Estimating Option Delta and Gamma With Bump-and-Reprice

Reprice the option at

S-h

S

, and

S+h

. Delta is the difference between the two bumped prices divided by

2h

. Gamma is the sum of the bumped prices less twice the centre price, divided by

h^2

. On a six-month call with

S=100

K=105

r=4\%

, and

\sigma=25\%

, a

1

-dollar bump yields delta about

0.470295

and gamma about

0.022499

See which Taylor terms the bumps cancel

Expanding $V(S+h)$ and $V(S-h)$ in Taylor series and subtracting cancels the even terms. Adding the expansions and subtracting $2V(S)$ cancels the odd terms. For a sufficiently smooth price,

\widehat{\Delta}_h=\frac{V(S+h)-V(S-h)}{2h},\qquad \widehat{\Gamma}_h=\frac{V(S+h)-2V(S)+V(S-h)}{h^2}.

Both have

O(h^2)

truncation error in this setting. For a Black–Scholes call with no dividends,

\Delta=N(d_1)

and

\Gamma=\phi(d_1)/(S\sigma\sqrt{\tau})

. When bumping spot, keep maturity, strike, rate and volatility fixed to measure the same model derivatives. If a hedging model specifies that implied volatility changes with spot, state and test that convention separately.

Delta divides the difference between two revaluations by

2h

, which is the distance between the bumped spots. Gamma divides the change in the two slopes by

h

again, so its denominator is

h^2

. Reprice all three points with the same strike, maturity, rate and volatility for a partial derivative with respect to spot. If you use a Monte Carlo pricer, common random numbers help remove noise that would otherwise be magnified by these divisions. Compare several bump sizes before interpreting a single error.

Change the bump and compare with the analytic Greeks

The analytic delta and gamma are $0.470317$ and $0.0225051$ . With $h=1$ , prices at $99$ , $100$ , and $101$ are about $5.320891$ , $5.779936$ , and $6.261481$ . With $h=5$ , the delta and gamma estimates are $0.469791$ and $0.0223602$ . The smaller bump comes closer here because analytic repricing has little noise at the dollar scale. With Monte Carlo repricing, reuse the same shocks at all three spots and estimate the uncertainty of the per-path price differences.

Central differences with a bump sweep

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

strike, maturity, rate, vol = 105, 0.5, 0.04, 0.25
spot = 100

def call_price(s):
    d1 = (log(s / strike) + (rate + vol**2 / 2) * maturity) / (vol * sqrt(maturity))
    return s * norm.cdf(d1) - strike * exp(-rate * maturity) * norm.cdf(
        d1 - vol * sqrt(maturity)
    )

center = call_price(spot)
for h in (5, 1, 0.1, 0.01):
    below, above = call_price(spot - h), call_price(spot + h)
    delta = (above - below) / (2 * h)
    gamma = (above - 2 * center + below) / h**2
    print(h, delta, gamma)

Explore the price curve in the options pricer to see where its slope and curvature are changing.

Estimating Option Delta and Gamma With Bump-and-Reprice

See which Taylor terms the bumps cancel

Change the bump and compare with the analytic Greeks

Central differences with a bump sweep

Move volatility and expiry in Black–Scholes

References and next steps