Moment Explosions and Long-Term Behavior of Affine Stochastic Volatility Models

Abstract

Abstract. We consider a class of asset pricing models, where the risk-neutral joint process of log-price and its stochastic variance is an affine process in the sense of Duffie, Filipovic, and Schachermayer [2003]. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long-term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff-Nielsen-Shephard model.

Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models

“…Section 2.2). Its proof goes beyond the analysis in [KR11] as one is forced to study the special Lévy-Khintchine form of the characteristics of the process, since their general convexity properties no longer suffice to establish the required behaviour of the limit.…”

“…It is shown in [KR11] that the function h can be obtained from the functions F and R without the explicit knowledge of φ and ψ (see Section 2 for definition of ψ, φ). Lemma 7 and Theorem 8, taken from [KR11, Lemma 3.2 and Theorem 3.4], describe certain properties of the limiting cumulant generating function h, which are needed in Section 5 but are insufficient to guarantee the essential smoothness of h. The main contribution of the present section is Theorem 10, which identifies sufficient conditions for the process (X, V ) that imply essential smoothness of the function h. The conditions in Theorem 10 are easy to apply to the models of Section 2.2, which will allow us to find their limiting implied volatility smiles.…”

Option Pricing in the Moderate Deviations Regime

“…In the Heston model, e.g., the critical moment is of order p + (t) ≈ 1/t t β−1 for small t, as follows from inverting (6.2) in [28]. On the other hand, we do not expect our results to be of much use in the presence of jumps.…”

Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion

“…Moreover every σ -finite invariant measure of the BAJD agrees with π up to a renormalization. The existence and uniqueness of an invariant probability measure for the BAJD has already been proved in [11] (see also [14]). Thus we can assume π to be a probability measure.…”

“…In particular they have found some sufficient conditions such that the affine process converges weakly to a limit distribution. This limit distribution was later shown in [11] as the unique invariant probability measure of the process. Under further sharper assumptions it was even shown in [15] that the convergence of the law of the process to its invariance probability measure under the total variation norm is exponentially fast, which is called the exponential ergodicity in the literature.…”