Analytic solutions and complete markets for the Heston model with stochastic volatility

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).