问一道题：NO.PZ2016070202000024 [ FRM II ]

What is the implied correlation between JPY/EUR and EUR/USD when given the following volatilities for foreign exchange rates? JPY/USD at 8%; JPY/EUR at 10%; EUR/USD at 6%.

选项：

A.

60%

B.

30%

C.

-30%

D.

-60%

解释：

D is correct. The logs of JPY/EUR and EUR/USD add up to that of JPY/USD：$;ln{\lbrack JPY/USD\rbrack}=ln{\lbrack JPY/EUR\rbrack}+ln{\lbrack EUR/USD\rbrack}$  So, $sigma^2{(JPY/USD)}=\sigma^2{(JPY/EUR)}+$  or $8^2=10^2+6^2+2\rho10\times6$, or $2\rho10\times6=-72$ $or\;\rho=-0.60$.

对数哪一部没问题可以理解，但对数等式两边取方差，为什么就可以把对数符号去掉直接是JPY/USD的方差了呢？不应该是JPY/USD的对数的方差吗？