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

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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