The Heston stochastic volatility model has a boundary trace at zero volatility

Abstract

We establish boundary regularity results in Hölder spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) H = R × (0, ∞) ⊂ R 2 . Starting with nonsmooth initial data u 0 ∈ H, we take advantage of smoothing properties of the parabolic semigroup e −tA : H → H, t ∈ R + , generated by the Heston model, to derive the smoothness of the solution u(t) = e −tA u 0 for all t > 0. The existence and uniqueness of a weak solution is obtained in a Hilbert space H = L 2 (H; w) with very weak growth restrictions at infinity and on the boundary ∂H = R × {0} ⊂ R 2 of the half-plane H. We investigate the influence of the boundary behavior of the initial data u 0 ∈ H on the boundary behavior of u(t) for t > 0.

Nonlinear diffusion with the p-Laplacian in a Black-Scholes-type model

“…(5.3) Some remarks on this asymptotic behavior for p = 2 can be found, e.g., in Alziary and Takáč [2,3], Björk [6], Heston [9], and Hull [10]. In principle, this asymptotic behavior (meaning "boundary conditions" near infinity ±∞) is determined by financial markets.…”

“…1/2 with a help from the second partial derivative ∂ 2 V ∂S 2 . Last but not least, a linear generalization of the classical (linear) Black-Scholes model (2.1), (2.2) with stochastic volatility, the so-called Heston model [9] has been treated mathematically in the works by Alziary and Takáč [2,3].…”

“…Analytical techniques of this kind play a crucial role in particular in the Heston model (see Heston [9]) treated in Alziary andTakáč [2,3]. There, the reader is referred especially to [2, Appendix: Sect.…”