eigenvalue and eigenvector of a matrix (and why we bother)

These 2 links give a good review on it:
http://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx
https://www.math.hmc.edu/calculus/tutorials/eigenstuff/

say we've a matrix A, if we can satisfy this:
A*v_e = lambda*v_e

v_e = eigen vector of matrix A
lambda = eigen value of matrix A

example
|  2  7 |   | -1 |         | -1 |
| -1 -6 | * |  1 | = -5 *  |  1 |

why do we even need this?
see http://math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors
in a nutshell, it allows us to transform from standard basis, which is sometimes computationally intensive to a different basis to work in, one which simplifies the calculations necessary"

taylor's series application

say we wanna know f(x1), but we only know

• x1-x0 is small,
• f(x0),
• f'(x0), ie first derivative
• f''(x0), ie 2nd derivative
• higher order of derivatives, etc.

, what do we do?

using taylor's series, we can estimate by
f(x1) = f(x0) + (x1-x0)*f'(x0)/1! + (x1-x0)*f''(x0)/2! + ...

:D

a concrete example, say,

• with the current underlying price, x0,
• we calculate an option's value, f(x0),
• the associated delta, f'(x0),
• gamma, f''(x0)

if the underlying price moves a little bit from x0 to x1, how do we estimate the new option price, f(x1), without going through the option pricing model?

Quantitative methods

Variance
Avg sq dev from mean

Bayes
P(A|B)*P(B) =P(B|A)*P(A)

Binomial distribution
Prob of x successes in n trials
Mean np
Var np (1-p)

Sampling distribution
Population of 1k bonds
Randomly pick 100 to get mean
Pick another 100 to get mean
Repeat x times
Now we have x means forming sampling distribution of the mean

Central limit theorem
Population mean = mean of sample means
Population variance = n * variance of sample means, with n = sample size

Std err of sample means
= Std dev of sample meanS
= population std dev / sqrt (n)
Think about it , more the observations vary, more likely u will get an inaccurate answer
Vice versa for sample size

Well, we don't have population std dev, so will use std dev of sample (NOT sample means)

Putting the above together, we get a point estimate of population mean from samplings.
We can get a confidence interval of our point estimate with the std error

Depending on the availability of population variance and sample size, we may use t distribution instead of normal, ie z.

Hypothesis testing
Is daily option return = 0?
Sample size of 250 days
Mean return = .1%
Sample std dev of return =.25%

Null hypothesis: population daily option return = 0
If the difference between the sample mean and population mean is big enough, then we can reject the null hypothesis and say mean return ! = 0.
How to quantity whether it's big enough?
We have to look at how accurate the sample mean is, ie how close sample mean is to population mean.
Say, the sample mean is 100% accurate,  ie sample size = population size, ANY difference between sample mean and the hypothesized population mean is sufficient for us to reject the null hypothesis.
We quantity the accuracy of the sample mean with std err, ie std dev of sample meanS, = population sd / sqrt (n)
=.25%/sqrt (250) = .000158

.1% divided by the above gives 6.33
Tells us that the difference is 6.33 std dev away, which is very unlikely
5% sig interval is at +- 1.96 sd with z distribution

Regression
R^2 = coefficient of determina
= explained variation / total variation
= (total - unexplained)/ total
With
Total = sum of sq dev from mean
Unexplained = sum of sq dev from predicted

Testing a regression coefficient for significance
There's a certain critical t stat value for the regression coefficient +- std err to be within. That value is a function of degree of freedom , n-k-1, sample size - independent variables - 1

Handbook of Exchange Rates - FX Options and Volatility Derivatives: An Overview from the Buy-Side Perspective

Handbook of Exchange Rates

24 FX Options and Volatility Derivatives: An Overview from the Buy-Side Perspective24.1 Introduction
24.2 Why Would One Bother with an Option?
0.4 rule of thumb: ATM call = ATM put =
ul price * 0.4 * vol * sqrt(t)

delta = how fast option price changes as UL price changes
gamma = how fast delta changes as UL price changes (always + for options)
vega = how fast option price changes as vol changes

option, in theory, can be replicated by continuously hedging delta, but for latter, during mkt crashes, no one is willing to buy UL and will suffer

gamma hedging (http://investorplace.com/2010/01/long-gamma-position/)
ex.  you're long gamma, ie u own option, since gamma always + for options
1) stock rallies, you get longer delta due to +gamma.  (so u sell the extra delta at a higher price to remain delta neutral)
2) stock drops,  you get shorter delta due to +gamma.  (so u buy the loss delta at a lower price to remain delta neutral)
buy low + sell high = $$$
there's no free lunch.  by owning option, you pay theta everyday.  u'd better hope vol is high so that u can make $ by gamma hedging.

24.3 Market for FX Options
assumptions to make:
expected realized vol
expected skewness
+ive skew - skew to the right - right has longer tail
+ive risk reversal: 2 OTM option - 25 risk reversal is the vol of the 25 delta call less the vol of the 25 delta put. The 25 delta put is the put whose strike has been chosen such that the delta is -25%.  it shows how much demand for upside relative to downside

can't we just use put call parity? no, b/c we have diff strikes
c + x/(1+rfr)^t = p + ul
@t, c + x = p + ul
c ITM
(ul - x) + x = ul
c OTM
x = (x -ul) + ul
expected kurtosis
A high kurtosis distribution - tall and skinny
low one is short w/ fat tails, ie extreme occurrences occur with a probability greater than normal.
expected term structure
PRINCIPAL COMPONENT ANALYSIS shows 3 main components
- // shift - high demand of specific stike/maturity shifts the whole surface
- steepener - changes relative price of short-term and long-term vol
- gull  - changes of relative vol of mid term vs short and long term
or diff strikes?


24.4 Volatility
variance swap - pays the diff b/w future realized variance & a predetermined variance strike

how does gamma affects the spot mkt?
say the mkt implied vol is getting higher than historical vol and the ones expecting it to get back to closer to the historical level will sell options and thus take on -ive gamma position
as UL goes up, u get less delta and have to buy spot to hedge and thus amplify the trend in the mkt.
it only applies when u assume the option buyer aren't gonna hedge.  that's a valid assumption if the buyers, like equity pm, are buying options to insure against their existing spot position, rather than option MM.
so, if we can identify the buyers, we can formulate a trend following strategy?
on the flip side, if implied vol is a lot lower than hist and we expect the implied vol will reverse, we'll buy options and thus hold a +ive gamma position.
as UL goes down, we have less delta and will buy and effectively dampens the spot trend.

black swan strategy
target fat tail risks

24.5 FX Options from the Buy-Side Perspective
corr swap - exchanges the realized correlation into strike corr multiplied by a notional