具有随机波动率的Heston模型：分析解与完全市场
On the Heston Model with Stochastic Volatility: Analytic Solutions and Complete Markets
摘要：
研究了具有随机波动率的股票欧式期权的Heston定价模型。这是一个Black-Scholes型方程，其对数股票价格$x\in \RR$和方差$v\in (0,\infty)$的空间域是半平面$\HH = \RR\times (0,\infty)$。{ \it volatility\/}则由$\sqrt{v}$给出。欧式看涨期权$p = p(x,v,t)$在时间$t\leq T$的价格的扩散方程是抛物线型的，并且在边界$\partial \HH = \RR\times \{0\}$退化为$v\to 0+$。我们的目标是用这个期权对冲波动性波动，即，函数$v\mapsto p(x,v,t)\colon (0,\infty)\to \RR$及其（局部）逆是特别感兴趣的。我们证明了$\frac{\partial p}{\partial v}(x,v,t) \not= 0$持有几乎无处不在$\HH\times (-\infty,T)$通过建立$p$在空间$(x,v)$和时间$t$变量的解析性。为此，我们证明了扩散方程中的Black-Scholes型算子在$\HH$上的一个适当的加权$L^2$ -空间中生成一个全纯$C^0$ -半群.证明了$C^0$ -半群解可以推广到$\CC^2\times \CC$中的复域上的全纯函数，通过对$C^0$-半群解的唯一全纯扩张，在$\HH$的某些复“移位”上建立了一些新的先验加权$L^2$ -估计.这些估计仅取决于终端数据在$\HH$上的加权$L^2$ -范数（在$t=T$处）。
英文摘要：
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 $t=T$).
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