Exercise regions of American options on several assets
The author considers American options written on \(n\) underlying assets. The logarithm of the stock prices has a form \[ (X_{t}^{s,x})_{i}= x_{i}+(r-\delta_{i}-{1\over2}\sum\limits_{j=1}^{n}\sigma_{ij}^{2}) (t-s)+\sum\limits_{j=1}^{n}\sigma_{ij}(W_{t}^{j}-W_{s}^{j}) \] for \(0\leq s\leq t\leq T\) and \(x=(x_1,\ldots,x_{n})\), where \((W^1_{t},\ldots,W_{t}^{n})\) is a standard \(n\)-dimensional Brownian motion, \(r\geq0\) is the interest rate, \(\delta_{i}\geq0\) is the dividend rate of the asset \(i\), the matrix \(\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}\) is invertible. The value of an American option with date of maturity \(T\) is given by \(V_{t}=C(t,X_{t})\), where \(C(t,x)=E(e^{-r\tau^{*}}\psi (X_{\tau^{*}}^{0,x}))\), \(\tau^{*}=\inf\{u\in[0,T-1]\mid C(t+u,X_{u}^{0,x})=\psi(X_{u}^{0,x})\}\), \(\psi(X_{t})\) is a payoff process, \(\psi(x)\) is some continuous nonnegative function. The set \({\mathcal E}= \{(t,x)\in [0,T[\times \mathbb R^{n}\mid C(t,x)=\psi(x)\}\) is called the exercise region of the American option. Let \({\mathcal A}\) is the differential operator associated to the considered diffusion model \[ {\mathcal A}F={1\over2}\sum\limits_{i,j=1}^{n} a_{ij}{\partial^{2}F\over \partial x_{i}\partial x_{j}}+ \sum\limits_{i=1}^{n}(r-\delta_{i}-{1\over2}\sum\limits_{j=1}^{n} \sigma_{ij}^{2}){\partial F\over\partial x_{i}}-rF,\;(a_{ij})_{1\leq i,j\leq n}=\Sigma^{*}\Sigma. \] The author proves the following general analytic result: The exercise region is empty if and only if \({\mathcal A}\psi\) is a nonzero positive measure on \(\mathbb R^{n}\). Then a particular class of payoff functions that are traded in financial markets is studied. Also the results concerning the valuation of various types of American options are presented.
scientific article; zbMATH DE number 1408674