Work out an option's delta.

FRM® Part I · Valuation and Risk Models · Option Sensitivity Measures: The “Greeks” · FRM1.VR.16.c

The idea

Delta is the sensitivity of an option's price to a small change in the price of the underlying. For a call, Delta = ∂c ÷ ∂S. For a European call on a stock with no dividends it equals N(d1), so it lies between 0 and 1. Here d1 = [ln(S ÷ K) + (r + σ²÷2) × T] ÷ (σ × √T). The put delta is N(d1) − 1, between −1 and 0. With a continuous dividend yield q, both deltas are scaled by a dividend factor, and d1 uses r − q in place of r. The call has Delta = e^(−qT) × N(d1). The put has Delta = e^(−qT) × [N(d1) − 1]. Do not confuse N(d1), the delta, with N(d2), the risk-neutral probability of exercise. The common errors are using N(d2), forgetting the dividend factor, and giving a put a positive delta. The exam usually asks you to compute N(d1) and then adjust for the put or the yield.

Formulas

Delta of a European call on an asset with a continuous yield

Δc = e^(−q × T) × N(d1)

q: the continuous dividend yield, zero for a stock with no dividends
T: the time to expiry in years
N: the standard normal cumulative distribution function
d1: [ln(S ÷ K) + (r − q + σ² ÷ 2) × T] ÷ (σ × √T)

With no dividends it is simply N(d1), between 0 and 1. Using N(d2), the exercise probability, is the usual mistake.

Delta of a European put on an asset with a continuous yield

Δp = e^(−q × T) × [N(d1) − 1]

q: the continuous dividend yield, zero for a stock with no dividends
T: the time to expiry in years
N: the standard normal cumulative distribution function
d1: [ln(S ÷ K) + (r − q + σ² ÷ 2) × T] ÷ (σ × √T)

Always negative. With a yield the put delta equals the call delta minus e^(−q × T), not minus one.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does the delta of a European call option measure, and over what range does it lie for a non-dividend-paying stock?

The change in the option price for a change in volatility; between 0 and 1.Why not: This is vega, not delta, and vega is not bounded by 1.
The change in the option price for a small change in the stock price; between 0 and 1. - correct
The change in the option price for a small change in the stock price; between -1 and 0.Why not: The measure is right but the range is that of a put delta.
The change in delta for a small change in the stock price; between 0 and 1.Why not: This describes gamma, the second derivative of the option price.

Delta = dC/dS. For a call it is N(d1) with no dividends, which lies between 0 and 1: near 0 for deep out-of-the-money options and near 1 for deep in-the-money options.

A harder one

A European call on a non-dividend-paying stock has S = 100, K = 95, r = 4% (continuously compounded), volatility 25% and T = 0.5 years. Its delta is closest to:

0.624Why not: N(d2) is the risk-neutral exercise probability, not the delta, which is N(d1).
0.689 - correct
-0.311Why not: N(d1) - 1 is the delta of the corresponding put.
0.647Why not: Drops the interest rate from d1; d1 includes (r + sigma²/2)T in the numerator.

d1 = [ln(100/95) + (0.04 + 0.25²/2) x 0.5] / (0.25 x sqrt(0.5)) = 0.4917. Call delta = N(d1) = 0.6885.

The hardest tier

A European call on an index paying a continuous dividend yield has S = 80, K = 85, r = 3%, q = 4% (both continuously compounded), volatility 22% and T = 1 year. Its delta, e^(-qT) N(d1), is closest to:

0.400 - correct
0.416Why not: Omits the e^(-qT) factor, which is needed when the underlying pays a dividend yield.
0.320Why not: Uses N(d2) in place of N(d1) before applying the yield factor.
0.488Why not: Ignores the dividend yield completely (in d1 and in the e^(-qT) factor).

d1 = [ln(80/85) + (0.03 - 0.04 + 0.22²/2) x 1] / 0.22 = -0.2110, N(d1) = 0.4164. Delta = e^(-0.04) x 0.4164 = 0.4001.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →