Electronic Journal of Differential Equations (Oct 2018)
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 $x\in \mathbb{R}$ and the variance $v\in (0,\infty)$ is the half-plane $\mathbb{H} = \mathbb{R}\times (0,\infty)$. The \emph{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 $t=T$).
