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Jeffrey Huang

第七讲
基于Monte Carlo模拟法的VaR计算

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VaR
ΔP分布的确定方法

收益率映射估值法

风险因子映射估值法

风险因子映射估值模拟法 (全部估值法)

风险因子映射估值分析法 (局部估值法)

历史模拟法

Monte Carlo模拟法

基于Delta、Gamma等 灵敏度指标的方法

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Monte Carlo
历史模拟法实质上是利用风险因子的历史数据序列模拟出资产组合的 未来损益分布,进而得到给定置信度下的VaR Monte Carlo模拟法不再借助于风险因子的历史数据,而是通过选择或 建立适当的随机模型模拟风险因子的未来变化路径,并利用估值公式 计算出对应路径的资产组合价值;不断重复上述模拟过程,最大限度 地获得风险因子的未来变化路径及其对应的资产组合价值在未来的可 能取值,以期更加准确地描绘出资产组合的未来损益分布,进而求得 VaR

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Jeffrey Huang

一、Monte Carlo模拟法

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Monte Carlo
随机抽样统计分析法的基本特点是对实际数据进行抽样分析
Motne Carlo模拟法是从计算机随机模拟出的而非实际存在的数据进行抽 样、统计,又称随机模拟方法 尽管抽样的数据来源不同,但采用Monte Carlo模拟法与随机抽样统计分 析法的重复抽样的原理相同,所以人们在很多情况下并不对两者加以区

分,而常常把这种一次又一次不断重复的随机抽样方法,统称为Monte
Carlo模拟法 Monte Carlo模拟法可以同时用于求解确定性问题和随机性问题

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确定性问题,指对那些已经存在的事实或现象进行研究的问题
这类问题往往很难直接求解,而借助于Monte Carlo模拟法对已经存在的 事实或现象进行模拟、观测、求解 首先,针对所要研究的确定性问题中已经存在的事实或现象,建立一个 概率模型或随机过程,使模型或过程的参数等于问题的解

然后,通过对模型或过程的反复观察或抽样试验来计算所求参数的统计
特征 最后,输出所求解的近似值,并估计解的精度

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随机性问题,指你要研究的问题中含有还未发生的随机性成分
这类问题一般须借助于随机数来对一些还没有发生的随机现象进行模拟 ,最后的求解结果也常常是对拟要研究的随机问题的未来变化分布的预 测 首先,针对待求解问题中的随机现象建立相应的随机模型

然后,对随机模型中的随机变量确定抽样方法,再通过计算机的模拟实
验产生所需要的随机数,得到模型中随机变量的有关特征数字 最后,根据随机模型所确定的解和相关随机变量的某些特征数字之间的 函数关系,计算出所求问题的近似解

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Monte Carlo

VaR

用Monte Carlo模拟法计算VaR,几乎都是随机性问题
用Monte Carlo模拟法计算VaR,是否成功取决于一下三个要素: 第一,用以模拟随机变量未来变化路径的随机模型的准确性 第二,每次模拟的独立性(时间区间分割数 n) 第三,足够多的模拟次数(模拟次数 N )

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Jeffrey Huang

二、单变量资产价格的随机 模拟

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Itô
Itô 过程,即随机变量﹛St﹜t≥0遵循:

dS  a( S , t )dt  b( S , t )d
其中,d   dt 随机变量 右边有两项组成: ,ε 是一个均值为0,方差为1的正态

第一项, a( S , t )dt 称为漂移项, a( S , t ) 称为漂移率 第二项, b( S , t )d 称为扩散项, b( S , t ) 称为扩散率或波动率

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特别地,当

a( S , t )  

, b( S , t )  S 时,上式变为:

dS  dt  d,dS  Sdt  Sd S μ 与 σ 为常数,此时的随机过程称为几何布朗运动 在Monte Carlo模拟中,几何布朗运动是最常见的用以描述股票价格未来 走势的随机模型 该模型也是Black-Scholes期权定价理论的基础

我们就以几何布朗运动为例,介绍在Monte Carlo模拟中如何采用离散化
方法模拟出股票价格未来变化的一条样本轨道或路径

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第一步,将连续布朗运动方程离散化 用 t 和 T 分别表示初始时刻和到期时刻,取任意的正整数 n,将区 间[ t,T ]均匀地分割为 n 等份,则每个小区间的长度为:

T t t  n
于是,离散型的几何布朗运动方程为:

St (i 1) t  St it  St it t  St it  t i  0,2, ,n  1 1,  
其中,St 是 t 时刻的股票价格,μ 与 σ 分别表示股票再 t 时刻变化 的数学期望和标准差, 是一个标准布朗运动过程,ε  t 是一个均值为0,方差为1的正态随机变量
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第二步,确定股票初始价格 St ;借助于股票价格的历史数据并采用合 适的时间序列分析方法估计出参数 μ 与 σ

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第三步,利用计算机生成 n 个相互独立的标准正态随机数,不妨记 为﹛εi:i =0, 1,2,…,n-1﹜,于是由上式可得:

St (i 1) t  St it  St it t  St it  i t i  0,2, ,n  1 1,  

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第四步,现在上述的迭代方程中,令 i = 0,于是由资产价格的初始 值 St 得到 St+Δt,再由 St+Δt生成 St+2Δt,以此递推,直到 St+nΔt = ST,
于是可生成资产价格离散时间序列﹛ St+iΔt :i = 1,2,…,n﹜,在 二维平面上可以绘出集合﹛ t+iΔt , St+iΔt :i = 0,1,2,…,n﹜中 的各点即可得到一个散点图,用线段依次将这些点连接起来就得到 连续几何布朗运动方程的一个近似的样本轨道,也可以说得到资产 价格未来变化的一个样本轨道

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第五步,不断重复第三步和第四步N次,就可以得到资产价格未来变
化的 N 条样本轨道以及资产价格在到期时刻T的变化分布S (i) :i = t 1,2,…,N, (i) 表示第 i 次模拟的资产在到期时刻的价格 S t Jeffrey Huang

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Jeffrey Huang

三、多变量资产价格的随机 模拟

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假设资产组合价值受 m 个风险因子Sj,t,的影响, Sj,t仍服从几何布 朗运动,j = 1,2,…,m
如果 m 个风险因子不相关,那么我们按照上文的单变量资产价格随 即模拟的步骤和方法分别独立地对每个风险因子变量进行模拟

S j ,t  (i 1) t  S j ,t it   j S j ,t it t   j S j ,t it  j ,i t j  1,   ,m; i  0,2, ,n  1 2, 1,  
其中,对于j = 1,2,…,m,εj 的取值相互独立,与时间次序也没 有关系

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遗憾的是,m 个风险因子相关的情况更加多见。那么,如何处理风 险因子相关的情况呢?
我们以均值向量为0,协方差矩阵为 R 的 m 维正态随机向量 ξ 为 例,说明产生多变量随机数的方法 首先,产生一个 m 维相互独立的标准正态随机向量 ε 然后,假设协方差矩阵 R 是正定的,则可用Cholecky因子分解法把 矩阵 R 表示为 R = T T′,其中T是下三角矩阵, T′是T的转置矩 阵 最后,由 m 维相互独立的标准正态随机向量 ε 和下三角矩阵 T ,可构造出随机向量为: η = T ε

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根据正态分布随机变量的性质可知,η 是均值向量为0的 m 维正态 随机向量,并且其协方差矩阵为:
E[(Tε)(Tε) ′] = TE [εε ′]T ′=
TT ′= R

由于新构造的随机向量 η 和随机向量 ξ 具有相同的联合分布函 数,从而就可以把随机变量 η 的模拟样本作为 ξ 的模拟样本

而要模拟随机向量 η 的样本,只需要首先独立地生成一个 N 维的 标准正态随机向量再乘以下三角矩阵 T 即可
于是,问题就变成了:如何将协方差矩阵 R 进行Cholesky因子分解 以确定矩阵 T

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R
以 m = 2 为例介绍Cholesky因子分解法 假设二维协方差矩阵: R    
1


1 
0 a T   11  a12 a22 

满足 R = TT ′的下三角矩阵:

其中,ρ为两个服从均值为0的正态随机向量的相关系数,于是,根 据Cholesky因子分解法,二维协方差矩阵 R 可以按照如下形式分解
1 R 



2  a11 0  a11 a12   a11  0 a      TT   a 1 22   12 a22   a11a22

a11a22  2 2  a12  a22 

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R
根据上式可得方程组:

 a11  1   a11a22   a 2  a 2  1  12 22
0   

由此求得:

1  1 T  2 (1   ) 2 
于是,得:

 1 1  1      T   2  (1   2 ) 2   

0  1       2  

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Jeffrey Huang

四、基于Monte Carlo模拟 法计算VaR的步骤

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Monte Carlo

VaR

第一步,识别风险因子变量Sj,t,其中j = 1,2,…,m;建立资产组 合价值V与风险因子变量 Sj,t 之间的映射关系 ,不妨设为: Vt = V( S1,t, S2,t ,…,Sm,t) 第二步,对风险因子未来变化进行随机模拟,得到各个风险因子 变量Sj,t 未来变化的一条样本轨道,并计算出 Sj,t, Sj,t+Δt, …, Sj,t+iΔt, …,Sj,t+nΔt = Sj,T,其中 j = 1,2,…,m 第三步,利用第一步给出的固执公式计算组合价值 Vt = V( S1,T, S2,T ,…,Sm,T)与ΔVT = VT - Vt 第四步,不断重复第三步和第四步N次,就可以得到资产组合损益分
(N) 布 V (1) , V (2) ,…, VT  T T

第五步,基于损益分布计算置信度 c 下的VaR,这与标准历史模拟法 相同
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Monte Carlo
第一,采用Monte Carlo模拟法可以产生大量的关于风险因子未来取值 的模拟样本,最大限度地将风险未来变化的各种可能情景模拟出来, 而且不必受到历史数据在数量与质量等方面所存在的种种制约,因此 ,与历史模拟法相比,基于该法所得的结果往往更加精确可靠 第二,Monte Carlo模拟法是一种完全估值法,可以处理非线性、非正 态问题

第三,Monte Carlo模拟法通过选择和建立随机模型,既可以模拟风险 因子未来变化的不同分布和不同行为特征,还可以深入、充分地挖掘 风险因子的历史数据中所包含的各种有益信息,并通过对模型中相关 参数的估计和修正反映到模型中去,从而使得随机模型对风险因子变 化的模拟更加贴近于现实
第四,Monte Carlo模拟法都可以借助于计算机来完成,从而可以大大 提高Monte Carlo模拟法的有效性和精确性
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Monte Carlo
第一,Monte Carlo模拟法的计算结果严重依赖于所选择或建立的随机模 型以及估计模型参数的历史数据,因此,这种方法容易存在模型风险和 参数估计误差 第二,在模拟过程中所使用的随机数序列一般地都是伪随机数,容易出 现循环和群聚效应,从而可能导致模拟错误和模拟失效 第三,由于一般的Monte Carlo模拟法模拟法的收敛速度与 1 / n 收敛于0 的速度相当,所以Monte Carlo模拟法的收敛速度慢、计算效率低,在加 上该法本身的计算量一般较大,往往需要花费大量时间,特别当风险因 子数量很多时,这种情况更为严重 第四,如果收敛速度慢、花费时间多等原因导致随机模拟的次数不够多 时,那么模拟样本的方差就会比较大,从而会降低该法的计算精确度 第五,采用Monte Carlo模拟法时,将风险因子随机模型离散化时,将时 间区间分割成若干小区间,每个小区间上风险因子不在随时间的变化而 变化,会造成误差,如果分割数过少,误差更大
Jeffrey Huang 25

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