters.

Pitfalls

• European Black-Scholes-Merton only. American puts, barriers and anything path dependent do not have this delta. At expiry the slope becomes a step. The Greeks calculator spells out that limit; this function requires T > 0 and σ > 0 so the formula stays the smooth one.
• Spot delta versus forward delta. The derivative here is with respect to spot, and it includes the dividend discount e^{−qT}. A forward delta omits that discount. Mixing them mis-hedges by that factor.
• A hedge ratio is not a hedge result. Replacing the option with −Δ shares cancels the first-order spot term in this model, at this instant. Gamma, vega and discrete rebalancing remain.

Python reference implementation

Requires scipy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download delta.py · All snippets

from math import exp, log, sqrt
from scipy.stats import norm

def bs_delta(S, K, T, r, sigma, call=True, q=0.0) -> float:
    """Black-Scholes-Merton spot delta. call=False returns the put delta.

    Requires positive S, K, T and sigma. q is a continuous dividend yield.
    """
    if S <= 0 or K <= 0 or T <= 0 or sigma <= 0:
        raise ValueError("S, K, T and sigma must be positive")
    d1 = (log(S / K) + (r - q + 0.5 * sigma * sigma) * T) / (sigma * sqrt(T))
    disc = exp(-q * T)
    return float(disc * norm.cdf(d1) if call else disc * (norm.cdf(d1) - 1.0))

FAQ

Why is call delta not exactly N(d1)?

N(d1) is the no-dividend formula. With a continuous yield q the spot delta of a European call is e^{−qT} N(d1). The two agree when q is 0, which is the default of this function.

Does delta tell me the probability the option finishes in the money?

No. Under the stock-numeraire measure, N(d1) is a probability related to the share, and N(d2) is the risk-neutral probability that a European call finishes in the money, in this model, with no dividends changing that statement. Neither one is a real-world probability, and neither one is a trading instruction.