Solution: Computing Vega Across Strikes and Expiries

Exercise: Computing Vega Across Strikes and Expiries

Computation

import numpy as np
from scipy.stats import norm

def bs_call(S, K, T, r, sigma):
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)

def vega_bs(S, K, T, r, sigma):
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    return S * norm.pdf(d1) * np.sqrt(T)

def gamma_bs(S, K, T, r, sigma):
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    return norm.pdf(d1) / (S*sigma*np.sqrt(T))

S, T, r, sigma = 100, 0.5, 0.05, 0.2

# Part 1: vega vs strike
for K in [80, 90, 100, 110, 120]:
    v = vega_bs(S, K, T, r, sigma)
    print(f"K={K}: vega={v:.4f} (per 1%: {v/100:.4f})")
# K=80:  vega=5.58  (per 1%: 0.0558)
# K=90:  vega=19.28 (per 1%: 0.1928)
# K=100: vega=27.34 (per 1%: 0.2734)
# K=110: vega=22.48 (per 1%: 0.2248)
# K=120: vega=11.10 (per 1%: 0.1110)

# Vega peaks ATM, falls off symmetrically in log-strike space.

# Part 2: vega vs expiry
K = 100
for T in [2.0, 1.0, 0.5, 0.1]:
    v = vega_bs(S, K, T, r, sigma)
    print(f"T={T}: vega={v:.2f}")
# T=2.0: vega=53.60
# T=1.0: vega=38.07
# T=0.5: vega=27.34
# T=0.1: vega=12.55

# Approximate sqrt(T) scaling: 53.60/38.07 ≈ 1.408 vs sqrt(2) = 1.414 ✓
# 38.07/27.34 ≈ 1.393 vs sqrt(2) ≈ 1.414 (small deviations due to d1 dependence on T)

# Part 3: gamma-vega identity
S, K, T, sigma = 100, 100, 0.5, 0.2
v = vega_bs(S, K, T, r, sigma)
g = gamma_bs(S, K, T, r, sigma)
rhs = sigma * S**2 * T * g
print(f"vega:                  {v:.4f}")
print(f"sigma*S^2*T*gamma:     {rhs:.4f}")
# vega:                  27.3383
# sigma*S^2*T*gamma:     27.3383   ✓ Equal to 4 decimal places

# Part 4: FD check
h = 0.001
fd = (bs_call(S, K, T, r, sigma+h) - bs_call(S, K, T, r, sigma-h)) / (2*h)
print(f"closed-form vega: {v:.6f}")
print(f"FD vega:          {fd:.6f}")
# closed-form vega: 27.338336
# FD vega:          27.338336

Interpretation

Vega peaks ATM. Like gamma, vega is maximised near the strike where the option has the most "optionality."
Vega grows as $\sqrt T$ . This makes long-dated options far more vol-sensitive than short-dated ones. A 2-year ATM option has ~4× the vega of a 0.1-year ATM option of the same strike and spot.
Gamma–vega identity $\mathcal{V} = \sigma S^2 T\Gamma$ is exact (verifiable symbolically). It's a useful back-of-envelope tool for vol traders: you can estimate vega knowing gamma, and vice versa.
Finite-difference vega agrees to 6 decimals — both methods converge to the same BS vega.

Takeaways

Vega is the flagship risk of a vol book. Every options trader watches vega across strikes (skew exposure) and maturities (term exposure).
Long-dated options dominate vega risk in a diversified book. If you have $1M vega of 1-month options and $1M vega of 2-year options, the 2-year options account for ~5-6× more P&L per point of vol move.
Skew and term-structure complicate vega. The flat-vega computed above assumes a parallel shift in implied vol; real vol surfaces can twist and steepen, requiring skew-vega and term-vega decompositions.
Numerical vega always works. For any model where you can price an option, you can compute vega by finite-differencing in $\sigma$ — essential for models like Heston or SABR where closed-form Greeks are complex.