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When Characteristic Functions Enable Fourier Option Pricing

Article Quant Q&A · Author: HSmile
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Summary

The note surveys when the characteristic function of a log asset price is available for Fourier-based option pricing. It identifies affine jump-diffusion models, including Black–Scholes, Heston, Bates, Merton, and Kou models, as well as exponential Lévy models, where the Lévy–Khintchine result supplies the characteristic function. Affine models can also include stochastic interest rates; the cited research describes their use in stochastic-volatility and jump settings. For rough volatility, the note mentions approximation methods.

The answer contrasts these cases with CEV, SABR, and local volatility models, for which it says the characteristic function is not known and simulation is a practical choice. Characteristic functions describe the asset-price distribution at a particular time, making them especially useful for path-independent options. Extensions exist for path-dependent and early-exercise contracts, though Monte Carlo is more commonly used there. The discussion is a model-level overview and does not compare computational speed or accuracy quantitatively.

Key ideas

  • Affine jump-diffusion and exponential Lévy models provide tractable characteristic functions for log prices.
  • Affine formulations can accommodate stochastic interest rates alongside stochastic volatility and jumps.
  • Approximation methods can extend characteristic-function techniques to rough volatility models.
  • The note identifies CEV, SABR, and local volatility as cases where simulation is a practical alternative.
  • Fourier methods are especially established for path-independent options, while simulation is common for path-dependent and early-exercise contracts.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# In what cases characteristic function of (log-)price process is known?


# In what cases characteristic function of (log-)price process is known?












Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant interest rate, but what if I want to add stochastic volatility and stochastic interest rate to my model? And can we in some way get the CF from SDEs (for example Heston model + stochastic interest rate)? I would like to know the pros and cons of using Fourier methods in option pricing instead of MC simulation and in what cases it's not possible to use Fourier methods (or when MC simualations are better choice).

## Answer by Kevin (score 4, accepted)

https://quant.stackexchange.com/a/66012

Duffie et al. (2000) show how to obtain the characteristic function of the log asset price in a fairly general affine jump diffusion model. Among others this includes the Black-Scholes (1973) model, the Heston (1993) model, the Bates (1996) model, the Merton (1976) model and the Kou (2002) model. This case also allows you to add stochastic interest rates. Bakshi, Cao and Chen (1997) use characteristic functions to price options in SVJ-I models (and find stochastic interest rates to be a minor feature of equity option pricing models).

The Lévy–Khintchine theorem tells us the characteristic function of the log asset price in exponential Lévy models. In addition to the previous jump diffusions, further examples include the variance gamma model, the CGMY model, and normal inverse Gaussian model and the Meixner model.

El Euch and Rosenbaum (2019) illustrate how to even approximate the characteristic function of the log asset price in rough volatility models.

The better question is for what models we do not know the characteristic function. Examples include the CEV model, the SABR model and local volatility models. In these cases, simulations are the way to go.

As a final note, characteristic functions capture the information about the probability distribution of the asset price at a particular point in time. They are therefore extremely popular to price path-independent options. Whilst there exist generalisations to handle path-dependency and early-exercise, Monte Carlo simulations are perhaps more popular in these cases.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.