Moment Explosions and Stationary Distributions in Affine Diffusion Models

Abstract

Many of the most widely used models in finance fall within the affine family of diffusion processes. The affine family combines modeling flexibility with substantial tractability, particularly through transform analysis; these models are used both for econometric modeling and for pricing and hedging of derivative securities. We analyze the tail behavior, the range of finite exponential moments, and the convergence to stationarity in affine models, focusing on the class of canonical models defined by Dai and Singleton (2000). We show that these models have limiting stationary distributions and characterize these limits. We show that the tails of both the transient and stationary distributions of these models are necessarily exponential or Gaussian; in the non-Gaussian case, we characterize the tail decay rate for any linear combination of factors. We also give necessary and sufficient conditions for a linear combination of factors to be Gaussian. Our results follow from an investigation into the stability properties of the systems of ordinary differential equations associated with affine diffusions.

Stability analysis of Riccati differential equations related to affine diffusion processes

“…This fact was proved in [16] for the Dai-Singleton canonical affine diffusions. However, the proof of this result is almost unchanged for the general canonical affine diffusions.…”

“…We omit its proof as the lemma is an extension of Lemma 4.1 in [16] and their proofs are almost identical except for some obvious changes.…”

Pricing and hedging of long dated variance swaps under a 3/2 volatility model

“…What really matters are the dynamics of the volatility process under the real world probability measure P. Under this probability measure the volatility should be realistically modeled and, thus, not explode. As mentioned in the introduction, also other volatility models, in particular those that model the leverage effect, may create similar volatility explosions under an assumed risk neutral probability measure potentially resulting in infinite prices, see [3][4][5]. The current paper aims to make the point that what counts is the real world volatility dynamics and one has to avoid a volatility explosion under P.…”

“…Several papers, which explore stochastic volatility models, have pointed at the seemingly undesirable property of some models, where the moments of squared volatility of higher order than one may become infinite in finite time. Examples are given in [3][4][5]. Furthermore, there exist various papers discussing the general problem of pricing and hedging variance swaps, including Brenner et al [6], Grünbuchler and Longstaff [7], Carr and Madan [8], Chriss and Morokoff (1999), Demeterfi et al [9], Brockhaus and Long [10], Matytsin [11], Javaheri et al [12], Swishchuk [13], Howison et al [14], Carr et al [15], Schoutens [16], Windcliff et al (2006), Zhang and Zhu (2006), Zhu and Zhang (2007), Sepp (2007), Elliott, Siu and Chan [17] and Carr and Lee [18].…”

Stationarity and Ergodicity for an Affine Two-Factor Model

“…A precise mathematical formulation and a complete characterization of regular affine processes are due to Duffie et al [12]. Later several authors have contributed to the theory of general affine processes: to name a few, Andersen and Piterbarg [1], Dawson and Li [11], Filipović and Mayerhofer [13], Glasserman and Kim [16], Jena et al [22] and Keller-Ressel et al [26].…”