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[[Monte Carlo methods in finance]] are often used to [[Corporate finance#Quantifying uncertainty|evaluate investments in projects]] at a business unit or corporate level, or other financial valuations. They can be used to model [[project management|project schedules]], where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[https://risk.octigo.pl/] Monte Carlo methods are also used in option pricing, default risk analysis.<ref>{{Cite book|title = An Introduction to Particle Methods with Financial Applications|publisher = Springer Berlin Heidelberg|journal = Numerical Methods in Finance|date = 2012|isbn = 978-3-642-25745-2|pages = 3–49|series = Springer Proceedings in Mathematics|volume = 12|first1 = René|last1 = Carmona|first2 = Pierre|last2 = Del Moral|first3 = Peng|last3 = Hu|first4 = Nadia|last4 = Oudjane|editor-first = René A.|editor-last = Carmona|editor2-first = Pierre Del|editor2-last = Moral|editor3-first = Peng|editor3-last = Hu|editor4-first = Nadia|display-editors = 3 |editor4-last = Oudjane|doi=10.1007/978-3-642-25746-9_1|citeseerx = 10.1.1.359.7957}}</ref><ref>{{Cite book |volume = 12|doi=10.1007/978-3-642-25746-9|series = Springer Proceedings in Mathematics|year = 2012|isbn = 978-3-642-25745-2|url = https://basepub.dauphine.fr/handle/123456789/11498|title=Numerical Methods in Finance|last1=Carmona|first1=René|last2=Del Moral|first2=Pierre|last3=Hu|first3=Peng|last4=Oudjane|first4=Nadia}}</ref><ref name="kr11">{{cite book|last1 = Kroese|first1 = D. P.|last2 = Taimre|first2 = T.|last3 = Botev|first3 = Z. I. |title = Handbook of Monte Carlo Methods|year = 2011|publisher = John Wiley & Sons}}</ref> Additionally, they can be used to estimate the financial impact of medical interventions.<ref>{{Cite journal |doi = 10.1371/journal.pone.0189718|pmid = 29284026|pmc = 5746244|title = A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic|journal = PLOS ONE|volume = 12|issue = 12|pages = e0189718|year = 2017|last1 = Arenas|first1 = Daniel J.|last2 = Lett|first2 = Lanair A.|last3 = Klusaritz|first3 = Heather|last4 = Teitelman|first4 = Anne M.|bibcode = 2017PLoSO..1289718A}}</ref>
[[Monte Carlo methods in finance]] are often used to [[Corporate finance#Quantifying uncertainty|evaluate investments in projects]] at a business unit or corporate level, or other financial valuations. They can be used to model [[project management|project schedules]], where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[https://risk.octigo.pl/] Monte Carlo methods are also used in option pricing, default risk analysis.<ref>{{Cite book|title = An Introduction to Particle Methods with Financial Applications|publisher = Springer Berlin Heidelberg|journal = Numerical Methods in Finance|date = 2012|isbn = 978-3-642-25745-2|pages = 3–49|series = Springer Proceedings in Mathematics|volume = 12|first1 = René|last1 = Carmona|first2 = Pierre|last2 = Del Moral|first3 = Peng|last3 = Hu|first4 = Nadia|last4 = Oudjane|editor-first = René A.|editor-last = Carmona|editor2-first = Pierre Del|editor2-last = Moral|editor3-first = Peng|editor3-last = Hu|editor4-first = Nadia|display-editors = 3 |editor4-last = Oudjane|doi=10.1007/978-3-642-25746-9_1|citeseerx = 10.1.1.359.7957}}</ref><ref>{{Cite book |volume = 12|doi=10.1007/978-3-642-25746-9|series = Springer Proceedings in Mathematics|year = 2012|isbn = 978-3-642-25745-2|url = https://basepub.dauphine.fr/handle/123456789/11498|title=Numerical Methods in Finance|last1=Carmona|first1=René|last2=Del Moral|first2=Pierre|last3=Hu|first3=Peng|last4=Oudjane|first4=Nadia}}</ref><ref name="kr11">{{cite book|last1 = Kroese|first1 = D. P.|last2 = Taimre|first2 = T.|last3 = Botev|first3 = Z. I. |title = Handbook of Monte Carlo Methods|year = 2011|publisher = John Wiley & Sons}}</ref> Additionally, they can be used to estimate the financial impact of medical interventions.<ref>{{Cite journal |doi = 10.1371/journal.pone.0189718|pmid = 29284026|pmc = 5746244|title = A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic|journal = PLOS ONE|volume = 12|issue = 12|pages = e0189718|year = 2017|last1 = Arenas|first1 = Daniel J.|last2 = Lett|first2 = Lanair A.|last3 = Klusaritz|first3 = Heather|last4 = Teitelman|first4 = Anne M.|bibcode = 2017PLoSO..1289718A}}</ref>
蒙特卡罗方法在金融经常被用于评估在一个业务单位或公司层面的项目投资，或其他金融估值。它们可以用来模拟项目进度，其中模拟汇总了对最坏情况、最好情况和每个任务最可能持续时间的估计，以确定整个项目的结果蒙特卡罗方法也被用于期权定价，违约风险分析。此外，它们还可以用来估计医疗干预的财务影响。
===Law===
===Law===
A refinement of this method, known as [[importance sampling]] in statistics, involves sampling the points randomly, but more frequently where the integrand is large. To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as [[stratified sampling]], [[Monte Carlo integration#Recursive stratified sampling|recursive stratified sampling]], adaptive umbrella sampling<ref>{{cite journal|last=MEZEI|first=M|title=Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias|journal=Journal of Computational Physics|date=31 December 1986|volume=68|issue=1|pages=237–248|doi=10.1016/0021-9991(87)90054-4|bibcode = 1987JCoPh..68..237M}}</ref><ref>{{cite journal|last1=Bartels|first1=Christian|last2=Karplus|first2=Martin|title=Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of the Potential Energy|journal=The Journal of Physical Chemistry B|date=31 December 1997|volume=102|issue=5|pages=865–880|doi=10.1021/jp972280j}}</ref> or the [[VEGAS algorithm]].
A refinement of this method, known as [[importance sampling]] in statistics, involves sampling the points randomly, but more frequently where the integrand is large. To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as [[stratified sampling]], [[Monte Carlo integration#Recursive stratified sampling|recursive stratified sampling]], adaptive umbrella sampling<ref>{{cite journal|last=MEZEI|first=M|title=Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias|journal=Journal of Computational Physics|date=31 December 1986|volume=68|issue=1|pages=237–248|doi=10.1016/0021-9991(87)90054-4|bibcode = 1987JCoPh..68..237M}}</ref><ref>{{cite journal|last1=Bartels|first1=Christian|last2=Karplus|first2=Martin|title=Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of the Potential Energy|journal=The Journal of Physical Chemistry B|date=31 December 1997|volume=102|issue=5|pages=865–880|doi=10.1021/jp972280j}}</ref> or the [[VEGAS algorithm]].
这种方法的改进，在统计学中称为重要抽样，涉及随机抽样点，但更频繁地在被积函数较大的地方。要精确地做到这一点，你必须已经知道积分，但你可以用一个类似函数的积分来近似这个积分，或者使用自适应例程，如分层抽样，递归分层抽样，自适应雨伞抽样或VEGAS算法。
A similar approach, the [[quasi-Monte Carlo method]], uses [[low-discrepancy sequence]]s. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly.
A similar approach, the [[quasi-Monte Carlo method]], uses [[low-discrepancy sequence]]s. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly.