convexity-lab

Option pricing toolkit with two complementary models:

• Black-Scholes-Merton (constant vol) — closed-form pricing, all first and second-order Greeks (including the Gamma convexity surface), Monte Carlo validation with antithetic variates, implied vol solver (Newton-Raphson + Brent fallback).
• Heston stochastic volatility — closed-form pricing via Fourier inversion of the characteristic function (with the little Heston trap to avoid branch cuts), implied vol smile generation.

Optional live spot via yfinance.

What is convexity?

For options, Gamma = ∂²V/∂S² — the second derivative of option value with respect to the underlying. It quantifies how curved the price function is, and is the dominant source of P&L for delta-hedged positions over short horizons:

dV ≈ Δ · dS + ½ · Γ · (dS)² + Θ · dt + ...

A long-options position is long convexity: every move in the underlying, up or down, produces gamma scalping P&L proportional to ½ · Γ · (dS)². The script also computes Volga (∂²V/∂σ²) — convexity in volatility itself.

Install

pip install -r requirements.txt

Dependencies: numpy, scipy, matplotlib, yfinance (the last is optional; the script falls back to synthetic data if it can't fetch live prices).

Run

Black-Scholes-Merton demo:

python stock_convexity.py                  # synthetic demo (no internet needed)
python stock_convexity.py NVDA             # live spot via yfinance
python stock_convexity.py NVDA --plot      # also save the gamma surface PNG
python stock_convexity.py --paths 1000000  # heavier Monte Carlo

Heston stochastic volatility demo:

python heston.py                           # print ATM Heston price + IV smile summary
python heston.py --plot                    # save heston_smile.png

Example output (NVDA spot pulled live):

============================================================================
 Black-Scholes-Merton analytics + convexity surface
============================================================================

[+] live spot from yfinance  (NVDA)  spot = $920.00

  Option contract: S=920.00  K=966.00  T=0.1644y  r=0.045  sigma=0.500  kind=call
  ------------------------------------------------------------
    Price                  56.421234
    Delta                   0.473219
    Gamma                   0.003912    <-- convexity in spot
    Vega                  146.882443
    Theta (annual)       -284.114502
    Rho                    61.224578
    Vanna                   0.024195    <-- delta vs vol
    Volga (vomma)          16.348822    <-- convexity in vol

  Monte Carlo (antithetic, n_paths = 200,000):
    MC price        = 56.428901  +- 0.064721  (95% CI)
    BSM closed-form = 56.421234
    abs residual    = 0.007667
    elapsed         = 28.4 ms

  Put-call parity check (residual should be ~1e-14):
    C - P                        =   -39.5234567890
    S e^{-qT} - K e^{-rT}        =   -39.5234567890
    residual                     =     0.0000000000

  Implied vol round-trip:  input = 0.500000   recovered = 0.500000

With --plot, you also get a 3D surface of Gamma over the (moneyness × time) grid — visually showing where convexity is concentrated (peaked at-the-money, exploding as expiration approaches).

Math

All formulas follow Hull, Options, Futures, and Other Derivatives, 11e.

BSM pricing (European call):

C = S · e^{-qT} · N(d₁) - K · e^{-rT} · N(d₂)

where

d₁ = [ln(S/K) + (r - q + ½σ²)T] / (σ√T), d₂ = d₁ - σ√T

Gamma (the convexity Greek):

Γ = e^{-qT} · φ(d₁) / (S · σ · √T)

Volga (convexity in vol):

Vomma = Vega · d₁ · d₂ / σ

Vanna (mixed second derivative):

Vanna = -e^{-qT} · φ(d₁) · d₂ / σ

Heston model

The Heston SDE is

dS_t = (r-q) S_t dt + √v_t · S_t · dW_t¹

dv_t = κ(θ - v_t) dt + σ_v · √v_t · dW_t²

d⟨W¹,W²⟩_t = ρ dt

Five parameters: v₀ (initial variance), κ (mean-reversion speed), θ (long-run variance), σ_v (vol-of-vol), ρ (asset-vol correlation). With ρ < 0 and σ_v > 0, the model produces the negative skew observed in equity index options (OTM puts more expensive than OTM calls).

Pricing uses Fourier inversion of two characteristic functions:

C = S · e^{-qT} · P₁ - K · e^{-rT} · P₂,

P_j = ½ + (1/π) ∫₀^∞ Re[ e^{-i u ln K} · f_j(u) / (i u) ] du

The "little trap" form (Albrecher et al. 2007) keeps g = (b - iρσu - d)/(b - iρσu + d) and uses exp(-dT) inside the log, eliminating the branch-cut discontinuity present in the original Heston (1993) formulation.

Sanity check: as σ_v → 0 with v₀ = θ, the Heston price degenerates to BSM with σ = √θ — verified by the test suite.

Validation

The test suite (pytest tests/) covers 21 tests:

BSM (14 tests):

1. Hull textbook reference values (example 15.6) matched to 1e-3
2. Put-call parity at machine precision (< 1e-12)
3. Gamma identical for call & put with same params (model property)
4. Gamma always positive (long convexity)
5. Deep-ITM call Δ → 1, deep-OTM call Δ → 0 (boundary behavior)
6. Monte Carlo matches closed-form within 4 standard errors
7. Antithetic variates produce strictly lower SE than plain MC
8. Implied vol round-trip exact to 1e-6 across σ ∈ [0.10, 0.90]

Heston (7 tests): 9. Heston with σ_v ≈ 0 and v₀ = θ degenerates to BSM (parametrized at 3 vols) 10. Put-call parity holds (model-free property) 11. Negative correlation produces negative skew (OTM put IV > OTM call IV) 12. Zero correlation produces approximately symmetric smile 13. Feller condition flag (2κθ > σ_v²) correctly identifies regimes

Limitations

• European exercise only (no early exercise / American options).
• Constant volatility (no local-vol, stochastic vol, or jumps).
• Constant dividend yield (continuous, not discrete dividends).
• Risk-free rate is flat (no term structure).

For exotic payoffs, stochastic vol (Heston), or American exercise, this needs binomial / PDE / LSM Monte Carlo extensions.

License

MIT — see LICENSE.