Exercise regions of American options on several assets
type: Item
instance of: scholarly article
title: Exercise regions of American options on several assets
zbMATH Open document ID: 1047.91037
published in: Finance and Stochastics
publication date: 2000-03-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 1408674
zbMATH Keywords: optimal stopping
zbMATH Keywords: free boundary problems
zbMATH Keywords: American options
zbMATH Keywords: exercise region
author: Stéphane Villeneuve
reviewed by: Aleksandr D. Borisenko
MaRDI profile type: Publication
full work available at URL: s007800050064
OpenAlex ID: W2169296382
DOI: 10.1007/S007800050064
description: scientific article; zbMATH DE number 1408674