Abstract

This thesis presents the balanced premium calibration method (BPCM), a three-layer framework for robustly fitting the Heston stochastic volatility model to large option datasets. The BPCM method involves three layers.

Layer 1 ensures market consistency by filtering and structurally repairing raw quotes via put–call parity and bid–ask bounds, separating stable observations from noise. Layer 2 performs daily least-squares calibration of the Heston parameters using closed-form characteristic function pricing and derived analytic gradients and Hessians, thereby achieving rapid convergence without finite-difference approximations. Layer 3 redistributes errors and allows for controlled adjustments to model inputs and outputs, absorbing residual pricing errors and restoring arbitrage-free consistency.

Working with 1.5 million call and put quotes on a major equity from 2018 to 2024, BPCM ensured that model prices closely adhere to market bid–ask spreads (91.58% adherence) for stable regimes while maintaining realistic spot price behaviour.

The calibrated model achieves high consistency with observed prices and reconstructs the underlying spot price trajectory with minimal deviation even during market crises. Moreover, residual error and correlation analyses reveal structural differences between stable and unstable data regimes. In the stable regime, implied option Greeks and calibration gradients behave smoothly and expectedly.

These findings confirm that BPCM delivers reliable calibration under normal conditions (consistent with classical theory) and expose predictable model stress under turbulent market regimes. In addition to BPCM, this thesis derives explicit closed-form expressions for the Heston model's Hessians. To the author's knowledge, these second-order derivatives lay the groundwork for future research in second-order optimisation techniques.

Acknowledgements

I would like to express my deepest gratitude to my supervisory team. I am particularly grateful to my first supervisor, Malham Simon, for his continuous encouragement, intelligent guidance, and generous patience during this research journey. I am deeply appreciative of my second supervisor, Anke Wiese, for her knowledge and support, which were useful in developing this work to its final version.

When I began this PhD, I came from a modest academic background in finance. Through their dedicated mentorship, my supervisors not only supported me technically but also helped raise my academic abilities to a level I could not have imagined at the outset. Their belief in my potential gave me the confidence to overcome challenges and grow as an independent researcher.

I would also like to thank Heriot-Watt University for providing an outstanding research environment and student support services that truly empower postgraduate researchers. From academic resources to technical and personal assistance, the infrastructure and atmosphere have played an essential role in enabling my development. My appreciation also extends to the Mathematics Department for their support and for fostering an intellectually rich environment that complemented my research in financial modelling.

I am profoundly grateful to Umm Al-Qura University for sponsoring my doctoral studies. I would also like to acknowledge the support of the Saudi Arabian Cultural Bureau in the UK and the Ministry of Education, Kingdom of Saudi Arabia for facilitating this sponsorship. Their investment in education and research has made this journey possible.

Finally, I am deeply thankful to my wife, Rafah Aldosari, whose unwavering support, patience, and belief in me have sustained me throughout this journey. Her encouragement behind the scenes has been invaluable. I am also profoundly grateful to my children, Ibrahim and Bina, whose pure hearts and natural disposition have reminded me of the essence of balance, sincerity, and presence.

Introduction

The accurate pricing and risk management of financial derivatives require models that capture the empirical dynamics of asset prices. The classical Black–Scholes framework, while elegant, assumes constant volatility — an assumption that is systematically violated in observed markets. This thesis addresses the problem of robustly calibrating a stochastic volatility model to large, real-world option datasets.

1.1 The Heston Model — Overview

The Heston (1993) model extends Black–Scholes by allowing instantaneous variance to follow a mean-reverting CIR process. Its semi-affine structure yields a characteristic function in closed form, making it tractable for large-scale calibration. The model is defined by five parameters:

Despite its popularity, calibrating the Heston model to large, noisy option data requires careful treatment of data quality, numerical stability, and computational efficiency — all addressed by the BPCM framework developed in this thesis.

1.2 Problem Statement & Motivation

Real-world option data suffers from bid–ask spread noise, missing quotes, violated no-arbitrage conditions, and regime changes (e.g., the 2020 COVID-19 crisis). A naive least-squares calibration applied to raw quotes produces unstable parameters and poor out-of-sample performance. This motivates a multi-layer framework that addresses data quality, calibration efficiency, and error correction in a unified, principled manner.

Option Price Model

2.1 Stochastic Differential Equations

Under the risk-neutral measure $\mathbb{Q}$, the Heston model specifies joint dynamics for the asset price $S_t$ and its instantaneous variance $v_t$:

where $dW_t^S \cdot dW_t^v = \rho\, dt$, and the Feller condition $2\kappa\theta \geq \sigma^2$ ensures $v_t > 0$ almost surely.

2.2 Characteristic Function

The log-price $x_t = \ln(S_t/S_0)$ has characteristic function:

where the auxiliary functions are:

with $d = \sqrt{(\rho\sigma i u - \kappa)^2 + \sigma^2(iu + u^2)}$ and $g = \frac{\kappa - \rho\sigma i u - d}{\kappa - \rho\sigma i u + d}$.

2.3 Closed-Form Pricing Formula

A European call option with strike $K$ and maturity $T$ is priced via the Carr–Madan (1999) formula using the characteristic function:

where $\Psi(u) = e^{rT}\,\varphi(u - (\alpha+1)i;\,T)$ and $\alpha > 0$ is a damping parameter.

Calibration Methods

3.1 Least-Squares Formulation

Given market prices $\{C_i^{\text{mkt}}\}_{i=1}^N$ for $N$ options, calibration minimises the weighted sum of squared deviations:

where $\Theta = (\kappa, \theta, \sigma, \rho, v_0)$ and weights $w_i$ are derived from inverse bid–ask spreads to penalise noisy quotes.

3.2 Analytic Gradients

Gradient-based optimisation requires $\partial C / \partial \Theta_j$. This thesis derives these analytically by differentiating the characteristic function with respect to each parameter, yielding closed-form expressions that avoid costly finite-difference approximations:

3.3 Closed-Form Hessians (Novel Contribution)

A key theoretical contribution of this thesis is the derivation of explicit closed-form second-order derivatives (Hessians) of the Heston price with respect to model parameters. To the author's knowledge, these expressions are not available in the existing literature and lay the groundwork for second-order optimisation methods (Newton-type) in stochastic volatility calibration.

Data Collection and Preparation

The empirical study uses 1.5 million call and put option quotes on a major equity index, covering the period January 2018 – December 2024. This span includes multiple market regimes: low-volatility pre-pandemic, the COVID-19 crisis (2020), and post-pandemic normalisation.

4.1 Data Source & Cleaning

Raw option data is subject to several pathologies: stale quotes, bid–ask inversions, zero-volume entries, and extreme moneyness. The pipeline applies:

• Moneyness filter: $0.7 \leq K/S \leq 1.3$
• Minimum bid price threshold
• Maturity filter: $7 \leq T \leq 365$ days
• Zero-volume elimination

4.2 Put–Call Parity Filtering

For each $(K, T)$ pair where both a call and a put exist, the put–call parity check verifies:

Quotes violating this relation beyond a tolerance $\varepsilon$ are flagged. Layer 1 of BPCM uses these violations to classify data into stable and unstable regimes, with separate handling for each.

Balanced Premium Calibration Method

The BPCM is the central contribution of this thesis. It is a three-layer framework that transforms raw, noisy market data into robustly calibrated Heston parameters while maintaining arbitrage-free consistency.

5.1 Layer 1: Data Engineering

Layer 1 constructs a clean option surface by: (i) removing quotes that violate static no-arbitrage conditions, (ii) repairing minor violations via bid–ask midpoint adjustments, and (iii) labelling each trading day as stable or unstable based on the proportion of violated parity conditions.

5.2 Layer 2: Daily Calibration

For each trading day $t$, Layer 2 solves the weighted least-squares problem using a trust-region method initialised from the previous day's parameters. The use of analytic gradients (Chapter 3) reduces per-iteration cost by 65% versus finite-difference methods.

5.3 Layer 3: Error Redistribution

After Layer 2 calibration, residual errors are redistributed across the option surface by solving a secondary linear programme that adjusts the calibrated prices within bid–ask bounds. This layer accounts for the structured residual patterns observed in unstable regimes.

5.4 Empirical Results (2018–2024)

Key empirical findings across 1.5 million quotes:

Conclusion and Future Research Directions

This thesis introduced the Balanced Premium Calibration Method (BPCM) — a principled, three-layer framework for robustly calibrating the Heston stochastic volatility model to large, real-world option datasets. The method achieves market-grade performance (91.58% bid–ask adherence) and produces stable, meaningful parameter estimates across six years of equity option data including the 2020 market crisis.

Beyond the empirical contributions, this thesis derives explicit closed-form Hessians for the Heston model — a theoretical result that, to the author's knowledge, has not previously appeared in the literature, and which opens the door to second-order calibration methods.

Future Research Directions

• Second-order optimisation: Apply the derived Hessians in Newton-type or quasi-Newton calibration algorithms to further accelerate convergence.
• Model extensions: Extend BPCM to rough volatility models (e.g., rough Heston) where characteristic functions remain tractable.
• Deep learning integration: Use BPCM as a data-cleaning pre-processing step for neural network implied-volatility surfaces.
• Multi-asset calibration: Extend the framework to cross-asset correlation calibration in multi-dimensional stochastic volatility models.

Bibliography

[Selected references — full bibliography in PDF version]

• Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility. Review of Financial Studies, 6(2), 327–343.
• Carr, P., & Madan, D. (1999). Option Valuation Using the Fast Fourier Transform. Journal of Computational Finance, 2(4), 61–73.
• Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A Theory of the Term Structure of Interest Rates. Econometrica, 53(2), 385–407.
• Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654.
• Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. John Wiley & Sons.
• Andersen, L., & Piterbarg, V. (2010). Interest Rate Modeling. Atlantic Financial Press.