Analytic solutions and complete markets for the Heston model with stochastic volatility
Date
2018-10-11
Authors
Alziary, Benedicte
Takac, Peter
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We study the Heston model for pricing European options on stocks with stochastic volatility. This is a Black-Scholes-type equation whose spatial domain for the logarithmic stock price x ∈ ℝ and the variance v ∈ (0, ∞) is the half-plane ℍ = ℝ x (0, ∞). The volatility is then given by √v. The diffusion equation for the price of the European call option p = p(x, v, t) at time t ≤ T is parabolic and degenerates at the boundary ∂ℍ = ℝ x {0} as v → 0+. The goal is to hedge with this option against volatility fluctuations, i.e., the function v ↦ p(x, v, t): (0, ∞) → ℝ and its (local) inverse are of particular interest. We prove that ∂p/∂v (x, v, t) ≠ 0 holds almost everywhere in ℍ x (-∞, T) by establishing the analyticity of p in both, space (x, v) and time t variables. To this end, we are able to show that the Black-Scholes-type operator, which appears in the diffusion equation, generates a holomorphic C0-semigroup in a suitable weighted L2-space over ℍ. We show that the C0-semigroup solution can be extended to a holomorphic function in a complex domain in ℂ2 x ℂ, by establishing some new a priori weighted L2-estimates over certain complex "shifts" of ℍ for the unique holomorphic extension. These estimates depend only on the weighted L2-norm of the terminal data over ℍ (at t = T).
Description
Keywords
Heston model, Stochastic volatility, Black-Scholes equation, European call option, Degenerate parabolic equation, Terminal value problem, Holomorphic extension, Analytic solution
Citation
Alziary, B., & Takác, P. (2018). Analytic solutions and complete markets for the Heston model with stochastic volatility. <i>Electronic Journal of Differential Equations, 2018</i>(168), pp. 1-54.
Rights
Attribution 4.0 International