AAPL Options Pricing Models

Implements and compares Black-Scholes, Heston, and Merton Jump-Diffusion models on real AAPL equity options data (2016–2020). The pipeline covers data ingestion, date-driven rate calibration, implied vol surface construction, multi-expiry calibration, analytical Greeks, and publication-quality visualisations.

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Models

Black-Scholes (1973)

Under the risk-neutral measure, the stock follows:

$$dS_t = r , S_t , dt + \sigma , S_t , dW_t$$

The call price is:

$$C = S \cdot N(d_1) - K e^{-rT} \cdot N(d_2), \quad d_1 = \frac{\ln(S/K) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$$

Implemented without dividends, consistent with the original formulation. A flat vol σ is calibrated per-expiry by minimising vega-weighted relative price error via scipy.optimize.minimize_scalar (bounded Brent). The vol surface is built with QuantLib's BlackVarianceSurface and bicubic interpolation.

Limitation: a single flat σ cannot reproduce the volatility smile. Average absolute error on the 1-year expiry: ~4.0%. Overall vega-weighted error across 17 expiries: 16.5%.

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Heston Stochastic Volatility (1993)

Variance evolves as a mean-reverting CIR process correlated with the spot:

$$dS_t = (r - q) S_t , dt + \sqrt{v_t} , S_t , dW_t^S$$

$$dv_t = \kappa(\theta - v_t) , dt + \sigma_v \sqrt{v_t} , dW_t^v, \quad \mathbb{E}[dW_t^S , dW_t^v] = \rho , dt$$

Parameter	Meaning
$v_0$	Initial instantaneous variance
$\kappa$	Mean-reversion speed
$\theta$	Long-run variance
$\sigma_v$	Vol-of-vol
$\rho$	Spot-vol correlation (negative = leverage effect)

The Feller condition $2\kappa\theta > \sigma_v^2$ ensures variance stays positive. In practice, calibration often violates it - the optimiser pushes toward high vol-of-vol, a known pathology handled by the CIR reflection boundary.

Priced via QuantLib's AnalyticHestonEngine (Fourier inversion), calibrated with LevenbergMarquardt against market call prices. Dividend yield included via the Merton (1973) continuous-dividend extension.

Calibrated parameters (2018-05-15): v₀ = 0.1704, κ = 9.54, θ = 0.0694, σᵥ = 3.36, ρ = −0.463. Average absolute error: 1.52%.

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Merton Jump-Diffusion (1976)

Augments BS diffusion with a compound Poisson jump process:

$$dS_t = (r - q - \lambda \bar{\kappa}) S_t , dt + \sigma S_t , dW_t + (J-1) S_t , dN_t$$

where $N_t$ is a Poisson process with intensity $\lambda$, and $J = e^Y$, $Y \sim \mathcal{N}(\mu_J, \sigma_J^2)$ is a lognormal jump size. The closed-form pricing series is:

$$C^{\text{Merton}} = \sum_{n=0}^{\infty} \frac{e^{-\lambda' T}(\lambda' T)^n}{n!} \cdot C^{\text{BS}}!\left(S, K, r_n, q, T, \sigma_n\right)$$

$$r_n = r - \lambda\bar{\kappa} + \frac{n(\mu_J + \frac{1}{2}\sigma_J^2)}{T}, \quad \sigma_n = \sqrt{\sigma^2 + \frac{n \sigma_J^2}{T}}, \quad \bar{\kappa} = e^{\mu_J + \frac{1}{2}\sigma_J^2} - 1$$

Truncated at 50 terms (early exit when weights drop below 1e-12). Calibrated via multi-start L-BFGS-B (5 random seeds, objective = IV-RMSE).

Calibrated parameters (2018-05-15): σ = 15.9%, λ = 0.57 jumps/yr, μ_J = −18.5%, σ_J = 22.6%. This implies roughly one crash-sized jump per year averaging a −14.7% return - economically consistent with AAPL tail risk pricing over 2018. IV-RMSE = 0.22%.

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Greeks - Merton Jump-Diffusion

Greeks are derived analytically by differentiating the infinite series term-by-term, applying the chain rule for the per-term vol:

$$\frac{\partial \sigma_n}{\partial \sigma} = \frac{\sigma}{\sigma_n}, \qquad \frac{\partial \sigma_n}{\partial \sigma_J} = \frac{n \sigma_J / T}{\sigma_n}$$

ATM Greeks at S = K = 186.44, T = 1.101 yr (2018-05-15):

Greek	Value	Interpretation
Price	$19.51	ATM call value
Delta Δ	0.6151	$0.615 gain per $1 spot move
Gamma Γ	0.00867	Delta change per $1 spot move
Vega σ	52.88	$52.88 per unit diffusion vol increase
Vega σ_J	26.90	$26.90 per unit jump vol increase
Theta Θ	−0.02864	~$0.029 loss per calendar day
Rho ρ	1.0481	$1.05 per 1 bp rise in r

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Volatility Surface & Results

The market vol surface is built from QuantLib BlackVarianceSurface with bicubic interpolation across 17 expiries, showing a pronounced left-wing skew and upward-sloping term structure (~18% short-dated to ~26% at 2 years).

Metric	Black-Scholes	Heston	Merton JDM
Parameters	1 per expiry	5	4
Avg price error (1yr)	3.97%	1.52%	0.34%
IV-RMSE (1yr)	~400 bp	~150 bp	14 bp
Smile shape	Flat	Monotone skew	Flexible skew + curvature
Greeks	Analytical	Numerical / QL	Analytical (series)

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Market Data & Rate Calibration

Data: AAPL options chain (CBOE-style CSV) from Kaggle by Kyle Graupe, 2016-01-04 to 2020-12-31. Strike filter: ±20% moneyness band around spot.

Risk-free rates from FRED H.15 (1-year Treasury CMT) and AAPL dividend yields are linearly interpolated by fractional year - no fixed arbitrary rate. Black-Scholes is implemented without dividends per its original formulation; Heston and Merton include continuous dividend yield.

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Installation & Usage

Python 3.9+ required. QuantLib can be tricky on Windows - use conda if pip fails.

pip install -r requirements.txt
# or
conda install -c conda-forge quantlib-python numpy pandas scipy matplotlib

Each script prompts for a quote date at runtime:

python models/BlackScholesOPM.py        # calibration table + 4 graphs
python models/HestonOPM.py              # Heston params + Feller check + 3 graphs
python models/MertonJumpDiffusionOPM.py # multi-start calibration + Greeks + 4 graphs

Interesting dates: 2020-03-16 (COVID crash), 2019-01-03 (AAPL profit warning), 2016-06-24 (Brexit vol spike).

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References

• Black & Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy.
• Heston (1993). A Closed-Form Solution for Options with Stochastic Volatility. Review of Financial Studies.
• Merton (1976). Option Prices When Underlying Stock Returns Are Discontinuous. Journal of Financial Economics.
• Gatheral (2006). The Volatility Surface: A Practitioner's Guide. Wiley Finance.
• Cont & Tankov (2004). Financial Modelling with Jump Processes. Chapman & Hall.
• QuantLib · FRED H.15

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Python 3.12 · QuantLib · NumPy · SciPy · Matplotlib · MIT License