Skip to content

Fourier Methods for Heston Model

We are extending our Fourier-based pricing methods — like we did in Black-Scholes — to a more realistic model of financial markets called the Heston Model, which allows volatility to be stochastic (i.e., random).

The Pricing Equation (Lewis 2001)

Under the Heston model, the European call option price is given by this integral (Lewis-style):

\[ C_0 = S_0 - \frac{\sqrt{S_0 K} \, e^{-rT}}{\pi} \int_0^{\infty} \text{Re} \left[ \frac{e^{i u \log(S_0/K)} \, \phi(u - i/2)}{u^2 + 1/4} \right] du \]

This formula has three key parts:

  • \(\phi(u)\): Characteristic function of the log-price under the Heston model.
  • \(\text{Re}[\cdot]\): Take only the real part.
  • \(\int_0^\infty \cdots du\): Numerically integrated.

The Heston model assumes:

  • The asset price \(S_t\) and its volatility \(\nu_t\) both follow stochastic (random) processes.
  • The volatility itself is mean-reverting — it wants to go back to some long-term average.

It adds realism, especially for modeling:

  • Volatility smiles/skews
  • Market shocks
  • Fat tails in return distribution

Characteristic Function in Heston (φH(u, T))

This is the most technical part: deriving the characteristic function under Heston dynamics. It’s a complex exponential with parts:

\[ \phi_H(u, T) = \exp \left( H_1(u, T) + H_2(u, T) \cdot \nu_0 \right) \]

Where:

  • \(H_1(u, T)\) and \(H_2(u, T)\) are expressions derived from solving the SDEs.

  • They depend on:

    • \(\kappa_\nu\): speed of mean reversion
    • \(\theta_\nu\): long-term mean of variance
    • \(\sigma_\nu\): volatility of variance (vol of vol)
    • \(\rho\): correlation between asset returns and variance
    • \(\nu_0\): initial variance level

“How does the distribution of log-prices change when both the price and its volatility are random?”

How to Price an Option with It?

Same idea as Black-Scholes Lewis-style pricing:

  1. Plug \(φH(u − i/2)\) into the integral formula.
  2. Use a quadrature method (like Simpson’s Rule or scipy.integrate.quad) to numerically integrate it.
  3. Done — you get a call price.

Calibration of Heston Model

Since the model has 5 unknown parameters: \(\kappa_\nu, \theta_\nu, \sigma_\nu, \rho, \nu_0\)

We calibrate it like this:

  1. Define a loss function = squared difference between market prices and model prices.

  2. Use optimization (scipy.optimize) to minimize that loss.

  3. Often we do it in 2 stages:

    • Coarse grid search (brute force)
    • Fine optimization (e.g., L-BFGS-B)

Coding

Python
import pandas as pd
from datetime import datetime
from wqu.sm import HestonCalibrator

# Load from CSV
data_path = "option_data.csv"
data = pd.read_csv(data_path)

# Set index level and tolerance
S0 = 3225.93
tol = 0.02

# Filter near-the-money European call options
options = data[(abs(data["Strike"] - S0) / S0) < tol].copy()
options["Date"] = pd.to_datetime(options["Date"])
options["Maturity"] = pd.to_datetime(options["Maturity"])
options["T"] = (options["Maturity"] - options["Date"]).dt.days / 365
options["r"] = 0.02  # constant short rate

# Initialize and run calibration
calibrator = HestonCalibrator(S0=S0, options_df=options)
optimal_params = calibrator.calibrate()

# Display results
print("Optimal Parameters:", optimal_params)
Text Only
>>> Starting brute-force search...
   0 | [ 2.5   0.01  0.05 -0.75  0.01] | MSE: 820.892168 | Min MSE: 820.892168
  25 | [ 2.5   0.02  0.05 -0.75  0.02] | MSE: 23.863624 | Min MSE: 21.567743
  50 | [ 2.5   0.02  0.25 -0.75  0.03] | MSE: 89.654952 | Min MSE: 21.567743
  75 | [ 2.5   0.03  0.15 -0.5   0.01] | MSE: 193.282610 | Min MSE: 21.567743
 100 | [ 2.5   0.04  0.05 -0.5   0.02] | MSE: 176.339739 | Min MSE: 21.567743
 125 | [ 2.5   0.04  0.25 -0.5   0.03] | MSE: 486.964581 | Min MSE: 21.567743
 150 | [ 7.5   0.01  0.15 -0.25  0.01] | MSE: 840.337090 | Min MSE: 21.567743
 175 | [ 7.5   0.02  0.05 -0.25  0.02] | MSE: 24.810371 | Min MSE: 21.567743
 200 | [ 7.5   0.02  0.25 -0.25  0.03] | MSE: 24.834228 | Min MSE: 21.567743
 225 | [7.5  0.03 0.15 0.   0.01] | MSE: 110.936421 | Min MSE: 21.567743
 250 | [7.5  0.04 0.05 0.   0.02] | MSE: 540.182792 | Min MSE: 21.567743
 275 | [7.5  0.04 0.25 0.   0.03] | MSE: 783.221740 | Min MSE: 21.567743
>>> Refining with local search...
 300 | [ 2.61379559  0.00992657  0.15610448 -0.76361614  0.02778356] | MSE: 8.045766 | Min MSE: 7.795453
 325 | [ 1.9152359   0.01257942  0.16036675 -0.91693167  0.0248233 ] | MSE: 6.300806 | Min MSE: 6.146318
 350 | [ 2.04831069  0.01215428  0.15832201 -0.89057611  0.02532865] | MSE: 6.150503 | Min MSE: 6.144623
 375 | [ 2.0376908   0.01207312  0.16816435 -0.86785979  0.02553311] | MSE: 6.097042 | Min MSE: 6.075189
 400 | [ 1.97316132  0.01247835  0.2032535  -0.83478704  0.0254568 ] | MSE: 6.001525 | Min MSE: 5.996115
 425 | [ 2.07617861  0.01268556  0.21375606 -0.83213139  0.02575716] | MSE: 5.947748 | Min MSE: 5.947748
 450 | [ 2.66904762  0.0147503   0.22891593 -0.87575667  0.02607257] | MSE: 5.683904 | Min MSE: 5.683904
 475 | [ 3.14901508  0.01554998  0.22701309 -0.86349685  0.02615741] | MSE: 5.357630 | Min MSE: 5.357630
 500 | [ 3.75301757  0.0179153   0.33579328 -0.72953664  0.02611118] | MSE: 4.602526 | Min MSE: 4.498631
 525 | [ 5.15416061  0.01857864  0.40457683 -0.45664742  0.0270172 ] | MSE: 3.894461 | Min MSE: 3.824773
 550 | [ 5.14078039  0.01861684  0.42859861 -0.43432012  0.02728158] | MSE: 3.769001 | Min MSE: 3.749195
 575 | [ 5.0507416   0.01868091  0.43385652 -0.44775932  0.02732562] | MSE: 3.718624 | Min MSE: 3.716546
 600 | [ 5.05235394  0.01871275  0.43473462 -0.44693948  0.02730235] | MSE: 3.716004 | Min MSE: 3.715792
 625 | [ 5.05371685  0.01872767  0.43504931 -0.44691718  0.0272971 ] | MSE: 3.715524 | Min MSE: 3.715392
 650 | [ 5.04446488  0.01872907  0.43469053 -0.44848558  0.02728228] | MSE: 3.715380 | Min MSE: 3.715376
 675 | [ 5.04382644  0.0187266   0.43463479 -0.44869357  0.02728557] | MSE: 3.715374 | Min MSE: 3.715374
 700 | [ 5.04426287  0.01872709  0.4346593  -0.44857764  0.02728496] | MSE: 3.715373 | Min MSE: 3.715373
 725 | [ 5.04476043  0.01872427  0.43464808 -0.44841756  0.02728689] | MSE: 3.715371 | Min MSE: 3.715371
 750 | [ 5.04505376  0.01872602  0.43468094 -0.44837446  0.02728584] | MSE: 3.715370 | Min MSE: 3.715370
 775 | [ 5.04707738  0.01872566  0.434764   -0.44801571  0.02728898] | MSE: 3.715367 | Min MSE: 3.715367
 800 | [ 5.04731712  0.01872561  0.43477373 -0.44796137  0.02728913] | MSE: 3.715367 | Min MSE: 3.715367
 825 | [ 5.04735223  0.01872569  0.43477611 -0.44795766  0.02728916] | MSE: 3.715367 | Min MSE: 3.715367
 850 | [ 5.04735429  0.0187257   0.43477637 -0.44795746  0.02728915] | MSE: 3.715367 | Min MSE: 3.715367
 875 | [ 5.04735384  0.01872571  0.43477645 -0.44795738  0.02728914] | MSE: 3.715367 | Min MSE: 3.715367
 900 | [ 5.04736306  0.01872573  0.43477709 -0.44795509  0.02728912] | MSE: 3.715367 | Min MSE: 3.715367
 925 | [ 5.04735787  0.01872573  0.43477684 -0.44795621  0.02728912] | MSE: 3.715367 | Min MSE: 3.715367
Optimization terminated successfully.
         Current function value: 3.715367
         Iterations: 479
         Function evaluations: 798
Optimal Parameters: [ 5.04735844  0.01872573  0.43477689 -0.44795617  0.02728912]