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

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 $x\in \text{RR}$ and the variance $v\in (0,\infty)$isthehalf\text{-}-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 $t\le T$ is parabolic and degenerates at the boundary $\mathrm{\partial }\text{HH}=\text{RR}\times \{0\}$ as $v\to 0+$. The goal is to hedge with this option against volatility fluctuations, i.e., the function $v\mapsto p(x,v,t)\text{\hspace{0.17em}⁣}:(0,\mathrm{\infty })\to \text{RR}$ and its (local) inverse are of particular interest. We prove that $\frac{\partial p}{\partial v}(x,v,t) \not= 0$holdsalmosteverywherein$\HH\times (-\infty,T)$ 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 $\text{HH}$. We show that the $C^0$-semigroup solution can be extended to a holomorphic function in a complex domain in ${\text{CC}}^2\times \text{CC}$, by establishing some new a~priori weighted $L^2$-estimates over certain complex “shifts” of $\text{HH}$ for the unique holomorphic extension. These estimates depend only on the weighted $L^2$-norm of the terminal data over $\text{HH}$ (at $t=T$).