On the Heston Model with Stochastic Volatility: Analytic Solutions and Complete Markets

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 $x\in \RR$ and the variance $v\in (0,\infty)$ is the half\--plane $\HH = \RR\times (0,\infty)$. The {\it 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 $\partial \HH = \RR\times \{0\}$ as $v\to 0+$. The goal is to hedge with this option against volatility fluctuations, i.e., the function $v\mapsto p(x,v,t)\colon (0,\infty)\to \RR$ and its (local) inverse are of particular interest. We prove that $\frac{\partial p}{\partial v}(x,v,t) \not= 0$ holds almost everywhere in $\HH\times (-\infty,T)$ 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 $\HH$. We show that the $C^0$-semigroup solution can be extended to a holomorphic function in a complex domain in $\CC^2\times \CC$, by establishing some new a~priori weighted $L^2$-estimates over certain complex "shifts" of $\HH$ for the unique holomorphic extension. These estimates depend only on the weighted $L^2$-norm of the terminal data over $\HH$ (at $t=T$).