64 Frequently Asked Questions In Quantitative Finance
column represents the probability of drawing 0.37 from
each of the hats, and the final column is the joint prob-
ability, that is, the probability of drawing both numbers
from each of the hats.
Using the information aboutbothdraws, we can see that
the most likely hat is now B.
Now let’s make this into precisely a quant finance
problem.
Find the volatility You have one hat containing normally
distributed random numbers, with a mean of zero and a
standard deviation ofσwhich is unknown. You drawN
numbersφifrom this hat. Estimateσ.
Q. What is the ‘probability’ of drawingφifrom a Normal
distribution with mean zero and standard deviationσ?
A. It is
1
√
2 πσ
e
−
φ^2 i
2 σ^2.
Q. What is the ‘probability’ of drawing all of the numbers
φ 1 ,φ 2 ,...,φNfrom independent Normal distributions
with mean zero and standard deviationσ?
A. It is
∏N
i= 1
1
√
2 πσ
e
−
φ^2 i
2 σ^2.
Now choose theσthat maximizes this quantity. This is
easy. First take logarithms of this expression, and the
differentiate with respect toσand set result equal to
zero:
d
dσ
(
−Nln(σ)−
1
2 σ^2
∑N
i= 1
φi^2
)
= 0.
