Monte Carlo valuation of American Options
Monte Carlo valuation of American Options
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DOI：
10.1007/978-3-662-09510-2_43
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
D. Lamper;S. Howison
中科院分区：
文献类型：
--
作者：
D. Lamper;S. Howison
An American option is a contract giving its holder the right to buy (call option) or sell (put option) one unit of an underlying security of valueSfor a prearranged amount. This right can be exercised at any time prior to the expiration dateT. In contrast, a European option can be exercised only at the expiry. Define the amount paid to the holder of an American option at the moment of exercise, the payoff, asΨ(S,t) ≥ 0; a standard contract is a put option whereΨ= max(K −S, 0) andKis the strike price. The discounted exercise value of the option isZ(t) =Ψ(t) /B(t), whereB(t) is the value at timetof $1 invested in a riskless money market account att =0. American option valuation can be characterised as an optimal stopping problem. The time 0 value of an American option is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$V(0) = \mathop {\sup }\limits_{0\tau T} E\left[ {Z\left( \tau \right)} \right]$$ \end{document} where the supremum is taken over all the possible stopping timesτless than the expiration dateT, and the expectation is taken over the risk-neutral probability density. This is theprimalproblem.
An American option is a contract giving its holder the right to buy (call option) or sell (put option) one unit of an underlying security of valueSfor a prearranged amount. This right can be exercised at any time prior to the expiration dateT. In contrast, a European option can be exercised only at the expiry. Define the amount paid to the holder of an American option at the moment of exercise, the payoff, asΨ(S,t) ≥ 0; a standard contract is a put option whereΨ= max(K −S, 0) andKis the strike price. The discounted exercise value of the option isZ(t) =Ψ(t) /B(t), whereB(t) is the value at timetof $1 invested in a riskless money market account att =0. American option valuation can be characterised as an optimal stopping problem. The time 0 value of an American option is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$V(0) = \mathop {\sup }\limits_{0\tau T} E\left[ {Z\left( \tau \right)} \right]$$ \end{document} where the supremum is taken over all the possible stopping timesτless than the expiration dateT, and the expectation is taken over the risk-neutral probability density. This is theprimalproblem.