Electron. J. Differential Equations, Vol. 2018 (2018), No. 168, pp. 1-54.

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

Benedicte Alziary, Peter Takac

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 $x\in \mathbb{R}$ and the variance $v\in (0,\infty)$ is the half-plane $\mathbb{H} = \mathbb{R}\times (0,\infty)$. The volatility is then given by $\sqrt{v}$. The diffusion equation for the price of the European call option p = p(x,v,t) at time $t\leq T$ is parabolic and degenerates at the boundary $\partial \mathbb{H} = \mathbb{R}\times \{ 0\}$ as $v\to 0+$. The goal is to hedge with this option against volatility fluctuations, i.e., the function $v\mapsto p(x,v,t)\colon (0,\infty)\to \mathbb{R}$ and its (local) inverse are of particular interest. We prove that $\frac{\partial p}{\partial v}(x,v,t) \neq 0$ holds almost everywhere in $\mathbb{H}\times (-\infty,T)$ 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 $C^0$-semigroup in a suitable weighted $L^2$-space over $\mathbb{H}$. We show that the $C^0$-semigroup solution can be extended to a holomorphic function in a complex domain in $\mathbb{C}^2\times \mathbb{C}$, by establishing some new a~priori weighted $L^2$-estimates over certain complex "shifts" of $\mathbb{H}$ for the unique holomorphic extension. These estimates depend only on the weighted $L^2$-norm of the terminal data over $\mathbb{H}$ (at t=T).

Submitted August 22, 2018. Published October 11, 2018.
Math Subject Classifications: 35B65, 91G80, 35K65, 35K15.
Key Words: Heston model; stochastic volatility; Black-Scholes equation; European call option; degenerate parabolic equation; terminal value problem; holomorphic extension; analytic solution.

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Bénédicte Alziary
Toulouse School of Economics, I.M.T.
Université de Toulouse - Capitole
21 Allées de Brienne
F-31000 Toulouse Cedex, France
email: benedicte.alziary@ut-capitole.fr
Peter Takác
Universität Rostock
Institut für Mathematik
Ulmenstrase~69, Haus 3
D-18057 Rostock, Germany
email: peter.takac@uni-rostock.de

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