Solution: Vega-Neutral Portfolio Construction

Exercise: Vega-Neutral Portfolio Construction

Part 1 — Individual Greeks

import numpy as np
from scipy.stats import norm

def bs_greeks(S, K, T, r, sigma, q=0):
    d1 = (np.log(S/K) + (r - q + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    C = S*np.exp(-q*T)*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
    delta = np.exp(-q*T)*norm.cdf(d1)
    gamma = np.exp(-q*T)*norm.pdf(d1) / (S*sigma*np.sqrt(T))
    vega = S*np.exp(-q*T)*norm.pdf(d1)*np.sqrt(T)
    return C, delta, gamma, vega

S, r, sigma = 100, 0.05, 0.2
C1, D1, G1, V1 = bs_greeks(S, 100, 0.25, r, sigma)
C2, D2, G2, V2 = bs_greeks(S, 100, 1.0, r, sigma)
print(f"C1 (3M ATM):   C={C1:.3f}, delta={D1:.4f}, gamma={G1:.4f}, vega={V1:.3f}")
print(f"C2 (12M ATM):  C={C2:.3f}, delta={D2:.4f}, gamma={G2:.4f}, vega={V2:.3f}")
# C1 (3M ATM):   C=4.615, delta=0.5596, gamma=0.0389, vega=19.426
# C2 (12M ATM):  C=10.451, delta=0.6368, gamma=0.0188, vega=37.524

Part 2 — Build portfolio

Fix $\alpha = 1$ . Vega neutrality:

19.426 + \beta\cdot 37.524 = 0 \implies \beta = -0.5177.

Delta neutrality:

0.5596 + \beta\cdot 0.6368 + \gamma = 0 \implies \gamma = -0.5596 - (-0.5177)(0.6368) = -0.5596 + 0.3297 = -0.2299.

alpha = 1
beta = -V1 / V2
gamma_stock = -(alpha*D1 + beta*D2)
print(f"alpha={alpha}, beta={beta:.4f}, stock shares={gamma_stock:.4f}")
# alpha=1.0000, beta=-0.5177, stock shares=-0.2299

Interpretation: Long 1 unit of 3-month ATM call, short 0.518 units of 12-month ATM call, short 0.230 shares of stock.

This is a calendar spread (long short-dated, short long-dated) with a delta hedge.

Part 3 — Portfolio gamma

\Gamma_P = 1\cdot 0.0389 + (-0.5177)\cdot 0.0188 = 0.0389 - 0.00975 = 0.0292.

Positive gamma — the portfolio is long gamma (long convexity). The calendar spread gains when the stock moves in either direction; it's a bet on "realized vol exceeds implied" on the short-dated tenor.

gamma_P = alpha*G1 + beta*G2
print(f"Portfolio gamma: {gamma_P:.4f}")
# Portfolio gamma: 0.0292

Part 4 — Parallel vol shift scenario

If both vols rise to 0.21:

C1_up, _, _, _ = bs_greeks(S, 100, 0.25, r, 0.21)
C2_up, _, _, _ = bs_greeks(S, 100, 1.0, r, 0.21)
pnl = alpha*(C1_up - C1) + beta*(C2_up - C2)
print(f"Parallel 1% IV shift P&L: {pnl:.4f}")
# Parallel 1% IV shift P&L: 0.0008   (≈ 0, vega-neutral as designed)

Verifying vega-neutrality empirically: P&L $\approx 0$ (second-order vomma residual).

Part 5 — Term-structure twist

Only 3-month vol rises to 0.21:

C1_up, _, _, _ = bs_greeks(S, 100, 0.25, r, 0.21)
# C2 unchanged
pnl_twist = alpha*(C1_up - C1) + beta*(C2_up_unchanged - C2)  # C2 unchanged
pnl_twist = alpha*(C1_up - C1)   # since C2 didn't move
print(f"Short-end-only 1% IV shift P&L: {pnl_twist:.4f}")
# Short-end-only 1% IV shift P&L: +0.1944   (≈ +$0.19)

The portfolio gains $0.19 because it's long 3-month vega ($19.43) and short 12-month vega (-0.518 × $37.52 = -$19.43). A unilateral rise in short-dated IV (which is what the position is net long of after netting) creates a positive P&L. Equivalently: the portfolio is long the short-end of the vol term structure.

Takeaways

Delta and vega neutrality can be imposed simultaneously with two option instruments of different maturities (or strikes), plus the stock to zero the delta. This gives a pure gamma / pure term-structure bet.
The resulting calendar spread is a classic structured-vol position: long gamma, flat vega to parallel shifts, exposed to vol term structure.
Vega-neutrality is a local concept. It holds at one point on the vol surface. A twist of the surface (skew steepening, term-structure flattening) produces P&L even with zero net parallel-vega.
Real-world vol trading books decompose vega into buckets: each strike × maturity gets its own P&L attribution. A "vega-neutral" position in this richer sense requires many more instruments to hedge.