Theory

QuantFlow is built around a unified mathematical framework based on characteristic functions and Lévy processes. This section introduces the core ideas that underpin the library's stochastic process models, Fourier inversion methods, and option pricing routines.

Characteristic Functions

The characteristic function of a random variable \(x\) is its Fourier transform under the probability measure:

\[ \Phi_{x,u} = \mathbb{E}\left[e^{iux}\right] \]

It is the central computational object throughout the library. Unlike the probability density function, the characteristic function is always well-defined, bounded, and closed under convolution of independent random variables. This makes it the natural tool for working with Lévy processes, where densities are often unavailable in closed form.

Lévy Processes

A Lévy process \(x_t\) has independent and stationary increments, and its characteristic function factors cleanly over time:

\[ \Phi_{x_t, u} = e^{-t\,\phi_{x_1, u}} \]

where \(\phi_{x_1,u}\) is the characteristic exponent at unit time, given by the Lévy-Khintchine formula.

The library extends this to time-changed Lévy processes \(y_t = x_{\tau_t}\), where \(\tau_t\) is a stochastic clock driven by an intensity process \(\lambda_t\). When \(\tau_t\) and \(x_t\) are independent, the characteristic function of \(y_t\) reduces to the Laplace transform of the integrated intensity:

\[ \Phi_{y_t, u} = \mathcal{L}_{\tau_t}\!\left(\phi_{x_1, u}\right) \]

This structure includes the Heston stochastic volatility model and its jump extensions as special cases, where the intensity process follows a CIR (Cox-Ingersoll-Ross) dynamics.

Fourier Inversion

Given the characteristic function, the probability density function is recovered via inverse Fourier transform. The library implements two numerical schemes:

• Trapezoidal / Simpson integration (default) using the Fractional FFT (FRFT), which allows the frequency and space domains to be discretized independently.
• Standard FFT, available as an alternative, with the constraint that \(\delta_u \cdot \delta_x = 2\pi / N\).

The FRFT is preferred in practice as it achieves higher accuracy with fewer points.

Option Pricing

European call options are priced by applying the Fourier inversion machinery to the damped call payoff. For an underlying \(S_t = S_0 e^{s_t}\) with log-price process \(s_t = x_t - c_t\) (where \(c_t\) is the convexity correction ensuring the forward is a martingale), the call price in log-moneyness \(k = \ln(K/S_0)\) is:

\[ c_k = \frac{e^{-\alpha k}}{\pi} \int_0^\infty e^{-ivk}\, \Psi(v - i\alpha)\, dv, \qquad \Psi_u = \frac{\Phi_{s_t}(u-i)}{iu(iu+1)} \]

The same numerical transforms used for PDF inversion are reused here, making option pricing computationally efficient across all supported models.