求助 - 在险价值 (VaR) 计算（紧迫！）

一个投资组合包含1000股A股，和2000股B股。A和B的当前价格分别为83元和110元。假设A和B的对未来十天的百分比变化是正态分布，平均值（mean)是3％和2％，标准偏差(standard deviation）分别是5％和7％。A和B之间的相关性（correlation）是0.7。
a）计算10天的99％，投资组合的在险价值(VaR)。
b）假设我们添加一看涨期权（call option）到投资组合，其标的资产（underlying asset)爲1000万股A股,期权执行价格为85元，到期（maturity)是六个月。无风险利率为5％。计算投资组合的10天99％的在险价值(VaR)。



原文
A portfolio consists of 1,000 shares of A stock, and 2,000 of B stock. Current prices of A and B are $ 83 and $110  respectively. Suppose that the percentage changes of A and B for the next ten days are normally distributed, with means 3% and 2%, and standard deviations 5% and 7% respectively. Let the correlation between A and B be 0.7.

a) Calculate the ten-day 99% Value-at-Risk (VaR) of the portfolio.
b) Suppose further that we add to the portfolio a call option underlying 1,000 shares of A stock with strike price being $85, maturity being six months. Let the risk-free interest rate be 5%. Calculate the ten-day 99% VaR of the portfolio.



Please provide a direct solution (with steps)
Thank you!!

我来说说我的想法吧，希望大家能够讨论。

对于99%VaR，其实就是P(X<x, Y<y) <= 1%，题目应该是P(Wealth < somevalue) <= 1%，我们的目标是求somevalue = 1000 * price(A) + 2000 * price(B)

我的方法是运用Monte Carlo来模拟，因为好像MultiNormal没有inverse吧，只有density function呀。

既然有了mean，variance，correlation，那么我们就可以模拟变量了。

生成uncorrelated variables (z1, z2)，对covariance进行Cholesky Decomposition，两者相乘得到相关变量，这里是改变百分比。然后再带入你的原来价格，最后算出总该变量。

Okay，这是第一个模拟，反复使用100000000000，你自己看看多少够就好。最后从模拟出的数据取最差1%的值，这就是你想要的结果了。

第二题同理，模拟出的价格带入bs公式，算出期权价格改变量即可。

题主还有问题请联系我。

对了，楼主想要稍微了解Cholesky的话，请参阅wiki，有比较简单的介绍。

no3621 发表于 2015-3-2 05:08
我来说说我的想法吧，希望大家能够讨论。

对于99%VaR，其实就是P(X

感謝你的回答，但本人程度有點低，看不懂

Actually, I am confusing about:
Standard deviation (SD) and mean are in percentage. I probably need to convert them to dollar value. But how?
How to calculate portfolio SD and mean?

According to your question, u wanna regard the portfolio as a variable follows univariate normal distribution, say wealth~N(mean, variance), then u wanna calculate inverseN(1%).

asset A percent change follows N(3%, 5%^2), asset B percent change follows N(2%, 7%)
ur wealth function is Wealth = 1000 * 83 * (1 + A percent) + 2000 * 110 * (1 + B percent)
To this point, u can easily calculate the mean and var for ur portfolio. Such as, x~N(0, 1), then 10x + 1 ~N(1, 100)Tips: var(x, y) = var(x) + var(y) + 2cov(x, y)

If you still have questions, plz feel free to ask me.

no3621 发表于 2015-3-2 23:15
According to your question, u wanna regard the portfolio as a variable follows univariate normal dis ...

"Wealth = 1000 * 83 * (1 + A percent) + 2000 * 110 * (1 + B percent)"
what do you mean by A/B percent? their percent change? or (3%, 2%)?
How the wealth calculation relates to VaR calculation

SD in terms of % need to convert to dollar value?

A percent and B percent are the variables that those assets price percent change after 10 days. They follows the normal distribution according to your question.

The wealth function I give you is in dollar value, like 1000 * 83 * (1 + A percent), it means the A asset dollar value after 10 days. Concretely, if after 10 days, A changes -10%, then A asset dollar value is 1000 * 83 * (1 - 10%).

However, A/B percent are correlated variables, so when u calculate wealth variance, u should consider covariance. And the unit in wealth function has already converted to dollar value rather than percent.

To connect wealth to VaR, u just need to calculate inverseN(1%), which represents the worst 1% circumstances in wealth function.

no3621 发表于 2015-3-3 00:01
A percent and B percent are the variables that those assets price percent change after 10 days. They ...

Thanks a lot Orz
But how can I know " assets price percent change after 10 days."?
And is there any formula to calculate inverseN(1%)? would u share as well

Thanks again for your kindness

fuuser 发表于 2015-3-3 00:38
Thanks a lot Orz
But how can I know " assets price percent change after 10 days."?
And is there  ...

inverse function of normal distribution.

u can use excel to do it.

the formula is NORMINV(). If you wanna use standard normal, u can use NORMSINV(). The parameter in these two function is different. Plz refer to the excel help. Baidu or Google is fine as well.

no3621 发表于 2015-3-3 00:41
inverse function of normal distribution.

u can use excel to do it.

请问一下，利用历史收益率计算单支股票的VaR计算出的结果大于100%，请问是什么原因导致的呢？