[Paper Review] A new look at the Heston characteristic function

Sebastián del Baño Rollin, Albert Ferreiro-Castilla+1 more|ArXiv.org|Feb 12, 2009

Stochastic processes and financial applications18 references19 citations

TL;DR

This paper presents a new analytical expression for the characteristic function of log-spot in the Heston stochastic volatility model, revealing its analyticity and enabling exact computation of the moment generating function's domain. The key contribution is a factorization of the moment generating function into Bessel-type components, which facilitates the construction of a convergent sequence of random variables approximating log-spot and provides deeper insight into the model's moment structure and extreme strike behavior.

ABSTRACT

A new expression for the characteristic function of log-spot in Heston model is presented. This expression more clearly exhibits its properties as an analytic characteristic function and allows us to compute the exact domain of the moment generating function. This result is then applied to the volatility smile at extreme strikes and to the control of the moments of spot. We also give a factorization of the moment generating function as product of Bessel type factors, and an approximating sequence to the law of log-spot is deduced.

Motivation & Objective

• To provide a new, analytically transparent expression for the Heston model's characteristic function that clearly exhibits its analytic properties.
• To determine the exact domain of the moment generating function (MGF) of log-spot, enabling precise analysis of moments and tail behavior.
• To factorize the MGF into Bessel-type components, revealing the underlying stochastic structure of log-spot as a sum of non-centered chi-squared variables.
• To derive a sequence of random variables converging in law to log-spot, offering a constructive approximation method.
• To apply the results to the volatility smile at extreme strikes, particularly refining Lee's formula for wing parameters.

Proposed method

• Derive the moment generating function (MGF) of log-spot by solving a real-valued PDE obtained via Itô's lemma, avoiding reliance on standard characteristic function forms.
• Use complex analysis techniques, including the Hadamard factorization theorem and Mittag-Leffler theorem, to analyze the poles and structure of the characteristic function.
• Factorize the MGF into products of Bessel-type moment generating functions by identifying the roots of an entire function derived from the characteristic function's denominator.
• Establish the convergence of a sequence of random variables to log-spot by leveraging the factorization and properties of Bessel distributions.
• Apply the MGF domain and pole structure to compute asymptotic parameters of the volatility smile at extreme strikes, particularly in the context of Lee's moment formula.
• Use continuity and Hurwitz's theorem to extend results to critical cases such as ρ = ±√2/2, where standard convergence fails.

Experimental results

Research questions

• RQ1 What is the exact domain of the moment generating function of log-spot in the Heston model, and how can it be computed numerically?
• RQ2 How can the characteristic function of log-spot be expressed in a form that explicitly reveals its analyticity and facilitates analysis?
• RQ3 Can the moment generating function of log-spot be factorized into components related to known distributions, such as Bessel or non-central chi-squared laws?
• RQ4 What is the limiting behavior of the volatility smile at extreme strikes, and how can it be quantified using the MGF's pole structure?
• RQ5 Can a sequence of random variables be constructed that converges in law to log-spot, and what is the probabilistic interpretation of this convergence?

Key findings

• The moment generating function of log-spot is analytic in a real interval around zero, and its exact domain is determined by the location of poles in the complex plane, which can be computed numerically via a simple procedure.
• The characteristic function is shown to be analytic by proving the existence of a real MGF in a neighborhood of zero, which is a stronger and more useful characterization than the standard form.
• The MGF of log-spot factorizes as a product of Bessel-type moment generating functions, identifying the underlying stochastic components as non-centered chi-squared variables.
• For certain parameter combinations, log-spot is distributed as a sum of independent non-centered chi-squared random variables, placing it in the second Wiener chaos, consistent with the CIR process being a sum of squared Ornstein-Uhlenbeck processes.
• A sequence of random variables converging in law to log-spot is explicitly constructed using the factorization, providing a new approximation method.
• The residue of the characteristic function's denominator at its poles is shown to be positive for large roots, and the asymptotic behavior of the residue is computed as 1/c² (or 2/c² if ρ²=1 and c≠2aρ), enabling precise tail analysis.