[Paper Review] FX Smile in the Heston Model

Agnieszka Janek, Tino Kluge+2 more|arXiv (Cornell University)|Oct 8, 2010

Stochastic processes and financial applications3 citations

TL;DR

This paper adapts the Heston stochastic volatility model to foreign exchange (FX) markets, demonstrating that the model can accurately reproduce the observed volatility smile in FX options through calibration of three key parameters. By leveraging semi-analytical pricing formulas and FFT-based numerical methods, the study shows the Heston model effectively captures market-implied volatility patterns while maintaining computational efficiency for practical front-office use.

ABSTRACT

Abstract: The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is nonnegative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.

Motivation & Objective

• To adapt the Heston stochastic volatility model to foreign exchange (FX) markets, where volatility smiles are prevalent.
• To demonstrate the computational feasibility of the Heston model in FX settings using semi-analytical formulas and FFT.
• To calibrate the model to real FX option market data and assess its ability to reproduce the observed volatility smile.
• To evaluate the role of key model parameters—especially mean reversion and volatility of volatility—in shaping the implied volatility surface.
• To compare the Heston model’s performance against the Black-Scholes model in capturing market dynamics in FX options.

Proposed method

• Adapts the original Heston (1993) model to FX by modeling the spot FX rate under stochastic volatility with mean-reverting CIR dynamics.
• Employs the characteristic function approach to derive a semi-analytical solution for European option prices via the inverse Fourier transform.
• Utilizes the Fast Fourier Transform (FFT) for efficient and accurate option pricing across a range of strikes and maturities.
• Applies Monte Carlo simulations to validate the semi-analytical results and assess model stability.
• Checks the Feller condition to ensure the variance process remains non-negative and avoids explosive behavior.
• Calibrates the model by minimizing the difference between model-implied and market-observed option prices using three key parameters: volatility of volatility (σ), mean reversion speed (κ), and long-run variance (θ).

Experimental results

Research questions

• RQ1 Can the Heston model effectively reproduce the volatility smile observed in FX options?
• RQ2 Which of the five Heston model parameters are most critical for calibrating to market-implied volatility surfaces in FX markets?
• RQ3 How does the semi-analytical solution combined with FFT improve computational efficiency compared to Monte Carlo simulation in the FX context?
• RQ4 Does the Heston model outperform the Black-Scholes model in pricing FX vanilla options with respect to market data?
• RQ5 What is the impact of the Feller condition on the stability and realism of the Heston model in FX applications?

Key findings

• The Heston model successfully reproduces the FX volatility smile using only three calibrated parameters: σ, κ, and θ, with the remaining two parameters (μ and ρ) having less impact on smile shape.
• The semi-analytical solution combined with FFT enables fast and accurate option pricing, making the model suitable for real-time calibration and risk management.
• Monte Carlo simulations confirm the stability and accuracy of the analytical results, particularly when the Feller condition is satisfied.
• The model captures the persistent smile structure not explained by local volatility models, supporting its use for both vanilla and exotic option pricing.
• The calibrated model produces implied volatility surfaces that closely match observed market data, validating its empirical relevance in FX markets.
• The Heston model’s ability to generate non-negative, mean-reverting volatility processes makes it more realistic than constant volatility models like Black-Scholes.